# Sci&Engr Guide DSP

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The Scientist and Engineer's Guide to Digital Signal Processing Second Edition Be sure to visit the book’s website at: www.DSPguide.com The Scientist and Engineer's Guide to Digital Signal Processing Second Edition by Steven W. Smith California Technical Publishing San Diego, California The Scientist and Engineer's Guide to Digital Signal Processing Second Edition by Steven W. Smith copyright © 1997-1999 by California Technical Publishing All rights reserved. No portion of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, without written permission of the publisher. ISBN 0-9660176-7-6 ISBN 0-9660176-4-1 ISBN 0-9660176-6-8 LCCN 97-80293 hardcover paperback electronic California Technical Publishing P.O. Box 502407 San Diego, CA 92150-2407 To contact the author or publisher through the internet: website: DSPguide.com e-mail: Smith@DSPguide.com Printed in the United States of America First Edition, 1997 Second Edition, 1999 Important Legal Information: Warning and Disclaimer This book presents the fundamentals of Digital Signal Processing using examples from common science and engineering problems. While the author believes that the concepts and data contained in this book are accurate and correct, they should not be used in any application without proper verification by the person making the application. Extensive and detailed testing is essential where incorrect functioning could result in personal injury or damage to property. The material in this book is intended solely as a teaching aid, and is not represented to be an appropriate or safe solution to any particular problem. For this reason, the author, publisher, and distributors make no warranties, express or implied, that the concepts, examples, data, algorithms, techniques, or programs contained in this book are free from error, conform to any industry standard, or are suitable for any application. The author, publisher, and distributors disclaim all liability and responsibility to any person or entity with respect to any loss or damage caused, or alleged to be caused, directly or indirectly, by the information contained in this book. If you do not wish to be bound by the above, you may return this book to the publisher for a full refund. Contents at a Glance FOUNDATIONS Chapter 1. The Breadth and Depth of DSP . . . . . . . . . . . . . . . . . . . 1 Chapter 2. Statistics, Probability and Noise . . . . . . . . . . . . . . . . . . 11 Chapter 3. ADC and DAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Chapter 4. DSP Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 FUNDAMENTALS Chapter 5. Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Chapter 6. Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Chapter 7. Properties of Convolution . . . . . . . . . . . . . . . . . . . . . . . 123 Chapter 8. The Discrete Fourier Transform . . . . . . . . . . . . . . . . . . 141 Chapter 9. Applications of the DFT . . . . . . . . . . . . . . . . . . . . . . . . 169 Chapter 10. Fourier Transform Properties . . . . . . . . . . . . . . . . . . . . 185 Chapter 11. Fourier Transform Pairs . . . . . . . . . . . . . . . . . . . . . . . . 209 Chapter 12. The Fast Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 225 Chapter 13. Continuous Signal Processing . . . . . . . . . . . . . . . . . . . . 243 DIGITAL FILTERS Chapter 14. Introduction to Digital Filters . . . . . . . . . . . . . . . . . . . . 261 Chapter 15. Moving Average Filters . . . . . . . . . . . . . . . . . . . . . . . . . 277 Chapter 16. Windowed-Sinc Filters . . . . . . . . . . . . . . . . . . . . . . . . . 285 Chapter 17. Custom Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 Chapter 18. FFT Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 Chapter 19. Recursive Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 Chapter 20. Chebyshev Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 Chapter 21. Filter Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 APPLICATIONS Chapter 22. Audio Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 Chapter 23. Image Formation and Display . . . . . . . . . . . . . . . . . . . . 373 Chapter 24. Linear Image Processing . . . . . . . . . . . . . . . . . . . . . . . . 397 Chapter 25. Special Imaging Techniques . . . . . . . . . . . . . . . . . . . . . 423 Chapter 26. Neural Networks (and more!) . . . . . . . . . . . . . . . . . . . . 451 Chapter 27. Data Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 Chapter 28. Digital Signal Processors . . . . . . . . . . . . . . . . . . . . . . . 503 Chapter 29. Getting Started with DSPs . . . . . . . . . . . . . . . . . . . . . . . 535 COMPLEX TECHNIQUES Chapter 30. Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551 Chapter 31. The Complex Fourier Transform . . . . . . . . . . . . . . . . . . 567 Chapter 32. The Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . 581 Chapter 33. The z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643 v Table of Contents FOUNDATIONS Chapter 1. The Breadth and Depth of DSP . . . . . . . . . . . . . 1 The Roots of DSP 1 Telecommunications 4 Audio Processing 5 Echo Location 7 Imaging Processing 9 Chapter 2. Statistics, Probability and Noise . . . . . . . . . . . . . 11 Signal and Graph Terminology 11 Mean and Standard Deviation 13 Signal vs. Underlying Process 17 The Histogram, Pmf and Pdf 19 The Normal Distribution 26 Digital Noise Generation 29 Precision and Accuracy 32 Chapter 3. ADC and DAC . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Quantization 35 The Sampling Theorem 39 Digital-to-Analog Conversion 44 Analog Filters for Data Conversion 48 Selecting the Antialias Filter 55 Multirate Data Conversion 58 Single Bit Data Conversion 60 Chapter 4. DSP Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Computer Numbers 67 Fixed Point (Integers) 68 Floating Point (Real Numbers) 70 Number Precision 72 Execution Speed: Program Language 76 Execution Speed: Hardware 80 Execution Speed: Programming Tips 84 vi FUNDAMENTALS Chapter 5. Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Signals and Systems 87 Requirements for Linearity 89 Static Linearity and Sinusoidal Fidelity 92 Examples of Linear and Nonlinear Systems 94 Special Properties of Linearity 96 Superposition: the Foundation of DSP 98 Common Decompositions 100 Alternatives to Linearity 104 Chapter 6. Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 The Delta Function and Impulse Response 107 Convolution 108 The Input Side Algorithm 112 The Output Side Algorithm 116 The Sum of Weighted Inputs 122 Chapter 7. Properties of Convolution . . . . . . . . . . . . . . . . . . 123 Common Impulse Responses 123 Mathematical Properties 132 Correlation 136 Speed 140 Chapter 8. The Discrete Fourier Transform . . . . . . . . . . . . 141 The Family of Fourier Transforms 141 Notation and Format of the real DFT 146 The Frequency Domain's Independent Variable 148 DFT Basis Functions 150 Synthesis, Calculating the Inverse DFT 152 Analysis, Calculating the DFT 156 Duality 161 Polar Notation 161 Polar Nuisances 164 Chapter 9. Applications of the DFT . . . . . . . . . . . . . . . . . . . 169 Spectral Analysis of Signals 169 Frequency Response of Systems 177 Convolution via the Frequency Domain 180 Chapter 10. Fourier Transform Properties . . . . . . . . . . . . . 185 Linearity of the Fourier Transform 185 Characteristics of the Phase 188 Periodic Nature of the DFT 194 Compression and Expansion, Multirate methods 200 vii Multiplying Signals (Amplitude Modulation) 204 The Discrete Time Fourier Transform 206 Parseval's Relation 208 Chapter 11. Fourier Transform Pairs . . . . . . . . . . . . . . . . . . 209 Delta Function Pairs 209 The Sinc Function 212 Other Transform Pairs 215 Gibbs Effect 218 Harmonics 220 Chirp Signals 222 Chapter 12. The Fast Fourier Transform . . . . . . . . . . . . . . . 225 Real DFT Using the Complex DFT 225 How the FFT Works 228 FFT Programs 233 Speed and Precision Comparisons 237 Further Speed Increases 238 Chapter 13. Continuous Signal Processing . . . . . . . . . . . . . . 243 The Delta Function 243 Convolution 246 The Fourier Transform 252 The Fourier Series 255 DIGITAL FILTERS Chapter 14. Introduction to Digital Filters . . . . . . . . . . . . . . 261 Filter Basics 261 How Information is Represented in Signals 265 Time Domain Parameters 266 Frequency Domain Parameters 268 High-Pass, Band-Pass and Band-Reject Filters 271 Filter Classification 274 Chapter 15. Moving Average Filters . . . . . . . . . . . . . . . . . . . 277 Implementation by Convolution 277 Noise Reduction vs. Step Response 278 Frequency Response 280 Relatives of the Moving Average Filter 280 Recursive Implementation 282 Chapter 16. Windowed-Sinc Filters . . . . . . . . . . . . . . . . . . . 285 Strategy of the Windowed-Sinc 285 Designing the Filter 288 Examples of Windowed-Sinc Filters 292 Pushing it to the Limit 293 viii Chapter 17. Custom Filters . . . . . . . . . . . . . . . . . . . . . . . . . . 297 Arbitrary Frequency Response 297 Deconvolution 300 Optimal Filters 307 Chapter 18. FFT Convolution . . . . . . . . . . . . . . . . . . . . . . . . 311 The Overlap-Add Method 311 FFT Convolution 312 Speed Improvements 316 Chapter 19. Recursive Filters . . . . . . . . . . . . . . . . . . . . . . . . 319 The Recursive Method 319 Single Pole Recursive Filters 322 Narrow-band Filters 326 Phase Response 328 Using Integers 332 Chapter 20. Chebyshev Filters . . . . . . . . . . . . . . . . . . . . . . . 333 The Chebyshev and Butterworth Responses 333 Designing the Filter 334 Step Response Overshoot 338 Stability 339 Chapter 21. Filter Comparison . . . . . . . . . . . . . . . . . . . . . . . 343 Match #1: Analog vs. Digital Filters 343 Match #2: Windowed-Sinc vs. Chebyshev 346 Match #3: Moving Average vs. Single Pole 348 APPLICATIONS Chapter 22. Audio Processing . . . . . . . . . . . . . . . . . . . . . . . . 351 Human Hearing 351 Timbre 355 Sound Quality vs. Data Rate 358 High Fidelity Audio 359 Companding 362 Speech Synthesis and Recognition 364 Nonlinear Audio Processing 368 Chapter 23. Image Formation and Display . . . . . . . . . . . . . . 373 Digital Image Structure 373 Cameras and Eyes 376 Television Video Signals 384 Other Image Acquisition and Display 386 Brightness and Contrast Adjustments 387 Grayscale Transforms 390 Warping 394 ix Chapter 24. Linear Image Processing . . . . . . . . . . . . . . . . . . 397 Convolution 397 3×3 Edge Modification 402 Convolution by Separability 404 Example of a Large PSF: Illumination Flattening 407 Fourier Image Analysis 410 FFT Convolution 416 A Closer Look at Image Convolution 418 Chapter 25. Special Imaging Techniques . . . . . . . . . . . . . . . 423 Spatial Resolution 423 Sample Spacing and Sampling Aperture 430 Signal-to-Noise Ratio 432 Morphological Image Processing 436 Computed Tomography 442 Chapter 26. Neural Networks (and more!) . . . . . . . . . . . . . . 451 Target Detection 451 Neural Network Architecture 458 Why Does it Work? 463 Training the Neural Network 465 Evaluating the Results 473 Recursive Filter Design 476 Chapter 27. Data Compression . . . . . . . . . . . . . . . . . . . . . . . 481 Data Compression Strategies 481 Run-Length Encoding 483 Huffman Encoding 484 Delta Encoding 486 LZW Compression 488 JPEG (Transform Compression) 494 MPEG 501 Chapter 28. Digital Signal Processors . . . . . . . . . . . . . . . . . . . 503 How DSPs are different 503 Circular Buffering 506 Architecture of the Digital Signal Processor 509 Fixed versus Floating Point 514 C versus Assembly 520 How Fast are DSPs? 526 The Digital Signal Processor Market 531 Chapter 29. Getting Started with DSPs . . . . . . . . . . . . . . . . 535 The ADSP-2106x family 535 The SHARC EZ-KIT Lite 537 Design Example: An FIR Audio Filter 538 Analog Measurements on a DSP System 542 x Another Look at Fixed versus Floating Point 544 Advanced Software Tools 546 COMPLEX TECHNIQUES Chapter 30. Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . 551 The Complex Number System 551 Polar Notation 555 Using Complex Numbers by Substitution 557 Complex Representation of Sinusoids 559 Complex Representation of Systems 561 Electrical Circuit Analysis 563 Chapter 31. The Complex Fourier Transform . . . . . . . . . . . 567 The Real DFT 567 Mathematical Equivalence 569 The Complex DFT 570 The Family of Fourier Transforms 575 Why the Complex Fourier Transform is Used 577 Chapter 32. The Laplace Transform . . . . . . . . . . . . . . . . . . . 581 The Nature of the s-Domain 581 Strategy of the Laplace Transform 588 Analysis of Electric Circuits 592 The Importance of Poles and Zeros 597 Filter Design in the s-Domain 600 Chapter 33. The z-Transform . . . . . . . . . . . . . . . . . . . . . . . . 605 The Nature of the z-Domain 605 Analysis of Recursive Systems 610 Cascade and Parallel Stages 616 Spectral Inversion 619 Gain Changes 621 Chebyshev-Butterworth Filter Design 623 The Best and Worst of DSP 630 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643 xi Preface Goals and Strategies of this Book The technical world is changing very rapidly. In only 15 years, the power of personal computers has increased by a factor of nearly one-thousand. By all accounts, it will increase by another factor of one-thousand in the next 15 years. This tremendous power has changed the way science and engineering is done, and there is no better example of this than Digital Signal Processing. In the early 1980s, DSP was taught as a graduate level course in electrical engineering. A decade later, DSP had become a standard part of the undergraduate curriculum. Today, DSP is a basic skill needed by scientists and engineers in many fields. Unfortunately, DSP education has been slow to adapt to this change. Nearly all DSP textbooks are still written in the traditional electrical engineering style of detailed and rigorous mathematics. DSP is incredibly powerful, but if you can't understand it, you can't use it! This book was written for scientists and engineers in a wide variety of fields: physics, bioengineering, geology, oceanography, mechanical and electrical engineering, to name just a few. The goal is to present practical techniques while avoiding the barriers of detailed mathematics and abstract theory. To achieve this goal, three strategies were employed in writing this book: First, the techniques are explained, not simply proven to be true through mathematical derivations. While much of the mathematics is included, it is not used as the primary means of conveying the information. Nothing beats a few well written paragraphs supported by good illustrations. Second, complex numbers are treated as an advanced topic, something to be learned after the fundamental principles are understood. Chapters 1-29 explain all the basic techniques using only algebra, and in rare cases, a small amount of elementary calculus. Chapters 30-33 show how complex math extends the power of DSP, presenting techniques that cannot be implemented with real numbers alone. Many would view this approach as heresy! Traditional DSP textbooks are full of complex math, often starting right from the first chapter. xii Third, very simple computer programs are used. Most DSP programs are written in C, Fortran, or a similar language. However, learning DSP has different requirements than using DSP. The student needs to concentrate on the algorithms and techniques, without being distracted by the quirks of a particular language. Power and flexibility aren't important; simplicity is critical. The programs in this book are written to teach DSP in the most straightforward way, with all other factors being treated as secondary. Good programming style is disregarded if it makes the program logic more clear. For instance: ‘ ‘ ‘ ‘ a simplified version of BASIC is used line numbers are included the only control structure used is the FOR-NEXT loop there are no I/O statements This is the simplest programming style I could find. Some may think that this book would be better if the programs had been written in C. I couldn't disagree more. The Intended Audience This book is primarily intended for a one year course in practical DSP, with the students being drawn from a wide variety of science and engineering fields. The suggested prerequisites are: ‘ A course in practical electronics: (op amps, RC circuits, etc.) ‘ A course in computer programming (Fortran or similar) ‘ One year of calculus This book was also written with the practicing professional in mind. Many everyday DSP applications are discussed: digital filters, neural networks, data compression, audio and image processing, etc. As much as possible, these chapters stand on their own, not requiring the reader to review the entire book to solve a specific problem. Support by Analog Devices The Second Edition of this book includes two new chapters on Digital Signal Processors, microprocessors specifically designed to carry out DSP tasks. Much of the information for these chapters was generously provided by Analog Devices, Inc., a world leader in the development and manufacturing of electronic components for signal processing. ADI's encouragement and support has significantly expanded the scope of this book, showing that DSP algorithms are only useful in conjunction with the appropriate hardware. xiii Acknowledgements A special thanks to the many reviewers who provided comments and suggestions on this book. Their generous donation of time and skill has made this a better work: Magnus Aronsson (Department of Electrical Engineering, University of Utah); Bruce B. Azimi (U.S. Navy); Vernon L. Chi (Department of Computer Science, University of North Carolina); Manohar Das, Ph.D. (Department of Electrical and Systems Engineering, Oakland University); Carol A. Dean (Analog Devices, Inc.); Fred DePiero, Ph.D. (Department of Electrical Engineering, CalPoly State University); Jose Fridman, Ph.D. (Analog Devices, Inc.); Frederick K. Duennebier, Ph. D. (Department of Geology and Geophysics, University of Hawaii, Manoa); D. Lee Fugal (Space & Signals Technologies); Filson H. Glanz, Ph.D. (Department of Electrical and Computer Engineering, University of New Hampshire); Kenneth H. Jacker, (Department of Computer Science, Appalachian State University); Rajiv Kapadia, Ph.D. (Department of Electrical Engineering, Mankato State University); Dan King (Analog Devices, Inc.); Kevin Leary (Analog Devices, Inc.); A. Dale Magoun, Ph.D. (Department of Computer Science, Northeast Louisiana University); Ben Mbugua (Analog Devices, Inc.); Bernard J. Maxum, Ph.D. (Department of Electrical Engineering, Lamar University); Paul Morgan, Ph.D. (Department of Geology, Northern Arizona University); Dale H. Mugler, Ph.D. (Department of Mathematical Science, University of Akron); Christopher L. Mullen, Ph.D. (Department of Civil Engineering, University of Mississippi); Cynthia L. Nelson, Ph.D. (Sandia National Laboratories); Branislava Perunicic-Drazenovic, Ph.D. (Department of Electrical Engineering, Lamar University); John Schmeelk, Ph.D. (Department of Mathematical Science, Virginia Commonwealth University); Richard R. Schultz, Ph.D. (Department of Electrical Engineering, University of North Dakota); David Skolnick (Analog Devices, Inc.); Jay L. Smith, Ph.D. (Center for Aerospace Technology, Weber State University); Jeffrey Smith, Ph.D. (Department of Computer Science, University of Georgia); Oscar Yanez Suarez, Ph.D. (Department of Electrical Engineering, Metropolitan University, Iztapalapa campus, Mexico City); and other reviewers who wish to remain anonymous. This book is now in the hands of the final reviewer, you. Please take the time to give me your comments and suggestions. This will allow future reprints and editions to serve your needs even better. All it takes is a two minute e-mail message to: Smith@DSPguide.com. Thanks; I hope you enjoy the book. Steve Smith January 1999 xiv CHAPTER 1 The Breadth and Depth of DSP Digital Signal Processing is one of the most powerful technologies that will shape science and engineering in the twenty-first century. Revolutionary changes have already been made in a broad range of fields: communications, medical imaging, radar & sonar, high fidelity music reproduction, and oil prospecting, to name just a few. Each of these areas has developed a deep DSP technology, with its own algorithms, mathematics, and specialized techniques. This combination of breath and depth makes it impossible for any one individual to master all of the DSP technology that has been developed. DSP education involves two tasks: learning general concepts that apply to the field as a whole, and learning specialized techniques for your particular area of interest. This chapter starts our journey into the world of Digital Signal Processing by describing the dramatic effect that DSP has made in several diverse fields. The revolution has begun. The Roots of DSP Digital Signal Processing is distinguished from other areas in computer science by the unique type of data it uses: signals. In most cases, these signals originate as sensory data from the real world: seismic vibrations, visual images, sound waves, etc. DSP is the mathematics, the algorithms, and the techniques used to manipulate these signals after they have been converted into a digital form. This includes a wide variety of goals, such as: enhancement of visual images, recognition and generation of speech, compression of data for storage and transmission, etc. Suppose we attach an analog-to-digital converter to a computer and use it to acquire a chunk of real world data. DSP answers the question: What next? The roots of DSP are in the 1960s and 1970s when digital computers first became available. Computers were expensive during this era, and DSP was limited to only a few critical applications. Pioneering efforts were made in four key areas: radar & sonar, where national security was at risk; oil exploration, where large amounts of money could be made; space exploration, where the 1 2 The Scientist and Engineer's Guide to Digital Signal Processing data are irreplaceable; and medical imaging, where lives could be saved. The personal computer revolution of the 1980s and 1990s caused DSP to explode with new applications. Rather than being motivated by military and government needs, DSP was suddenly driven by the commercial marketplace. Anyone who thought they could make money in the rapidly expanding field was suddenly a DSP vender. DSP reached the public in such products as: mobile telephones, compact disc players, and electronic voice mail. Figure 1-1 illustrates a few of these varied applications. This technological revolution occurred from the top-down. In the early 1980s, DSP was taught as a graduate level course in electrical engineering. A decade later, DSP had become a standard part of the undergraduate curriculum. Today, DSP is a basic skill needed by scientists and engineers DSP Space -Space photograph enhancement -Data compression -Intelligent sensory analysis by remote space probes Medical -Diagnostic imaging (CT, MRI, ultrasound, and others) -Electrocardiogram analysis -Medical image storage/retrieval Commercial -Image and sound compression for multimedia presentation -Movie special effects -Video conference calling Telephone -Voice and data compression -Echo reduction -Signal multiplexing -Filtering Military -Radar -Sonar -Ordnance guidance -Secure communication Industrial -Oil and mineral prospecting -Process monitoring & control -Nondestructive testing -CAD and design tools Scientific -Earthquake recording & analysis -Data acquisition -Spectral analysis -Simulation and modeling FIGURE 1-1 DSP has revolutionized many areas in science and engineering. A few of these diverse applications are shown here. Chapter 1- The Breadth and Depth of DSP 3 in many fields. As an analogy, DSP can be compared to a previous technological revolution: electronics. While still the realm of electrical engineering, nearly every scientist and engineer has some background in basic circuit design. Without it, they would be lost in the technological world. DSP has the same future. This recent history is more than a curiosity; it has a tremendous impact on your ability to learn and use DSP. Suppose you encounter a DSP problem, and turn to textbooks or other publications to find a solution. What you will typically find is page after page of equations, obscure mathematical symbols, and unfamiliar terminology. It's a nightmare! Much of the DSP literature is baffling even to those experienced in the field. It's not that there is anything wrong with this material, it is just intended for a very specialized audience. State-of-the-art researchers need this kind of detailed mathematics to understand the theoretical implications of the work. A basic premise of this book is that most practical DSP techniques can be learned and used without the traditional barriers of detailed mathematics and theory. The Scientist and Engineer’s Guide to Digital Signal Processing is written for those who want to use DSP as a tool, not a new career. The remainder of this chapter illustrates areas where DSP has produced revolutionary changes. As you go through each application, notice that DSP is very interdisciplinary, relying on the technical work in many adjacent fields. As Fig. 1-2 suggests, the borders between DSP and other technical disciplines are not sharp and well defined, but rather fuzzy and overlapping. If you want to specialize in DSP, these are the allied areas you will also need to study. Digital Signal Processing Communication Theory Numerical Analysis Probability and Statistics Analog Signal Processing Analog Electronics Digital Electronics Decision Theory FIGURE 1-2 Digital Signal Processing has fuzzy and overlapping borders with many other areas of science, engineering and mathematics. 4 The Scientist and Engineer's Guide to Digital Signal Processing Telecommunications Telecommunications is about transferring information from one location to another. This includes many forms of information: telephone conversations, television signals, computer files, and other types of data. To transfer the information, you need a channel between the two locations. This may be a wire pair, radio signal, optical fiber, etc. Telecommunications companies receive payment for transferring their customer's information, while they must pay to establish and maintain the channel. The financial bottom line is simple: the more information they can pass through a single channel, the more money they make. DSP has revolutionized the telecommunications industry in many areas: signaling tone generation and detection, frequency band shifting, filtering to remove power line hum, etc. Three specific examples from the telephone network will be discussed here: multiplexing, compression, and echo control. Multiplexing There are approximately one billion telephones in the world. At the press of a few buttons, switching networks allow any one of these to be connected to any other in only a few seconds. The immensity of this task is mind boggling! Until the 1960s, a connection between two telephones required passing the analog voice signals through mechanical switches and amplifiers. One connection required one pair of wires. In comparison, DSP converts audio signals into a stream of serial digital data. Since bits can be easily intertwined and later separated, many telephone conversations can be transmitted on a single channel. For example, a telephone standard known as the T-carrier system can simultaneously transmit 24 voice signals. Each voice signal is sampled 8000 times per second using an 8 bit companded (logarithmic compressed) analog-to-digital conversion. This results in each voice signal being represented as 64,000 bits/sec, and all 24 channels being contained in 1.544 megabits/sec. This signal can be transmitted about 6000 feet using ordinary telephone lines of 22 gauge copper wire, a typical interconnection distance. The financial advantage of digital transmission is enormous. Wire and analog switches are expensive; digital logic gates are cheap. Compression When a voice signal is digitized at 8000 samples/sec, most of the digital information is redundant. That is, the information carried by any one sample is largely duplicated by the neighboring samples. Dozens of DSP algorithms have been developed to convert digitized voice signals into data streams that require fewer bits/sec. These are called data compression algorithms. Matching uncompression algorithms are used to restore the signal to its original form. These algorithms vary in the amount of compression achieved and the resulting sound quality. In general, reducing the data rate from 64 kilobits/sec to 32 kilobits/sec results in no loss of sound quality. When compressed to a data rate of 8 kilobits/sec, the sound is noticeably affected, but still usable for long distance telephone networks. The highest achievable compression is about 2 kilobits/sec, resulting in Chapter 1- The Breadth and Depth of DSP 5 sound that is highly distorted, but usable for some applications such as military and undersea communications. Echo control Echoes are a serious problem in long distance telephone connections. When you speak into a telephone, a signal representing your voice travels to the connecting receiver, where a portion of it returns as an echo. If the connection is within a few hundred miles, the elapsed time for receiving the echo is only a few milliseconds. The human ear is accustomed to hearing echoes with these small time delays, and the connection sounds quite normal. As the distance becomes larger, the echo becomes increasingly noticeable and irritating. The delay can be several hundred milliseconds for intercontinental communications, and is particularity objectionable. Digital Signal Processing attacks this type of problem by measuring the returned signal and generating an appropriate antisignal to cancel the offending echo. This same technique allows speakerphone users to hear and speak at the same time without fighting audio feedback (squealing). It can also be used to reduce environmental noise by canceling it with digitally generated antinoise. Audio Processing The two principal human senses are vision and hearing. Correspondingly, much of DSP is related to image and audio processing. People listen to both music and speech. DSP has made revolutionary changes in both these areas. Music The path leading from the musician's microphone to the audiophile's speaker is remarkably long. Digital data representation is important to prevent the degradation commonly associated with analog storage and manipulation. This is very familiar to anyone who has compared the musical quality of cassette tapes with compact disks. In a typical scenario, a musical piece is recorded in a sound studio on multiple channels or tracks. In some cases, this even involves recording individual instruments and singers separately. This is done to give the sound engineer greater flexibility in creating the final product. The complex process of combining the individual tracks into a final product is called mix down. DSP can provide several important functions during mix down, including: filtering, signal addition and subtraction, signal editing, etc. One of the most interesting DSP applications in music preparation is artificial reverberation. If the individual channels are simply added together, the resulting piece sounds frail and diluted, much as if the musicians were playing outdoors. This is because listeners are greatly influenced by the echo or reverberation content of the music, which is usually minimized in the sound studio. DSP allows artificial echoes and reverberation to be added during mix down to simulate various ideal listening environments. Echoes with delays of a few hundred milliseconds give the impression of cathedral like 6 The Scientist and Engineer's Guide to Digital Signal Processing locations. Adding echoes with delays of 10-20 milliseconds provide the perception of more modest size listening rooms. Speech generation Speech generation and recognition are used to communicate between humans and machines. Rather than using your hands and eyes, you use your mouth and ears. This is very convenient when your hands and eyes should be doing something else, such as: driving a car, performing surgery, or (unfortunately) firing your weapons at the enemy. Two approaches are used for computer generated speech: digital recording and vocal tract simulation. In digital recording, the voice of a human speaker is digitized and stored, usually in a compressed form. During playback, the stored data are uncompressed and converted back into an analog signal. An entire hour of recorded speech requires only about three megabytes of storage, well within the capabilities of even small computer systems. This is the most common method of digital speech generation used today. Vocal tract simulators are more complicated, trying to mimic the physical mechanisms by which humans create speech. The human vocal tract is an acoustic cavity with resonate frequencies determined by the size and shape of the chambers. Sound originates in the vocal tract in one of two basic ways, called voiced and fricative sounds. With voiced sounds, vocal cord vibration produces near periodic pulses of air into the vocal cavities. In comparison, fricative sounds originate from the noisy air turbulence at narrow constrictions, such as the teeth and lips. Vocal tract simulators operate by generating digital signals that resemble these two types of excitation. The characteristics of the resonate chamber are simulated by passing the excitation signal through a digital filter with similar resonances. This approach was used in one of the very early DSP success stories, the Speak & Spell, a widely sold electronic learning aid for children. Speech recognition The automated recognition of human speech is immensely more difficult than speech generation. Speech recognition is a classic example of things that the human brain does well, but digital computers do poorly. Digital computers can store and recall vast amounts of data, perform mathematical calculations at blazing speeds, and do repetitive tasks without becoming bored or inefficient. Unfortunately, present day computers perform very poorly when faced with raw sensory data. Teaching a computer to send you a monthly electric bill is easy. Teaching the same computer to understand your voice is a major undertaking. Digital Signal Processing generally approaches the problem of voice recognition in two steps: feature extraction followed by feature matching. Each word in the incoming audio signal is isolated and then analyzed to identify the type of excitation and resonate frequencies. These parameters are then compared with previous examples of spoken words to identify the closest match. Often, these systems are limited to only a few hundred words; can only accept speech with distinct pauses between words; and must be retrained for each individual speaker. While this is adequate for many commercial Chapter 1- The Breadth and Depth of DSP 7 applications, these limitations are humbling when compared to the abilities of human hearing. There is a great deal of work to be done in this area, with tremendous financial rewards for those that produce successful commercial products. Echo Location A common method of obtaining information about a remote object is to bounce a wave off of it. For example, radar operates by transmitting pulses of radio waves, and examining the received signal for echoes from aircraft. In sonar, sound waves are transmitted through the water to detect submarines and other submerged objects. Geophysicists have long probed the earth by setting off explosions and listening for the echoes from deeply buried layers of rock. While these applications have a common thread, each has its own specific problems and needs. Digital Signal Processing has produced revolutionary changes in all three areas. Radar Radar is an acronym for RAdio Detection And Ranging. In the simplest radar system, a radio transmitter produces a pulse of radio frequency energy a few microseconds long. This pulse is fed into a highly directional antenna, where the resulting radio wave propagates away at the speed of light. Aircraft in the path of this wave will reflect a small portion of the energy back toward a receiving antenna, situated near the transmission site. The distance to the object is calculated from the elapsed time between the transmitted pulse and the received echo. The direction to the object is found more simply; you know where you pointed the directional antenna when the echo was received. The operating range of a radar system is determined by two parameters: how much energy is in the initial pulse, and the noise level of the radio receiver. Unfortunately, increasing the energy in the pulse usually requires making the pulse longer. In turn, the longer pulse reduces the accuracy and precision of the elapsed time measurement. This results in a conflict between two important parameters: the ability to detect objects at long range, and the ability to accurately determine an object's distance. DSP has revolutionized radar in three areas, all of which relate to this basic problem. First, DSP can compress the pulse after it is received, providing better distance determination without reducing the operating range. Second, DSP can filter the received signal to decrease the noise. This increases the range, without degrading the distance determination. Third, DSP enables the rapid selection and generation of different pulse shapes and lengths. Among other things, this allows the pulse to be optimized for a particular detection problem. Now the impressive part: much of this is done at a sampling rate comparable to the radio frequency used, at high as several hundred megahertz! When it comes to radar, DSP is as much about high-speed hardware design as it is about algorithms. 8 The Scientist and Engineer's Guide to Digital Signal Processing Sonar Sonar is an acronym for SOund NAvigation and Ranging. It is divided into two categories, active and passive. In active sonar, sound pulses between 2 kHz and 40 kHz are transmitted into the water, and the resulting echoes detected and analyzed. Uses of active sonar include: detection & localization of undersea bodies, navigation, communication, and mapping the sea floor. A maximum operating range of 10 to 100 kilometers is typical. In comparison, passive sonar simply listens to underwater sounds, which includes: natural turbulence, marine life, and mechanical sounds from submarines and surface vessels. Since passive sonar emits no energy, it is ideal for covert operations. You want to detect the other guy, without him detecting you. The most important application of passive sonar is in military surveillance systems that detect and track submarines. Passive sonar typically uses lower frequencies than active sonar because they propagate through the water with less absorption. Detection ranges can be thousands of kilometers. DSP has revolutionized sonar in many of the same areas as radar: pulse generation, pulse compression, and filtering of detected signals. In one view, sonar is simpler than radar because of the lower frequencies involved. In another view, sonar is more difficult than radar because the environment is much less uniform and stable. Sonar systems usually employ extensive arrays of transmitting and receiving elements, rather than just a single channel. By properly controlling and mixing the signals in these many elements, the sonar system can steer the emitted pulse to the desired location and determine the direction that echoes are received from. To handle these multiple channels, sonar systems require the same massive DSP computing power as radar. Reflection seismology As early as the 1920s, geophysicists discovered that the structure of the earth's crust could be probed with sound. Prospectors could set off an explosion and record the echoes from boundary layers more than ten kilometers below the surface. These echo seismograms were interpreted by the raw eye to map the subsurface structure. The reflection seismic method rapidly became the primary method for locating petroleum and mineral deposits, and remains so today. In the ideal case, a sound pulse sent into the ground produces a single echo for each boundary layer the pulse passes through. Unfortunately, the situation is not usually this simple. Each echo returning to the surface must pass through all the other boundary layers above where it originated. This can result in the echo bouncing between layers, giving rise to echoes of echoes being detected at the surface. These secondary echoes can make the detected signal very complicated and difficult to interpret. Digital Signal Processing has been widely used since the 1960s to isolate the primary from the secondary echoes in reflection seismograms. How did the early geophysicists manage without DSP? The answer is simple: they looked in easy places, where multiple reflections were minimized. DSP allows oil to be found in difficult locations, such as under the ocean. Chapter 1- The Breadth and Depth of DSP 9 Image Processing Images are signals with special characteristics. First, they are a measure of a parameter over space (distance), while most signals are a measure of a parameter over time. Second, they contain a great deal of information. For example, more than 10 megabytes can be required to store one second of television video. This is more than a thousand times greater than for a similar length voice signal. Third, the final judge of quality is often a subjective human evaluation, rather than an objective criteria. These special characteristics have made image processing a distinct subgroup within DSP. Medical In 1895, Wilhelm Conrad Röntgen discovered that x-rays could pass through substantial amounts of matter. Medicine was revolutionized by the ability to look inside the living human body. Medical x-ray systems spread throughout the world in only a few years. In spite of its obvious success, medical x-ray imaging was limited by four problems until DSP and related techniques came along in the 1970s. First, overlapping structures in the body can hide behind each other. For example, portions of the heart might not be visible behind the ribs. Second, it is not always possible to distinguish between similar tissues. For example, it may be able to separate bone from soft tissue, but not distinguish a tumor from the liver. Third, x-ray images show anatomy, the body's structure, and not physiology, the body's operation. The x-ray image of a living person looks exactly like the x-ray image of a dead one! Fourth, x-ray exposure can cause cancer, requiring it to be used sparingly and only with proper justification. The problem of overlapping structures was solved in 1971 with the introduction of the first computed tomography scanner (formerly called computed axial tomography, or CAT scanner). Computed tomography (CT) is a classic example of Digital Signal Processing. X-rays from many directions are passed through the section of the patient's body being examined. Instead of simply forming images with the detected x-rays, the signals are converted into digital data and stored in a computer. The information is then used to calculate images that appear to be slices through the body. These images show much greater detail than conventional techniques, allowing significantly better diagnosis and treatment. The impact of CT was nearly as large as the original introduction of x-ray imaging itself. Within only a few years, every major hospital in the world had access to a CT scanner. In 1979, two of CT's principle contributors, Godfrey N. Hounsfield and Allan M. Cormack, shared the Nobel Prize in Medicine. That's good DSP! The last three x-ray problems have been solved by using penetrating energy other than x-rays, such as radio and sound waves. DSP plays a key role in all these techniques. For example, Magnetic Resonance Imaging (MRI) uses magnetic fields in conjunction with radio waves to probe the interior of the human body. Properly adjusting the strength and frequency of the fields cause the atomic nuclei in a localized region of the body to resonate between quantum energy states. This resonance results in the emission of a secondary radio 10 The Scientist and Engineer's Guide to Digital Signal Processing wave, detected with an antenna placed near the body. The strength and other characteristics of this detected signal provide information about the localized region in resonance. Adjustment of the magnetic field allows the resonance region to be scanned throughout the body, mapping the internal structure. This information is usually presented as images, just as in computed tomography. Besides providing excellent discrimination between different types of soft tissue, MRI can provide information about physiology, such as blood flow through arteries. MRI relies totally on Digital Signal Processing techniques, and could not be implemented without them. Space Sometimes, you just have to make the most out of a bad picture. This is frequently the case with images taken from unmanned satellites and space exploration vehicles. No one is going to send a repairman to Mars just to tweak the knobs on a camera! DSP can improve the quality of images taken under extremely unfavorable conditions in several ways: brightness and contrast adjustment, edge detection, noise reduction, focus adjustment, motion blur reduction, etc. Images that have spatial distortion, such as encountered when a flat image is taken of a spherical planet, can also be warped into a correct representation. Many individual images can also be combined into a single database, allowing the information to be displayed in unique ways. For example, a video sequence simulating an aerial flight over the surface of a distant planet. Commercial Imaging Products The large information content in images is a problem for systems sold in mass quantity to the general public. Commercial systems must be cheap, and this doesn't mesh well with large memories and high data transfer rates. One answer to this dilemma is image compression. Just as with voice signals, images contain a tremendous amount of redundant information, and can be run through algorithms that reduce the number of bits needed to represent them. Television and other moving pictures are especially suitable for compression, since most of the image remain the same from frame-to-frame. Commercial imaging products that take advantage of this technology include: video telephones, computer programs that display moving pictures, and digital television. CHAPTER 2 Statistics, Probability and Noise Statistics and probability are used in Digital Signal Processing to characterize signals and the processes that generate them. For example, a primary use of DSP is to reduce interference, noise, and other undesirable components in acquired data. These may be an inherent part of the signal being measured, arise from imperfections in the data acquisition system, or be introduced as an unavoidable byproduct of some DSP operation. Statistics and probability allow these disruptive features to be measured and classified, the first step in developing strategies to remove the offending components. This chapter introduces the most important concepts in statistics and probability, with emphasis on how they apply to acquired signals. Signal and Graph Terminology A signal is a description of how one parameter is related to another parameter. For example, the most common type of signal in analog electronics is a voltage that varies with time. Since both parameters can assume a continuous range of values, we will call this a continuous signal. In comparison, passing this signal through an analog-to-digital converter forces each of the two parameters to be quantized. For instance, imagine the conversion being done with 12 bits at a sampling rate of 1000 samples per second. The voltage is curtailed to 4096 (212) possible binary levels, and the time is only defined at one millisecond increments. Signals formed from parameters that are quantized in this manner are said to be discrete signals or digitized signals. For the most part, continuous signals exist in nature, while discrete signals exist inside computers (although you can find exceptions to both cases). It is also possible to have signals where one parameter is continuous and the other is discrete. Since these mixed signals are quite uncommon, they do not have special names given to them, and the nature of the two parameters must be explicitly stated. Figure 2-1 shows two discrete signals, such as might be acquired with a digital data acquisition system. The vertical axis may represent voltage, light 11 12 The Scientist and Engineer's Guide to Digital Signal Processing intensity, sound pressure, or an infinite number of other parameters. Since we don't know what it represents in this particular case, we will give it the generic label: amplitude. This parameter is also called several other names: the yaxis, the dependent variable, the range, and the ordinate. The horizontal axis represents the other parameter of the signal, going by such names as: the x-axis, the independent variable, the domain, and the abscissa. Time is the most common parameter to appear on the horizontal axis of acquired signals; however, other parameters are used in specific applications. For example, a geophysicist might acquire measurements of rock density at equally spaced distances along the surface of the earth. To keep things general, we will simply label the horizontal axis: sample number. If this were a continuous signal, another label would have to be used, such as: time, distance, x, etc. The two parameters that form a signal are generally not interchangeable. The parameter on the y-axis (the dependent variable) is said to be a function of the parameter on the x-axis (the independent variable). In other words, the independent variable describes how or when each sample is taken, while the dependent variable is the actual measurement. Given a specific value on the x-axis, we can always find the corresponding value on the y-axis, but usually not the other way around. Pay particular attention to the word: domain, a very widely used term in DSP. For instance, a signal that uses time as the independent variable (i.e., the parameter on the horizontal axis), is said to be in the time domain. Another common signal in DSP uses frequency as the independent variable, resulting in the term, frequency domain. Likewise, signals that use distance as the independent parameter are said to be in the spatial domain (distance is a measure of space). The type of parameter on the horizontal axis is the domain of the signal; it's that simple. What if the x-axis is labeled with something very generic, such as sample number? Authors commonly refer to these signals as being in the time domain. This is because sampling at equal intervals of time is the most common way of obtaining signals, and they don't have anything more specific to call it. Although the signals in Fig. 2-1 are discrete, they are displayed in this figure as continuous lines. This is because there are too many samples to be distinguishable if they were displayed as individual markers. In graphs that portray shorter signals, say less than 100 samples, the individual markers are usually shown. Continuous lines may or may not be drawn to connect the markers, depending on how the author wants you to view the data. For instance, a continuous line could imply what is happening between samples, or simply be an aid to help the reader's eye follow a trend in noisy data. The point is, examine the labeling of the horizontal axis to find if you are working with a discrete or continuous signal. Don't rely on an illustrator's ability to draw dots. The variable, N, is widely used in DSP to represent the total number of samples in a signal. For example, N ' 512 for the signals in Fig. 2-1. To Chapter 2- Statistics, Probability and Noise 8 8 b. Mean = 3.0, F = 0.2 a. Mean = 0.5, F = 1 6 6 4 Amplitude Amplitude 13 2 0 4 2 0 -2 -2 -4 -4 0 64 128 192 256 320 Sample number 384 448 512 511 0 64 128 192 256 320 384 448 512 511 Sample number FIGURE 2-1 Examples of two digitized signals with different means and standard deviations. keep the data organized, each sample is assigned a sample number or index. These are the numbers that appear along the horizontal axis. Two notations for assigning sample numbers are commonly used. In the first notation, the sample indexes run from 1 to N (e.g., 1 to 512). In the second notation, the sample indexes run from 0 to N & 1 (e.g., 0 to 511). Mathematicians often use the first method (1 to N), while those in DSP commonly uses the second (0 to N & 1 ). In this book, we will use the second notation. Don't dismiss this as a trivial problem. It will confuse you sometime during your career. Look out for it! Mean and Standard Deviation The mean, indicated by µ (a lower case Greek mu), is the statistician's jargon for the average value of a signal. It is found just as you would expect: add all of the samples together, and divide by N. It looks like this in mathematical form: EQUATION 2-1 Calculation of a signal's mean. The signal is contained in x0 through xN-1, i is an index that runs through these values, and µ is the mean. µ ' 1 j xi N &1 N i'0 In words, sum the values in the signal, xi , by letting the index, i, run from 0 to N & 1 . Then finish the calculation by dividing the sum by N. This is identical to the equation: µ ' (x0% x1% x2% þ% xN & 1) /N . If you are not already familiar with E (upper case Greek sigma) being used to indicate summation, study these equations carefully, and compare them with the computer program in Table 2-1. Summations of this type are abundant in DSP, and you need to understand this notation fully. 14 The Scientist and Engineer's Guide to Digital Signal Processing In electronics, the mean is commonly called the DC (direct current) value. Likewise, AC (alternating current) refers to how the signal fluctuates around the mean value. If the signal is a simple repetitive waveform, such as a sine or square wave, its excursions can be described by its peak-to-peak amplitude. Unfortunately, most acquired signals do not show a well defined peak-to-peak value, but have a random nature, such as the signals in Fig. 2-1. A more generalized method must be used in these cases, called the standard deviation, denoted by F (a lower case Greek sigma). As a starting point, the expression,*xi & µ*, describes how far the i th sample deviates (differs) from the mean. The average deviation of a signal is found by summing the deviations of all the individual samples, and then dividing by the number of samples, N. Notice that we take the absolute value of each deviation before the summation; otherwise the positive and negative terms would average to zero. The average deviation provides a single number representing the typical distance that the samples are from the mean. While convenient and straightforward, the average deviation is almost never used in statistics. This is because it doesn't fit well with the physics of how signals operate. In most cases, the important parameter is not the deviation from the mean, but the power represented by the deviation from the mean. For example, when random noise signals combine in an electronic circuit, the resultant noise is equal to the combined power of the individual signals, not their combined amplitude. The standard deviation is similar to the average deviation, except the averaging is done with power instead of amplitude. This is achieved by squaring each of the deviations before taking the average (remember, power % voltage2). To finish, the square root is taken to compensate for the initial squaring. In equation form, the standard deviation is calculated: EQUATION 2-2 Calculation of the standard deviation of a signal. The signal is stored in xi , µ is the mean found from Eq. 2-1, N is the number of samples, and σ is the standard deviation. F ' 2 1 2 j(xi & µ ) N &1 N& 1 i ' 0 In the alternative notation: F ' (x0& µ)2 % (x1& µ)2 % þ% (xN & 1& µ)2 / (N & 1) . Notice that the average is carried out by dividing by N & 1 instead of N. This is a subtle feature of the equation that will be discussed in the next section. The term, F2 , occurs frequently in statistics and is given the name variance. The standard deviation is a measure of how far the signal fluctuates from the mean. The variance represents the power of this fluctuation. Another term you should become familiar with is the rms (root-mean-square) value, frequently used in electronics. By definition, the standard deviation only measures the AC portion of a signal, while the rms value measures both the AC and DC components. If a signal has no DC component, its rms value is identical to its standard deviation. Figure 2-2 shows the relationship between the standard deviation and the peak-to-peak value of several common waveforms. Chapter 2- Statistics, Probability and Noise b. Triangle wave, Vpp = 12 F a. Square Wave, Vpp = 2F F Vpp F Vpp c. Sine wave, Vpp = 2 2 F Vpp 15 d. Random noise, Vpp . 6-8 F F Vpp F FIGURE 2-2 Ratio of the peak-to-peak amplitude to the standard deviation for several common waveforms. For the square wave, this ratio is 2; for the triangle wave it is 12 ' 3.46 ; for the sine wave it is 2 2 ' 2.83 . While random noise has no exact peak-to-peak value, it is approximately 6 to 8 times the standard deviation. Table 2-1 lists a computer routine for calculating the mean and standard deviation using Eqs. 2-1 and 2-2. The programs in this book are intended to convey algorithms in the most straightforward way; all other factors are treated as secondary. Good programming techniques are disregarded if it makes the program logic more clear. For instance: a simplified version of BASIC is used, line numbers are included, the only control structure allowed is the FOR-NEXT loop, there are no I/O statements, etc. Think of these programs as an alternative way of understanding the equations used 100 CALCULATION OF THE MEAN AND STANDARD DEVIATION 110 ' 120 DIM X[511] 'The signal is held in X[0] to X[511] 130 N% = 512 'N% is the number of points in the signal 140 ' 150 GOSUB XXXX 'Mythical subroutine that loads the signal into X[ ] 160 ' 170 MEAN = 0 'Find the mean via Eq. 2-1 180 FOR I% = 0 TO N%-1 190 MEAN = MEAN + X[I%] 200 NEXT I% 210 MEAN = MEAN/N% 220 ' 230 VARIANCE = 0 'Find the standard deviation via Eq. 2-2 240 FOR I% = 0 TO N%-1 250 VARIANCE = VARIANCE + ( X[I%] - MEAN )^2 260 NEXT I% 270 VARIANCE = VARIANCE/(N%-1) 280 SD = SQR(VARIANCE) 290 ' 300 PRINT MEAN SD 'Print the calculated mean and standard deviation 310 ' 320 END TABLE 2-1 16 The Scientist and Engineer's Guide to Digital Signal Processing in DSP. If you can't grasp one, maybe the other will help. In BASIC, the % character at the end of a variable name indicates it is an integer. All other variables are floating point. Chapter 4 discusses these variable types in detail. This method of calculating the mean and standard deviation is adequate for many applications; however, it has two limitations. First, if the mean is much larger than the standard deviation, Eq. 2-2 involves subtracting two numbers that are very close in value. This can result in excessive round-off error in the calculations, a topic discussed in more detail in Chapter 4. Second, it is often desirable to recalculate the mean and standard deviation as new samples are acquired and added to the signal. We will call this type of calculation: running statistics. While the method of Eqs. 2-1 and 2-2 can be used for running statistics, it requires that all of the samples be involved in each new calculation. This is a very inefficient use of computational power and memory. A solution to these problems can be found by manipulating Eqs. 2-1 and 2-2 to provide another equation for calculating the standard deviation: EQUATION 2-3 Calculation of the standard deviation using running statistics. This equation provides the same result as Eq. 2-2, but with less roundoff noise and greater computational efficiency. The signal is expressed in terms of three accumulated parameters: N, the total number of samples; sum, the sum of these samples; and sum of squares, the sum of the squares of the samples. The mean and standard deviation are then calculated from these three accumulated parameters. F 2 ' 1 N& 1 j N&1 i' 0 2 xi & j xi N&1 1 N 2 i '0 or using a simpler notation, F2 ' 1 N& 1 sum of squares & sum 2 N While moving through the signal, a running tally is kept of three parameters: (1) the number of samples already processed, (2) the sum of these samples, and (3) the sum of the squares of the samples (that is, square the value of each sample and add the result to the accumulated value). After any number of samples have been processed, the mean and standard deviation can be efficiently calculated using only the current value of the three parameters. Table 2-2 shows a program that reports the mean and standard deviation in this manner as each new sample is taken into account. This is the method used in hand calculators to find the statistics of a sequence of numbers. Every time you enter a number and press the E (summation) key, the three parameters are updated. The mean and standard deviation can then be found whenever desired, without having to recalculate the entire sequence. Chapter 2- Statistics, Probability and Noise 17 100 'MEAN AND STANDARD DEVIATION USING RUNNING STATISTICS 110 ' 120 DIM X[511] 'The signal is held in X[0] to X[511] 130 ' 140 GOSUB XXXX 'Mythical subroutine that loads the signal into X[ ] 150 ' 160 N% = 0 'Zero the three running parameters 170 SUM = 0 180 SUMSQUARES = 0 190 ' 200 FOR I% = 0 TO 511 'Loop through each sample in the signal 210 ' 220 N% = N%+1 'Update the three parameters 230 SUM = SUM + X(I%) 240 SUMSQUARES = SUMSQUARES + X(I%)^2 250 ' 260 MEAN = SUM/N% 'Calculate mean and standard deviation via Eq. 2-3 270 VARIANCE = (SUMSQUARES - SUM^2/N%) / (N%-1) 280 SD = SQR(VARIANCE) 290 ' 300 PRINT MEAN SD 'Print the running mean and standard deviation 310 ' 320 NEXT I% 330 ' 340 END TABLE 2-2 Before ending this discussion on the mean and standard deviation, two other terms need to be mentioned. In some situations, the mean describes what is being measured, while the standard deviation represents noise and other interference. In these cases, the standard deviation is not important in itself, but only in comparison to the mean. This gives rise to the term: signal-to-noise ratio (SNR), which is equal to the mean divided by the standard deviation. Another term is also used, the coefficient of variation (CV). This is defined as the standard deviation divided by the mean, multiplied by 100 percent. For example, a signal (or other group of measure values) with a CV of 2%, has an SNR of 50. Better data means a higher value for the SNR and a lower value for the CV. Signal vs. Underlying Process Statistics is the science of interpreting numerical data, such as acquired signals. In comparison, probability is used in DSP to understand the processes that generate signals. Although they are closely related, the distinction between the acquired signal and the underlying process is key to many DSP techniques. For example, imagine creating a 1000 point signal by flipping a coin 1000 times. If the coin flip is heads, the corresponding sample is made a value of one. On tails, the sample is set to zero. The process that created this signal has a mean of exactly 0.5, determined by the relative probability of each possible outcome: 50% heads, 50% tails. However, it is unlikely that the actual 1000 point signal will have a mean of exactly 0.5. Random chance 18 The Scientist and Engineer's Guide to Digital Signal Processing will make the number of ones and zeros slightly different each time the signal is generated. The probabilities of the underlying process are constant, but the statistics of the acquired signal change each time the experiment is repeated. This random irregularity found in actual data is called by such names as: statistical variation, statistical fluctuation, and statistical noise. This presents a bit of a dilemma. When you see the terms: mean and standard deviation, how do you know if the author is referring to the statistics of an actual signal, or the probabilities of the underlying process that created the signal? Unfortunately, the only way you can tell is by the context. This is not so for all terms used in statistics and probability. For example, the histogram and probability mass function (discussed in the next section) are matching concepts that are given separate names. Now, back to Eq. 2-2, calculation of the standard deviation. As previously mentioned, this equation divides by N-1 in calculating the average of the squared deviations, rather than simply by N. To understand why this is so, imagine that you want to find the mean and standard deviation of some process that generates signals. Toward this end, you acquire a signal of N samples from the process, and calculate the mean of the signal via Eq. 2.1. You can then use this as an estimate of the mean of the underlying process; however, you know there will be an error due to statistical noise. In particular, for random signals, the typical error between the mean of the N points, and the mean of the underlying process, is given by: EQUATION 2-4 Typical error in calculating the mean of an underlying process by using a finite number of samples, N. The parameter, σ , is the standard deviation. Typical error ' F N 1/2 If N is small, the statistical noise in the calculated mean will be very large. In other words, you do not have access to enough data to properly characterize the process. The larger the value of N, the smaller the expected error will become. A milestone in probability theory, the Strong Law of Large Numbers, guarantees that the error becomes zero as N approaches infinity. In the next step, we would like to calculate the standard deviation of the acquired signal, and use it as an estimate of the standard deviation of the underlying process. Herein lies the problem. Before you can calculate the standard deviation using Eq. 2-2, you need to already know the mean, µ. However, you don't know the mean of the underlying process, only the mean of the N point signal, which contains an error due to statistical noise. This error tends to reduce the calculated value of the standard deviation. To compensate for this, N is replaced by N-1. If N is large, the difference doesn't matter. If N is small, this replacement provides a more accurate Chapter 2- Statistics, Probability and Noise 8 8 a. Changing mean and standard deviation b. Changing mean, constant standard deviation 6 6 4 4 Amplitude Amplitude 19 2 0 -2 2 0 -2 -4 -4 0 64 128 192 256 320 Sample number 384 448 512 511 0 64 128 192 256 320 Sample number 384 448 512 511 FIGURE 2-3 Examples of signals generated from nonstationary processes. In (a), both the mean and standard deviation change. In (b), the standard deviation remains a constant value of one, while the mean changes from a value of zero to two. It is a common analysis technique to break these signals into short segments, and calculate the statistics of each segment individually. estimate of the standard deviation of the underlying process. In other words, Eq. 2-2 is an estimate of the standard deviation of the underlying process. If we divided by N in the equation, it would provide the standard deviation of the acquired signal. As an illustration of these ideas, look at the signals in Fig. 2-3, and ask: are the variations in these signals a result of statistical noise, or is the underlying process changing? It probably isn't hard to convince yourself that these changes are too large for random chance, and must be related to the underlying process. Processes that change their characteristics in this manner are called nonstationary. In comparison, the signals previously presented in Fig. 2-1 were generated from a stationary process, and the variations result completely from statistical noise. Figure 2-3b illustrates a common problem with nonstationary signals: the slowly changing mean interferes with the calculation of the standard deviation. In this example, the standard deviation of the signal, over a short interval, is one. However, the standard deviation of the entire signal is 1.16. This error can be nearly eliminated by breaking the signal into short sections, and calculating the statistics for each section individually. If needed, the standard deviations for each of the sections can be averaged to produce a single value. The Histogram, Pmf and Pdf Suppose we attach an 8 bit analog-to-digital converter to a computer, and acquire 256,000 samples of some signal. As an example, Fig. 2-4a shows 128 samples that might be a part of this data set. The value of each sample will be one of 256 possibilities, 0 through 255. The histogram displays the number of samples there are in the signal that have each of these possible values. Figure (b) shows the histogram for the 128 samples in (a). For 20 The Scientist and Engineer's Guide to Digital Signal Processing 9 255 a. 128 samples of 8 bit signal Number of occurences 192 Amplitude b. 128 point histogram 8 128 64 7 6 5 4 3 2 1 0 0 0 16 32 48 64 80 Sample number 96 112 128 127 90 100 110 120 130 140 Value of sample 150 160 170 150 160 170 10000 c. 256,000 point histogram 8000 Number of occurences FIGURE 2-4 Examples of histograms. Figure (a) shows 128 samples from a very long signal, with each sample being an integer between 0 and 255. Figures (b) and (c) shows histograms using 128 and 256,000 samples from the signal, respectively. As shown, the histogram is smoother when more samples are used. 6000 4000 2000 0 90 100 110 120 130 140 Value of sample example, there are 2 samples that have a value of 110, 8 samples that have a value of 131, 0 samples that have a value of 170, etc. We will represent the histogram by Hi, where i is an index that runs from 0 to M-1, and M is the number of possible values that each sample can take on. For instance, H50 is the number of samples that have a value of 50. Figure (c) shows the histogram of the signal using the full data set, all 256k points. As can be seen, the larger number of samples results in a much smoother appearance. Just as with the mean, the statistical noise (roughness) of the histogram is inversely proportional to the square root of the number of samples used. From the way it is defined, the sum of all of the values in the histogram must be equal to the number of points in the signal: EQUATION 2-5 The sum of all of the values in the histogram is equal to the number of points in the signal. In this equation, H i is the histogram, N is the number of points in the signal, and M is the number of points in the histogram. N ' j Hi M&1 i '0 The histogram can be used to efficiently calculate the mean and standard deviation of very large data sets. This is especially important for images, which can contain millions of samples. The histogram groups samples Chapter 2- Statistics, Probability and Noise 21 together that have the same value. This allows the statistics to be calculated by working with a few groups, rather than a large number of individual samples. Using this approach, the mean and standard deviation are calculated from the histogram by the equations: EQUATION 2-6 Calculation of the mean from the histogram. This can be viewed as combining all samples having the same value into groups, and then using Eq. 2-1 on each group. EQUATION 2-7 Calculation of the standard deviation from the histogram. This is the same concept as Eq. 2-2, except that all samples having the same value are operated on at once. µ ' F2 ' 1 1 j i Hi M&1 N i '0 2 j (i & µ ) Hi M &1 N & 1 i '0 Table 2-3 contains a program for calculating the histogram, mean, and standard deviation using these equations. Calculation of the histogram is very fast, since it only requires indexing and incrementing. In comparison, 100 'CALCULATION OF THE HISTOGRAM, MEAN, AND STANDARD DEVIATION 110 ' 120 DIM X%[25000] 'X%[0] to X%[25000] holds the signal being processed 130 DIM H%[255] 'H%[0] to H%[255] holds the histogram 140 N% = 25001 'Set the number of points in the signal 150 ' 160 FOR I% = 0 TO 255 'Zero the histogram, so it can be used as an accumulator 170 H%[I%] = 0 180 NEXT I% 190 ' 200 GOSUB XXXX 'Mythical subroutine that loads the signal into X%[ ] 210 ' 220 FOR I% = 0 TO 25000 'Calculate the histogram for 25001 points 230 H%[ X%[I%] ] = H%[ X%[I%] ] + 1 240 NEXT I% 250 ' 260 MEAN = 0 'Calculate the mean via Eq. 2-6 270 FOR I% = 0 TO 255 280 MEAN = MEAN + I% * H%[I%] 290 NEXT I% 300 MEAN = MEAN / N% 310 ' 320 VARIANCE = 0 'Calculate the standard deviation via Eq. 2-7 330 FOR I% = 0 TO 255 340 VARIANCE = VARIANCE + H[I%] * (I%-MEAN)^2 350 NEXT I% 360 VARIANCE = VARIANCE / (N%-1) 370 SD = SQR(VARIANCE) 380 ' 390 PRINT MEAN SD 'Print the calculated mean and standard deviation. 400 ' 410 END TABLE 2-3 22 The Scientist and Engineer's Guide to Digital Signal Processing calculating the mean and standard deviation requires the time consuming operations of addition and multiplication. The strategy of this algorithm is to use these slow operations only on the few numbers in the histogram, not the many samples in the signal. This makes the algorithm much faster than the previously described methods. Think a factor of ten for very long signals with the calculations being performed on a general purpose computer. The notion that the acquired signal is a noisy version of the underlying process is very important; so important that some of the concepts are given different names. The histogram is what is formed from an acquired signal. The corresponding curve for the underlying process is called the probability mass function (pmf). A histogram is always calculated using a finite number of samples, while the pmf is what would be obtained with an infinite number of samples. The pmf can be estimated (inferred) from the histogram, or it may be deduced by some mathematical technique, such as in the coin flipping example. Figure 2-5 shows an example pmf, and one of the possible histograms that could be associated with it. The key to understanding these concepts rests in the units of the vertical axis. As previously described, the vertical axis of the histogram is the number of times that a particular value occurs in the signal. The vertical axis of the pmf contains similar information, except expressed on a fractional basis. In other words, each value in the histogram is divided by the total number of samples to approximate the pmf. This means that each value in the pmf must be between zero and one, and that the sum of all of the values in the pmf will be equal to one. The pmf is important because it describes the probability that a certain value will be generated. For example, imagine a signal with the pmf of Fig. 2-5b, such as previously shown in Fig. 2-4a. What is the probability that a sample taken from this signal will have a value of 120? Figure 2-5b provides the answer, 0.03, or about 1 chance in 34. What is the probability that a randomly chosen sample will have a value greater than 150? Adding up the values in the pmf for: 151, 152, 153, @@@ , 255, provides the answer, 0.0122, or about 1 chance in 82. Thus, the signal would be expected to have a value exceeding 150 on an average of every 82 points. What is the probability that any one sample will be between 0 and 255? Summing all of the values in the histogram produces the probability of 1.00, a certainty that this will occur. The histogram and pmf can only be used with discrete data, such as a digitized signal residing in a computer. A similar concept applies to continuous signals, such as voltages appearing in analog electronics. The probability density function (pdf), also called the probability distribution function, is to continuous signals what the probability mass function is to discrete signals. For example, imagine an analog signal passing through an analog-to-digital converter, resulting in the digitized signal of Fig. 2-4a. For simplicity, we will assume that voltages between 0 and 255 millivolts become digitized into digital numbers between 0 and 255. The pmf of this digital Chapter 2- Statistics, Probability and Noise 10000 0.060 b. Probability Mass Function (pmf) Probability of occurence a. Histogram Number of occurences 23 8000 6000 4000 2000 0 0.050 0.040 0.030 0.020 0.010 0.000 100 110 120 130 140 Value of sample 150 160 170 FIGURE 2-5 The relationship between (a) the histogram, (b) the probability mass function (pmf), and (c) the probability density function (pdf). The histogram is calculated from a finite number of samples. The pmf describes the probabilities of the underlying process. The pdf is similar to the pmf, but is used with continuous rather than discrete signals. Even though the vertical axis of (b) and (c) have the same values (0 to 0.06), this is only a coincidence of this example. The amplitude of these three curves is determined by: (a) the sum of the values in the histogram being equal to the number of samples in the signal; (b) the sum of the values in the pmf being equal to one, and (c) the area under the pdf curve being equal to one. 90 100 110 120 130 140 Value of sample 150 160 170 0.060 c. Probability Density Function (pdf) 0.050 Probability density 90 0.040 0.030 0.020 0.010 0.000 90 100 110 120 130 140 150 Signal level (millivolts) 160 170 signal is shown by the markers in Fig. 2-5b. Similarly, the pdf of the analog signal is shown by the continuous line in (c), indicating the signal can take on a continuous range of values, such as the voltage in an electronic circuit. The vertical axis of the pdf is in units of probability density, rather than just probability. For example, a pdf of 0.03 at 120.5 does not mean that the a voltage of 120.5 millivolts will occur 3% of the time. In fact, the probability of the continuous signal being exactly 120.5 millivolts is infinitesimally small. This is because there are an infinite number of possible values that the signal needs to divide its time between: 120.49997, 120.49998, 120.49999, etc. The chance that the signal happens to be exactly 120.50000þ is very remote indeed! To calculate a probability, the probability density is multiplied by a range of values. For example, the probability that the signal, at any given instant, will be between the values of 120 and 121 is: 121&120 × 0.03 ' 0.03 . The probability that the signal will be between 120.4 and 120.5 is: 120.5&120.4 × 0.03 ' 0.003 , etc. If the pdf is not constant over the range of interest, the multiplication becomes the integral of the pdf over that range. In other words, the area under the pdf bounded by the specified values. Since the value of the signal must always be something, the total area under the pdf 24 The Scientist and Engineer's Guide to Digital Signal Processing curve, the integral from &4 to %4, will always be equal to one. This is analogous to the sum of all of the pmf values being equal to one, and the sum of all of the histogram values being equal to N. The histogram, pmf, and pdf are very similar concepts. Mathematicians always keep them straight, but you will frequently find them used interchangeably (and therefore, incorrectly) by many scientists and 2 pdf a. Square wave Amplitude 1 0 -1 -2 0 16 32 48 64 80 96 112 128 127 Time (or other variable) 2 pdf b. Triangle wave Amplitude 1 0 -1 -2 0 16 32 48 64 80 96 112 127 128 Time (or other variable) 2 pdf c. Random noise 1 Amplitude FIGURE 2-6 Three common waveforms and their probability density functions. As in these examples, the pdf graph is often rotated one-quarter turn and placed at the side of the signal it describes. The pdf of a square wave, shown in (a), consists of two infinitesimally narrow spikes, corresponding to the signal only having two possible values. The pdf of the triangle wave, (b), has a constant value over a range, and is often called a uniform distribution. The pdf of random noise, as in (c), is the most interesting of all, a bell shaped curve known as a Gaussian. 0 -1 -2 0 16 32 48 64 80 96 Time (or other variable) 112 128 127 Chapter 2- Statistics, Probability and Noise 25 engineers. Figure 2-6 shows three continuous waveforms and their pdfs. If these were discrete signals, signified by changing the horizontal axis labeling to "sample number," pmfs would be used. A problem occurs in calculating the histogram when the number of levels each sample can take on is much larger than the number of samples in the signal. This is always true for signals represented in floating point notation, where each sample is stored as a fractional value. For example, integer representation might require the sample value to be 3 or 4, while floating point allows millions of possible fractional values between 3 and 4. The previously described approach for calculating the histogram involves counting the number of samples that have each of the possible quantization levels. This is not possible with floating point data because there are billions of possible levels that would have to be taken into account. Even worse, nearly all of these possible levels would have no samples that correspond to them. For example, imagine a 10,000 sample signal, with each sample having one billion possible values. The conventional histogram would consist of one billion data points, with all but about 10,000 of them having a value of zero. The solution to these problems is a technique called binning. This is done by arbitrarily selecting the length of the histogram to be some convenient number, such as 1000 points, often called bins. The value of each bin represent the total number of samples in the signal that have a value within a certain range. For example, imagine a floating point signal that contains values from 0.0 to 10.0, and a histogram with 1000 bins. Bin 0 in the histogram is the number of samples in the signal with a value between 0 and 0.01, bin 1 is the number of samples with a value between 0.01 and 0.02, and so forth, up to bin 999 containing the number of samples with a value between 9.99 and 10.0. Table 2-4 presents a program for calculating a binned histogram in this manner. 100 'CALCULATION OF BINNED HISTOGRAM 110 ' 120 DIM X[25000] 'X[0] to X[25000] holds the floating point signal, 130 ' 'with each sample being in the range: 0.0 to 10.0 140 DIM H%[999] 'H%[0] to H%[999] holds the binned histogram 150 ' 160 FOR I% = 0 TO 999 'Zero the binned histogram for use as an accumulator 170 H%[I%] = 0 180 NEXT I% 190 ' 200 GOSUB XXXX 'Mythical subroutine that loads the signal into X%[ ] 210 ' 220 FOR I% = 0 TO 25000 'Calculate the binned histogram for 25001 points 230 BINNUM% = INT( X[I%] * .01 ) 240 H%[ BINNUM%] = H%[ BINNUM%] + 1 250 NEXT I% 260 ' 270 END TABLE 2-4 26 The Scientist and Engineer's Guide to Digital Signal Processing 0.8 4 b. Histogram of 601 bins Number of occurences a. Example signal Amplitude 3 2 1 0 0.6 0.4 0.2 0 0 50 100 150 200 Sample number 250 0 300 150 300 450 600 6 8 Bin number in histogram 160 c. Histogram of 9 bins Number of occurences FIGURE 2-7 Example of binned histograms. As shown in (a), the signal used in this example is 300 samples long, with each sample a floating point number uniformly distributed between 1 and 3. Figures (b) and (c) show binned histograms of this signal, using 601 and 9 bins, respectively. As shown, a large number of bins results in poor resolution along the vertical axis, while a small number of bins provides poor resolution along the horizontal axis. Using more samples makes the resolution better in both directions. 120 80 40 0 0 2 4 Bin number in histogram How many bins should be used? This is a compromise between two problems. As shown in Fig. 2-7, too many bins makes it difficult to estimate the amplitude of the underlying pmf. This is because only a few samples fall into each bin, making the statistical noise very high. At the other extreme, too few of bins makes it difficult to estimate the underlying pmf in the horizontal direction. In other words, the number of bins controls a tradeoff between resolution in along the y-axis, and resolution along the x-axis. The Normal Distribution Signals formed from random processes usually have a bell shaped pdf. This is called a normal distribution, a Gauss distribution, or a Gaussian, after the great German mathematician, Karl Friedrich Gauss (1777-1855). The reason why this curve occurs so frequently in nature will be discussed shortly in conjunction with digital noise generation. The basic shape of the curve is generated from a negative squared exponent: y (x) ' e & x 2 Chapter 2- Statistics, Probability and Noise 27 This raw curve can be converted into the complete Gaussian by adding an adjustable mean, µ, and standard deviation, F. In addition, the equation must be normalized so that the total area under the curve is equal to one, a requirement of all probability distribution functions. This results in the general form of the normal distribution, one of the most important relations in statistics and probability: EQUATION 2-8 Equation for the normal distribution, also called the Gauss distribution, or simply a Gaussian. In this relation, P(x) is the probability distribution function, µ is the mean, and σ is the standard deviation. 1 P (x) ' 2B F e & (x & µ ) 2 / 2F2 Figure 2-8 shows several examples of Gaussian curves with various means and standard deviations. The mean centers the curve over a particular value, while the standard deviation controls the width of the bell shape. An interesting characteristic of the Gaussian is that the tails drop toward zero very rapidly, much faster than with other common functions such as decaying exponentials or 1/x. For example, at two, four, and six standard 1.5 0.6 b. Mean = 0, F = 1 a. Raw shape, no normalization y (x ) ' e & x 2 0.4 P(x) y(x) 1.0 0.5 0.2 0.0 0.0 -5 -4 -3 -2 -1 0 x 1 2 3 4 5 -5 -4 -3 -2 -1 0 x 1 2 3 4 5 0.3 c. Mean = 20, F = 3 -4F -3F -2F -1F µ 1F 2F 3F 4F 0.2 P(x) FIGURE 2-8 Examples of Gaussian curves. Figure (a) shows the shape of the raw curve without normalization or the addition of adjustable parameters. In (b) and (c), the complete Gaussian curve is shown for various means and standard deviations. 0.1 0.0 0 5 10 15 20 x 25 30 35 40 28 The Scientist and Engineer's Guide to Digital Signal Processing deviations from the mean, the value of the Gaussian curve has dropped to about 1/19, 1/7563, and 1/166,666,666, respectively. This is why normally distributed signals, such as illustrated in Fig. 2-6c, appear to have an approximate peak-to-peak value. In principle, signals of this type can experience excursions of unlimited amplitude. In practice, the sharp drop of the Gaussian pdf dictates that these extremes almost never occur. This results in the waveform having a relatively bounded appearance with an apparent peakto-peak amplitude of about 6-8F. As previously shown, the integral of the pdf is used to find the probability that a signal will be within a certain range of values. This makes the integral of the pdf important enough that it is given its own name, the cumulative distribution function (cdf). An especially obnoxious problem with the Gaussian is that it cannot be integrated using elementary methods. To get around this, the integral of the Gaussian can be calculated by numerical integration. This involves sampling the continuous Gaussian curve very finely, say, a few million points between -10F and +10F. The samples in this discrete signal are then added to simulate integration. The discrete curve resulting from this simulated integration is then stored in a table for use in calculating probabilities. The cdf of the normal distribution is shown in Fig. 2-9, with its numeric values listed in Table 2-5. Since this curve is used so frequently in probability, it is given its own symbol: M(x) (upper case Greek phi). For example, M(&2) has a value of 0.0228. This indicates that there is a 2.28% probability that the value of the signal will be between -4 and two standard deviations below the mean, at any randomly chosen time. Likewise, the value: M(1) ' 0.8413 , means there is an 84.13% chance that the value of the signal, at a randomly selected instant, will be between -4 and one standard deviation above the mean. To calculate the probability that the signal will be will be between two values, it is necessary to subtract the appropriate numbers found in the M(x) table. For example, the probability that the value of the signal, at some randomly chosen time, will be between two standard deviations below the mean and one standard deviation above the mean, is given by: M(1) & M(&2) ' 0.8185 , or 81.85% Using this method, samples taken from a normally distributed signal will be within ±1F of the mean about 68% of the time. They will be within ±2F about 95% of the time, and within ±3F about 99.75% of the time. The probability of the signal being more than 10 standard deviations from the mean is so minuscule, it would be expected to occur for only a few microseconds since the beginning of the universe, about 10 billion years! Equation 2-8 can also be used to express the probability mass function of normally distributed discrete signals. In this case, x is restricted to be one of the quantized levels that the signal can take on, such as one of the 4096 binary values exiting a 12 bit analog-to-digital converter. Ignore the 1/ 2BF term, it is only used to make the total area under the pdf curve equal to one. Instead, you must include whatever term is needed to make the sum of all the values in the pmf equal to one. In most cases, this is done by Chapter 2- Statistics, Probability and Noise 1.0 0.9 0.8 0.7 M(x) 0.6 0.5 0.4 0.3 0.2 0.1 0 -4 -3 -2 -1 0 x 1 2 3 FIGURE 2-9 & TABLE 2-5 M (x), the cumulative distribution function of the normal distribution (mean = 0, standard deviation = 1). These values are calculated by numerically integrating the normal distribution shown in Fig. 2-8b. In words, M (x) is the probability that the value of a normally distributed signal, at some randomly chosen time, will be less than x. In this table, the value of x is expressed in units of standard deviations referenced to the mean. 4 29 x M(x) x M(x) -3.4 -3.3 -3.2 -3.1 -3.0 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 -2.2 -2.1 -2.0 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 .0003 .0005 .0007 .0010 .0013 .0019 .0026 .0035 .0047 .0062 .0082 .0107 .0139 .0179 .0228 .0287 .0359 .0446 .0548 .0668 .0808 .0968 .1151 .1357 .1587 .1841 .2119 .2420 .2743 .3085 .3446 .3821 .4207 .4602 .5000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 .5000 .5398 .5793 .6179 .6554 .6915 .7257 .7580 .7881 .8159 .8413 .8643 .8849 .9032 .9192 .9332 .9452 .9554 .9641 .9713 .9772 .9821 .9861 .9893 .9918 .9938 .9953 .9965 .9974 .9981 .9987 .9990 .9993 .9995 .9997 generating the curve without worrying about normalization, summing all of the unnormalized values, and then dividing all of the values by the sum. Digital Noise Generation Random noise is an important topic in both electronics and DSP. For example, it limits how small of a signal an instrument can measure, the distance a radio system can communicate, and how much radiation is required to produce an xray image. A common need in DSP is to generate signals that resemble various types of random noise. This is required to test the performance of algorithms that must work in the presence of noise. The heart of digital noise generation is the random number generator. Most programming languages have this as a standard function. The BASIC statement: X = RND, loads the variable, X, with a new random number each time the command is encountered. Each random number has a value between zero and one, with an equal probability of being anywhere between these two extremes. Figure 2-10a shows a signal formed by taking 128 samples from this type of random number generator. The mean of the underlying process that generated this signal is 0.5, the standard deviation is 1 / 12 ' 0.29 , and the distribution is uniform between zero and one. 30 The Scientist and Engineer's Guide to Digital Signal Processing Algorithms need to be tested using the same kind of data they will encounter in actual operation. This creates the need to generate digital noise with a Gaussian pdf. There are two methods for generating such signals using a random number generator. Figure 2-10 illustrates the first method. Figure (b) shows a signal obtained by adding two random numbers to form each sample, i.e., X = RND+RND. Since each of the random numbers can run from zero to one, the sum can run from zero to two. The mean is now one, and the standard deviation is 1 / 6 (remember, when independent random signals are added, the variances also add). As shown, the pdf has changed from a uniform d i s t r i b u t i o n t o a triangular distribution. That is, the signal spends more of its time around a value of one, with less time spent near zero or two. Figure (c) takes this idea a step further by adding twelve random numbers to produce each sample. The mean is now six, and the standard deviation is one. What is most important, the pdf has virtually become a Gaussian. This procedure can be used to create a normally distributed noise signal with an arbitrary mean and standard deviation. For each sample in the signal: (1) add twelve random numbers, (2) subtract six to make the mean equal to zero, (3) multiply by the standard deviation desired, and (4) add the desired mean. The mathematical basis for this algorithm is contained in the Central Limit Theorem, one of the most important concepts in probability. In its simplest form, the Central Limit Theorem states that a sum of random numbers becomes normally distributed as more and more of the random numbers are added together. The Central Limit Theorem does not require the individual random numbers be from any particular distribution, or even that the random numbers be from the same distribution. The Central Limit Theorem provides the reason why normally distributed signals are seen so widely in nature. Whenever many different random forces are interacting, the resulting pdf becomes a Gaussian. In the second method for generating normally distributed random numbers, the random number generator is invoked twice, to obtain R1 and R2. A normally distributed random number, X, can then be found: EQUATION 2-9 Generation of normally distributed random numbers. R 1 and R 2 are random numbers with a uniform distribution between zero and one. This results in X being normally distributed with a mean of zero, and a standard deviation of one. The log is base e, and the cosine is in radians. X ' (& 2 log R1)1/2 cos(2BR2 ) Just as before, this approach can generate normally distributed random signals with an arbitrary mean and standard deviation. Take each number generated by this equation, multiply it by the desired standard deviation, and add the desired mean. Chapter 2- Statistics, Probability and Noise 31 12 11 pdf a. X = RND mean = 0.5, F = 1/%12 10 9 Amplitude 8 7 6 5 4 3 2 1 0 0 16 32 48 64 80 Sample number 96 112 128 127 12 11 pdf b. X = RND+RND mean = 1.0, F = 1/%6 10 9 Amplitude 8 7 6 5 4 3 2 1 0 0 16 32 48 64 80 Sample number 96 112 128 127 12 11 pdf c. X = RND+RND+ ... +RND (12 times) mean = 6.0, F = 1 10 9 Amplitude 8 7 6 5 4 3 2 1 0 0 16 32 48 64 80 Sample number 96 112 128 127 FIGURE 2-10 Converting a uniform distribution to a Gaussian distribution. Figure (a) shows a signal where each sample is generated by a random number generator. As indicated by the pdf, the value of each sample is uniformly distributed between zero and one. Each sample in (b) is formed by adding two values from the random number generator. In (c), each sample is created by adding twelve values from the random number generator. The pdf of (c) is very nearly Gaussian, with a mean of six, and a standard deviation of one. 32 The Scientist and Engineer's Guide to Digital Signal Processing Random number generators operate by starting with a seed, a number between zero and one. When the random number generator is invoked, the seed is passed through a fixed algorithm, resulting in a new number between zero and one. This new number is reported as the random number, and is then internally stored to be used as the seed the next time the random number generator is called. The algorithm that transforms the seed into the new random number is often of the form: EQUATION 2-10 Common algorithm for generating uniformly distributed random numbers between zero and one. In this method, S is the seed, R is the new random number, and a,b,& c are appropriately chosen constants. In words, the quantity aS+b is divided by c, and the remainder is taken as R. R ' (aS % b) modulo c In this manner, a continuous sequence of random numbers can be generated, all starting from the same seed. This allows a program to be run multiple times using exactly the same random number sequences. If you want the random number sequence to change, most languages have a provision for reseeding the random number generator, allowing you to choose the number first used as the seed. A common technique is to use the time (as indicated by the system's clock) as the seed, thus providing a new sequence each time the program is run. From a pure mathematical view, the numbers generated in this way cannot be absolutely random since each number is fully determined by the previous number. The term pseudo-random is often used to describe this situation. However, this is not something you should be concerned with. The sequences generated by random number generators are statistically random to an exceedingly high degree. It is very unlikely that you will encounter a situation where they are not adequate. Precision and Accuracy Precision and accuracy are terms used to describe systems and methods that measure, estimate, or predict. In all these cases, there is some parameter you wish to know the value of. This is called the true value, or simply, truth. The method provides a measured value, that you want to be as close to the true value as possible. Precision and accuracy are ways of describing the error that can exist between these two values. Unfortunately, precision and accuracy are used interchangeably in non-technical settings. In fact, dictionaries define them by referring to each other! In spite of this, science and engineering have very specific definitions for each. You should make a point of using the terms correctly, and quietly tolerate others when they used them incorrectly. Chapter 2- Statistics, Probability and Noise 33 As an example, consider an oceanographer measuring water depth using a sonar system. Short bursts of sound are transmitted from the ship, reflected from the ocean floor, and received at the surface as an echo. Sound waves travel at a relatively constant velocity in water, allowing the depth to be found from the elapsed time between the transmitted and received pulses. As with all empirical measurements, a certain amount of error exists between the measured and true values. This particular measurement could be affected by many factors: random noise in the electronics, waves on the ocean surface, plant growth on the ocean floor, variations in the water temperature causing the sound velocity to change, etc. To investigate these effects, the oceanographer takes many successive readings at a location known to be exactly 1000 meters deep (the true value). These measurements are then arranged as the histogram shown in Fig. 2-11. As would be expected from the Central Limit Theorem, the acquired data are normally distributed. The mean occurs at the center of the distribution, and represents the best estimate of the depth based on all of the measured data. The standard deviation defines the width of the distribution, describing how much variation occurs between successive measurements. This situation results in two general types of error that the system can experience. First, the mean may be shifted from the true value. The amount of this shift is called the accuracy of the measurement. Second, individual measurements may not agree well with each other, as indicated by the width of the distribution. This is called the precision of the measurement, and is expressed by quoting the standard deviation, the signal-to-noise ratio, or the CV. Consider a measurement that has good accuracy, but poor precision; the histogram is centered over the true value, but is very broad. Although the measurements are correct as a group, each individual reading is a poor measure of the true value. This situation is said to have poor repeatability; measurements taken in succession don't agree well. Poor precision results from random errors. This is the name given to errors that change each mean true value 140 Number of occurences 120 FIGURE 2-11 Definitions of accuracy and precision. Accuracy is the difference between the true value and the mean of the under-lying process that generates the data. Precision is the spread of the values, specified by the standard deviation, the signal-to-noise ratio, or the CV. Accuracy 100 80 60 Precision 40 20 0 500 600 700 800 900 1000 1100 1200 Ocean depth (meters) 1300 1400 1500 34 The Scientist and Engineer's Guide to Digital Signal Processing time the measurement is repeated. Averaging several measurements will always improve the precision. In short, precision is a measure of random noise. Now, imagine a measurement that is very precise, but has poor accuracy. This makes the histogram very slender, but not centered over the true value. Successive readings are close in value; however, they all have a large error. Poor accuracy results from systematic errors. These are errors that become repeated in exactly the same manner each time the measurement is conducted. Accuracy is usually dependent on how you calibrate the system. For example, in the ocean depth measurement, the parameter directly measured is elapsed time. This is converted into depth by a calibration procedure that relates milliseconds to meters. This may be as simple as multiplying by a fixed velocity, or as complicated as dozens of second order corrections. Averaging individual measurements does nothing to improve the accuracy. In short, accuracy is a measure of calibration. In actual practice there are many ways that precision and accuracy can become intertwined. For example, imagine building an electronic amplifier from 1% resistors. This tolerance indicates that the value of each resistor will be within 1% of the stated value over a wide range of conditions, such as temperature, humidity, age, etc. This error in the resistance will produce a corresponding error in the gain of the amplifier. Is this error a problem of accuracy or precision? The answer depends on how you take the measurements. For example, suppose you build one amplifier and test it several times over a few minutes. The error in gain remains constant with each test, and you conclude the problem is accuracy. In comparison, suppose you build one thousand of the amplifiers. The gain from device to device will fluctuate randomly, and the problem appears to be one of precision. Likewise, any one of these amplifiers will show gain fluctuations in response to temperature and other environmental changes. Again, the problem would be called precision. When deciding which name to call the problem, ask yourself two questions. First: Will averaging successive readings provide a better measurement? If yes, call the error precision; if no, call it accuracy. Second: Will calibration correct the error? If yes, call it accuracy; if no, call it precision. This may require some thought, especially related to how the device will be calibrated, and how often it will be done. CHAPTER 3 ADC and DAC Most of the signals directly encountered in science and engineering are continuous: light intensity that changes with distance; voltage that varies over time; a chemical reaction rate that depends on temperature, etc. Analog-to-Digital Conversion (ADC) and Digital-to-Analog Conversion (DAC) are the processes that allow digital computers to interact with these everyday signals. Digital information is different from its continuous counterpart in two important respects: it is sampled, and it is quantized. Both of these restrict how much information a digital signal can contain. This chapter is about information management: understanding what information you need to retain, and what information you can afford to lose. In turn, this dictates the selection of the sampling frequency, number of bits, and type of analog filtering needed for converting between the analog and digital realms. Quantization First, a bit of trivia. As you know, it is a digital computer, not a digit computer. The information processed is called digital data, not digit data. Why then, is analog-to-digital conversion generally called: digitize and digitization, rather than digitalize and digitalization? The answer is nothing you would expect. When electronics got around to inventing digital techniques, the preferred names had already been snatched up by the medical community nearly a century before. Digitalize and digitalization mean to administer the heart stimulant digitalis. Figure 3-1 shows the electronic waveforms of a typical analog-to-digital conversion. Figure (a) is the analog signal to be digitized. As shown by the labels on the graph, this signal is a voltage that varies over time. To make the numbers easier, we will assume that the voltage can vary from 0 to 4.095 volts, corresponding to the digital numbers between 0 and 4095 that will be produced by a 12 bit digitizer. Notice that the block diagram is broken into two sections, the sample-and-hold (S/H), and the analog-to-digital converter (ADC). As you probably learned in electronics classes, the sample-and-hold is required to keep the voltage entering the ADC constant while the 35 36 The Scientist and Engineer's Guide to Digital Signal Processing conversion is taking place. However, this is not the reason it is shown here; breaking the digitization into these two stages is an important theoretical model for understanding digitization. The fact that it happens to look like common electronics is just a fortunate bonus. As shown by the difference between (a) and (b), the output of the sample-andhold is allowed to change only at periodic intervals, at which time it is made identical to the instantaneous value of the input signal. Changes in the input signal that occur between these sampling times are completely ignored. That is, sampling converts the independent variable (time in this example) from continuous to discrete. As shown by the difference between (b) and (c), the ADC produces an integer value between 0 and 4095 for each of the flat regions in (b). This introduces an error, since each plateau can be any voltage between 0 and 4.095 volts. For example, both 2.56000 volts and 2.56001 volts will be converted into digital number 2560. In other words, quantization converts the dependent variable (voltage in this example) from continuous to discrete. Notice that we carefully avoid comparing (a) and (c), as this would lump the sampling and quantization together. It is important that we analyze them separately because they degrade the signal in different ways, as well as being controlled by different parameters in the electronics. There are also cases where one is used without the other. For instance, sampling without quantization is used in switched capacitor filters. First we will look at the effects of quantization. Any one sample in the digitized signal can have a maximum error of ±½ LSB (Least Significant Bit, jargon for the distance between adjacent quantization levels). Figure (d) shows the quantization error for this particular example, found by subtracting (b) from (c), with the appropriate conversions. In other words, the digital output (c), is equivalent to the continuous input (b), plus a quantization error (d). An important feature of this analysis is that the quantization error appears very much like random noise. This sets the stage for an important model of quantization error. In most cases, quantization results in nothing more than the addition of a specific amount of random noise to the signal. The additive noise is uniformly distributed between ±½ LSB, has a mean of zero, and a standard deviation of 1/ 12 LSB (-0.29 LSB). For example, passing an analog signal through an 8 bit digitizer adds an rms noise of: 0.29 /256 , or about 1/900 of the full scale value. A 12 bit conversion adds a noise of: 0.29 /4096 . 1 /14,000 , while a 16 bit conversion adds: 0.29 /65536 . 1 /227,000 . Since quantization error is a random noise, the number of bits determines the precision of the data. For example, you might make the statement: "We increased the precision of the measurement from 8 to 12 bits." This model is extremely powerful, because the random noise generated by quantization will simply add to whatever noise is already present in the Chapter 3- ADC and DAC 3.025 FIGURE 3-1 Waveforms illustrating the digitization process. The conversion is broken into two stages to allow the effects of sampling to be separated from the effects of quantization. The first stage is the sample-and-hold (S/H), where the only information retained is the instantaneous value of the signal when the periodic sampling takes place. In the second stage, the ADC converts the voltage to the nearest integer number. This results in each sample in the digitized signal having an error of up to ±½ LSB, as shown in (d). As a result, quantization can usually be modeled as simply adding noise to the signal. Amplitude (in volts) a. Original analog signal 3.020 3.015 3.010 3.005 3.000 0 5 10 15 20 25 Time 30 35 40 45 37 50 digital output analog input ADC S/H 3.025 3025 c. Digitized signal 3020 Digital number 3.020 3.015 3.010 3.005 3015 3010 3005 3.000 3000 0 5 10 15 20 25 Time 30 35 40 45 50 0 5 10 15 20 25 d. Quantization error pdf 0.5 0.0 -0.5 -1.0 0 5 10 15 20 25 30 Sample number 35 40 45 50 30 Sample number 1.0 Error (in LSBs) Amplitude (in volts) b. Sampled analog signal 35 40 45 50 38 The Scientist and Engineer's Guide to Digital Signal Processing analog signal. For example, imagine an analog signal with a maximum amplitude of 1.0 volt, and a random noise of 1.0 millivolt rms. Digitizing this signal to 8 bits results in 1.0 volt becoming digital number 255, and 1.0 millivolt becoming 0.255 LSB. As discussed in the last chapter, random noise signals are combined by adding their variances. That is, the signals are added in quadrature: A 2 %B 2 ' C . The total noise on the digitized signal is therefore given by: 0.2552 % 0.292 ' 0.386 LSB. This is an increase of about 50% over the noise already in the analog signal. Digitizing this same signal to 12 bits would produce virtually no increase in the noise, and nothing would be lost due to quantization. When faced with the decision of how many bits are needed in a system, ask two questions: (1) How much noise is already present in the analog signal? (2) How much noise can be tolerated in the digital signal? When isn't this model of quantization valid? Only when the quantization error cannot be treated as random. The only common occurrence of this is when the analog signal remains at about the same value for many consecutive samples, as is illustrated in Fig. 3-2a. The output remains stuck on the same digital number for many samples in a row, even though the analog signal may be changing up to ±½ LSB. Instead of being an additive random noise, the quantization error now looks like a thresholding effect or weird distortion. Dithering is a common technique for improving the digitization of these slowly varying signals. As shown in Fig. 3-2b, a small amount of random noise is added to the analog signal. In this example, the added noise is normally distributed with a standard deviation of 2/3 LSB, resulting in a peakto-peak amplitude of about 3 LSB. Figure (c) shows how the addition of this dithering noise has affected the digitized signal. Even when the original analog signal is changing by less than ±½ LSB, the added noise causes the digital output to randomly toggle between adjacent levels. To understand how this improves the situation, imagine that the input signal is a constant analog voltage of 3.0001 volts, making it one-tenth of the way between the digital levels 3000 and 3001. Without dithering, taking 10,000 samples of this signal would produce 10,000 identical numbers, all having the value of 3000. Next, repeat the thought experiment with a small amount of dithering noise added. The 10,000 values will now oscillate between two (or more) levels, with about 90% having a value of 3000, and 10% having a value of 3001. Taking the average of all 10,000 values results in something close to 3000.1. Even though a single measurement has the inherent ±½ LSB limitation, the statistics of a large number of the samples can do much better. This is quite a strange situation: adding noise provides more information. Circuits for dithering can be quite sophisticated, such as using a computer to generate random numbers, and then passing them through a DAC to produce the added noise. After digitization, the computer can subtract Chapter 3- ADC and DAC 3005 a. Digitization of a small amplitude signal b. Dithering noise added 3004 3004 3003 Millivolts original analog signal analog signal 3002 3003 3002 3001 3001 with added noise digital signal 3000 0 5 10 15 20 25 30 35 Time (or sample number) 3000 40 45 FIGURE 3-2 Illustration of dithering. Figure (a) shows how an analog signal that varies less than ±½ LSB can become stuck on the same quantization level during digitization. Dithering improves this situation by adding a small amount of random noise to the analog signal, such as shown in (b). In this example, the added noise is normally distributed with a standard deviation of 2/3 LSB. As shown in (c), the added noise causes the digitized signal to toggle between adjacent quantization levels, providing more information about the original signal. 50 0 5 10 15 20 25 Time 30 35 40 45 50 40 45 50 3005 Millivolts (or digital number) Millivolts (or digital number) 3005 39 c. Digitization of dithered signal 3004 original analog signal 3003 3002 3001 digital signal 3000 0 5 10 15 20 25 30 35 Time (or sample number) the random numbers from the digital signal using floating point arithmetic. This elegant technique is called subtractive dither, but is only used in the most elaborate systems. The simplest method, although not always possible, is to use the noise already present in the analog signal for dithering. The Sampling Theorem The definition of proper sampling is quite simple. Suppose you sample a continuous signal in some manner. If you can exactly reconstruct the analog signal from the samples, you must have done the sampling properly. Even if the sampled data appears confusing or incomplete, the key information has been captured if you can reverse the process. Figure 3-3 shows several sinusoids before and after digitization. The continious line represents the analog signal entering the ADC, while the square markers are the digital signal leaving the ADC. In (a), the analog signal is a constant DC value, a cosine wave of zero frequency. Since the analog signal is a series of straight lines between each of the samples, all of the information needed to reconstruct the analog signal is contained in the digital data. According to our definition, this is proper sampling. 40 The Scientist and Engineer's Guide to Digital Signal Processing The sine wave shown in (b) has a frequency of 0.09 of the sampling rate. This might represent, for example, a 90 cycle/second sine wave being sampled at 1000 samples/second. Expressed in another way, there are 11.1 samples taken over each complete cycle of the sinusoid. This situation is more complicated than the previous case, because the analog signal cannot be reconstructed by simply drawing straight lines between the data points. Do these samples properly represent the analog signal? The answer is yes, because no other sinusoid, or combination of sinusoids, will produce this pattern of samples (within the reasonable constraints listed below). These samples correspond to only one analog signal, and therefore the analog signal can be exactly reconstructed. Again, an instance of proper sampling. In (c), the situation is made more difficult by increasing the sine wave's frequency to 0.31 of the sampling rate. This results in only 3.2 samples per sine wave cycle. Here the samples are so sparse that they don't even appear to follow the general trend of the analog signal. Do these samples properly represent the analog waveform? Again, the answer is yes, and for exactly the same reason. The samples are a unique representation of the analog signal. All of the information needed to reconstruct the continuous waveform is contained in the digital data. How you go about doing this will be discussed later in this chapter. Obviously, it must be more sophisticated than just drawing straight lines between the data points. As strange as it seems, this is proper sampling according to our definition. In (d), the analog frequency is pushed even higher to 0.95 of the sampling rate, with a mere 1.05 samples per sine wave cycle. Do these samples properly represent the data? No, they don't! The samples represent a different sine wave from the one contained in the analog signal. In particular, the original sine wave of 0.95 frequency misrepresents itself as a sine wave of 0.05 frequency in the digital signal. This phenomenon of sinusoids changing frequency during sampling is called aliasing. Just as a criminal might take on an assumed name or identity (an alias), the sinusoid assumes another frequency that is not its own. Since the digital data is no longer uniquely related to a particular analog signal, an unambiguous reconstruction is impossible. There is nothing in the sampled data to suggest that the original analog signal had a frequency of 0.95 rather than 0.05. The sine wave has hidden its true identity completely; the perfect crime has been committed! According to our definition, this is an example of improper sampling. This line of reasoning leads to a milestone in DSP, the sampling theorem. Frequently this is called the Shannon sampling theorem, or the Nyquist sampling theorem, after the authors of 1940s papers on the topic. The sampling theorem indicates that a continuous signal can be properly sampled, only if it does not contain frequency components above one-half of the sampling rate. For instance, a sampling rate of 2,000 samples/second requires the analog signal to be composed of frequencies below 1000 cycles/second. If frequencies above this limit are present in the signal, they will be aliased to frequencies between 0 and 1000 cycles/second, combining with whatever information that was legitimately there. Chapter 3- ADC and DAC 3 3 b. Analog frequency = 0.09 of sampling rate 2 2 1 1 Amplitude Amplitude a. Analog frequency = 0.0 (i.e., DC) 0 0 -1 -1 -2 -2 -3 -3 Time (or sample number) 3 d. Analog frequency = 0.95 of sampling rate 2 2 1 1 Amplitude Amplitude Time (or sample number) 3 c. Analog frequency = 0.31 of sampling rate 0 0 -1 -1 -2 -2 -3 41 Time (or sample number) -3 Time (or sample number) FIGURE 3-3 Illustration of proper and improper sampling. A continuous signal is sampled properly if the samples contain all the information needed to recreated the original waveform. Figures (a), (b), and (c) illustrate proper sampling of three sinusoidal waves. This is certainly not obvious, since the samples in (c) do not even appear to capture the shape of the waveform. Nevertheless, each of these continuous signals forms a unique one-to-one pair with its pattern of samples. This guarantees that reconstruction can take place. In (d), the frequency of the analog sine wave is greater than the Nyquist frequency (one-half of the sampling rate). This results in aliasing, where the frequency of the sampled data is different from the frequency of the continuous signal. Since aliasing has corrupted the information, the original signal cannot be reconstructed from the samples. Two terms are widely used when discussing the sampling theorem: the Nyquist frequency and the Nyquist rate. Unfortunately, their meaning is not standardized. To understand this, consider an analog signal composed of frequencies between DC and 3 kHz. To properly digitize this signal it must be sampled at 6,000 samples/sec (6 kHz) or higher. Suppose we choose to sample at 8,000 samples/sec (8 kHz), allowing frequencies between DC and 4 kHz to be properly represented. In this situation their are four important frequencies: (1) the highest frequency in the signal, 3 kHz; (2) twice this frequency, 6 kHz; (3) the sampling rate, 8 kHz; and (4) one-half the sampling rate, 4 kHz. Which of these four is the Nyquist frequency and which is the Nyquist rate? It depends who you ask! All of the possible combinations are 42 The Scientist and Engineer's Guide to Digital Signal Processing used. Fortunately, most authors are careful to define how they are using the terms. In this book, they are both used to mean one-half the sampling rate. Figure 3-4 shows how frequencies are changed during aliasing. The key point to remember is that a digital signal cannot contain frequencies above one-half the sampling rate (i.e., the Nyquist frequency/rate). When the frequency of the continuous wave is below the Nyquist rate, the frequency of the sampled data is a match. However, when the continuous signal's frequency is above the Nyquist rate, aliasing changes the frequency into something that can be represented in the sampled data. As shown by the zigzagging line in Fig. 3-4, every continuous frequency above the Nyquist rate has a corresponding digital frequency between zero and one-half the sampling rate. If there happens to be a sinusoid already at this lower frequency, the aliased signal will add to it, resulting in a loss of information. Aliasing is a double curse; information can be lost about the higher and the lower frequency. Suppose you are given a digital signal containing a frequency of 0.2 of the sampling rate. If this signal were obtained by proper sampling, the original analog signal must have had a frequency of 0.2. If aliasing took place during sampling, the digital frequency of 0.2 could have come from any one of an infinite number of frequencies in the analog signal: 0.2, 0.8, 1.2, 1.8, 2.2, þ . Just as aliasing can change the frequency during sampling, it can also change the phase. For example, look back at the aliased signal in Fig. 3-3d. The aliased digital signal is inverted from the original analog signal; one is a sine wave while the other is a negative sine wave. In other words, aliasing has changed the frequency and introduced a 180E phase shift. Only two phase shifts are possible: 0E (no phase shift) and 180E (inversion). The zero phase shift occurs for analog frequencies of 0 to 0.5, 1.0 to 1.5, 2.0 to 2.5, etc. An inverted phase occurs for analog frequencies of 0.5 to 1.0, 1.5 to 2.0, 2.5 to 3.0, and so on. Now we will dive into a more detailed analysis of sampling and how aliasing occurs. Our overall goal is to understand what happens to the information when a signal is converted from a continuous to a discrete form. The problem is, these are very different things; one is a continuous waveform while the other is an array of numbers. This "apples-to-oranges" comparison makes the analysis very difficult. The solution is to introduce a theoretical concept called the impulse train. Figure 3-5a shows an example analog signal. Figure (c) shows the signal sampled by using an impulse train. The impulse train is a continuous signal consisting of a series of narrow spikes (impulses) that match the original signal at the sampling instants. Each impulse is infinitesimally narrow, a concept that will be discussed in Chapter 13. Between these sampling times the value of the waveform is zero. Keep in mind that the impulse train is a theoretical concept, not a waveform that can exist in an electronic circuit. Since both the original analog signal and the impulse train are continuous waveforms, we can make an "apples-apples" comparison between the two. Chapter 3- ADC and DAC DC 43 Nyquist Frequency GOOD ALIASED Digital frequency 0.5 0.4 0.3 0.2 0.1 0.0 Digital phase (degrees) 0.0 0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 2.0 2.5 Continuous frequency (as a fraction of the sampling rate) 270 180 90 0 -90 0.0 Continuous frequency (as a fraction of the sampling rate) FIGURE 3-4 Conversion of analog frequency into digital frequency during sampling. Continuous signals with a frequency less than one-half of the sampling rate are directly converted into the corresponding digital frequency. Above one-half of the sampling rate, aliasing takes place, resulting in the frequency being misrepresented in the digital data. Aliasing always changes a higher frequency into a lower frequency between 0 and 0.5. In addition, aliasing may also change the phase of the signal by 180 degrees. Now we need to examine the relationship between the impulse train and the discrete signal (an array of numbers). This one is easy; in terms of information content, they are identical. If one is known, it is trivial to calculate the other. Think of these as different ends of a bridge crossing between the analog and digital worlds. This means we have achieved our overall goal once we understand the consequences of changing the waveform in Fig. 3-5a into the waveform in Fig. 3.5c. Three continuous waveforms are shown in the left-hand column in Fig. 3-5. The corresponding frequency spectra of these signals are displayed in the righthand column. This should be a familiar concept from your knowledge of electronics; every waveform can be viewed as being composed of sinusoids of varying amplitude and frequency. Later chapters will discuss the frequency domain in detail. (You may want to revisit this discussion after becoming more familiar with frequency spectra). Figure (a) shows an analog signal we wish to sample. As indicated by its frequency spectrum in (b), it is composed only of frequency components between 0 and about 0.33 fs, where fs is the sampling frequency we intend to 44 The Scientist and Engineer's Guide to Digital Signal Processing use. For example, this might be a speech signal that has been filtered to remove all frequencies above 3.3 kHz. Correspondingly, fs would be 10 kHz (10,000 samples/second), our intended sampling rate. Sampling the signal in (a) by using an impulse train produces the signal shown in (c), and its frequency spectrum shown in (d). This spectrum is a duplication of the spectrum of the original signal. Each multiple of the sampling frequency, fs, 2f s, 3f s, 4f s, etc., has received a copy and a left-forright flipped copy of the original frequency spectrum. The copy is called the upper sideband, while the flipped copy is called the lower sideband. Sampling has generated new frequencies. Is this proper sampling? The answer is yes, because the signal in (c) can be transformed back into the signal in (a) by eliminating all frequencies above ½f s. That is, an analog low-pass filter will convert the impulse train, (b), back into the original analog signal, (a). If you are already familiar with the basics of DSP, here is a more technical explanation of why this spectral duplication occurs. (Ignore this paragraph if you are new to DSP). In the time domain, sampling is achieved by multiplying the original signal by an impulse train of unity amplitude spikes. The frequency spectrum of this unity amplitude impulse train is also a unity amplitude impulse train, with the spikes occurring at multiples of the sampling frequency, f s, 2f s, 3f s, 4f s, etc. When two time domain signals are multiplied, their frequency spectra are convolved. This results in the original spectrum being duplicated to the location of each spike in the impulse train's spectrum. Viewing the original signal as composed of both positive and negative frequencies accounts for the upper and lower sidebands, respectively. This is the same as amplitude modulation, discussed in Chapter 10. Figure (e) shows an example of improper sampling, resulting from too low of sampling rate. The analog signal still contains frequencies up to 3.3 kHz, but the sampling rate has been lowered to 5 kHz. Notice that fS , 2f S , 3fS þ along the horizontal axis are spaced closer in (f) than in (d). The frequency spectrum, (f), shows the problem: the duplicated portions of the spectrum have invaded the band between zero and one-half of the sampling frequency. Although (f) shows these overlapping frequencies as retaining their separate identity, in actual practice they add together forming a single confused mess. Since there is no way to separate the overlapping frequencies, information is lost, and the original signal cannot be reconstructed. This overlap occurs when the analog signal contains frequencies greater than one-half the sampling rate, that is, we have proven the sampling theorem. Digital-to-Analog Conversion In theory, the simplest method for digital-to-analog conversion is to pull the samples from memory and convert them into an impulse train. This is Chapter 3- ADC and DAC 45 Time Domain Frequency Domain 3 3 a. Original analog signal b. Original signal's spectrum 2 1 Amplitude Amplitude 2 0 1 -1 -2 0 -3 0 1 2 3 4 0 5 100 fs 200 Time 2fs 400 500 3fs 600 3 3 c. Sampling at 3 times highest frequency 2 d. Duplicated spectrum from sampling lower upper sideband sideband original signal 1 2 impulse train Amplitude Amplitude 300 Frequency 0 1 -1 -2 0 -3 0 1 2 Time 3 4 5 0 100 fs 200 300 Frequency 2fs 400 500 3fs 600 3 3 e. Sampling at 1.5 times highest frequency 2 f. Overlapping spectra causing aliasing 2 impulse train Amplitude Amplitude original signal 1 0 1 -1 -2 0 -3 0 1 2 Time 3 4 5 0 fs 100 2fs 200 3fs 300 Frequency 4fs 400 5fs 500 6fs 600 FIGURE 3-5 The sampling theorem in the time and frequency domains. Figures (a) and (b) show an analog signal composed of frequency components between zero and 0.33 of the sampling frequency, fs. In (c), the analog signal is sampled by converting it to an impulse train. In the frequency domain, (d), this results in the spectrum being duplicated into an infinite number of upper and lower sidebands. Since the original frequencies in (b) exist undistorted in (d), proper sampling has taken place. In comparison, the analog signal in (e) is sampled at 0.66 of the sampling frequency, a value exceeding the Nyquist rate. This results in aliasing, indicated by the sidebands in (f) overlapping. 46 The Scientist and Engineer's Guide to Digital Signal Processing illustrated in Fig. 3-6a, with the corresponding frequency spectrum in (b). As just described, the original analog signal can be perfectly reconstructed by passing this impulse train through a low-pass filter, with the cutoff frequency equal to one-half of the sampling rate. In other words, the original signal and the impulse train have identical frequency spectra below the Nyquist frequency (one-half the sampling rate). At higher frequencies, the impulse train contains a duplication of this information, while the original analog signal contains nothing (assuming aliasing did not occur). While this method is mathematically pure, it is difficult to generate the required narrow pulses in electronics. To get around this, nearly all DACs operate by holding the last value until another sample is received. This is called a zeroth-order hold, the DAC equivalent of the sample-and-hold used during ADC. (A first-order hold is straight lines between the points, a second-order hold uses parabolas, etc.). The zeroth-order hold produces the staircase appearance shown in (c). In the frequency domain, the zeroth-order hold results in the spectrum of the impulse train being multiplied by the dark curve shown in (d), given by the equation: EQUATION 3-1 High frequency amplitude reduction due to the zeroth-order hold. This curve is plotted in Fig. 3-6d. The sampling frequency is represented by fS . For f ' 0, H ( f ) ' 1 . H( f ) ' / 00 sin(Bf /f s ) Bf /f s /0 0 This is of the general form: sin (Bx) /(Bx) , called the sinc function or sinc(x). The sinc function is very common in DSP, and will be discussed in more detail in later chapters. If you already have a background in this material, the zerothorder hold can be understood as the convolution of the impulse train with a rectangular pulse, having a width equal to the sampling period. This results in the frequency domain being multiplied by the Fourier transform of the rectangular pulse, i.e., the sinc function. In Fig. (d), the light line shows the frequency spectrum of the impulse train (the "correct" spectrum), while the dark line shows the sinc. The frequency spectrum of the zeroth order hold signal is equal to the product of these two curves. The analog filter used to convert the zeroth-order hold signal, (c), into the reconstructed signal, (f), needs to do two things: (1) remove all frequencies above one-half of the sampling rate, and (2) boost the frequencies by the reciprocal of the zeroth-order hold's effect, i.e., 1/sinc(x). This amounts to an amplification of about 36% at one-half of the sampling frequency. Figure (e) shows the ideal frequency response of this analog filter. This 1/sinc(x) frequency boost can be handled in four ways: (1) ignore it and accept the consequences, (2) design an analog filter to include the 1/sinc(x) Chapter 3- ADC and DAC 47 Time Domain Frequency Domain 2 3 a. Impulse train b. Spectrum of impulse train 1 Amplitude Amplitude 2 0 -1 1 -2 0 -3 0 1 2 Time 3 4 5 0 100 fs 200 300 Frequency 2fs 400 500 3fs 600 500 3fs 600 500 3fs 600 500 3fs 600 2 3 c. Zeroth-order hold 2 d. Spectrum multiplied by sinc Amplitude Amplitude "correct" spectrum 1 0 -1 sinc 1 -2 0 -3 0 1 2 3 4 0 5 100 fs 200 Time e. Ideal reconstruction filter 1 0 0 100 fs 200 300 Frequency 2fs 400 2 3 f. Reconstructed analog signal 2 g. Reconstructed spectrum 1 Amplitude Amplitude 2fs 400 2 Amplitude FIGURE 3-6 Analysis of digital-to-analog conversion. In (a), the digital data are converted into an impulse train, with the spectrum in (b). This is changed into the reconstructed signal, (f), by using an electronic low-pass filter to remove frequencies above one-half the sampling rate [compare (b) and (g)]. However, most electronic DACs create a zeroth-order hold waveform, (c), instead of an impulse train. The spectrum of the zeroth-order hold is equal to the spectrum of the impulse train multiplied by the sinc function shown in (d). To convert the zeroth-order hold into the reconstructed signal, the analog filter must remove all frequencies above the Nyquist rate, and correct for the sinc, as shown in (e). 300 Frequency 0 -1 1 -2 0 -3 0 1 2 3 Time 4 5 0 100 fs 200 300 Frequency 2fs 400 48 The Scientist and Engineer's Guide to Digital Signal Processing response, (3) use a fancy multirate technique described later in this chapter, or (4) make the correction in software before the DAC (see Chapter 24). Before leaving this section on sampling, we need to dispel a common myth about analog versus digital signals. As this chapter has shown, the amount of information carried in a digital signal is limited in two ways: First, the number of bits per sample limits the resolution of the dependent variable. That is, small changes in the signal's amplitude may be lost in the quantization noise. Second, the sampling rate limits the resolution of the independent variable, i.e., closely spaced events in the analog signal may be lost between the samples. This is another way of saying that frequencies above one-half the sampling rate are lost. Here is the myth: "Since analog signals use continuous parameters, they have infinitely good resolution in both the independent and the dependent variables." Not true! Analog signals are limited by the same two problems as digital signals: noise and bandwidth (the highest frequency allowed in the signal). The noise in an analog signal limits the measurement of the waveform's amplitude, just as quantization noise does in a digital signal. Likewise, the ability to separate closely spaced events in an analog signal depends on the highest frequency allowed in the waveform. To understand this, imagine an analog signal containing two closely spaced pulses. If we place the signal through a low-pass filter (removing the high frequencies), the pulses will blur into a single blob. For instance, an analog signal formed from frequencies between DC and 10 kHz will have exactly the same resolution as a digital signal sampled at 20 kHz. It must, since the sampling theorem guarantees that the two contain the same information. Analog Filters for Data Conversion Figure 3-7 shows a block diagram of a DSP system, as the sampling theorem dictates it should be. Before encountering the analog-to-digital converter, antialias filter Analog Filter Analog Input reconstruction filter Digital Processing ADC Filtered Analog Input Digitized Input Analog Filter DAC Digitized Output S/H Analog Output Analog Output FIGURE 3-7 Analog electronic filters used to comply with the sampling theorem. The electronic filter placed before an ADC is called an antialias filter. It is used to remove frequency components above one-half of the sampling rate that would alias during the sampling. The electronic filter placed after a DAC is called a reconstruction filter. It also eliminates frequencies above the Nyquist rate, and may include a correction for the zeroth-order hold. Chapter 3- ADC and DAC 49 the input signal is processed with an electronic low-pass filter to remove all frequencies above the Nyquist frequency (one-half the sampling rate). This is done to prevent aliasing during sampling, and is correspondingly called an antialias filter. On the other end, the digitized signal is passed through a digital-to-analog converter and another low-pass filter set to the Nyquist frequency. This output filter is called a reconstruction filter, and may include the previously described zeroth-order-hold frequency boost. Unfortunately, there is a serious problem with this simple model: the limitations of electronic filters can be as bad as the problems they are trying to prevent. If your main interest is in software, you are probably thinking that you don't need to read this section. Wrong! Even if you have vowed never to touch an oscilloscope, an understanding of the properties of analog filters is important for successful DSP. First, the characteristics of every digitized signal you encounter will depend on what type of antialias filter was used when it was acquired. If you don't understand the nature of the antialias filter, you cannot understand the nature of the digital signal. Second, the future of DSP is to replace hardware with software. For example, the multirate techniques presented later in this chapter reduce the need for antialias and reconstruction filters by fancy software tricks. If you don't understand the hardware, you cannot design software to replace it. Third, much of DSP is related to digital filter design. A common strategy is to start with an equivalent analog filter, and convert it into software. Later chapters assume you have a basic knowledge of analog filter techniques. Three types of analog filters are commonly used: Chebyshev, Butterworth, and Bessel (also called a Thompson filter). Each of these is designed to optimize a different performance parameter. The complexity of each filter can be adjusted by selecting the number of poles and zeros, mathematical terms that will be discussed in later chapters. The more poles in a filter, the more electronics it requires, and the better it performs. Each of these names describe what the filter does, not a particular arrangement of resistors and capacitors. For example, a six pole Bessel filter can be implemented by many different types of circuits, all of which have the same overall characteristics. For DSP purposes, the characteristics of these filters are more important than how they are constructed. Nevertheless, we will start with a short segment on the electronic design of these filters to provide an overall framework. Figure 3-8 shows a common building block for analog filter design, the modified Sallen-Key circuit. This is named after the authors of a 1950s paper describing the technique. The circuit shown is a two pole low-pass filter that can be configured as any of the three basic types. Table 3-1 provides the necessary information to select the appropriate resistors and capacitors. For example, to design a 1 kHz, 2 pole Butterworth filter, Table 3-1 provides the parameters: k1 = 0.1592 and k2 = 0.586. Arbitrarily selecting R1 = 10K and C = 0.01uF (common values for op amp circuits), R and Rf can be calculated as 15.95K and 5.86K, respectively. Rounding these last two values to the nearest 1% standard resistors, results in R = 15.8K and Rf = 5.90K All of the components should be 1% precision or better. 50 The Scientist and Engineer's Guide to Digital Signal Processing C FIGURE 3-8 The modified Sallen-Key circuit, a building block for active filter design. The circuit shown implements a 2 pole low-pass filter. Higher order filters (more poles) can be formed by cascading stages. Find k1 and k2 from Table 3-1, arbitrarily select R1 and C (try 10K and 0.01µF), and then calculate R and Rf from the equations in the figure. The parameter, fc, is the cutoff frequency of the filter, in hertz. R R C Rf R ' k1 R1 C fc R f ' R1 k2 TABLE 3-1 Parameters for designing Bessel, Butterworth, and Chebyshev (6% ripple) filters. Bessel k1 k2 # poles Butterworth k1 k2 Chebyshev k1 k2 2 stage 1 0.1251 0.268 0.1592 0.586 0.1293 0.842 4 stage 1 stage 2 0.1111 0.0991 0.084 0.759 0.1592 0.1592 0.152 1.235 0.2666 0.1544 0.582 1.660 6 stage 1 stage 2 stage 3 0.0990 0.0941 0.0834 0.040 0.364 1.023 0.1592 0.1592 0.1592 0.068 0.586 1.483 0.4019 0.2072 0.1574 0.537 1.448 1.846 8 stage 1 stage 2 stage 3 stage 4 0.0894 0.0867 0.0814 0.0726 0.024 0.213 0.593 1.184 0.1592 0.1592 0.1592 0.1592 0.038 0.337 0.889 1.610 0.5359 0.2657 0.1848 0.1582 0.522 1.379 1.711 1.913 The particular op amp used isn't critical, as long as the unity gain frequency is more than 30 to 100 times higher than the filter's cutoff frequency. This is an easy requirement as long as the filter's cutoff frequency is below about 100 kHz. Four, six, and eight pole filters are formed by cascading 2,3, and 4 of these circuits, respectively. For example, Fig. 3-9 shows the schematic of a 6 pole 0.01µF 0.01µF 0.01µF 10K 10K 9.53K 9.53K 8.25K 0.01µF 402S 3.65K stage 1 k1 = 0.0990 k2 = 0.040 8.25K 0.01µF 0.01µF 10.2K 10K stage 2 k1 = 0.0941 k2 = 0.364 10K stage 3 k1 = 0.0834 k2 = 1.023 FIGURE 3-9 A six pole Bessel filter formed by cascading three Sallen-Key circuits. This is a low-pass filter with a cutoff frequency of 1 kHz. 10K Chapter 3- ADC and DAC 51 Bessel filter created by cascading three stages. Each stage has different values for k1 and k2 as provided by Table 3-1, resulting in different resistors and capacitors being used. Need a high-pass filter? Simply swap the R and C components in the circuits (leaving Rf and R1 alone). This type of circuit is very common for small quantity manufacturing and R&D applications; however, serious production requires the filter to be made as an integrated circuit. The problem is, it is difficult to make resistors directly in silicon. The answer is the switched capacitor filter. Figure 3-10 illustrates its operation by comparing it to a simple RC network. If a step function is fed into an RC low-pass filter, the output rises exponentially until it matches the input. The voltage on the capacitor doesn't change instantaneously, because the resistor restricts the flow of electrical charge. The switched capacitor filter operates by replacing the basic resistorcapacitor network with two capacitors and an electronic switch. The newly added capacitor is much smaller in value than the already existing capacitor, say, 1% of its value. The switch alternately connects the small capacitor between the input and the output at a very high frequency, typically 100 times faster than the cutoff frequency of the filter. When the switch is connected to the input, the small capacitor rapidly charges to whatever voltage is presently on the input. When the switch is connected to the output, the charge on the small capacitor is transferred to the large capacitor. In a resistor, the rate of charge transfer is determined by its resistance. In a switched capacitor circuit, the rate of charge transfer is determined by the value of the small capacitor and by the switching frequency. This results in a very useful feature of switched capacitor voltage voltage Resistor-Capacitor R C time low R high R time Switched Capacitor time voltage voltage f C/100 C high f low f time FIGURE 3-10 Switched capacitor filter operation. Switched capacitor filters use switches and capacitors to mimic resistors. As shown by the equivalent step responses, two capacitors and one switch can perform the same function as a resistor-capacitor network. 52 The Scientist and Engineer's Guide to Digital Signal Processing filters: the cutoff frequency of the filter is directly proportional to the clock frequency used to drive the switches. This makes the switched capacitor filter ideal for data acquisition systems that operate with more than one sampling rate. These are easy-to-use devices; pay ten bucks and have the performance of an eight pole filter inside a single 8 pin IC. Now for the important part: the characteristics of the three classic filter types. The first performance parameter we want to explore is cutoff frequency sharpness. A low-pass filter is designed to block all frequencies above the cutoff frequency (the stopband), while passing all frequencies below (the passband). Figure 3-11 shows the frequency response of these three filters on a logarithmic (dB) scale. These graphs are shown for filters with a one hertz cutoff frequency, but they can be directly scaled to whatever cutoff frequency you need to use. How do these filters rate? The Chebyshev is clearly the best, the Butterworth is worse, and the Bessel is absolutely ghastly! As you probably surmised, this is what the Chebyshev is designed to do, roll-off (drop in amplitude) as rapidly as possible. Unfortunately, even an 8 pole Chebyshev isn't as good as you would like for an antialias filter. For example, imagine a 12 bit system sampling at 10,000 samples per second. The sampling theorem dictates that any frequency above 5 kHz will be aliased, something you want to avoid. With a little guess work, you decide that all frequencies above 5 kHz must be reduced in amplitude by a factor of 100, insuring that any aliased frequencies will have an amplitude of less than one percent. Looking at Fig. 3-11c, you find that an 8 pole Chebyshev filter, with a cutoff frequency of 1 hertz, doesn't reach an attenuation (signal reduction) of 100 until about 1.35 hertz. Scaling this to the example, the filter's cutoff frequency must be set to 3.7 kHz so that everything above 5 kHz will have the required attenuation. This results in the frequency band between 3.7 kHz and 5 kHz being wasted on the inadequate roll-off of the analog filter. A subtle point: the attenuation factor of 100 in this example is probably sufficient even though there are 4096 steps in 12 bits. From Fig. 3-4, 5100 hertz will alias to 4900 hertz, 6000 hertz will alias to 4000 hertz, etc. You don't care what the amplitudes of the signals between 5000 and 6300 hertz are, because they alias into the unusable region between 3700 hertz and 5000 hertz. In order for a frequency to alias into the filter's passband (0 to 3.7 kHz), it must be greater than 6300 hertz, or 1.7 times the filter's cutoff frequency of 3700 hertz. As shown in Fig. 3-11c, the attenuation provided by an 8 pole Chebyshev filter at 1.7 times the cutoff frequency is about 1300, much more adequate than the 100 we started the analysis with. The moral to this story: In most systems, the frequency band between about 0.4 and 0.5 of the sampling frequency is an unusable wasteland of filter roll-off and aliased signals. This is a direct result of the limitations of analog filters. The frequency response of the perfect low-pass filter is flat across the entire passband. All of the filters look great in this respect in Fig. 3-11, but only because the vertical axis is displayed on a logarithmic scale. Another story is told when the graphs are converted to a linear vertical scale, as is shown Chapter 3- ADC and DAC 53 Log scale Linear scale 10 1.6 a. Bessel 1.2 2 pole 0.1 Amplitude Amplitude a. Bessel 1.4 1 4 ideal 8 0.01 1 ideal 0.8 0.6 0.4 0.001 2 pole 0.2 0 1 2 3 Frequency (hertz) 4 0 5 10 1 2 4 5 3 4 5 4 5 b. Butterworth 1.4 1 1.2 0.1 Amplitude 2 pole 4 0.01 8 1 0.8 0.6 0.4 0.001 2 pole 0.2 0 1 2 3 Frequency (hertz) 4 5 0 10 4 8 0 0.0001 1 2 Frequency (hertz) 1.6 c. Chebyshev (6% ripple) c. Chebyshev (6% ripple) 1.4 1 1.2 2 pole Amplitude Amplitude 3 Frequency (hertz) 1.6 b. Butterworth Amplitude 8 0 0.0001 4 0.1 4 0.01 1 0.8 0.6 8 0.4 0.001 2 pole 0.2 0.0001 0 8 0 1 2 3 Frequency (hertz) 4 FIGURE 3-11 Frequency response of the three filters on a logarithmic scale. The Chebyshev filter has the sharpest roll-off. 5 0 1 4 2 3 Frequency (hertz) FIGURE 3-12 Frequency response of the three filters on a linear scale. The Butterworth filter provides the flattest passband. in Fig. 3-12. Passband ripple can now be seen in the Chebyshev filter (wavy variations in the amplitude of the passed frequencies). In fact, the Chebyshev filter obtains its excellent roll-off by allowing this passband ripple. When more passband ripple is allowed in a filter, a faster roll-off 54 The Scientist and Engineer's Guide to Digital Signal Processing can be achieved. All the Chebyshev filters designed by using Table 3-1 have a passband ripple of about 6% (0.5 dB), a good compromise, and a common choice. A similar design, the elliptic filter, allows ripple in both the passband and the stopband. Although harder to design, elliptic filters can achieve an even better tradeoff between roll-off and passband ripple. In comparison, the Butterworth filter is optimized to provide the sharpest rolloff possible without allowing ripple in the passband. It is commonly called the maximally flat filter, and is identical to a Chebyshev designed for zero passband ripple. The Bessel filter has no ripple in the passband, but the rolloff far worse than the Butterworth. The last parameter to evaluate is the step response, how the filter responds when the input rapidly changes from one value to another. Figure 3-13 shows the step response of each of the three filters. The horizontal axis is shown for filters with a 1 hertz cutoff frequency, but can be scaled (inversely) for higher cutoff frequencies. For example, a 1000 hertz cutoff frequency would show a step response in milliseconds, rather than seconds. The Butterworth and Chebyshev filters overshoot and show ringing (oscillations that slowly decreasing in amplitude). In comparison, the Bessel filter has neither of these nasty problems. 1.6 1.6 1.4 1.4 a. Bessel b. Butterworth 1.2 Amplitude 1.0 2 0.8 4 0.6 8 pole 0.4 1.0 2 0.8 4 8 pole 0.6 0.4 0.2 0.2 0.0 0.0 0 1 2 Time (seconds) 3 4 0 1 2 Time (seconds) 3 4 3 4 1.6 1.4 FIGURE 3-13 Step response of the three filters. The times shown on the horizontal axis correspond to a one hertz cutoff frequency. The Bessel is the optimum filter when overshoot and ringing must be minimized. c. Chebyshev (6% ripple) 1.2 Amplitude Amplitude 1.2 1.0 2 0.8 4 0.6 8 pole 0.4 0.2 0.0 0 1 2 Time (seconds) Chapter 3- ADC and DAC 1.5 1.5 a. Pulse waveform b. After Bessel filter 1.0 Amplitude 1.0 Amplitude 55 0.5 0.0 0.5 0.0 -0.5 -0.5 100 200 Time 300 400 500 FIGURE 3-14 Pulse response of the Bessel and Chebyshev filters. A key property of the Bessel filter is that the rising and falling edges in the filter's output looking similar. In the jargon of the field, this is called linear phase. Figure (b) shows the result of passing the pulse waveform in (a) through a 4 pole Bessel filter. Both edges are smoothed in a similar manner. Figure (c) shows the result of passing (a) through a 4 pole Chebyshev filter. The left edge overshoots on the top, while the right edge overshoots on the bottom. Many applications cannot tolerate this distortion. 0 100 200 Time 300 400 500 400 500 1.5 c. After Chebyshev filter 1.0 Amplitude 0 0.5 0.0 -0.5 0 100 200 Time 300 Figure 3-14 further illustrates this very favorable characteristic of the Bessel filter. Figure (a) shows a pulse waveform, which can be viewed as a rising step followed by a falling step. Figures (b) and (c) show how this waveform would appear after Bessel and Chebyshev filters, respectively. If this were a video signal, for instance, the distortion introduced by the Chebyshev filter would be devastating! The overshoot would change the brightness of the edges of objects compared to their centers. Worse yet, the left side of objects would look bright, while the right side of objects would look dark. Many applications cannot tolerate poor performance in the step response. This is where the Bessel filter shines; no overshoot and symmetrical edges. Selecting The Antialias Filter Table 3-2 summarizes the characteristics of these three filters, showing how each optimizes a particular parameter at the expense of everything else. The Chebyshev optimizes the roll-off, the Butterworth optimizes the passband flatness, and the Bessel optimizes the step response. The selection of the antialias filter depends almost entirely on one issue: how information is represented in the signals you intend to process. While 56 The Scientist and Engineer's Guide to Digital Signal Processing Step Response Frequency Response Voltage gain at DC Overshoot Time to settle to 1% Time to settle to 0.1% Ripple in passband Frequency for x100 attenuation Frequency for x1000 attenuation Bessel 2 pole 4 pole 6 pole 8 pole 1.27 1.91 2.87 4.32 0.4% 0.9% 0.7% 0.4% 0.60 0.66 0.74 0.80 1.12 1.20 1.18 1.16 0% 0% 0% 0% 12.74 4.74 3.65 3.35 40.4 8.45 5.43 4.53 Butterworth 2 pole 4 pole 6 pole 8 pole 1.59 2.58 4.21 6.84 4.3% 10.9% 14.3% 16.4% 1.06 1.68 2.74 3.50 1.66 2.74 3.92 5.12 0% 0% 0% 0% 10.0 3.17 2.16 1.78 31.6 5.62 3.17 2.38 Chebyshev 2 pole 4 pole 6 pole 8 pole 1.84 4.21 10.71 28.58 10.8% 18.2% 21.3% 23.0% 1.10 3.04 5.86 8.34 1.62 5.42 10.4 16.4 6% 6% 6% 6% 12.33 2.59 1.63 1.34 38.9 4.47 2.26 1.66 TABLE 3-2 Characteristics of the three classic filters. The Bessel filter provides the best step response, making it the choice for time domain encoded signals. The Chebyshev and Butterworth filters are used to eliminate frequencies in the stopband, making them ideal for frequency domain encoded signals. Values in this table are in the units of seconds and hertz, for a one hertz cutoff frequency. there are many ways for information to be encoded in an analog waveform, only two methods are common, time domain encoding, and frequency domain encoding. The difference between these two is critical in DSP, and will be a reoccurring theme throughout this book. In frequency domain encoding, the information is contained in sinusoidal waves that combine to form the signal. Audio signals are an excellent example of this. When a person hears speech or music, the perceived sound depends on the frequencies present, and not on the particular shape of the waveform. This can be shown by passing an audio signal through a circuit that changes the phase of the various sinusoids, but retains their frequency and amplitude. The resulting signal looks completely different on an oscilloscope, but sounds identical. The pertinent information has been left intact, even though the waveform has been significantly altered. Since aliasing misplaces and overlaps frequency components, it directly destroys information encoded in the frequency domain. Consequently, digitization of these signals usually involves an antialias filter with a sharp cutoff, such as a Chebyshev, Elliptic, or Butterworth. What about the nasty step response of these filters? It doesn't matter; the encoded information isn't affected by this type of distortion. In contrast, time domain encoding uses the shape of the waveform to store information. For example, physicians can monitor the electrical activity of a Chapter 3- ADC and DAC 57 person's heart by attaching electrodes to their chest and arms (an electrocardiogram or EKG). The shape of the EKG waveform provides the information being sought, such as when the various chambers contract during a heartbeat. Images are another example of this type of signal. Rather than a waveform that varies over time, images encode information in the shape of a waveform that varies over distance. Pictures are formed from regions of brightness and color, and how they relate to other regions of brightness and color. You don't look at the Mona Lisa and say, "My, what an interesting collection of sinusoids." Here's the problem: The sampling theorem is an analysis of what happens in the frequency domain during digitization. This makes it ideal to under-stand the analog-to-digital conversion of signals having their information encoded in the frequency domain. However, the sampling theorem is little help in understanding how time domain encoded signals should be digitized. Let's take a closer look. Figure 3-15 illustrates the choices for digitizing a time domain encoded signal. Figure (a) is an example analog signal to be digitized. In this case, the information we want to capture is the shape of the rectangular pulses. A short burst of a high frequency sine wave is also included in this example signal. This represents wideband noise, interference, and similar junk that always appears on analog signals. The other figures show how the digitized signal would appear with different antialias filter options: a Chebyshev filter, a Bessel filter, and no filter. It is important to understand that none of these options will allow the original signal to be reconstructed from the sampled data. This is because the original signal inherently contains frequency components greater than one-half of the sampling rate. Since these frequencies cannot exist in the digitized signal, the reconstructed signal cannot contain them either. These high frequencies result from two sources: (1) noise and interference, which you would like to eliminate, and (2) sharp edges in the waveform, which probably contain information you want to retain. The Chebyshev filter, shown in (b), attacks the problem by aggressively removing all high frequency components. This results in a filtered analog signal that can be sampled and later perfectly reconstructed. However, the reconstructed analog signal is identical to the filtered signal, not the original signal. Although nothing is lost in sampling, the waveform has been severely distorted by the antialias filter. As shown in (b), the cure is worse than the disease! Don't do it! The Bessel filter, (c), is designed for just this problem. Its output closely resembles the original waveform, with only a gentle rounding of the edges. By adjusting the filter's cutoff frequency, the smoothness of the edges can be traded for elimination of high frequency components in the signal. Using more poles in the filter allows a better tradeoff between these two parameters. A common guideline is to set the cutoff frequency at about one-quarter of the sampling frequency. This results in about two samples 58 The Scientist and Engineer's Guide to Digital Signal Processing along the rising portion of each edge. Notice that both the Bessel and the Chebyshev filter have removed the burst of high frequency noise present in the original signal. The last choice is to use no antialias filter at all, as is shown in (d). This has the strong advantage that the value of each sample is identical to the value of the original analog signal. In other words, it has perfect edge sharpness; a change in the original signal is immediately mirrored in the digital data. The disadvantage is that aliasing can distort the signal. This takes two different forms. First, high frequency interference and noise, such as the example sinusoidal burst, will turn into meaningless samples, as shown in (d). That is, any high frequency noise present in the analog signal will appear as aliased noise in the digital signal. In a more general sense, this is not a problem of the sampling, but a problem of the upstream analog electronics. It is not the ADC's purpose to reduce noise and interference; this is the responsibility of the analog electronics before the digitization takes place. It may turn out that a Bessel filter should be placed before the digitizer to control this problem. However, this means the filter should be viewed as part of the analog processing, not something that is being done for the sake of the digitizer. The second manifestation of aliasing is more subtle. When an event occurs in the analog signal (such as an edge), the digital signal in (d) detects the change on the next sample. There is no information in the digital data to indicate what happens between samples. Now, compare using no filter with using a Bessel filter for this problem. For example, imagine drawing straight lines between the samples in (c). The time when this constructed line crosses one-half the amplitude of the step provides a subsample estimate of when the edge occurred in the analog signal. When no filter is used, this subsample information is completely lost. You don't need a fancy theorem to evaluate how this will affect your particular situation, just a good understanding of what you plan to do with the data once is it acquired. Multirate Data Conversion There is a strong trend in electronics to replace analog circuitry with digital algorithms. Data conversion is an excellent example of this. Consider the design of a digital voice recorder, a system that will digitize a voice signal, store the data in digital form, and later reconstruct the signal for playback. To recreate intelligible speech, the system must capture the frequencies between about 100 and 3000 hertz. However, the analog signal produced by the microphone also contains much higher frequencies, say to 40 kHz. The brute force approach is to pass the analog signal through an eight pole low-pass Chebyshev filter at 3 kHz, and then sample at 8 kHz. On the other end, the DAC reconstructs the analog signal at 8 kHz with a zeroth order hold. Another Chebyshev filter at 3 kHz is used to produce the final voice signal. Chapter 3- ADC and DAC 3 3 a. Analog waveform b. With Chebyshev filter waveform to be captured 2 Amplitude Amplitude 2 1 0 1 0 high-frequency noise to be rejected -1 0 100 200 300 Time 400 -1 500 600 0 3 10 20 30 40 50 60 30 40 50 60 Sample number 3 c. With Bessel filter d. No analog filter 2 Amplitude 2 Amplitude 59 1 0 1 0 -1 -1 0 10 20 30 Sample number 40 50 60 0 10 20 Sample number FIGURE 3-15 Three antialias filter options for time domain encoded signals. The goal is to eliminate high frequencies (that will alias during sampling), while simultaneously retaining edge sharpness (that carries information). Figure (a) shows an example analog signal containing both sharp edges and a high frequency noise burst. Figure (b) shows the digitized signal using a Chebyshev filter. While the high frequencies have been effectively removed, the edges have been grossly distorted. This is usually a terrible solution. The Bessel filter, shown in (c), provides a gentle edge smoothing while removing the high frequencies. Figure (d) shows the digitized signal using no antialias filter. In this case, the edges have retained perfect sharpness; however, the high frequency burst has aliased into several meaningless samples. There are many useful benefits in sampling faster than this direct analysis. For example, imagine redesigning the digital voice recorder using a 64 kHz sampling rate. The antialias filter now has an easier task: pass all freq-uencies below 3 kHz, while rejecting all frequencies above 32 kHz. A similar simplification occurs for the reconstruction filter. In short, the higher sampling rate allows the eight pole filters to be replaced with simple resistor-capacitor (RC) networks. The problem is, the digital system is now swamped with data from the higher sampling rate. The next level of sophistication involves multirate techniques, using more than one sampling rate in the same system. It works like this for the digital voice recorder example. First, pass the voice signal through a simple RC low- 60 The Scientist and Engineer's Guide to Digital Signal Processing pass filter and sample the data at 64 kHz. The resulting digital data contains the desired voice band between 100 and 3000 hertz, but also has an unusable band between 3 kHz and 32 kHz. Second, remove these unusable frequencies in software, by using a digital low-pass filter at 3 kHz. Third, resample the digital signal from 64 kHz to 8 kHz by simply discarding every seven out of eight samples, a procedure called decimation. The resulting digital data is equivalent to that produced by aggressive analog filtering and direct 8 kHz sampling. Multirate techniques can also be used in the output portion of our example system. The 8 kHz data is pulled from memory and converted to a 64 kHz sampling rate, a procedure called interpolation. This involves placing seven samples, with a value of zero, between each of the samples obtained from memory. The resulting signal is a digital impulse train, containing the desired voice band between 100 and 3000 hertz, plus spectral duplications between 3 kHz and 32 kHz. Refer back to Figs. 3-6 a&b to understand why this it true. Everything above 3 kHz is then removed with a digital low-pass filter. After conversion to an analog signal through a DAC, a simple RC network is all that is required to produce the final voice signal. Multirate data conversion is valuable for two reasons: (1) it replaces analog components with software, a clear economic advantage in massproduced products, and (2) it can achieve higher levels of performance in critical applications. For example, compact disc audio systems use techniques of this type to achieve the best possible sound quality. This increased performance is a result of replacing analog components (1% precision), with digital algorithms (0.0001% precision from round-off error). As discussed in upcoming chapters, digital filters outperform analog filters by hundreds of times in key areas. Single Bit Data Conversion A popular technique in telecommunications and high fidelity music reproduction is single bit ADC and DAC. These are multirate techniques where a higher sampling rate is traded for a lower number of bits. In the extreme, only a single bit is needed for each sample. While there are many different circuit configurations, most are based on the use of delta modulation. Three example circuits will be presented to give you a flavor of the field. All of these circuits are implemented in IC's, so don't worry where all of the individual transistors and op amps should go. No one is going to ask you to build one of these circuits from basic components. Figure 3-16 shows the block diagram of a typical delta modulator. The analog input is a voice signal with an amplitude of a few volts, while the output signal is a stream of digital ones and zeros. A comparator decides which has the greater voltage, the incoming analog signal, or the voltage stored on the capacitor. This decision, in the form of a digital one or zero, is applied to the input of the latch. At each clock pulse, typically at a few hundred kilohertz, the latch transfers whatever digital state appears on its Chapter 3- ADC and DAC 61 clock analog input digital latch comparator positive charge injector clock negative charge injector clock delta modulated output FIGURE 3-16 Block diagram of a delta modulation circuit. The input voltage is compared with the voltage stored on the capacitor, resulting in a digital zero or one being applied to the input of the latch. The output of the latch is updated in synchronization with the clock, and used in a feedback loop to cause the capacitor voltage to track the input voltage. input, to its output. This latch insures that the output is synchronized with the clock, thereby defining the sampling rate, i.e., the rate at which the 1 bit output can update itself. A feedback loop is formed by taking the digital output and using it to drive an electronic switch. If the output is a digital one, the switch connects the capacitor to a positive charge injector. This is a very loose term for a circuit that increases the voltage on the capacitor by a fixed amount, say 1 millivolt per clock cycle. This may be nothing more than a resistor connected to a large positive voltage. If the output is a digital zero, the switch is connected to a negative charge injector. This decreases the voltage on the capacitor by the same fixed amount. Figure 3-17 illustrates the signals produced by this circuit. At time equal zero, the analog input and the voltage on the capacitor both start with a voltage of zero. As shown in (a), the input signal suddenly increases to 9.5 volts on the eighth clock cycle. Since the input signal is now more positive than the voltage on the capacitor, the digital output changes to a one, as shown in (b). This results in the switch being connected to the positive charge injector, and the voltage on the capacitor increasing by a small amount on each clock cycle. Although an increment of 1 volt per clock cycle is shown in (a), this is only for illustration, and a value of 1 millivolt is more typical. This staircase increase in the capacitor voltage continues until it exceeds the voltage of the input signal. Here the system reached an equilibrium with the output oscillating between a digital one and zero, causing the voltage on the capacitor to oscillate between 9 volts and 10 62 The Scientist and Engineer's Guide to Digital Signal Processing volts. In this manner, the feedback of the circuit forces the capacitor voltage to track the voltage of the input signal. If the input signal changes very rapidly, the voltage on the capacitor changes at a constant rate until a match is obtained. This constant rate of change is called the slew rate, just as in other electronic devices such as op amps. Now, consider the characteristics of the delta modulated output signal. If the analog input is increasing in value, the output signal will consist of more ones than zeros. Likewise, if the analog input is decreasing in value, the output will consist of more zeros than ones. If the analog input is constant, the digital output will alternate between zero and one with an equal number of each. Put in more general terms, the relative number of ones versus zeros is directly proportional to the slope (derivative) of the analog input. This circuit is a cheap method of transforming an analog signal into a serial stream of ones and zeros for transmission or digital storage. An especially attractive feature is that all the bits have the same meaning, unlike the conventional serial format: start bit, LSB, @ @ @ ,MSB, stop bit. The circuit at the receiver is identical to the feedback portion of the transmitting circuit. Just as the voltage on the capacitor in the transmitting circuit follows the analog input, so does the voltage on the capacitor in the receiving circuit. That is, the capacitor voltage shown in (a) also represents how the reconstructed signal would appear. A critical limitation of this circuit is the unavoidable tradeoff between (1) maximum slew rate, (2) quantization size, and (3) data rate. In particular, if the maximum slew rate and quantization size are adjusted to acceptable values for voice communication, the data rate ends up in the MHz range. This is too high to be of commercial value. For instance, conventional sampling of a voice signal requires only about 64,000 bits per second. A solution to this problem is shown in Fig. 3-18, the Continuously Variable Slope Delta (CVSD) modulator, a technique implemented in the Motorola MC3518 family. In this approach, the clock rate and the quantization size are set to something acceptable, say 30 kHz, and 2000 levels. This results in a terrible slew rate, which you correct with additional circuitry. In operation, a shift resister continually looks at the last four bits that the system has produced. If the circuit is in a slew rate limited condition, the last four bits will be all ones (positive slope) or all zeros (negative slope). A logic circuit detects this situation and produces an analog signal that increase the level of charge produced by the charge injectors. This boosts the slew rate by increasing the size of the voltage steps being applied to the capacitor. An analog filter is usually placed between the logic circuitry and the charge injectors. This allows the step size to depend on how long the circuit has been in a slew limited condition. As long as the circuit is slew limited, the step size keeps getting larger and larger. This is often called a syllabic filter, since its characteristics depend on the average length of the syllables making up speech. With proper optimization (from the chip manufacturer's Chapter 3- ADC and DAC 63 15 a. Analog signals capacitor voltage Amplitude (volts) 10 5 signal voltage 0 -5 -10 0 10 20 30 40 50 60 70 80 90 100 110 120 50 60 70 80 90 100 110 120 Time (clock cycles) 3 b. Digital output Digital logic state 2 slew rate limited 1 0 -1 0 10 20 30 40 Time (clock cycles) FIGURE 3-17 Example of signals produced by the delta modulator in Fig. 3-16. Figure (a) shows the analog input signal, and the corresponding voltage on the capacitor. Figure (b) shows the delta modulated output, a digital stream of ones and zeros. spec sheet, not your own work), data rates of 16 to 32 kHz produce acceptable quality speech. The continually changing step size makes the digital data difficult to understand, but fortunately, you don't need to. At the receiver, the analog signal is reconstructed by incorporating a syllabic filter that is identical to the one in the transmission circuit. If the two filters are matched, little distortion results from the CVSD modulation. CVSD is probably the easiest way to digitally transmit a voice signal. While CVSD modulation is great for encoding voice signals, it cannot be used for general purpose analog-to-digital conversion. Even if you get around the fact that the digital data is related to the derivative of the input signal, the changing step size will confuse things beyond repair. In addition, the DC level of the analog signal is usually not captured in the digital data. The delta-sigma converter, shown in Fig. 3-19, eliminates these problems by cleverly combining analog electronics with DSP algorithms. Notice that the voltage on the capacitor is now being compared with ground potential. The feedback loop has also been modified so that the voltage on the 64 The Scientist and Engineer's Guide to Digital Signal Processing clock analog input digital latch comparator 1 delta modulated output positive clock charge injector syllabic filter logic negative 0 charge injector clock all 1's, all 0's detect 4 bit shift register CVSD Modifications FIGURE 3-18 CVSD modulation block diagram. A logic circuit is added to the basic delta modulator to improve the slew rate. capacitor is decreased when the circuit's output is a digital one, and increased when it is a digital zero. As the input signal increases and decreases in voltage, it tries to raise and lower the voltage on the capacitor. This change in voltage is detected by the comparator, resulting in the charge injectors producing a counteracting charge to keep the capacitor at zero volts. If the input voltage is positive, the digital output will be composed of more ones than zeros. The excess number of ones being needed to generate the negative charge that cancels with the positive input signal. Likewise, if the input voltage is negative, the digital output will be composed of more zeros than ones, providing a net positive charge injection. If the input signal is equal to zero volts, an equal number of ones and zeros will be generated in the output, providing an overall charge injection of zero. The relative number of ones and zeros in the output is now related to the level of the input voltage, not the slope as in the previous circuit. This is much simpler. For instance, you could form a 12 bit ADC by feeding the digital output into a counter, and counting the number of ones over 4096 clock cycles. A digital number of 4095 would correspond to the maximum positive input voltage. Likewise, digital number 0 would correspond to the maximum negative input voltage, and 2048 would correspond to an input voltage of zero. This also shows the origin of the name, delta-sigma: delta modulation followed by summation (sigma). The ones and zeros produced by this type of delta modulator are very easy to transform back into an analog signal. All that is required is an analog lowpass filter, which might be as simple as a single RC network. The high Chapter 3- ADC and DAC 65 clock analog input Digital Latch comparator 0 1 positive charge injector clock negative clock charge injector delta modulated signal digital output Counter Latch RESET to LATCH to begin ADC cycle end ADC cycle Digital low-pass filter digital output Decimate FIGURE 3-19 Block diagram of a delta-sigma analog-to-digital converter. In the simplest case, the pulses from a delta modulator are counted for a predetermined number of clock cycles. The output of the counter is then latched to complete the conversion. In a more sophisticated circuit, the pulses are passed through a digital low-pass filter and then resampled (decimated) to a lower sampling rate. and low voltages corresponding to the digital ones and zeros average out to form the correct analog voltage. For example, suppose that the ones and zeros are represented by 5 volts and 0 volts, respectively. If 80% of the bits in the data stream are ones, and 20% are zeros, the output of the low-pass filter will be 4 volts. This method of transforming the single bit data stream back into the original waveform is important for several reasons. First, it describes a slick way to replace the counter in the delta-sigma ADC circuit. Instead of simply counting the pulses from the delta modulator, the binary signal is passed through a digital low-pass filter, and then decimated to reduce the sampling rate. For example, this procedure might start by changing each of the ones and zeros in the digital stream into a 12 bit sample; ones become a value of 4095, while zeros become a value of 0. Using a digital low-pass filter on this signal produces a digitized version of the original waveform, just as an analog lowpass filter would form an analog recreation. Decimation then reduces the sampling rate by discarding most of the samples. This results in a digital signal that is equivalent to direct sampling of the original waveform. This approach is used in many commercial ADC's for digitizing voice and other audio signals. An example is the National Semiconductor ADC16071, which provides 16 bit analog-to-digital conversion at sampling rates up to 192 kHz. At a sampling rate of 100 kHz, the delta modulator operates with a clock frequency of 6.4 MHz. The low-pass digital filter is a 246 point FIR, such as described in Chapter 16. This removes all frequencies in the digital data above 50 kHz, ½ of the eventual sampling rate. Conceptually, this can be 66 The Scientist and Engineer's Guide to Digital Signal Processing viewed as forming a digital signal at 6.4 MHz, with each sample represented by 16 bits. The signal is then decimated from 6.4 MHz to 100 kHz, accomplished by deleting every 63 out of 64 samples. In actual operation, much more goes on inside of this device than described by this simple discussion. Delta-sigma converters can also be used for digital-to-analog conversion of voice and audio signals. The digital signal is retrieved from memory, and converted into a delta modulated stream of ones and zeros. As mentioned above, this single bit signal can easily be changed into the reconstructed analog signal with a simple low-pass analog filter. As with the antialias filter, usually only a single RC network is required. This is because the majority of the filtration is handled by the high-performance digital filters. Delta-sigma ADC's have several quirks that limit their use to specific applications. For example, it is difficult to multiplex their inputs. When the input is switched from one signal to another, proper operation is not established until the digital filter can clear itself of data from the previous signal. Deltasigma converters are also limited in another respect: you don't know exactly when each sample was taken. Each acquired sample is a composite of the one bit information taken over a segment of the input signal. This is not a problem for signals encoded in the frequency domain, such as audio, but it is a significant limitation for time domain encoded signals. To understand the shape of a signal's waveform, you often need to know the precise instant each sample was taken. Lastly, most of these devices are specifically designed for audio applications, and their performance specifications are quoted accordingly. For example, a 16 bit ADC used for voice signals does not necessarily mean that each sample has 16 bits of precision. Much more likely, the manufacturer is stating that voice signals can be digitized to 16 bits of dynamic range. Don't expect to get a full 16 bits of useful information from this device for general purpose data acquisition. While these explanations and examples provide an introduction to single bit ADC and DAC, it must be emphasized that they are simplified descriptions of sophisticated DSP and integrated circuit technology. You wouldn't expect the manufacturer to tell their competitors all the internal workings of their chips, so don't expect them to tell you. CHAPTER DSP Software 4 DSP applications are usually programmed in the same languages as other science and engineering tasks, such as: C, BASIC and assembly. The power and versatility of C makes it the language of choice for computer scientists and other professional programmers. On the other hand, the simplicity of BASIC makes it ideal for scientists and engineers who only occasionally visit the programming world. Regardless of the language you use, most of the important DSP software issues are buried far below in the realm of whirling ones and zeros. This includes such topics as: how numbers are represented by bit patterns, round-off error in computer arithmetic, the computational speed of different types of processors, etc. This chapter is about the things you can do at the high level to avoid being trampled by the low level internal workings of your computer. Computer Numbers Digital computers are very proficient at storing and recalling numbers; unfortunately, this process isn't without error. For example, you instruct your computer to store the number: 1.41421356. The computer does its best, storing the closest number it can represent: 1.41421354. In some cases this error is quite insignificant, while in other cases it is disastrous. As another illustration, a classic computational error results from the addition of two numbers with very different values, for example, 1 and 0.00000001. We would like the answer to be 1.00000001, but the computer replies with 1. An understanding of how computers store and manipulate numbers allows you to anticipate and correct these problems before your program spits out meaningless data. These problems arise because a fixed number of bits are allocated to store each number, usually 8, 16, 32 or 64. For example, consider the case where eight bits are used to store the value of a variable. Since there are 2 8 = 256 possible bit patterns, the variable can only take on 256 different values. This is a fundamental limitation of the situation, and there is nothing we can do about it. The part we can control is what value we declare each bit pattern 67 68 The Scientist and Engineer's Guide to Digital Signal Processing to represent. In the simplest cases, the 256 bit patterns might represent the integers from 0 to 255, 1 to 256, -127 to 128, etc. In a more unusual scheme, the 256 bit patterns might represent 256 exponentially related numbers: 1, 10, 100, 1000, þ, 10254, 10255. Everyone accessing the data must understand what value each bit pattern represents. This is usually provided by an algorithm or formula for converting between the represented value and the corresponding bit pattern, and back again. While many encoding schemes are possible, only two general formats have become common, fixed point (also called integer numbers) and floating point (also called real numbers). In this book's BASIC programs, fixed point variables are indicated by the % symbol as the last character in the name, such as: I%, N%, SUM%, etc. All other variables are floating point, for example: X, Y, MEAN, etc. When you evaluate the formats presented in the next few pages, try to understand them in terms of their range (the largest and smallest numbers they can represent) and their precision (the size of the gaps between numbers). Fixed Point (Integers) Fixed point representation is used to store integers, the positive and negative whole numbers: þ &3,&2, &1, 0, 1, 2, 3,þ . High level programs, such as C and BASIC, usually allocate 16 bits to store each integer. In the simplest case, the 216 ' 65,536 possible bit patterns are assigned to the numbers 0 through 65,535. This is called unsigned integer format, and a simplified example is shown in Fig. 4-1 (using only 4 bits per number). Conversion between the bit pattern and the number being represented is nothing more than changing between base 2 (binary) and base 10 (decimal). The disadvantage of unsigned integer is that negative numbers cannot be represented. Offset binary is similar to unsigned integer, except the decimal values are shifted to allow for negative numbers. In the 4 bit example of Fig. 4-1, the decimal numbers are offset by seven, resulting in the 16 bit patterns corresponding to the integer numbers -7 through 8. In this same manner, a 16 bit representation would use 32,767 as an offset, resulting in a range between -32,767 and 32,768. Offset binary is not a standardized format, and you will find other offsets used, such 32,768. The most important use of offset binary is in ADC and DAC. For example, the input voltage range of -5v to 5v might be mapped to the digital numbers 0 to 4095, for a 12 bit conversion. Sign and magnitude is another simple way of representing negative integers. The far left bit is called the sign bit, and is made a zero for positive numbers, and a one for negative numbers. The other bits are a standard binary representation of the absolute value of the number. This results in one wasted bit pattern, since there are two representations for zero, 0000 (positive zero) and 1000 (negative zero). This encoding scheme results in 16 bit numbers having a range of -32,767 to 32,767. Chapter 4- DSP Software UNSIGNED INTEGER Decimal 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 OFFSET BINARY 69 SIGN AND MAGNITUDE TWO'S COMPLEMENT Bit Pattern Decimal Bit Pattern Decimal Bit Pattern Decimal Bit Pattern 1111 1110 1101 1100 1011 1010 1001 1000 0111 0110 0101 0100 0011 0010 0001 0000 8 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 1111 1110 1101 1100 1011 1010 1001 1000 0111 0110 0101 0100 0011 0010 0001 0000 7 6 5 4 3 2 1 0 0 -1 -2 -3 -4 -5 -6 -7 0111 0110 0101 0100 0011 0010 0001 0000 1000 1001 1010 1011 1100 1101 1110 1111 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 0111 0110 0101 0100 0011 0010 0001 0000 1111 1110 1101 1100 1011 1010 1001 1000 16 bit range: 0 to 65,535 16 bit range -32,767 to 32,768 16 bit range -32,767 to 32,767 16 bit range -32,768 to 32,767 FIGURE 4-1 Common formats for fixed point (integer) representation. Unsigned integer is a simple binary format, but cannot represent negative numbers. Offset binary and sign & magnitude allow negative numbers, but they are difficult to implement in hardware. Two's complement is the easiest to design hardware for, and is the most common format for general purpose computing. These first three representations are conceptually simple, but difficult to implement in hardware. Remember, when A=B+C is entered into a computer program, some hardware engineer had to figure out how to make the bit pattern representing B, combine with the bit pattern representing C, to form the bit pattern representing A. Two's complement is the format loved by hardware engineers, and is how integers are usually represented in computers. To understand the encoding pattern, look first at decimal number zero in Fig. 4-1, which corresponds to a binary zero, 0000. As we count upward, the decimal number is simply the binary equivalent (0 = 0000, 1 = 0001, 2 = 0010, 3 = 0011, etc.). Now, remember that these four bits are stored in a register consisting of 4 flip-flops. If we again start at 0000 and begin subtracting, the digital hardware automatically counts in two's complement: 0 = 0000, -1 = 1111, -2 = 1110, -3 = 1101, etc. This is analogous to the odometer in a new automobile. If driven forward, it changes: 00000, 00001, 00002, 00003, and so on. When driven backwards, the odometer changes: 00000, 99999, 99998, 99997, etc. Using 16 bits, two's complement can represent numbers from -32,768 to 32,767. The left most bit is a 0 if the number is positive or zero, and a 1 if the number is negative. Consequently, the left most bit is called the sign bit, just as in sign & magnitude representation. Converting between decimal and two's complement is straightforward for positive numbers, a simple decimal to binary 70 The Scientist and Engineer's Guide to Digital Signal Processing conversion. For negative numbers, the following algorithm is often used: (1) take the absolute value of the decimal number, (2) convert it to binary, (3) complement all of the bits (ones become zeros and zeros become ones), (4) add 1 to the binary number. For example: -5 6 5 6 0101 6 1010 6 1011. Two's complement is hard for humans, but easy for digital electronics. Floating Point (Real Numbers) The encoding scheme for floating point numbers is more complicated than for fixed point. The basic idea is the same as used in scientific notation, where a mantissa is multiplied by ten raised to some exponent. For instance, 5.4321 × 106, where 5.4321 is the mantissa and 6 is the exponent. Scientific notation is exceptional at representing very large and very small numbers. For example: 1.2 × 1050, the number of atoms in the earth, or 2.6 × 10& 23, the distance a turtle crawls in one second, compared to the diameter of our galaxy. Notice that numbers represented in scientific notation are normalized so that there is only a single nonzero digit left of the decimal point. This is achieved by adjusting the exponent as needed. Floating point representation is similar to scientific notation, except everything is carried out in base two, rather than base ten. While several similar formats are in use, the most common is ANSI/IEEE Std. 754-1985. This standard defines the format for 32 bit numbers called single precision, as well as 64 bit numbers called double precision. As shown in Fig. 4-2, the 32 bits used in single precision are divided into three separate groups: bits 0 through 22 form the mantissa, bits 23 through 30 form the exponent, and bit 31 is the sign bit. These bits form the floating point number, v, by the following relation: EQUATION 4-1 Equation for converting a bit pattern into a floating point number. The number is represented by v, S is the value of the sign bit, M is the value of the mantissa, and E is the value of the exponent. v ' (& 1)S × M × 2E & 127 The term: (&1) S , simply means that the sign bit, S, is 0 for a positive number and 1 for a negative number. The variable, E, is the number between 0 and 255 represented by the eight exponent bits. Subtracting 127 from this number allows the exponent term to run from 2& 127 to 2128. In other words, the exponent is stored in offset binary with an offset of 127. The mantissa, M, is formed from the 23 bits as a binary fraction. For example, the decimal fraction: 2.783, is interpreted: 2 % 7/10 % 8/100 % 3/1000 . The binary fraction: 1.0101, means: 1 % 0/2 % 1/4 % 0/8 % 1/16 . Floating point numbers are normalized in the same way as scientific notation, that is, there is only one nonzero digit left of the decimal point (called a binary point in Chapter 4- DSP Software 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 MSB 8 7 6 5 LSB MSB 4 3 2 1 0 LSB EXPONENT 8 bits SIGN 1 bit 71 MANTISSA 23 bits Example 1 0 00000111 11000000000000000000000 % 1.75 × 2(7& 127) ' % 1.316554 × 10& 36 + 7 0.75 Example 2 1 10000001 01100000000000000000000 & 1.375 × 2(129& 127) ' & 5.500000 129 0.375 FIGURE 4-2 Single precision floating point storage format. The 32 bits are broken into three separate parts, the sign bit, the exponent and the mantissa. Equations 4-1 and 4-2 shows how the represented number is found from these three parts. MSB and LSB refer to “most significant bit” and “least significant bit,” respectively. base 2). Since the only nonzero number that exists in base two is 1, the leading digit in the mantissa will always be a 1, and therefore does not need to be stored. Removing this redundancy allows the number to have an additional one bit of precision. The 23 stored bits, referred to by the notation: m22, m21, m21,þ, m0 , form the mantissa according to: EQUATION 4-2 Algorithm for converting the bit pattern into the mantissa, M, used in Eq. 4-1. M ' 1.m22m21m20m19 @@@ m2m1m0 In other words, M ' 1 % m22 2& 1 % m21 2& 2 % m20 2& 3@@@ . If bits 0 through 22 are all zeros, M takes on the value of one. If bits 0 through 22 are all ones, M is just a hair under two, i.e., 2 &2& 23. Using this encoding scheme, the largest number that can be represented is: ±(2&2& 23) × 2128 ' ±6.8 × 1038. Likewise, the smallest number that can be represented is: ±1.0 × 2& 127 ' ±5.9 × 10& 39. The IEEE standard reduces this range slightly to free bit patterns that are assigned special meanings. In particular, the largest and smallest numbers allowed in the standard are 72 The Scientist and Engineer's Guide to Digital Signal Processing ±3.4 × 1038 and ±1.2 × 10& 38, respectively. The freed bit patterns allow three special classes of numbers: (1) ±0 is defined as all of the mantissa and exponent bits being zero. (2) ±4 is defined as all of the mantissa bits being zero, and all of the exponent bits being one. (3) A group of very small unnormalized numbers between ±1.2 × 10& 38 and ±1.4 × 10& 45 . These are lower precision numbers obtained by removing the requirement that the leading digit in the mantissa be a one. Besides these three special classes, there are bit patterns that are not assigned a meaning, commonly referred to as NANs (Not A Number). The IEEE standard for double precision simply adds more bits to both the mantissa and exponent. Of the 64 bits used to store a double precision number, bits 0 through 51 are the mantissa, bits 52 through 62 are the exponent, and bit 63 is the sign bit. As before, the mantissa is between one and just under two, i.e., M ' 1 % m51 2& 1 % m50 2& 2 % m49 2& 3@@@ . The 11 exponent bits form a number between 0 and 2047, with an offset of 1023, allowing exponents from 2& 1023 to 21024 . The largest and smallest numbers allowed are ±1.8 × 10308 and ±2.2 × 10& 308 , respectively. These are incredibly large and small numbers! It is quite uncommon to find an application where single precision is not adequate. You will probably never find a case where double precision limits what you want to accomplish. Number Precision The errors associated with number representation are very similar to quantization errors during ADC. You want to store a continuous range of values; however, you can represent only a finite number of quantized levels. Every time a new number is generated, after a math calculation for example, it must be rounded to the nearest value that can be stored in the format you are using. As an example, imagine that you allocate 32 bits to store a number. Since there are exactly 232 ' 4,294,967,296 different bit patterns possible, you can represent exactly 4,294,967,296 different numbers. Some programming languages allow a variable called a long integer, stored as 32 bits, fixed point, two's complement. This means that the 4,294,967,296 possible bit patterns represent the integers between -2,147,483,648 and 2,147,483,647. In comparison, single precision floating point spreads these 4,294,967,296 bit patterns over the much larger range: &3.4 × 1038 to 3.4 × 1038 . With fixed point variables, the gaps between adjacent numbers are always exactly one. In floating point notation, the gaps between adjacent numbers vary over the represented number range. If we randomly pick a floating point number, the gap next to that number is approximately ten million times smaller than the number itself (to be exact, 2& 24 to 2& 23 times the number). This is a key concept of floating point notation: large numbers have large gaps between them, while small numbers have small gaps. Figure 4-3 illustrates this by showing consecutive floating point numbers, and the gaps that separate them. Chapter 4- DSP Software 73 0.00001233862713 0.00001233862804 0.00001233862895 0.00001233862986 spacing = 0.00000000000091 (1 part in 13 million) 1.000000000 1.000000119 1.000000238 1.000000358 spacing = 0.000000119 (1 part in 8 million) 1.996093750 1.996093869 1.996093988 1.996094108 spacing = 0.000000119 (1 part in 17 million) 636.0312500 636.0313110 636.0313720 636.0314331 spacing = 0.0000610 (1 part in 10 million) 217063424.0 217063440.0 217063456.0 217063472.0 spacing = 16.0 (1 part in 14 million) ! FIGURE 4-3 Examples of the spacing between single precision floating point numbers. The spacing between adjacent numbers is always between about 1 part in 8 million and 1 part in 17 million of the value of the number. ! ! ! The program in Table 4-1 illustrates how round-off error (quantization error in math calculations) causes problems in DSP. Within the program loop, two random numbers are added to the floating point variable X, and then subtracted back out again. Ideally, this should do nothing. In reality, the round-off error from each of the arithmetic operations causes the value of X to gradually drift away from its initial value. This drift can take one of two forms depending on how the errors add together. If the round-off errors are randomly positive and negative, the value of the variable will randomly increase and decrease. If the errors are predominately of the same sign, the value of the variable will drift away much more rapidly and uniformly. TABLE 4-1 Program for demonstrating floating point error accumulation. This program initially sets the value of X to 1.000000, and then runs through a loop that should ideally do nothing. During each loop, two random numbers, A and B, are added to X, and then subtracted back out. The accumulated error from these additions and subtraction causes X to wander from its initial value. As Fig. 44 shows, the error may be random or additive. 100 X = 1 'initialize X 110 ' 120 FOR I% = 0 TO 2000 130 A = RND 'load random numbers 140 B = RND 'into A and B 150 ' 160 X = X + A 'add A and B to X 170 X = X + B 180 X = X ! A 'undo the additions 190 X = X ! B 200 ' 210 PRINT X 'ideally, X should be 1 220 NEXT I% 230 END 74 The Scientist and Engineer's Guide to Digital Signal Processing 1.0002 Additive error 1.0001 Value of X FIGURE 4-4 Accumulation of round-off error in floating point variables. These curves are generated by the program shown in Table 4-1. When a floating point variable is repeatedly used in arithmetic operations, accumulated round-off error causes the variable's value to drift. If the errors are both positive and negative, the value will increase and decrease in a random fashion. If the round-off errors are predominately of the same sign, the value will change in a much more rapid and uniform manner. 1 Random error 0.9999 0.9998 0 500 1000 Number of loops 1500 2000 Figure 4-4 shows how the variable, X, in this example program drifts in value. An obvious concern is that additive error is much worse than random error. This is because random errors tend to cancel with each other, while the additive errors simply accumulate. The additive error is roughly equal to the round-off error from a single operation, multiplied by the total number of operations. In comparison, the random error only increases in proportion to the square root of the number of operations. As shown by this example, additive error can be hundreds of times worse than random error for common DSP algorithms. Unfortunately, it is nearly impossible to control or predict which of these two behaviors a particular algorithm will experience. For example, the program in Table 4-1 generates an additive error. This can be changed to a random error by merely making a slight modification to the numbers being added and subtracted. In particular, the random error curve in Fig. 4-4 was generated by defining: A ' EXP(RND) and B ' EXP(RND) , rather than: A ' RND and B ' RND . Instead of A and B being randomly distributed numbers between 0 and 1, they become exponentially distributed values between 1 and 2.718. Even this small change is sufficient to toggle the mode of error accumulation. Since we can't control which way the round-off errors accumulate, keep in mind the worse case scenario. Expect that every single precision number will have an error of about one part in forty million, multiplied by the number of operations it has been through. This is based on the assumption of additive error, and the average error from a single operation being one-quarter of a quantization level. Through the same analysis, every double precision number has an error of about one part in forty quadrillion, multiplied by the number of operations. Chapter 4- DSP Software 100 'Floating Point Loop Control 110 FOR X = 0 TO 10 STEP 0.01 120 PRINT X 130 NEXT X 100 'Integer Loop Control 110 FOR I% = 0 TO 1000 120 X = I%/100 130 PRINT X 140 NEXT I% Program Output: Program Output: 0.00 0.01 0.02 0.03 ! 9.960132 9.970133 9.980133 9.990133 75 0.00 0.01 0.02 0.03 ! 9.96 9.97 9.98 9.99 10.00 TABLE 4-2 Comparison of floating point and integer variables for loop control. The left hand program controls the FOR-NEXT loop with a floating point variable, X. This results in an accumulated round-off error of 0.000133 by the end of the program, causing the last loop value, X = 10.0, to be omitted. In comparison, the right hand program uses an integer, I%, for the loop index. This provides perfect precision, and guarantees that the proper number of loop cycles will be completed. Table 4-2 illustrates a particularly annoying problem of round-off error. Each of the two programs in this table perform the same task: printing 1001 numbers equally spaced between 0 and 10. The left-hand program uses the floating point variable, X, as the loop index. When instructed to execute a loop, the computer begins by setting the index variable to the starting value of the loop (0 in this example). At the end of each loop cycle, the step size (0.01 in the case) is added to the index. A decision is then made: are more loops cycles required, or is the loop completed? The loop ends when the computer finds that the value of the index is greater than the termination value (in this example, 10.0). As shown by the generated output, round-off error in the additions cause the value of X to accumulate a significant discrepancy over the course of the loop. In fact, the accumulated error prevents the execution of the last loop cycle. Instead of X having a value of 10.0 on the last cycle, the error makes the last value of X equal to 10.000133. Since X is greater than the termination value, the computer thinks its work is done, and the loop prematurely ends. This missing last value is a common bug in many computer programs. In comparison, the program on the right uses an integer variable, I%, to control the loop. The addition, subtraction, or multiplication of two integers always produces another integer. This means that fixed point notation has absolutely no round-off error with these operations. Integers are ideal for controlling loops, as well as other variables that undergo multiple mathematical operations. The last loop cycle is guaranteed to execute! Unless you have some strong motivation to do otherwise, always use integers for loop indexes and counters. 76 The Scientist and Engineer's Guide to Digital Signal Processing If you must use a floating point variable as a loop index, try to use fractions that are a power of two (such as: 1/2, 1/4, 3/8, 27/16 ), instead of a power of ten (such as: 0.1, 0.6, 1.4, 2.3 , etc.). For instance, it would be better to use: FOR X = 1 TO 10 STEP 0.125, rather than: FOR X = 1 to 10 STEP 0.1. This allows the index to always have an exact binary representation, thereby reducing round-off error. For example, the decimal number: 1.125, can be represented exactly in binary notation: 1.001000000000000000000000×20. In comparison, the decimal number: 1.1, falls between two floating point numbers: 1.0999999046 and 1.1000000238 (in binary these numbers are: 1.00011001100110011001100×20 and 1.00011001100110011001101×20). This results in an inherent error each time 1.1 is encountered in a program. A useful fact to remember: single precision floating point has an exact binary representation for every whole number between ±16.8 million (to be exact, ±224). Above this value, the gaps between the levels are larger than one, causing some whole number values to be missed. This allows floating point whole numbers (between ±16.8 million) to be added, subtracted and multiplied, with no round-off error. Execution Speed: Program Language DSP programming can be loosely divided into three levels of sophistication: Assembly, Compiled, and Application Specific. To understand the difference between these three, we need to start with the very basics of digital electronics. All microprocessors are based around a set of internal binary registers, that is, a group of flip-flops that can store a series of ones and zeros. For example, the 8088 microprocessor, the core of the original IBM PC, has four general purpose registers, each consisting of 16 bits. These are identified by the names: AX, BX, CX, and DX. There are also nine additional registers with special purposes, called: SI, DI, SP, BP, CS, DS, SS, ES, and IP. For example, IP, the Instruction Pointer, keeps track of where in memory the next instruction resides. Suppose you write a program to add the numbers: 1234 and 4321. When the program begins, IP contains the address of a section of memory that contains a pattern of ones and zeros, as shown in Table 4-3. Although it looks meaningless to most humans, this pattern of ones and zeros contains all of the commands and data required to complete the task. For example, when the microprocessor encounters the bit pattern: 00000011 11000011, it interpreters it as a command to take the 16 bits stored in the BX register, add them in binary to the 16 bits stored in the AX register, and store the result in the AX register. This level of programming is called machine code, and is only a hair above working with the actual electronic circuits. Since working in binary will eventually drive even the most patient engineer crazy, these patterns of ones and zeros are assigned names according to the function they perform. This level of programming is called assembly, and an example is shown in Table 4-4. Although an assembly program is much easier to understand, it is fundamentally the same as programming in Chapter 4- DSP Software TABLE 4-3 A machine code program for adding 1234 and 4321. This is the lowest level of programming: direct manipulation of the digital electronics. (The right column is a continuation of the left column). 10111001 11010010 00000100 10001001 00001110 00000000 00000000 10111001 11100001 00010000 10001001 00001110 00000010 77 00000000 10100001 00000000 00000000 10001011 00011110 00000010 00000000 00000011 11000011 10100011 00000100 00000000 machine code, since there is a one-to-one correspondence between the program commands and the action taken in the microprocessor. For example: ADD AX, BX translates to: 00000011 11000011. A program called an assembler is used to convert the assembly code in Table 4-4 (called the source code) into the patterns of ones and zeros shown in Table 4-3 (called the object code or executable code). This executable code can be directly run on the microprocessor. Obviously, assembly programming requires an extensive understanding of the internal construction of the particular microprocessor you intend to use. TABLE 4-4 An assembly program for adding 1234 and 4321. An assembler is a program that converts an assembly program into machine code. MOV CX,1234 MOV DS:[0],CX ;store 1234 in register CX, and then ;transfer it to memory location DS:[0] MOV CX,4321 MOV DS:[2],CX ;store 4321 in register CX, and then ;transfer it to memory location DS:[0] MOV AX,DS:[0] MOV BX,DS:[2] ;move variables stored in memory at ;DS:[0] and DS:[2] into AX & BX ADD AX,BX ;add AX and BX, store sum in AX MOV DS:[4],AX ;move the sum into memory at DS:[4] Assembly programming involves the direct manipulation of the digital electronics: registers, memory locations, status bits, etc. The next level of sophistication can manipulate abstract variables without any reference to the particular hardware. These are called compiled or high-level languages. A dozen or so are in common use, such as: C, BASIC, FORTRAN, PASCAL, APL, COBOL, LISP, etc. Table 4-5 shows a BASIC program for adding 1234 and 4321. The programmer only knows about the variables A, B, and C, and nothing about the hardware. TABLE 4-5 A BASIC program for adding 1234 and 4321. A compiler is a program that converts this type of high-level source code into machine code. 100 110 120 130 A = 1234 B = 4321 C = A+B END 78 The Scientist and Engineer's Guide to Digital Signal Processing A program called a compiler is used to transform the high-level source code directly into machine code. This requires the compiler to assign hardware memory locations to each of the abstract variables being referenced. For example, the first time the compiler encounters the variable A in Table 4-5 (line 100), it understands that the programmer is using this symbol to mean a single precision floating point variable. Correspondingly, the compiler designates four bytes of memory that will be used for nothing but to hold the value of this variable. Each subsequent time that an A appears in the program, the computer knows to update the value of the four bytes as needed. The compiler also breaks complicated mathematical expressions, such as: Y = LOG(XCOS(Z)), into more basic arithmetic. Microprocessors only know how to add, subtract, multiply and divide. Anything more complicated must be done as a series of these four elementary operations. High-level languages isolate the programmer from the hardware. This makes the programming much easier and allows the source code to be transported between different types of microprocessors. Most important, the programmer who uses a compiled language needs to know nothing about the internal workings of the computer. Another programmer has assumed this responsibility, the one who wrote the compiler. Most compilers operate by converting the entire program into machine code before it is executed. An exception to this is a type of compiler called an interpreter, of which interpreter BASIC is the most common example. An interpreter converts a single line of source code into machine code, executes that machine code, and then goes on to the next line of source code. This provides an interactive environment for simple programs, although the execution speed is extremely slow (think a factor of 100). The highest level of programming sophistication is found in applications packages for DSP. These come in a variety of forms, and are often provided to support specific hardware. Suppose you buy a newly developed DSP microprocessor to embed in your current project. These devices often have lots of built-in features for DSP: analog inputs, analog outputs, digital I/O, antialias and reconstruction filters, etc. The question is: how do you program it? In the worst case, the manufacturer will give you an assembler, and expect you to learn the internal architecture of the device. In a more typical scenario, a C compiler will be provided, allowing you to program without being bothered by how the microprocessor actually operates. In the best case, the manufacturer will provide a sophisticated software package to help in the programming: libraries of algorithms, prewritten routines for I/O, debugging tools, etc. You might simply connect icons to form the desired system in an easy-to-use graphical display. The things you manipulate are signal pathways, algorithms for processing signals, analog I/O parameters, etc. When you are satisfied with the design, it is transformed into suitable machine code for execution in the hardware. Other types of applications packages are used with image processing, spectral analysis, instrumentation and control, digital filter design, etc. This is the shape of the future. Chapter 4- DSP Software 79 The distinction between these three levels can be very fuzzy. For example, most complied languages allow you to directly manipulate the hardware. Likewise, a high-level language with a well stocked library of DSP functions is very close to being an applications package. The point of these three catagories is understand what you are manipulating: (1) hardware, (2) abstract variables, or (3) entire procedures and algorithms. There is also another important concept behind these classifications. When you use a high-level language, you are relying on the programmer who wrote the compiler to understand the best techniques for hardware manipulation. Similarly, when you use an applications package, you are relying on the programmer who wrote the package to understand the best DSP techniques. Here's the rub: these programmers have never seen the particular problem you are dealing with. Therefore, they cannot always provide you with an optimal solution. As you operate on a higher level, expect that the final machine code will be less efficient in terms of memory usage, speed, and precision. Which programming language should you use? That depends on who you are and what you plan to do. Most computer scientists and programmers use C (or the more advanced C++). Power, flexibility, modularity; C has it all. C is so popular, the question becomes: Why would anyone program their DSP application in something other than C? Three answers come to mind. First, DSP has grown so rapidly that some organizations and individuals are stuck in the mode of other languages, such as FORTRAN and PASCAL. This is especially true of military and government agencies that are notoriously slow to change. Second, some applications require the utmost efficiency, only achievable by assembly programming. This falls into the category of "a little more speed for a lot more work." Third, C is not an especially easy language to master, especially for part time programmers. This includes a wide range of engineers and scientists who occasionally need DSP techniques to assist in their research or design activities. This group often turns to BASIC because of its simplicity. Why was BASIC chosen for this book? This book is about algorithms, not programming style. You should be concentrating on DSP techniques, and not be distracted by the quirks of a particular language. For instance, all the programs in this book have line numbers. This makes it easy to describe how the program operates: "line 100 does such-and-such, line 110 does this-and that," etc. Of course, you will probably never use line numbers in your actual programs. The point is, learning DSP has different requirements than using DSP. There are many books on the market that provide exquisite source code for DSP algorithms. If you are simply looking for prewritten code to copy into your program, you are in the wrong place. Comparing the execution speed of hardware or software is a thankless task; no matter what the result, the loser will cry that the match was unfair! Programmers who like high-level languages (such as traditional computer scientists), will argue that assembly is only 50% faster than compiled code, but five times more trouble. Those who like assembly (typically, scientists and 80 The Scientist and Engineer's Guide to Digital Signal Processing hardware engineers) will claim the reverse: assembly is five times faster, but only 50% more difficult to use. As in most controversies, both sides can provide selective data to support their claims. As a rule-of-thumb, expect that a subroutine written in assembly will be between 1.5 and 3.0 times faster than the comparable high-level program. The only way to know the exact value is to write the code and conduct speed tests. Since personal computers are increasing in speed about 40% every year, writing a routine in assembly is equivalent to about a two year jump in hardware technology. Most professional programmers are rather offended at the idea of using assembly, and gag if you suggest BASIC. Their rational is quite simple: assembly and BASIC discourage the use of good software practices. Good code should be portable (able to move from one type of computer to another), modular (broken into a well defined subroutine structure), and easy to understand (lots of comments and descriptive variable names). The weak structure of assembly and BASIC makes it difficult to achieve these standards. This is compounded by the fact that the people who are attracted to assembly and BASIC often have little formal training in proper software structure and documentation. Assembly lovers respond to this attack with a zinger of their own. Suppose you write a program in C, and your competitor writes the same program in assembly. The end user's first impression will be that your program is junk because it is twice as slow. No one would suggest that you write large programs in assembly, only those portions of the program that need rapid execution. For example, many functions in DSP software libraries are written in assembly, and then accessed from larger programs written in C. Even the staunchest software purist will use assembly code, as long as they don't have to write it. Execution Speed: Hardware Computing power is increasing so rapidly, any book on the subject will be obsolete before it is published. It's an author's nightmare! The original IBM PC was introduced in 1981, based around the 8088 microprocessor with a 4.77 MHz clock and an 8 bit data bus. This was followed by a new generation of personal computers being introduced every 3-4 years: 8088 ! 80286 ! 80386 ! 80486 ! 80586 (Pentium). Each of these new systems boosted the computing speed by a factor of about five over the previous technology. By 1996, the clock speed had increased to 200 MHz, and the data bus to 32 bits. With other improvements, this resulted in an increase in computing power of nearly one thousand in only 15 years! You should expect another factor of one thousand in the next 15 years. The only way to obtain up-to-date information in this rapidly changing field is directly from the manufacturers: advertisements, specification sheets, price lists, etc. Forget books for performance data, look in magazines and your daily Chapter 4- DSP Software 81 newspaper. Expect that raw computational speed will more than double each two years. Learning about the current state of computer power is simply not enough; you need to understand and track how it is evolving. Keeping this in mind, we can jump into an overview of how execution speed is limited by computer hardware. Since computers are composed of many subsystems, the time required to execute a particular task will depend on two primary factors: (1) the speed of the individual subsystems, and (2) the time it takes to transfer data between these blocks. Figure 4-5 shows a simplified diagram of the most important speed limiting components in a typical personnel computer. The Central Processing Unit (CPU) is the heart of the system. As previously described, it consists of a dozen or so registers, each capable of holding 32 bits (in present generation personnel computers). Also included in the CPU is the digital electronics needed for rudimentary operations, such as moving bits around and fixed point arithmetic. More involved mathematics is handled by transferring the data to a special hardware circuit called a math coprocessor (also called an arithmetic logic unit, or ALU). The math coprocessor may be contained in the same chip as the CPU, or it may be a separate electronic device. For example, the addition of two floating point numbers would require the CPU to transfer 8 bytes (4 for each number) to the math coprocessor, and several bytes that describe what to do with the data. After a short computational time, the math coprocessor would pass four bytes back to the CPU, containing the floating point number that is the sum. The most inexpensive computer systems don't have a math coprocessor, or provide it only as an option. For example, the 80486DX microprocessor has an internal math coprocessor, while the 80486SX does not. These lower performance systems replace hardware with software. Each of the mathematical functions is broken into Math Coprocessor FIGURE 4-5 Architecture of a typical computer system. The computational speed is limited by: (1) the speed of the individual subsystems, and (2) the rate at which data can be transferred between these subsystems. Central Processing Unit (CPU) Cache Memory Main Memory (program and data) 82 The Scientist and Engineer's Guide to Digital Signal Processing elementary binary operations that can be handled directly within the CPU. While this provides the same result, the execution time is much slower, say, a factor of 10 to 20. Most personal computer software can be used with or without a math coprocessor. This is accomplished by having the compiler generate machine code to handle both cases, all stored in the final executable program. If a math coprocessor is present on the particular computer being used, one section of the code will be run. If a math coprocessor is not present, the other section of the code will be used. The compiler can also be directed to generate code for only one of these situations. For example, you will occasionally find a program that requires that a math coprocessor be present, and will crash if run on a computer that does not have one. Applications such as word processing usually do not benefit from a math coprocessor. This is because they involve moving data around in memory, not the calculation of mathematical expressions. Likewise, calculations involving fixed point variables (integers) are unaffected by the presence of a math coprocessor, since they are handled within the CPU. On the other hand, the execution speed of DSP and other computational programs using floating point calculations can be an order of magnitude different with and without a math coprocessor. The CPU and main memory are contained in separate chips in most computer systems. For obvious reasons, you would like the main memory to be very large and very fast. Unfortunately, this makes the memory very expensive. The transfer of data between the main memory and the CPU is a very common bottleneck for speed. The CPU asks the main memory for the binary information at a particular memory address, and then must wait to receive the information. A common technique to get around this problem is to use a memory cache. This is a small amount of very fast memory used as a buffer between the CPU and the main memory. A few hundred kilobytes is typical. When the CPU requests the main memory to provide the binary data at a particular address, high speed digital electronics copies a section of the main memory around this address into the memory cache. The next time that the CPU requests memory information, it is very likely that it will already be contained in the memory cache, making the retrieval very rapid. This is based on the fact that programs tend to access memory locations that are nearby neighbors of previously accessed data. In typical personnel computer applications, the addition of a memory cache can improve the overall speed by several times. The memory cache may be in the same chip as the CPU, or it may be an external electronic device. The rate at which data can be transferred between subsystems depends on the number of parallel data lines provided, and the maximum rate that digital signals that can be passed along each line. Digital data can generally be transferred at a much higher rate within a single chip as compared to transferring data between chips. Likewise, data paths that must pass through electrical connectors to other printed circuit boards (i.e., a bus structure) will be slower still. This is a strong motivation for stuffing as much electronics as possible inside the CPU. Chapter 4- DSP Software 83 A particularly nasty problem for computer speed is backward compatibility. When a computer company introduces a new product, say a data acquisition card or a software program, they want to sell it into the largest possible market. This means that it must be compatible with most of the computers currently in use, which could span several generations of technology. This frequently limits the performance of the hardware or software to that of a much older system. For example, suppose you buy an I/O card that plugs into the bus of your 200 MHz Pentium personal computer, providing you with eight digital lines that can transmit and receive data one byte at a time. You then write an assembly program to rapidly transfer data between your computer and some external device, such as a scientific experiment or another computer. Much to your surprise, the maximum data transfer rate is only about 100,000 bytes per second, more than one thousand times slower than the microprocessor clock rate! The villain is the ISA bus, a technology that is backward compatible to the computers of the early 1980s. Table 4-6 provides execution times for several generations of computers. Obviously, you should treat these as very rough approximations. If you want to understand your system, take measurements on your system. It's quite easy; write a loop that executes a million of some operation, and use your watch to time how long it takes. The first three systems, the 80286, 80486, and Pentium, are the standard desk-top personal computers of 1986, 1993 and 1996, respectively. The fourth is a 1994 microprocessor designed especially for DSP tasks, the Texas Instruments TMS320C40. 80286 (12 MHz) 80486 (33 MHz) PENTIUM (100 MHz) INTEGER A% = B%+C% A% = B%!C% A% = B%×C% A% = B%÷C% 1.6 1.6 2.7 64 0.12 0.12 0.59 9.2 0.04 0.04 0.13 1.5 FLOATING POINT A = B+C A = B! C A = B×C A = B÷C A = SQR(B) A = LOG(B) A = EXP(B) A = B^C A = SIN(B) A = ARCTAN(B) 33 35 35 49 45 186 246 311 262 168 2.5 2.5 2.5 4.5 5.3 19 25 31 30 21 0.50 0.50 0.50 0.87 1.3 3.4 5.5 5.3 6.6 4.4 TMS320C40 (40 MHz) 0.10 0.10 0.10 0.80 0.90 1.7 1.7 2.4 1.1 2.2 TABLE 4-6 Measured execution times for various computers. Times are in microseconds. The 80286, 80486, and Pentium are three generations of personal computers, while the TMS320C40 is a microprocessor specifically designed for DSP tasks. All of the personal computers include a math coprocessor. Use these times only as a general estimate; times on your computer will vary according to the particular hardware and software used. 84 The Scientist and Engineer's Guide to Digital Signal Processing The Pentium is faster than the 80286 system for four reasons, (1) the greater clock speed, (2) more lines in the data bus, (3) the addition of a memory cache, and (4) a more efficient internal design, requiring fewer clock cycles per instruction. If the Pentium was a Cadillac, the TMS320C40 would be a Ferrari: less comfort, but blinding speed. This chip is representative of several microprocessors specifically designed to decrease the execution time of DSP algorithms. Others in this category are the Intel i860, AT&T DSP3210, Motorola DSP96002, and the Analog Devices ADSP-2171. These often go by the names: DSP microprocessor, Digital Signal Processor, and RISC (Reduced Instruction Set Computer). This last name reflects that the increased speed results from fewer assembly level instructions being made available to the programmer. In comparison, more traditional microprocessors, such as the Pentium, are called CISC (Complex Instruction Set Computer). DSP microprocessors are used in two ways: as slave modules under the control of a more conventional computer, or as an imbedded processor in a dedicated application, such as a cellular telephone. Some models only handle fixed point numbers, while others can work with floating point. The internal architecture used to obtain the increased speed includes: (1) lots of very fast cache memory contained within the chip, (2) separate buses for the program and data, allowing the two to be accessed simultaneously (called a Harvard Architecture), (3) fast hardware for math calculations contained directly in the microprocessor, and (4) a pipeline design. A pipeline architecture breaks the hardware required for a certain task into several successive stages. For example, the addition of two numbers may be done in three pipeline stages. The first stage of the pipeline does nothing but fetch the numbers to be added from memory. The only task of the second stage is to add the two numbers together. The third stage does nothing but store the result in memory. If each stage can complete its task in a single clock cycle, the entire procedure will take three clock cycles to execute. The key feature of the pipeline structure is that another task can be started before the previous task is completed. In this example, we could begin the addition of another two numbers as soon as the first stage is idle, at the end of the first clock cycle. For a large number of operations, the speed of the system will be quoted as one addition per clock cycle, even though the addition of any two numbers requires three clock cycles to complete. Pipelines are great for speed, but they can be difficult to program. The algorithm must allow a new calculation to begin, even though the results of previous calculations are unavailable (because they are still in the pipeline). Chapters 28 and 29 discuss DSP microprocessors in much more detail. These are amazing devices; their high-power and low-cost will bring DSP to a wide range of consumer and scientific applications. This is one of the technologies that will shape the twenty-first century. Chapter 4- DSP Software 85 Execution Speed: Programming Tips While computer hardware and programming languages are important for maximizing execution speed, they are not something you change on a day-to day basis. In comparison, how you program can be changed at any time, and will drastically affect how long the program will require to execute. Here are three suggestions. First, use integers instead of floating point variables whenever possible. Conventional microprocessors, such as used in personal computers, process integers 10 to 20 times faster than floating point numbers. On systems without a math coprocessor, the difference can be 200 to 1. An exception to this is integer division, which is often accomplished by converting the values into floating point. This makes the operation ghastly slow compared to other integer calculations. See Table 4-6 for details. Second, avoid using functions such as: sin(x ), log(x ), y x , etc. These transcendental functions are calculated as a series of additions, subtractions and multiplications. For example, the Maclaurin power series provides: EQUATION 4-3 Maclaurin power series expansion for three transcendental functions. This is how computers calculate functions of this type, and why they execute so slowly. sin(x ) ' x & x 3 x 5 x 7 x 9 x 11 % & % & % @@@ 3! 5! 7! 9! 11! cos(x ) ' 1 & x 2 x 4 x 6 x 8 x 10 % & % & % @@@ 2! 4! 6! 8! 10! ex ' 1 % x % x2 x3 x4 x5 % % % % @@@ 2! 3! 4! 5! While these relations are infinite in length, the terms rapidly become small enough to be ignored. For example: sin (1 ) ' 1 & 0.166666 % 0.008333 & 0.000198 % 0.000002 & @@@ These functions require about ten times longer to calculate than a single addition or multiplication (see Table 4-6). Several tricks can be used to bypass these calculations, such as: x 3 ' x @ x @ x ; sin(x ) . x , when x is very small; sin(& x ) ' sin(x) , where you already know one of the values and need to find the other, etc. Most languages only provide a few transcendental functions, and expect you to derive the others by means of the relations in Table 4-7. Not surprisingly, these derived calculations are even slower. 86 The Scientist and Engineer's Guide to Digital Signal Processing FUNCTION EQUATION FOR CALCULATING Secant (X) = Cosecant (X) = Cotangent (X) = 1/COS(X) 1/SIN(X) 1/TAN(X) Arc Sine (X) = Arc Cosine (X) = Arc Secant (X) = Arc Cosecant (X) = Arc Cotangent (X) = ATN(X/SQR(1-X*X)) -ATN(X/SQR(1-X*X)) + PI/2 ATN(SQR(X*X-1)) + (SGN(X)-1) * PI/2 ATN(1/SQR(X*X-1)) + (SGN(X)-1) * PI/2 -ATN(X) + PI/2 Hyperbolic Sine (X) = Hyperbolic Cosine (X) = Hyperbolic Tangent (X) = Hyperbolic Secant (X) = Hyperbolic Cosecant (X) = Hyperbolic Cotangent (X) = (EXP(X)-EXP(-X))/2 (EXP(X)+EXP(-X))/2 (EXP(X)-EXP(-X))/(EXP(X)+EXP(-X)) 1/HYPERBOLIC COSINE 1/HYPERBOLIC SINE 1/HYPERBOLIC TANGENT Arc Hyperbolic Sine (X) = Arc Hyperbolic Cosine (X) = Arc Hyperbolic Tangent (X) = Arc Hyperbolic Secant (X) = Arc Hyperbolic Cosecant (X) = Arc Hyperbolic Cotangent (X) = LOG(X+SQR(X*X+1)) LOG(X+SQR(X*X-1)) LOG((1+X) /(1-X))/2 LOG((SQR(1-X*X)+1)/X) LOG(1+SGN(X)*SQR(1+X*X))/X LOG((X+1)/(X-1))/2 LOG10(X) = PI = LOG(X)/LOG(10) = 0.4342945 LOG(X) 4*ATN(1) = 3.141592653589794 TABLE 4-7 Calculating rarely used functions from more common ones. All angles are in radians, ATN(X) is the arctangent, LOG(X) is the natural logarithm, SGN(X) is the sign of X (i.e., -1 for X#0, 1 for X>0), EXP(X) is eX. Another option is to precalculate these slow functions, and store the values in a look-up table (LUT). For example, imagine an 8 bit data acquisition system used to continually monitor the voltage across a resistor. If the parameter of interest is the power being dissipated in the resistor, the measured voltage can be used to calculate: P ' V 2/R . As a faster alternative, the power corresponding to each of the possible 256 voltage measurements can be calculated beforehand, and stored in a LUT. When the system is running, the measured voltage, a digital number between 0 and 255, becomes an index in the LUT to find the corresponding power. Look-up tables can be hundreds of times faster than direct calculation. Third, learn what is fast and what is slow on your particular system. This comes with experience and testing, and there will always be surprises. Pay particular attention to graphics commands and I/O. There are usually several ways to handle these requirements, and the speeds can be tremendously different. For example, the BASIC command: BLOAD, transfers a data file directly into a section of memory. Reading the same file into memory byte-bybyte (in a loop) can be 100 times as slow. As another example, the BASIC command: LINE, can be used to draw a colored box on the video screen. Drawing the same box pixel-by-pixel can also take 100 times as long. Even putting a print statement within a loop (to keep track of what it is doing) can slow the operation by thousands! CHAPTER Linear Systems 5 Most DSP techniques are based on a divide-and-conquer strategy called superposition. The signal being processed is broken into simple components, each component is processed individually, and the results reunited. This approach has the tremendous power of breaking a single complicated problem into many easy ones. Superposition can only be used with linear systems, a term meaning that certain mathematical rules apply. Fortunately, most of the applications encountered in science and engineering fall into this category. This chapter presents the foundation of DSP: what it means for a system to be linear, various ways for breaking signals into simpler components, and how superposition provides a variety of signal processing techniques. Signals and Systems A signal is a description of how one parameter varies with another parameter. For instance, voltage changing over time in an electronic circuit, or brightness varying with distance in an image. A system is any process that produces an output signal in response to an input signal. This is illustrated by the block diagram in Fig. 5-1. Continuous systems input and output continuous signals, such as in analog electronics. Discrete systems input and output discrete signals, such as computer programs that manipulate the values stored in arrays. Several rules are used for naming signals. These aren't always followed in DSP, but they are very common and you should memorize them. The mathematics is difficult enough without a clear notation. First, continuous signals use parentheses, such as: x(t) and y(t) , while discrete signals use brackets, as in: x[n] and y[n] . Second, signals use lower case letters. Upper case letters are reserved for the frequency domain, discussed in later chapters. Third, the name given to a signal is usually descriptive of the parameters it represents. For example, a voltage depending on time might be called: v(t) , or a stock market price measured each day could be: p[d] . 87 88 The Scientist and Engineer's Guide to Digital Signal Processing x(t) Continuous System y(t) x[n] Discrete System y[n] FIGURE 5-1 Terminology for signals and systems. A system is any process that generates an output signal in response to an input signal. Continuous signals are usually represented with parentheses, while discrete signals use brackets. All signals use lower case letters, reserving the upper case for the frequency domain (presented in later chapters). Unless there is a better name available, the input signal is called: x(t) or x[n], while the output is called: y(t) or y[n]. Signals and systems are frequently discussed without knowing the exact parameters being represented. This is the same as using x and y in algebra, without assigning a physical meaning to the variables. This brings in a fourth rule for naming signals. If a more descriptive name is not available, the input signal to a discrete system is usually called: x[n] , and the output signal: y[n] . For continuous systems, the signals: x(t) and y(t) are used. There are many reasons for wanting to understand a system. For example, you may want to design a system to remove noise in an electrocardiogram, sharpen an out-of-focus image, or remove echoes in an audio recording. In other cases, the system might have a distortion or interfering effect that you need to characterize or measure. For instance, when you speak into a telephone, you expect the other person to hear something that resembles your voice. Unfortunately, the input signal to a transmission line is seldom identical to the output signal. If you understand how the transmission line (the system) is changing the signal, maybe you can compensate for its effect. In still other cases, the system may represent some physical process that you want to study or analyze. Radar and sonar are good examples of this. These methods operate by comparing the transmitted and reflected signals to find the characteristics of a remote object. In terms of system theory, the problem is to find the system that changes the transmitted signal into the received signal. At first glance, it may seem an overwhelming task to understand all of the possible systems in the world. Fortunately, most useful systems fall into a category called linear systems. This fact is extremely important. Without the linear system concept, we would be forced to examine the individual Chapter 5- Linear Systems 89 characteristics of many unrelated systems. With this approach, we can focus on the traits of the linear system category as a whole. Our first task is to identify what properties make a system linear, and how they fit into the everyday notion of electronics, software, and other signal processing systems. Requirements for Linearity A system is called linear if it has two mathematical properties: homogeneity (hÇma-gen-~-ity) and additivity. If you can show that a system has both properties, then you have proven that the system is linear. Likewise, if you can show that a system doesn't have one or both properties, you have proven that it isn't linear. A third property, shift invariance, is not a strict requirement for linearity, but it is a mandatory property for most DSP techniques. When you see the term linear system used in DSP, you should assume it includes shift invariance unless you have reason to believe otherwise. These three properties form the mathematics of how linear system theory is defined and used. Later in this chapter we will look at more intuitive ways of understanding linearity. For now, let's go through these formal mathematical properties. As illustrated in Fig. 5-2, homogeneity means that a change in the input signal's amplitude results in a corresponding change in the output signal's amplitude. In mathematical terms, if an input signal of x[n] results in an output signal of y[n] , an input of k x[n] results in an output of k y[n] , for any input signal and constant, k. IF x[n] System y[n] THEN k x[n] System k y[n] FIGURE 5-2 Definition of homogeneity. A system is said to be homogeneous if an amplitude change in the input results in an identical amplitude change in the output. That is, if x[n] results in y[n], then kx[n] results in ky[n], for any signal, x[n], and any constant, k. 90 The Scientist and Engineer's Guide to Digital Signal Processing A simple resistor provides a good example of both homogenous and nonhomogeneous systems. If the input to the system is the voltage across the resistor, v(t) , and the output from the system is the current through the resistor, i(t) , the system is homogeneous. Ohm's law guarantees this; if the voltage is increased or decreased, there will be a corresponding increase or decrease in the current. Now, consider another system where the input signal is the voltage across the resistor, v(t) , but the output signal is the power being dissipated in the resistor, p(t) . Since power is proportional to the square of the voltage, if the input signal is increased by a factor of two, the output signal is increase by a factor of four. This system is not homogeneous and therefore cannot be linear. The property of additivity is illustrated in Fig. 5-3. Consider a system where an input of x1[n] produces an output of y1[n] . Further suppose that a different input, x2[n] , produces another output, y2[n] . The system is said to be additive, if an input of x1[n] % x2[n] results in an output of y1[n] % y2[n] , for all possible input signals. In words, signals added at the input produce signals that are added at the output. IF x1[n] System y1[n] AND IF x2[n] System y2[n] THEN System x1[n]+x2[n] y1[n]+y2[n] FIGURE 5-3 Definition of additivity. A system is said to be additive if added signals pass through it without interacting. Formally, if x1[n] results in y1[n], and if x2[n] results in y2[n], then x1[n]+x2[n] results in y2[n]+y2[n]. Chapter 5- Linear Systems 91 The important point is that added signals pass through the system without interacting. As an example, think about a telephone conversation with your Aunt Edna and Uncle Bernie. Aunt Edna begins a rather lengthy story about how well her radishes are doing this year. In the background, Uncle Bernie is yelling at the dog for having an accident in his favorite chair. The two voice signals are added and electronically transmitted through the telephone network. Since this system is additive, the sound you hear is the sum of the two voices as they would sound if transmitted individually. You hear Edna and Bernie, not the creature, Ednabernie. A good example of a nonadditive circuit is the mixer stage in a radio transmitter. Two signals are present: an audio signal that contains the voice or music, and a carrier wave that can propagate through space when applied to an antenna. The two signals are added and applied to a nonlinearity, such as a pn junction diode. This results in the signals merging to form a third signal, a modulated radio wave capable of carrying the information over great distances. As shown in Fig. 5-4, shift invariance means that a shift in the input signal will result in nothing more than an identical shift in the output signal. In more formal terms, if an input signal of x[n] results in an output of y [n] , an input signal of x[n % s] results in an output of y[n % s] , for any input signal and any constant, s. Pay particular notice to how the mathematics of this shift is written, it will be used in upcoming chapters. By adding a constant, s, to the independent variable, n, the waveform can be advanced or retarded in the horizontal direction. For example, when s ' 2 , the signal is shifted left by two samples; when s ' & 2 , the signal is shifted right by two samples. IF x[n] System y[n] THEN x[n+s] System y[n+s] FIGURE 5-4 Definition of shift invariance. A system is said to be shift invariant if a shift in the input signal causes an identical shift in the output signal. In mathematical terms, if x[n] produces y[n], then x[n+s] produces y[n+s], for any signal, x[n], and any constant, s. 92 The Scientist and Engineer's Guide to Digital Signal Processing Shift invariance is important because it means the characteristics of the system do not change with time (or whatever the independent variable happens to be). If a blip in the input causes a blop in the output, you can be assured that another blip will cause an identical blop. Most of the systems you encounter will be shift invariant. This is fortunate, because it is difficult to deal with systems that change their characteristics while in operation. For example, imagine that you have designed a digital filter to compensate for the degrading effects of a telephone transmission line. Your filter makes the voices sound more natural and easier to understand. Much to your surprise, along comes winter and you find the characteristics of the telephone line have changed with temperature. Your compensation filter is now mismatched and doesn't work especially well. This situation may require a more sophisticated algorithm that can adapt to changing conditions. Why do homogeneity and additivity play a critical role in linearity, while shift invariance is something on the side? This is because linearity is a very broad concept, encompassing much more than just signals and systems. For example, consider a farmer selling oranges for $2 per crate and apples for $5 per crate. If the farmer sells only oranges, he will receive $20 for 10 crates, and $40 for 20 crates, making the exchange homogenous. If he sells 20 crates of oranges and 10 crates of apples, the farmer will receive: 20 ×$2 % 10 ×$5 ' $90 . This is the same amount as if the two had been sold individually, making the transaction additive. Being both homogenous and additive, this sale of goods is a linear process. However, since there are no signals involved, this is not a system, and shift invariance has no meaning. Shift invariance can be thought of as an additional aspect of linearity needed when signals and systems are involved. Static Linearity and Sinusoidal Fidelity Homogeneity, additivity, and shift invariance are important because they provide the mathematical basis for defining linear systems. Unfortunately, these properties alone don't provide most scientists and engineers with an intuitive feeling of what linear systems are about. The properties of static linearity and sinusoidal fidelity are often of help here. These are not especially important from a mathematical standpoint, but relate to how humans think about and understand linear systems. You should pay special attention to this section. Static linearity defines how a linear system reacts when the signals aren't changing, i.e., when they are DC or static. The static response of a linear system is very simple: the output is the input multiplied by a constant. That is, a graph of the possible input values plotted against the corresponding output values is a straight line that passes through the origin. This is shown in Fig. 5-5 for two common linear systems: Ohm's law for resistors, and Hooke's law for springs. For comparison, Fig. 5-6 shows the static relationship for two nonlinear systems: a pn junction diode, and the magnetic properties of iron. low resistance 93 Elongation Current Chapter 5- Linear Systems weak spring strong spring high resistance Voltage Force a. Ohm's law b. Hooke's law FIGURE 5-5 Two examples of static linearity. In (a), Ohm's law: the current through a resistor is equal to the voltage across the resistor divided by the resistance. In (b), Hooke's law: The elongation of a spring is equal to the applied force multiplied by the spring stiffness coefficient. B Current All linear systems have the property of static linearity. The opposite is usually true, but not always. There are systems that show static linearity, but are not linear with respect to changing signals. However, a very common class of systems can be completely understood with static linearity alone. In these systems it doesn't matter if the input signal is static or changing. These are called memoryless systems, because the output depends only on the present state of the input, and not on its history. For example, the instantaneous current in a resistor depends only on the instantaneous voltage across it, and not on how the signals came to be the value they are. If a system has static linearity, and is memoryless, then the system must be linear. This provides an important way to understand (and prove) the linearity of these simple systems. 0.6 v Voltage a. Silicon diode B b. Iron FIGURE 5-6 Two examples of DC nonlinearity. In (a), a silicon diode has an exponential relationship between voltage and current. In (b), the relationship between magnetic intensity, H, and flux density, B, in iron depends on the history of the sample, a behavior called hysteresis. 94 The Scientist and Engineer's Guide to Digital Signal Processing An important characteristic of linear systems is how they behave with sinusoids, a property we will call sinusoidal fidelity: If the input to a linear system is a sinusoidal wave, the output will also be a sinusoidal wave, and at exactly the same frequency as the input. Sinusoids are the only waveform that have this property. For instance, there is no reason to expect that a square wave entering a linear system will produce a square wave on the output. Although a sinusoid on the input guarantees a sinusoid on the output, the two may be different in amplitude and phase. This should be familiar from your knowledge of electronics: a circuit can be described by its frequency response, graphs of how the circuit's gain and phase vary with frequency. Now for the reverse question: If a system always produces a sinusoidal output in response to a sinusoidal input, is the system guaranteed to be linear? The answer is no, but the exceptions are rare and usually obvious. For example, imagine an evil demon hiding inside a system, with the goal of trying to mislead you. The demon has an oscilloscope to observe the input signal, and a sine wave generator to produce an output signal. When you feed a sine wave into the input, the demon quickly measures the frequency and adjusts his signal generator to produce a corresponding output. Of course, this system is not linear, because it is not additive. To show this, place the sum of two sine waves into the system. The demon can only respond with a single sine wave for the output. This example is not as contrived as you might think; phase lock loops operate in much this way. To get a better feeling for linearity, think about a technician trying to determine if an electronic device is linear. The technician would attach a sine wave generator to the input of the device, and an oscilloscope to the output. With a sine wave input, the technician would look to see if the output is also a sine wave. For example, the output cannot be clipped on the top or bottom, the top half cannot look different from the bottom half, there must be no distortion where the signal crosses zero, etc. Next, the technician would vary the amplitude of the input and observe the effect on the output signal. If the system is linear, the amplitude of the output must track the amplitude of the input. Lastly, the technician would vary the input signal's frequency, and verify that the output signal's frequency changes accordingly. As the frequency is changed, there will likely be amplitude and phase changes seen in the output, but these are perfectly permissible in a linear system. At some frequencies, the output may even be zero, that is, a sinusoid with zero amplitude. If the technician sees all these things, he will conclude that the system is linear. While this conclusion is not a rigorous mathematical proof, the level of confidence is justifiably high. Examples of Linear and Nonlinear Systems Table 5-1 provides examples of common linear and nonlinear systems. As you go through the lists, keep in mind the mathematician's view of linearity (homogeneity, additivity, and shift invariance), as well as the informal way most scientists and engineers use (static linearity and sinusoidal fidelity). Chapter 5- Linear Systems Examples of Linear Systems Wave propagation such as sound and electromagnetic waves Electrical circuits composed of resistors, capacitors, and inductors Electronic circuits, such as amplifiers and filters Mechanical motion from the interaction of masses, springs, and dashpots (dampeners) Systems described by differential equations such as resistor-capacitor-inductor networks Multiplication by a constant, that is, amplification or attenuation of the signal Signal changes, such as echoes, resonances, and image blurring The unity system where the output is always equal to the input The null system where the output is always equal to the zero, regardless of the input Differentiation and integration, and the analogous operations of first difference and running sum for discrete signals Small perturbations in an otherwise nonlinear system, for instance, a small signal being amplified by a properly biased transistor Convolution, a mathematical operation where each value in the output is expressed as the sum of values in the input multiplied by a set of weighing coefficients. Recursion, a technique similar to convolution, except previously calculated values in the output are used in addition to values from the input Examples of Nonlinear Systems Systems that do not have static linearity, for instance, the voltage and power in a resistor: P ' V 2R , the radiant energy emission of a hot object depending on its temperature: R ' kT 4 , the intensity of light transmitted through a thickness of translucent material: I ' e & "T , etc. Systems that do not have sinusoidal fidelity, such as electronics circuits for: peak detection, squaring, sine wave to square wave conversion, frequency doubling, etc. Common electronic distortion, such as clipping, crossover distortion and slewing Multiplication of one signal by another signal, such as in amplitude modulation and automatic gain controls Hysteresis phenomena, such as magnetic flux density versus magnetic intensity in iron, or mechanical stress versus strain in vulcanized rubber Saturation, such as electronic amplifiers and transformers driven too hard Systems with a threshold, for example, digital logic gates, or seismic vibrations that are strong enough to pulverize the intervening rock Table 5-1 Examples of linear and nonlinear systems. Formally, linear systems are defined by the properties of homogeneity, additivity, and shift invariance. Informally, most scientists and engineers think of linear systems in terms of static linearity and sinusoidal fidelity. 95 96 The Scientist and Engineer's Guide to Digital Signal Processing Special Properties of Linearity Linearity is commutative, a property involving the combination of two or more systems. Figure 5-10 shows the general idea. Imagine two systems combined in a cascade, that is, the output of one system is the input to the next. If each system is linear, then the overall combination will also be linear. The commutative property states that the order of the systems in the cascade can be rearranged without affecting the characteristics of the overall combination. You probably have used this principle in electronic circuits. For example, imagine a circuit composed of two stages, one for amplification, and one for filtering. Which is best, amplify and then filter, or filter and then amplify? If both stages are linear, the order doesn't make any difference and the overall result is the same. Keep in mind that actual electronics has nonlinear effects that may make the order important, for instance: interference, DC offsets, internal noise, slew rate distortion, etc. IF FIGURE 5-7 The commutative property for linear systems. When two or more linear systems are arranged in a cascade, the order of the systems does not affect the characteristics of the overall combination. x[n] System A System B System B System A y[n] THEN x[n] y[n] Figure 5-8 shows the next step in linear system theory: multiple inputs and outputs. A system with multiple inputs and/or outputs will be linear if it is composed of linear subsystems and additions of signals. The complexity does not matter, only that nothing nonlinear is allowed inside of the system. To understand what linearity means for systems with multiple inputs and/or outputs, consider the following thought experiment. Start by placing a signal on one input while the other inputs are held at zero. This will cause the multiple outputs to respond with some pattern of signals. Next, repeat the procedure by placing another signal on a different input. Just as before, keep all of the other inputs at zero. This second input signal will result in another pattern of signals appearing on the multiple outputs. To finish the experiment, place both signals on their respective inputs simultaneously. The signals appearing on the outputs will simply be the superposition (sum) of the output signals produced when the input signals were applied separately. Chapter 5- Linear Systems x1[n] FIGURE 5-8 Any system with multiple inputs and/or outputs will be linear if it is composed of linear systems and signal additions. System A x2[n] x3[n] 97 System B y2[n] System C System D y1[n] System E y3[n] The use of multiplication in linear systems is frequently misunderstood. This is because multiplication can be either linear or nonlinear, depending on what the signal is multiplied by. Figure 5-9 illustrates the two cases. A system that multiplies the input signal by a constant, is linear. This system is an amplifier or an attenuator, depending if the constant is greater or less than one, respectively. In contrast, multiplying a signal by another signal is nonlinear. Imagine a sinusoid multiplied by another sinusoid of a different frequency; the resulting waveform is clearly not sinusoidal. Another commonly misunderstood situation relates to parasitic signals added in electronics, such as DC offsets and thermal noise. Is the addition of these extraneous signals linear or nonlinear? The answer depends on where the contaminating signals are viewed as originating. If they are viewed as coming from within the system, the process is nonlinear. This is because a sinusoidal input does not produce a pure sinusoidal output. Conversely, the extraneous signal can be viewed as externally entering the system on a separate input of a multiple input system. This makes the process linear, since only a signal addition is required within the system. x1[n] x[n] y[n] y[n] constant x2[n] Linear Nonlinear a. Multiplication by a constant b. Multiplication of two signals FIGURE 5-9 Linearity of multiplication. Multiplying a signal by a constant is a linear operation. In contrast, the multiplication of two signals is nonlinear. 98 The Scientist and Engineer's Guide to Digital Signal Processing Superposition: the Foundation of DSP When we are dealing with linear systems, the only way signals can be combined is by scaling (multiplication of the signals by constants) followed by addition. For instance, a signal cannot be multiplied by another signal. Figure 5-10 shows an example: three signals: x0[n] , x1[n] , and x2[n] are added to form a fourth signal, x [n] . This process of combining signals through scaling and addition is called synthesis. Decomposition is the inverse operation of synthesis, where a single signal is broken into two or more additive components. This is more involved than synthesis, because there are infinite possible decompositions for any given signal. For example, the numbers 15 and 25 can only be synthesized (added) into the number 40. In comparison, the number 40 can be decomposed into: 1 % 39 or 2 % 38 or & 30.5 % 60 % 10.5 , etc. Now we come to the heart of DSP: superposition, the overall strategy for understanding how signals and systems can be analyzed. Consider an input x0[n] synthesis x[n] decomposition x1[n] FIGURE 5-10 Illustration of synthesis and decomposition of signals. In synthesis, two or more signals are added to form another signal. Decomposition is the opposite process, breaking one signal into two or more additive component signals. x2[n] Chapter 5- Linear Systems 99 x[n] decomposition The Fundamental Concept of DSP x0[n] System y0[n] x1[n] System y1[n] System FIGURE 5-11 The fundamental concept in DSP. Any signal, such as x [n] , can be decomposed into a group of additive components, shown here by the signals: x1[n], x2[n], and x3[n] . Passing these components through a linear system produces the signals, y1[n], y2[n], and y3[n] . The synthesis (addition) of these output signals forms y [n] , the same signal produced when x [n] is passed through the system. y2[n] synthesis x2[n] y[n] signal, called x[n] , passing through a linear system, resulting in an output signal, y[n] . As illustrated in Fig. 5-11, the input signal can be decomposed into a group of simpler signals: x0[n] , x1[n] , x2[n] , etc. We will call these the input signal components. Next, each input signal component is individually passed through the system, resulting in a set of output signal components: y0[n] , y1[n] , y2[n] , etc. These output signal components are then synthesized into the output signal, y[n] . 100 The Scientist and Engineer's Guide to Digital Signal Processing Here is the important part: the output signal obtained by this method is identical to the one produced by directly passing the input signal through the system. This is a very powerful idea. Instead of trying to understanding how complicated signals are changed by a system, all we need to know is how simple signals are modified. In the jargon of signal processing, the input and output signals are viewed as a superposition (sum) of simpler waveforms. This is the basis of nearly all signal processing techniques. As a simple example of how superposition is used, multiply the number 2041 by the number 4, in your head. How did you do it? You might have imagined 2041 match sticks floating in your mind, quadrupled the mental image, and started counting. Much more likely, you used superposition to simplify the problem. The number 2041 can be decomposed into: 2000 % 40 % 1 . Each of these components can be multiplied by 4 and then synthesized to find the final answer, i.e., 8000 % 160 % 4 ' 8164 . Common Decompositions Keep in mind that the goal of this method is to replace a complicated problem with several easy ones. If the decomposition doesn't simplify the situation in some way, then nothing has been gained. There are two main ways to decompose signals in signal processing: impulse decomposition and Fourier decomposition. They are described in detail in the next several chapters. In addition, several minor decompositions are occasionally used. Here are brief descriptions of the two major decompositions, along with three of the minor ones. Impulse Decomposition As shown in Fig. 5-12, impulse decomposition breaks an N samples signal into N component signals, each containing N samples. Each of the component signals contains one point from the original signal, with the remainder of the values being zero. A single nonzero point in a string of zeros is called an impulse. Impulse decomposition is important because it allows signals to be examined one sample at a time. Similarly, systems are characterized by how they respond to impulses. By knowing how a system responds to an impulse, the system's output can be calculated for any given input. This approach is called convolution, and is the topic of the next two chapters. Step Decomposition Step decomposition, shown in Fig. 5-13, also breaks an N sample signal into N component signals, each composed of N samples. Each component signal is a step, that is, the first samples have a value of zero, while the last samples are some constant value. Consider the decomposition of an N point signal, x[n] , into the components: x0[n], x1[n], x2[n], þ xN& 1[n] . The k th component signal, xk[n] , is composed of zeros for points 0 through k& 1 , while the remaining points have a value of: x[k] & x[k& 1] . For example, the 5 th component signal, x5[n] , is composed of zeros for points 0 through 4, while the remaining samples have a value of: x[5] & x[4] (the difference between Chapter 5- Linear Systems x[n] 101 x[n] Impulse Decomposition Step Decomposition x0[n] x0[n] x1[n] x1[n] x2[n] x2[n] x27[n] x27[n] FIGURE 5-12 Example of impulse decomposition. An N point signal is broken into N components, each consisting of a single nonzero point. FIGURE 5-13 Example of step decomposition. An N point signal is broken into N signals, each consisting of a step function sample 4 and 5 of the original signal). As a special case, x0[n] has all of its samples equal to x[0] . Just as impulse decomposition looks at signals one point at a time, step decomposition characterizes signals by the difference between adjacent samples. Likewise, systems are characterized by how they respond to a change in the input signal. 102 The Scientist and Engineer's Guide to Digital Signal Processing Even/Odd Decomposition The even/odd decomposition, shown in Fig. 5-14, breaks a signal into two component signals, one having even symmetry and the other having odd symmetry. An N point signal is said to have even symmetry if it is a mirror image around point N/2 . That is, sample x[N/2 % 1] must equal x[N/2 & 1] , sample x[N/2 % 2] must equal x[N/2 & 2] , etc. Similarly, odd symmetry occurs when the matching points have equal magnitudes but are opposite in sign, such as: x[N/2 % 1] ' & x[N/2 & 1] , x[N/2 % 2] ' & x[N/2 & 2] , etc. These definitions assume that the signal is composed of an even number of samples, and that the indexes run from 0 to N& 1 . The decomposition is calculated from the relations: EQUATION 5-1 Equations for even/odd decomposition. These equations separate a signal, x [n] , into its even part, xE [n] , and its odd part, xO [n] . Since this decomposition is based on circularly symmetry, the zeroth samples in the even and odd signals are calculated: xE [0] ' x [0] , and xO [0] ' 0 . All of the signals are N samples long, with indexes running from 0 to N-1 xE [n ] ' x [n] % x [N & n] 2 x O [n ] ' x [n ] & x [N& n ] 2 This may seem a strange definition of left-right symmetry, since N/2 & ½ (between two samples) is the exact center of the signal, not N/2 . Likewise, this off-center symmetry means that sample zero needs special handling. What's this all about? This decomposition is part of an important concept in DSP called circular symmetry. It is based on viewing the end of the signal as connected to the beginning of the signal. Just as point x[4] is next to point x[5] , point x[N& 1] is next to point x[0] . Picture a snake biting its own tail. When even and odd signals are viewed in this circular manner, there are actually two lines of symmetry, one at point x[N/2] and another at point x[0] . For example, in an even signal, this symmetry around x[0] means that point x[1] equals point x[N& 1] , point x[2] equals point x[N& 2] , etc. In an odd signal, point 0 and point N/2 always have a value of zero. In an even signal, point 0 and point N /2 are equal to the corresponding points in the original signal. What is the motivation for viewing the last sample in a signal as being next to the first sample? There is nothing in conventional data acquisition to support this circular notion. In fact, the first and last samples generally have less in common than any other two points in the sequence. It's common sense! The missing piece to this puzzle is a DSP technique called Fourier analysis. The mathematics of Fourier analysis inherently views the signal as being circular, although it usually has no physical meaning in terms of where the data came from. We will look at this in more detail in Chapter 10. For now, the important thing to understand is that Eq. 5-1 provides a valid decomposition, simply because the even and odd parts can be added together to reconstruct the original signal. Chapter 5- Linear Systems x[n] 103 x[n] Even/Odd Decomposition Interlaced Decomposition even symmetry even samples xE[n] xE[n] odd symmetry odd samples xO[n] xO[n] odd symmetry 0 N/2 N FIGURE 5-14 Example of even/odd decomposition. An N point signal is broken into two N point signals, one with even symmetry, and the other with odd symmetry. FIGURE 5-15 Example of interlaced decomposition. An N point signal is broken into two N point signals, one with the odd samples set to zero, the other with the even samples set to zero. Interlaced Decomposition As shown in Fig. 5-15, the interlaced decomposition breaks the signal into two component signals, the even sample signal and the odd sample signal (not to be confused with even and odd symmetry signals). To find the even sample signal, start with the original signal and set all of the odd numbered samples to zero. To find the odd sample signal, start with the original signal and set all of the even numbered samples to zero. It's that simple. At first glance, this decomposition might seem trivial and uninteresting. This is ironic, because the interlaced decomposition is the basis for an extremely important algorithm in DSP, the Fast Fourier Transform (FFT). The procedure for calculating the Fourier decomposition has been known for several hundred years. Unfortunately, it is frustratingly slow, often requiring minutes or hours to execute on present day computers. The FFT is a family of algorithms developed in the 1960s to reduce this computation time. The strategy is an exquisite example of DSP: reduce the signal to elementary components by repeated use of the interlace transform; calculate the Fourier decomposition of the individual components; synthesize the results into the final answer. The 104 The Scientist and Engineer's Guide to Digital Signal Processing results are dramatic; it is common for the speed to be improved by a factor of hundreds or thousands. Fourier Decomposition Fourier decomposition is very mathematical and not at all obvious. Figure 5-16 shows an example of the technique. Any N point signal can be decomposed into N% 2 signals, half of them sine waves and half of them cosine waves. The lowest frequency cosine wave (called xC0 [n] in this illustration), makes zero complete cycles over the N samples, i.e., it is a DC signal. The next cosine components: xC1 [n] , xC2 [n] , and xC3 [n] , make 1, 2, and 3 complete cycles over the N samples, respectively. This pattern holds for the remainder of the cosine waves, as well as for the sine wave components. Since the frequency of each component is fixed, the only thing that changes for different signals being decomposed is the amplitude of each of the sine and cosine waves. Fourier decomposition is important for three reasons. First, a wide variety of signals are inherently created from superimposed sinusoids. Audio signals are a good example of this. Fourier decomposition provides a direct analysis of the information contained in these types of signals. Second, linear systems respond to sinusoids in a unique way: a sinusoidal input always results in a sinusoidal output. In this approach, systems are characterized by how they change the amplitude and phase of sinusoids passing through them. Since an input signal can be decomposed into sinusoids, knowing how a system will react to sinusoids allows the output of the system to be found. Third, the Fourier decomposition is the basis for a broad and powerful area of mathematics called Fourier analysis, and the even more advanced Laplace and z-transforms. Most cutting-edge DSP algorithms are based on some aspect of these techniques. Why is it even possible to decompose an arbitrary signal into sine and cosine waves? How are the amplitudes of these sinusoids determined for a particular signal? What kinds of systems can be designed with this technique? These are the questions to be answered in later chapters. The details of the Fourier decomposition are too involved to be presented in this brief overview. For now, the important idea to understand is that when all of the component sinusoids are added together, the original signal is exactly reconstructed. Much more on this in Chapter 8. Alternatives to Linearity To appreciate the importance of linear systems, consider that there is only one major strategy for analyzing systems that are nonlinear. That strategy is to make the nonlinear system resemble a linear system. There are three common ways of doing this: First, ignore the nonlinearity. If the nonlinearity is small enough, the system can be approximated as linear. Errors resulting from the original assumption are tolerated as noise or simply ignored. Chapter 5- Linear Systems 105 x[n] Fourier Decomposition cosine waves sine waves xC0[n] xS0[n] xC1[n] xS1[n] xC2[n] xS2[n] xC3[n] xS3[n] xC8[n] xS8[n] FIGURE 5-16 Illustration of Fourier decomposition. An N point signal is decomposed into N+2 signals, each having N points. Half of these signals are cosine waves, and half are sine waves. The frequencies of the sinusoids are fixed; only the amplitudes can change. 106 The Scientist and Engineer's Guide to Digital Signal Processing Second, keep the signals very small. Many nonlinear systems appear linear if the signals have a very small amplitude. For instance, transistors are very nonlinear over their full range of operation, but provide accurate linear amplification when the signals are kept under a few millivolts. Operational amplifiers take this idea to the extreme. By using very high open-loop gain together with negative feedback, the input signal to the op amp (i.e., the difference between the inverting and noninverting inputs) is kept to only a few microvolts. This minuscule input signal results in excellent linearity from an otherwise ghastly nonlinear circuit. Third, apply a linearizing transform. For example, consider two signals being multiplied to make a third: a[n] ' b[n] × c[n] . Taking the logarithm of the signals changes the nonlinear process of multiplication into the linear process of addition: log( a[n]) ' log( b[n]) % log(c[n] ) . The fancy name for this approach is homomorphic signal processing. For example, a visual image can be modeled as the reflectivity of the scene (a two-dimensional signal) being multiplied by the ambient illumination (another two-dimensional signal). Homomorphic techniques enable the illumination signal to be made more uniform, thereby improving the image. In the next chapters we examine the two main techniques of signal processing: convolution and Fourier analysis. Both are based on the strategy presented in this chapter: (1) decompose signals into simple additive components, (2) process the components in some useful manner, and (3) synthesize the components into a final result. This is DSP. CHAPTER 6 Convolution Convolution is a mathematical way of combining two signals to form a third signal. It is the single most important technique in Digital Signal Processing. Using the strategy of impulse decomposition, systems are described by a signal called the impulse response. Convolution is important because it relates the three signals of interest: the input signal, the output signal, and the impulse response. This chapter presents convolution from two different viewpoints, called the input side algorithm and the output side algorithm. Convolution provides the mathematical framework for DSP; there is nothing more important in this book. The Delta Function and Impulse Response The previous chapter describes how a signal can be decomposed into a group of components called impulses. An impulse is a signal composed of all zeros, except a single nonzero point. In effect, impulse decomposition provides a way to analyze signals one sample at a time. The previous chapter also presented the fundamental concept of DSP: the input signal is decomposed into simple additive components, each of these components is passed through a linear system, and the resulting output components are synthesized (added). The signal resulting from this divide-and-conquer procedure is identical to that obtained by directly passing the original signal through the system. While many different decompositions are possible, two form the backbone of signal processing: impulse decomposition and Fourier decomposition. When impulse decomposition is used, the procedure can be described by a mathematical operation called convolution. In this chapter (and most of the following ones) we will only be dealing with discrete signals. Convolution also applies to continuous signals, but the mathematics is more complicated. We will look at how continious signals are processed in Chapter 13. Figure 6-1 defines two important terms used in DSP. The first is the delta function, symbolized by the Greek letter delta, *[n] . The delta function is a normalized impulse, that is, sample number zero has a value of one, while 107 108 The Scientist and Engineer's Guide to Digital Signal Processing all other samples have a value of zero. For this reason, the delta function is frequently called the unit impulse. The second term defined in Fig. 6-1 is the impulse response. As the name suggests, the impulse response is the signal that exits a system when a delta function (unit impulse) is the input. If two systems are different in any way, they will have different impulse responses. Just as the input and output signals are often called x[n] and y[n] , the impulse response is usually given the symbol, h[n] . Of course, this can be changed if a more descriptive name is available, for instance, f [n] might be used to identify the impulse response of a filter. Any impulse can be represented as a shifted and scaled delta function. Consider a signal, a[n] , composed of all zeros except sample number 8, which has a value of -3. This is the same as a delta function shifted to the right by 8 samples, and multiplied by -3. In equation form: a[n] ' & 3*[n& 8] . Make sure you understand this notation, it is used in nearly all DSP equations. If the input to a system is an impulse, such as & 3*[n& 8] , what is the system's output? This is where the properties of homogeneity and shift invariance are used. Scaling and shifting the input results in an identical scaling and shifting of the output. If *[n] results in h[n] , it follows that & 3*[n& 8] results in & 3h[n& 8] . In words, the output is a version of the impulse response that has been shifted and scaled by the same amount as the delta function on the input. If you know a system's impulse response, you immediately know how it will react to any impulse. Convolution Let's summarize this way of understanding how a system changes an input signal into an output signal. First, the input signal can be decomposed into a set of impulses, each of which can be viewed as a scaled and shifted delta function. Second, the output resulting from each impulse is a scaled and shifted version of the impulse response. Third, the overall output signal can be found by adding these scaled and shifted impulse responses. In other words, if we know a system's impulse response, then we can calculate what the output will be for any possible input signal. This means we know everything about the system. There is nothing more that can be learned about a linear system's characteristics. (However, in later chapters we will show that this information can be represented in different forms). The impulse response goes by a different name in some applications. If the system being considered is a filter, the impulse response is called the filter kernel, the convolution kernel, or simply, the kernel. In image processing, the impulse response is called the point spread function. While these terms are used in slightly different ways, they all mean the same thing, the signal produced by a system when the input is a delta function. Chapter 6- Convolution 109 Impulse Response Delta Function 2 2 1 1 0 0 -1 -1 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 6 *[n] Linear System h[n] FIGURE 6-1 Definition of delta function and impulse response. The delta function is a normalized impulse. All of its samples have a value of zero, except for sample number zero, which has a value of one. The Greek letter delta, *[n] , is used to identify the delta function. The impulse response of a linear system, usually denoted by h[n] , is the output of the system when the input is a delta function. Convolution is a formal mathematical operation, just as multiplication, addition, and integration. Addition takes two numbers and produces a third number, while convolution takes two signals and produces a third signal. Convolution is used in the mathematics of many fields, such as probability and statistics. In linear systems, convolution is used to describe the relationship between three signals of interest: the input signal, the impulse response, and the output signal. Figure 6-2 shows the notation when convolution is used with linear systems. An input signal, x[n] , enters a linear system with an impulse response, h[n] , resulting in an output signal, y[n] . In equation form: x[n] t h[n] ' y[n] . Expressed in words, the input signal convolved with the impulse response is equal to the output signal. Just as addition is represented by the plus, +, and multiplication by the cross, ×, convolution is represented by the star, t. It is unfortunate that most programming languages also use the star to indicate multiplication. A star in a computer program means multiplication, while a star in an equation means convolution. FIGURE 6-2 How convolution is used in DSP. The output signal from a linear system is equal to the input signal convolved with the system's impulse response. Convolution is denoted by a star when writing equations. Linear System h[n] x[n] x[n] h[n] = y[n] y[n] 110 The Scientist and Engineer's Guide to Digital Signal Processing 0.08 4 3 0.06 3 2 1 0 Amplitude 4 Amplitude Amplitude a. Low-pass Filter 0.04 0.02 0.00 -1 -2 10 20 30 40 50 60 70 80 1 0 -1 -2 -0.02 0 2 0 10 20 30 Sample Snumber Sample number 0 10 20 30 0 10 20 30 40 50 60 70 80 90 100 110 40 50 60 70 80 90 100 110 Sample number 1.25 4 3 1.00 3 2 1 0 -1 Amplitude 4 Amplitude Amplitude b. High-pass Filter 0.75 0.50 0.25 10 20 30 40 50 60 Sample number Input Signal 70 80 0 -2 -0.25 0 1 -1 0.00 -2 2 0 10 20 30 Sample Snumber Impulse Response Sample number Output Signal FIGURE 6-3 Examples of low-pass and high-pass filtering using convolution. In this example, the input signal is a few cycles of a sine wave plus a slowly rising ramp. These two components are separated by using properly selected impulse responses. Figure 6-3 shows convolution being used for low-pass and high-pass filtering. The example input signal is the sum of two components: three cycles of a sine wave (representing a high frequency), plus a slowly rising ramp (composed of low frequencies). In (a), the impulse response for the low-pass filter is a smooth arch, resulting in only the slowly changing ramp waveform being passed to the output. Similarly, the high-pass filter, (b), allows only the more rapidly changing sinusoid to pass. Figure 6-4 illustrates two additional examples of how convolution is used to process signals. The inverting attenuator, (a), flips the signal top-for-bottom, and reduces its amplitude. The discrete derivative (also called the first difference), shown in (b), results in an output signal related to the slope of the input signal. Notice the lengths of the signals in Figs. 6-3 and 6-4. The input signals are 81 samples long, while each impulse response is composed of 31 samples. In most DSP applications, the input signal is hundreds, thousands, or even millions of samples in length. The impulse response is usually much shorter, say, a few points to a few hundred points. The mathematics behind convolution doesn't restrict how long these signals are. It does, however, specify the length of the output signal. The length of the output signal is Chapter 6- Convolution 111 a. Inverting Attenuator 4 2.00 4 3 2 1 0 1.00 Amplitude Amplitude Amplitude 3 0.00 -1.00 10 20 30 40 50 60 70 80 0 -2 -2.00 0 1 -1 -1 -2 2 0 10 20 0 30 10 20 30 40 50 60 70 80 90 100 110 80 90 100 110 Sample number Sample Snumber Sample number b. Discrete Derivative 2.00 4 4 3 2 1 0 1.00 Amplitude Amplitude Amplitude 3 0.00 -1.00 2 1 0 -1 -1 -2.00 -2 0 10 20 30 40 50 60 Sample number Input Signal 70 80 -2 0 10 20 30 Sample Snumber Impulse Response 0 10 20 30 40 50 60 70 Sample number Output Signal FIGURE 6-4 Examples of signals being processed using convolution. Many signal processing tasks use very simple impulse responses. As shown in these examples, dramatic changes can be achieved with only a few nonzero points. equal to the length of the input signal, plus the length of the impulse response, minus one. For the signals in Figs. 6-3 and 6-4, each output signal is: 81 % 31 & 1 ' 111 samples long. The input signal runs from sample 0 to 80, the impulse response from sample 0 to 30, and the output signal from sample 0 to 110. Now we come to the detailed mathematics of convolution. As used in Digital Signal Processing, convolution can be understood in two separate ways. The first looks at convolution from the viewpoint of the input signal. This involves analyzing how each sample in the input signal contributes to many points in the output signal. The second way looks at convolution from the viewpoint of the output signal. This examines how each sample in the output signal has received information from many points in the input signal. Keep in mind that these two perspectives are different ways of thinking about the same mathematical operation. The first viewpoint is important because it provides a conceptual understanding of how convolution pertains to DSP. The second viewpoint describes the mathematics of convolution. This typifies one of the most difficult tasks you will encounter in DSP: making your conceptual understanding fit with the jumble of mathematics used to communicate the ideas. 112 The Scientist and Engineer's Guide to Digital Signal Processing The Input Side Algorithm Figure 6-5 shows a simple convolution problem: a 9 point input signal, x[n] , is passed through a system with a 4 point impulse response, h[n] , resulting in a 9 % 4 & 1 ' 12 point output signal, y[n] . In mathematical terms, x[n] is convolved with h[n] to produce y[n] . This first viewpoint of convolution is based on the fundamental concept of DSP: decompose the input, pass the components through the system, and synthesize the output. In this example, each of the nine samples in the input signal will contribute a scaled and shifted version of the impulse response to the output signal. These nine signals are shown in Fig. 6-6. Adding these nine signals produces the output signal, y[n] . Let's look at several of these nine signals in detail. We will start with sample number four in the input signal, i.e., x[4] . This sample is at index number four, and has a value of 1.4. When the signal is decomposed, this turns into an impulse represented as: 1.4 *[n& 4] . After passing through the system, the resulting output component will be: 1.4 h[n& 4] . This signal is shown in the center box of the nine signals in Fig. 6-6. Notice that this is the impulse response, h[n] , multiplied by 1.4, and shifted four samples to the right. Zeros have been added at samples 0-3 and at samples 8-11 to serve as place holders. To make this more clear, Fig. 6-6 uses squares to represent the data points that come from the shifted and scaled impulse response, and diamonds for the added zeros. Now examine sample x[8] , the last point in the input signal. This sample is at index number eight, and has a value of -0.5. As shown in the lower-right graph of Fig. 6-6, x[8] results in an impulse response that has been shifted to the right by eight points and multiplied by -0.5. Place holding zeros have been added at points 0-7. Lastly, examine the effect of points x[0] and x[7] . Both these samples have a value of zero, and therefore produce output components consisting of all zeros. x[n] h[n] y[n] 3 3 3 2 2 2 1 1 1 0 0 0 -1 -1 -1 -2 -2 -2 -3 -3 -3 0 1 2 3 4 5 6 7 8 0 1 2 3 0 1 2 3 4 5 6 7 8 9 10 11 FIGURE 6-5 Example convolution problem. A nine point input signal, convolved with a four point impulse response, results in a twelve point output signal. Each point in the input signal contributes a scaled and shifted impulse response to the output signal. These nine scaled and shifted impulse responses are shown in Fig. 6-6. Chapter 6- Convolution 113 3 3 3 2 2 2 1 1 1 0 0 0 -1 -1 -1 contribution from x[0 ] h[n- 0] -2 -3 -2 contribution from x[ 1] h[n- 1] -3 0 1 2 3 4 5 6 7 8 9 10 11 -2 -3 0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11 3 3 3 2 2 2 1 1 1 0 0 0 -1 -1 -1 -2 contribution from x[ 3] h[n- 3] -3 -2 contribution from x[ 4] h[n- 4] -3 0 1 2 3 4 5 6 7 8 9 10 11 -2 0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11 3 3 2 2 2 1 1 1 0 0 0 -1 -1 -1 contribution from x[6] h[n- 6] -3 -2 contribution from x[ 7] h[n- 7] -3 0 1 2 3 4 5 6 7 8 9 10 11 contribution from x[ 5] h[n- 5] -3 3 -2 contribution from x[ 2] h[n- 2] -2 contribution from x[ 8] h[n- 8] -3 0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11 FIGURE 6-6 Output signal components for the convolution in Fig. 6-5. In these signals, each point that results from a scaled and shifted impulse response is represented by a square marker. The remaining data points, represented by diamonds, are zeros that have been added as place holders. In this example, x[n] is a nine point signal and h[n] is a four point signal. In our next example, shown in Fig. 6-7, we will reverse the situation by making x[n] a four point signal, and h[n] a nine point signal. The same two waveforms are used, they are just swapped. As shown by the output signal components, the four samples in x[n] result in four shifted and scaled versions of the nine point impulse response. Just as before, leading and trailing zeros are added as place holders. But wait just one moment! The output signal in Fig. 6-7 is identical to the output signal in Fig. 6-5. This isn't a mistake, but an important property. Convolution is commutative: a[n] t b[n] ' b[n] t a[n] . The mathematics does not care which is the input signal and which is the impulse response, only that two signals are convolved with each other. Although the mathematics may allow it, exchanging the two signals has no physical meaning in system theory. The input signal and impulse response are two totally different things and exchanging them doesn't make sense. What the commutative property provides is a mathematical tool for manipulating equations to achieve various results. 114 The Scientist and Engineer's Guide to Digital Signal Processing A program for calculating convolutions using the input side algorithm is shown in Table 6-1. Remember, the programs in this book are meant to convey algorithms in the simplest form, even at the expense of good programming style. For instance, all of the input and output is handled in mythical subroutines (lines 160 and 280), meaning we do not define how these operations are conducted. Do not skip over these programs; they are a key part of the material and you need to understand them in detail. The program convolves an 81 point input signal, held in array X[ ], with a 31 point impulse response, held in array H[ ], resulting in a 111 point output signal, held in array Y[ ]. These are the same lengths shown in Figs. 6-3 and 6-4. Notice that the names of these arrays use upper case letters. This is a violation of the naming conventions previously discussed, because upper case letters are reserved for frequency domain signals. Unfortunately, the simple BASIC used in this book does not allow lower case variable names. Also notice that line 240 uses a star for multiplication. Remember, a star in a program means multiplication, while a star in an equation means convolution. A star in text (such as documentation or program comments) can mean either. The mythical subroutine in line 160 places the input signal into X[ ] and the impulse response into H[ ]. Lines 180-200 set all of the values in Y[ ] to zero. This is necessary because Y[ ] is used as an accumulator to sum the output components as they are calculated. Lines 220 to 260 are the heart of the program. The FOR statement in line 220 controls a loop that steps through each point in the input signal, X[ ]. For each sample in the input signal, an inner loop (lines 230-250) calculates a scaled and shifted version of the impulse response, and adds it to the array accumulating the output signal, Y[ ]. This nested loop structure (one loop within another loop) is a key characteristic of convolution programs; become familiar with it. 100 'CONVOLUTION USING THE INPUT SIDE ALGORITHM 110 ' 120 DIM X[80] 'The input signal, 81 points 130 DIM H[30] 'The impulse response, 31 points 140 DIM Y[110] 'The output signal, 111 points 150 ' 160 GOSUB XXXX 'Mythical subroutine to load X[ ] and H[ ] 170 ' 180 FOR I% = 0 TO 110 'Zero the output array 190 Y(I%) = 0 200 NEXT I% 210 ' 220 FOR I% = 0 TO 80 'Loop for each point in X[ ] 230 FOR J% = 0 TO 30 'Loop for each point in H[ ] 240 Y[I%+J%] = Y[I%+J%] + X[I%] tH[J%] 250 NEXT J% 260 NEXT I% '(remember, t is multiplication in programs!) 270 ' 280 GOSUB XXXX 'Mythical subroutine to store Y[ ] 290 ' 300 END TABLE 6-1 Chapter 6- Convolution x[n] 115 y[n] h[n] 3 3 3 2 2 2 1 1 1 0 0 0 -1 -1 -1 -2 -2 -2 -3 -3 0 1 2 3 -3 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 8 8 9 10 11 Output signal components 3 3 2 2 1 1 0 0 -1 -1 contribution from x[0] h[n- 0] -2 -3 0 1 -3 2 3 4 5 6 7 8 9 10 11 0 1 3 3 2 2 1 1 0 0 -1 -1 contribution from x[ 2] h[n- 2] -2 -3 0 1 2 3 4 5 6 7 8 9 10 11 contribution from x[ 1] h[n- 1] -2 2 3 4 5 6 7 8 9 10 11 contribution from x[ 3] h[n- 3] -2 -3 0 1 2 3 4 5 6 7 8 9 10 11 FIGURE 6-7 A second example of convolution. The waveforms for the input signal and impulse response are exchanged from the example of Fig. 6-5. Since convolution is commutative, the output signals for the two examples are identical. Keeping the indexing straight in line 240 can drive you crazy! Let's say we are halfway through the execution of this program, so that we have just begun action on sample X[40], i.e., I% = 40. The inner loop runs through each point in the impulse response doing three things. First, the impulse response is scaled by multiplying it by the value of the input sample. If this were the only action taken by the inner loop, line 240 could be written, Y[J%] = X[40] t H[J%]. Second, the scaled impulse is shifted 40 samples to the right by adding this number to the index used in the output signal. This second action would change line 240 to: Y[40+J%] = X[40] t H[J%]. Third, Y[ ] must accumulate (synthesize) all the signals resulting from each sample in the input signal. Therefore, the new information must be added to the information that is already in the array. This results in the final command: Y[40+J%] = Y[40+J%] + X[40] t H[J%]. Study this carefully; it is very confusing, but very important. 116 The Scientist and Engineer's Guide to Digital Signal Processing The Output Side Algorithm The first viewpoint of convolution analyzes how each sample in the input signal affects many samples in the output signal. In this second viewpoint, we reverse this by looking at individual samples in the output signal, and finding the contributing points from the input. This is important from both mathematical and practical standpoints. Suppose that we are given some input signal and impulse response, and want to find the convolution of the two. The most straightforward method would be to write a program that loops through the output signal, calculating one sample on each loop cycle. Likewise, equations are written in the form: y[n] ' some combination of other variables. That is, sample n in the output signal is equal to some combination of the many values in the input signal and impulse response. This requires a knowledge of how each sample in the output signal can be calculated independently of all other samples in the output signal. The output side algorithm provides this information. Let's look at an example of how a single point in the output signal is influenced by several points from the input. The example point we will use is y[6] in Fig. 6-5. This point is equal to the sum of all the sixth points in the nine output components, shown in Fig. 6-6. Now, look closely at these nine output components and identify which can affect y[6] . That is, find which of these nine signals contains a nonzero sample at the sixth position. Five of the output components only have added zeros (the diamond markers) at the sixth sample, and can therefore be ignored. Only four of the output components are capable of having a nonzero value in the sixth position. These are the output components generated from the input samples: x[3], x[4], x[5], and x[6] . By adding the sixth sample from each of these output components, y[6] is determined as: y[6] ' x[3] h[3] % x[4 ]h[2] % x[5] h[1] % x[6]h[0] . That is, four samples from the input signal are multiplied by the four samples in the impulse response, and the products added. Figure 6-8 illustrates the output side algorithm as a convolution machine, a flow diagram of how convolution occurs. Think of the input signal, x[n] , and the output signal, y[n] , as fixed on the page. The convolution machine, everything inside the dashed box, is free to move left and right as needed. The convolution machine is positioned so that its output is aligned with the output sample being calculated. Four samples from the input signal fall into the inputs of the convolution machine. These values are multiplied by the indicated samples in the impulse response, and the products are added. This produces the value for the output signal, which drops into its proper place. For example, y[6] i s s h o w n b e i n g c a l c u l a t e d f r o m t h e f o u r i n p u t s a m p l e s : x[3], x[4], x[5], and x[6] . To calculate y[7] , the convolution machine moves one sample to the right. This results in another four samples entering the machine, x[4] through x[7] , and the value for y[7] dropping into the proper place. This process is repeated for all points in the output signal needing to be calculated. Chapter 6- Convolution 117 3 2 x[n] 1 0 -1 -2 -3 0 1 2 3 4 5 6 7 8 3.0 3 2.0 2 1.0 1 h[n] 0.0 0 -1 -1.0 (flipped) -2 -2.0 -3 -3.0 -3 -2 -1 0 3 2 y[n] 1 0 -1 -2 -3 0 1 2 3 4 5 6 7 8 9 10 11 FIGURE 6-8 The convolution machine. This is a flow diagram showing how each sample in the output signal is influenced by the input signal and impulse response. See the text for details. The arrangement of the impulse response inside the convolution machine is very important. The impulse response is flipped left-for-right. This places sample number zero on the right, and increasingly positive sample numbers running to the left. Compare this to the normal impulse response in Fig. 6-5 to understand the geometry of this flip. Why is this flip needed? It simply falls out of the mathematics. The impulse response describes how each point in the input signal affects the output signal. This results in each point in the output signal being affected by points in the input signal weighted by a flipped impulse response. 118 The Scientist and Engineer's Guide to Digital Signal Processing x[n] 3 3 2 2 1 x[n] 0 -1 -2 1 0 -1 -2 -3 -3 0 1 2 3 4 5 6 7 8 0 3 3.0 3 2.0 2 7 8 -1.0 -1 -2 -2.0 -3 -3.0 y[n] 6 0.0 0 (flipped) -2 -2.0 5 1.0 1 h[n] -1.0 -1 -3 -2 -1 4 2.0 2 0.0 0 (flipped) 2 3.0 3 1.0 1 h[n] 1 -3 -3.0 0 -3 -2 -1 0 3 3 2 2 1 y[n] 0 -1 1 0 -1 -2 -2 -3 -3 0 1 2 3 4 5 6 7 8 9 10 11 a. Set to calculate y[0] 0 1 2 3 4 5 6 7 8 9 10 11 b. Set to calculate y[3] FIGURE 6-9 The convolution machine in action. Figures (a) through (d) show the convolution machine set to calculate four different output signal samples, y[0], y[3], y[8], and y[11]. Figure 6-9 shows the convolution machine being used to calculate several samples in the output signal. This diagram also illustrates a real nuisance in convolution. In (a), the convolution machine is located fully to the left with its output aimed at y[0] . In this position, it is trying to receive input from samples: x[& 3], x[& 2], x[& 1], and x[0] . The problem is, three of these samples: x[& 3], x[& 2], and x[& 1] , do not exist! This same dilemma arises in (d), where the convolution machine tries to accept samples to the right of the defined input signal, points x[9], x[10], and x[11] . One way to handle this problem is by inventing the nonexistent samples. This involves adding samples to the ends of the input signal, with each of the added samples having a value of zero. This is called padding the signal with zeros. Instead of trying to access a nonexistent value, the convolution machine receives a sample that has a value of zero. Since this zero is eliminated during the multiplication, the result is mathematically the same as ignoring the nonexistent inputs. Chapter 6- Convolution x[n] 119 3 3 2 2 1 1 x[n] 0 -1 0 -1 -2 -2 -3 -3 0 1 2 3 4 5 6 7 8 0 1 2 3 6 7 8 3.0 3 2.0 2 2.0 2 1.0 1 h[n] 0.0 0 -1.0 -1 (flipped) 5 3.0 3 1.0 1 h[n] 4 0.0 0 -1 -1.0 (flipped) -2 -2.0 -2 -2.0 -3.0 -3 -3 -3.0 -3 -2 -1 0 y[n] -3 -2 -1 3 3 2 2 1 y[n] 0 -1 0 1 0 -1 -2 -2 -3 -3 0 1 2 3 4 5 6 7 8 0 9 10 11 1 2 3 4 5 6 7 8 9 10 11 d. Set to calculate y[11] c. Set to calculate y[8] Figure 6-9 (continued) The important part is that the far left and far right samples in the output signal are based on incomplete information. In DSP jargon, the impulse response is not fully immersed in the input signal. If the impulse response is M points in length, the first and last M& 1 samples in the output signal are based on less information than the samples between. This is analogous to an electronic circuit requiring a certain amount of time to stabilize after the power is applied. The difference is that this transient is easy to ignore in electronics, but very prominent in DSP. Figure 6-10 shows an example of the trouble these end effects can cause. The input signal is a sine wave plus a DC component. The desire is to remove the DC part of the signal, while leaving the sine wave intact. This calls for a highpass filter, such as the impulse response shown in the figure. The problem is, the first and last 30 points are a mess! The shape of these end regions can be understood by imagining the input signal padded with 30 zeros on the left side, samples x[& 1] through x[& 30] , and 30 zeros on the right, samples x[81] through x[110] . The output signal can then be viewed as a filtered version of this longer waveform. These "end effect" problems are widespread in 120 The Scientist and Engineer's Guide to Digital Signal Processing DSP. As a general rule, expect that the beginning and ending samples in processed signals will be quite useless. Now the math. Using the convolution machine as a guideline, we can write the standard equation for convolution. If x[n] is an N point signal running from 0 to N-1, and h[n] is an M point signal running from 0 to M-1, the convolution of the two: y[n] ' x[n] t h[n] , is an N+M-1 point signal running from 0 to N+M-2, given by: EQUATION 6-1 The convolution summation. This is the formal definition of convolution, written in the shorthand: y [n] ' x [n] t h [n] . In this equation, h [n] is an M point signal with indexes running from 0 to M-1. y [i ] ' j h [ j ] x [i & j ] M&1 j '0 This equation is called the convolution sum. It allows each point in the output signal to be calculated independently of all other points in the output signal. The index, i, determines which sample in the output signal is being calculated, and therefore corresponds to the left-right position of the convolution machine. In computer programs performing convolution, a loop makes this index run through each sample in the output signal. To calculate one of the output samples, the index, j, is used inside of the convolution machine. As j runs through 0 to M-1, each sample in the impulse response, h[ j], is multiplied by the proper sample from the input signal, x[i& j]. All these products are added to produce the output sample being calculated. Study Eq. 6-1 until you fully understand how it is implemented by the convolution machine. Much of DSP is based on this equation. (Don't be confused by the n in y[n] ' x[n] t h[n] . This is merely a place holder to indicate that some variable is the index into the array. Sometimes the equations are written: y[ ] ' x[ ] t h[ ] , just to avoid having to bring in a meaningless symbol). Table 6-2 shows a program for performing convolutions using the output side algorithm, a direct use of Eq. 6-1. This program produces the same output signal as the program for the input side algorithm, shown previously in Table 6-1. Notice the main difference between these two programs: the input side algorithm loops through each sample in the input signal (line 220 of Table 61), while the output side algorithm loops through each sample in the output signal (line 180 of Table 6-2). Here is a detailed operation of this program. The FOR-NEXT loop in lines 180 to 250 steps through each sample in the output signal, using I% as the index. For each of these values, an inner loop, composed of lines 200 to 230, calculates the value of the output sample, Y[I%]. The value of Y[I%] is set to zero in line 190, allowing it to accumulate the products inside of the convolution machine. The FOR-NEXT loop in lines 200 to 240 provide a direct implementation of Eq. 6-1. The index, J%, steps through each Chapter 6- Convolution Input signal 121 Impulse response Output signal 4 1.5 4 2 1.0 2 0 -2 Amplitude Amplitude Amplitude unusable 0.5 0.0 -0.5 -4 0 10 20 30 40 50 60 Sample number 70 80 usable unusable 0 -2 -4 0 10 20 30 Sample Snumber 0 10 20 30 40 50 60 70 80 90 100 110 Sample number FIGURE 6-10 End effects in convolution. When an input signal is convolved with an M point impulse response, the first and last M-1 points in the output signal may not be usable. In this example, the impulse response is a high-pass filter used to remove the DC component from the input signal. sample in the impulse response. Line 230 provides the multiplication of each sample in the impulse response, H[J%], with the appropriate sample from the input signal, X[I%-J%], and adds the result to the accumulator. In line 230, the sample taken from the input signal is: X[I%-J%]. Lines 210 and 220 prevent this from being outside the defined array, X[0] to X[80]. In other words, this program handles undefined samples in the input signal by ignoring them. Another alternative would be to define the input signal's array from X[-30] to X[110], allowing 30 zeros to be padded on each side of the true data. As a third alternative, the FOR-NEXT loop in line 180 could be changed to run from 30 to 80, rather than 0 to 110. That is, the program would only calculate the samples in the output signal where the impulse response is fully immersed in the input signal. The important thing is that you must use one of these three techniques. If you don't, the program will crash when it tries to read the out-of-bounds data. 100 'CONVOLUTION USING THE OUTPUT SIDE ALGORITHM 110 ' 120 DIM X[80] 'The input signal, 81 points 130 DIM H[30] 'The impulse response, 31 points 140 DIM Y[110] 'The output signal, 111 points 150 ' 160 GOSUB XXXX 'Mythical subroutine to load X[ ] and H[ ] 170 ' 180 FOR I% = 0 TO 110 'Loop for each point in Y[ ] 190 Y[I%] = 0 'Zero the sample in the output array 200 FOR J% = 0 TO 30 'Loop for each point in H[ ] 210 IF (I%-J% < 0) THEN GOTO 240 220 IF (I%-J% > 80) THEN GOTO 240 230 Y(I%) = Y(I%) + H(J%) t X(I%-J%) 240 NEXT J% 250 NEXT I% 260 ' 270 GOSUB XXXX 'Mythical subroutine to store Y[ ] 280 ' 290 END TABLE 6-2 122 The Scientist and Engineer's Guide to Digital Signal Processing The Sum of Weighted Inputs The characteristics of a linear system are completely described by its impulse response. This is the basis of the input side algorithm: each point in the input signal contributes a scaled and shifted version of the impulse response to the output signal. The mathematical consequences of this lead to the output side algorithm: each point in the output signal receives a contribution from many points in the input signal, multiplied by a flipped impulse response. While this is all true, it doesn't provide the full story on why convolution is important in signal processing. Look back at the convolution machine in Fig. 6-8, and ignore that the signal inside the dotted box is an impulse response. Think of it as a set of weighing coefficients that happen to be embedded in the flow diagram. In this view, each sample in the output signal is equal to a sum of weighted inputs. Each sample in the output is influenced by a region of samples in the input signal, as determined by what the weighing coefficients are chosen to be. For example, imagine there are ten weighing coefficients, each with a value of onetenth. This makes each sample in the output signal the average of ten samples from the input. Taking this further, the weighing coefficients do not need to be restricted to the left side of the output sample being calculated. For instance, Fig. 6-8 shows y[6] being calculated from: x[3], x[4], x[5], and x[6] . Viewing the convolution machine as a sum of weighted inputs, the weighing coefficients could be chosen symmetrically around the output sample. For example, y[6] might receive contributions from: x[4], x[5], x[6], x[7], and x[8] . Using the same indexing notation as in Fig. 6-8, the weighing coefficients for these five inputs would be held in: h[2], h[1], h[0], h[& 1], and h[& 2] . In other words, the impulse response that corresponds to our selection of symmetrical weighing coefficients requires the use of negative indexes. We will return to this in the next chapter. Mathematically, there is only one concept here: convolution as defined by Eq. 6-1. However, science and engineering problems approach this single concept from two distinct directions. Sometimes you will want to think of a system in terms of what its impulse response looks like. Other times you will understand the system as a set of weighing coefficients. You need to become familiar with both views, and how to toggle between them. CHAPTER 7 Properties of Convolution A linear system's characteristics are completely specified by the system's impulse response, as governed by the mathematics of convolution. This is the basis of many signal processing techniques. For example: Digital filters are created by designing an appropriate impulse response. Enemy aircraft are detected with radar by analyzing a measured impulse response. Echo suppression in long distance telephone calls is accomplished by creating an impulse response that counteracts the impulse response of the reverberation. The list goes on and on. This chapter expands on the properties and usage of convolution in several areas. First, several common impulse responses are discussed. Second, methods are presented for dealing with cascade and parallel combinations of linear systems. Third, the technique of correlation is introduced. Fourth, a nasty problem with convolution is examined, the computation time can be unacceptably long using conventional algorithms and computers. Common Impulse Responses Delta Function The simplest impulse response is nothing more that a delta function, as shown in Fig. 7-1a. That is, an impulse on the input produces an identical impulse on the output. This means that all signals are passed through the system without change. Convolving any signal with a delta function results in exactly the same signal. Mathematically, this is written: EQUATION 7-1 The delta function is the identity for convolution. Any signal convolved with a delta function is left unchanged. x[n ] ( *[n ] ' x[n ] This property makes the delta function the identity for convolution. This is analogous to zero being the identity for addition ( a % 0 ' a ) , and one being the identity for multiplication ( a × 1 ' a ) . At first glance, this type of system 123 124 The Scientist and Engineer's Guide to Digital Signal Processing may seem trivial and uninteresting. Not so! Such systems are the ideal for data storage, communication and measurement. Much of DSP is concerned with passing information through systems without change or degradation. Figure 7-1b shows a slight modification to the delta function impulse response. If the delta function is made larger or smaller in amplitude, the resulting system is an amplifier or attenuator, respectively. In equation form, amplification results if k is greater than one, and attenuation results if k is less than one: EQUATION 7-2 A system that amplifies or attenuates has a scaled delta function for an impulse response. In this equation, k determines the amplification or attenuation. x [n] ( k*[n] ' k x[n ] The impulse response in Fig. 7-1c is a delta function with a shift. This results in a system that introduces an identical shift between the input and output signals. This could be described as a signal delay, or a signal advance, depending on the direction of the shift. Letting the shift be represented by the parameter, s, this can be written as the equation: EQUATION 7-3 A relative shift between the input and output signals corresponds to an impulse response that is a shifted delta function. The variable, s, determines the amount of shift in this equation. x [n] ( *[n% s ] ' x[n % s] Science and engineering are filled with cases where one signal is a shifted version of another. For example, consider a radio signal transmitted from a remote space probe, and the corresponding signal received on the earth. The time it takes the radio wave to propagate over the distance causes a delay between the transmitted and received signals. In biology, the electrical signals in adjacent nerve cells are shifted versions of each other, as determined by the time it takes an action potential to cross the synaptic junction that connects the two. Figure 7-1d shows an impulse response composed of a delta function plus a shifted and scaled delta function. By superposition, the output of this system is the input signal plus a delayed version of the input signal, i.e., an echo. Echoes are important in many DSP applications. The addition of echoes is a key part in making audio recordings sound natural and pleasant. Radar and sonar analyze echoes to detect aircraft and submarines. Geophysicists use echoes to find oil. Echoes are also very important in telephone networks, because you want to avoid them. Chapter 7- Properties of Convolution 125 a. Identity The delta function is the identity for convolution. Convolving a signal with the delta function leaves the signal unchanged. This is the goal of systems that transmit or store signals. Amplitude 2 1 0 -1 -2 -2 -1 0 -2 -1 0 -2 -1 0 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 5 6 Sample number b. Amplification & Attenuation Increasing or decreasing the amplitude of the delta function forms an impulse response that amplifies or attenuates, respectively. This impulse response will amplify the signal by 1.6. Amplitude 2 1 0 -1 -2 Sample number Amplitude 2 c. Shift Shifting the delta function produces a corresponding shift between the input and output signals. Depending on the direction, this can be called a delay or an advance. This impulse response delays the signal by four samples. 1 0 -1 -2 Sample number Amplitude 2 d. Echo A delta function plus a shifted and scaled delta function results in an echo being added to the original signal. In this example, the echo is delayed by four samples and has an amplitude of 60% of the original signal. 1 0 -1 -2 -2 -1 0 1 2 3 4 Sample number FIGURE 7-1 Simple impulse responses using shifted and scaled delta functions. Calculus-like Operations Convolution can change discrete signals in ways that resemble integration and differentiation. Since the terms "derivative" and "integral" specifically refer to operations on continuous signals, other names are given to their discrete counterparts. The discrete operation that mimics the first derivative is called the first difference. Likewise, the discrete form of the integral is called the 126 The Scientist and Engineer's Guide to Digital Signal Processing running sum. It is also common to hear these operations called the discrete derivative and the discrete integral, although mathematicians frown when they hear these informal terms used. Figure 7-2 shows the impulse responses that implement the first difference and the running sum. Figure 7-3 shows an example using these operations. In 73a, the original signal is composed of several sections with varying slopes. Convolving this signal with the first difference impulse response produces the signal in Fig. 7-3b. Just as with the first derivative, the amplitude of each point in the first difference signal is equal to the slope at the corresponding location in the original signal. The running sum is the inverse operation of the first difference. That is, convolving the signal in (b), with the running sum's impulse response, produces the signal in (a). These impulse responses are simple enough that a full convolution program is usually not needed to implement them. Rather, think of them in the alternative mode: each sample in the output signal is a sum of weighted samples from the input. For instance, the first difference can be calculated: EQUATION 7-4 Calculation of the first difference. In this relation, x [n] is the original signal, and y [n] is the first difference. y[n ] ' x [n] & x [n& 1] That is, each sample in the output signal is equal to the difference between two adjacent samples in the input signal. For instance, y[40] ' x[40] & x[39] . It should be mentioned that this is not the only way to define a discrete derivative. Another common method is to define the slope symmetrically around the point being examined, such as: y[n] ' ( x[n% 1] & x[n& 1] )/ 2 . Amplitude 2 a. First Difference This is the discrete version of the first derivative. Each sample in the output signal is equal to the difference between adjacent samples in the input signal. In other words, the output signal is the slope of the input signal. 1 0 -1 -2 -2 -1 0 -2 -1 0 1 2 3 4 5 6 1 2 3 4 5 6 Sample number Amplitude 2 b. Running Sum The running sum is the discrete version of the integral. Each sample in the output signal is equal to the sum of all samples in the input signal to the left. Note that the impulse response extends to infinity, a rather nasty feature. 1 0 -1 -2 Sample number FIGURE 7-2 Impulse responses that mimic calculus operations. Chapter 7- Properties of Convolution 127 2.0 a. Original signal Amplitude 1.0 0.0 -1.0 FIGURE 7-3 Example of calculus-like operations. The signal in (b) is the first difference of the signal in (a). Correspondingly, the signal is (a) is the running sum of the signal in (b). These processing methods are used with discrete signals the same as differentiation and integration are used with continuous signals. -2.0 0 10 20 30 40 50 Sample number First Difference 60 70 80 70 80 Running Sum 0.2 b. First difference Amplitude 0.1 0.0 -0.1 -0.2 0 10 20 30 40 50 Sample number 60 Using this same approach, each sample in the running sum can be calculated by summing all points in the original signal to the left of the sample's location. For instance, if y[n] is the running sum of x[n] , then sample y[40] is found by adding samples x[0] through x[40] . Likewise, sample y[41] is found by adding samples x[0] through x[41] . Of course, it would be very inefficient to calculate the running sum in this manner. For example, if y[40] has already been c a l c u l a t e d , y[41] c a n b e c a l c u l a t e d w i t h o n l y a s i n g l e a d d i t i o n : y[41] ' x[41] % y[40] . In equation form: EQUATION 7-5 Calculation of the running sum. In this relation, x [n] is the original signal, and y [n] is the running sum. y[n ] ' x [n] % y [n& 1] Relations of this type are called recursion equations or difference equations. We will revisit them in Chapter 19. For now, the important idea to understand is that these relations are identical to convolution using the impulse responses of Fig. 7-2. Table 7-1 provides computer programs that implement these calculus-like operations. 128 The Scientist and Engineer's Guide to Digital Signal Processing 100 110 110 120 130 'Calculation of the First Difference Y[0] = 0 FOR I% = 1 TO N%-1 Y[I%] = X[I%] - Y[I%-1] NEXT I% 100 110 120 120 130 'Calculation of the running sum Y[0] = X[0] FOR I% = 1 TO N%-1 Y[I%] = Y[I%-1] + X[I%] NEXT I% Table 7-1 Programs for calculating the first difference and running sum. The original signal is held in X[ ], and the processed signal (the first difference or running sum) is held in Y[ ]. Both arrays run from 0 to N%-1. Low-pass and High-pass Filters The design of digital filters is covered in detail in later chapters. For now, be satisfied to understand the general shape of low-pass and high-pass filter kernels (another name for a filter's impulse response). Figure 7-4 shows several common low-pass filter kernels. In general, low-pass filter kernels are composed of a group of adjacent positive points. This results in each sample in the output signal being a weighted average of many adjacent points from the input signal. This averaging smoothes the signal, thereby removing highfrequency components. As shown by the sinc function in (c), some low-pass filter kernels include a few negative valued samples in the tails. Just as in analog electronics, digital low-pass filters are used for noise reduction, signal separation, wave shaping, etc. 0.4 0.4 a. Exponential b. Square pulse Amplitude 0.3 0.2 0.1 0.0 -0.1 -20 0.2 0.1 0.0 -15 -10 -5 0 5 Sample number 10 15 20 -0.1 -20 -15 -10 -5 0 5 10 15 20 -5 0 5 10 15 20 Sample number 0.4 FIGURE 7-4 Typical low-pass filter kernels. Low-pass filter kernels are formed from a group of adjacent positive points that provide an averaging (smoothing) of the signal. As discussed in later chapters, each of these filter kernels is best for a particular purpose. The exponential, (a), is the simplest recursive filter. The rectangular pulse, (b), is best at reducing noise while maintaining edge sharpness. The sinc function in (c), a curve of the form: sin(x)/(x) , is used to separate one band of frequencies from another. c. Sinc 0.3 Amplitude Amplitude 0.3 0.2 0.1 0.0 -0.1 -20 -15 -10 Sample number Chapter 7- Properties of Convolution 1.5 1.50 a. Exponential b. Square pulse Amplitude 1.00 0.5 0.0 0.50 0.00 -15 -10 -5 0 5 Sample number 10 15 20 -0.50 -20 -15 -10 -5 0 5 10 15 20 -5 0 5 10 15 20 Sample number 1.50 c. Sinc FIGURE 7-5 Typical high-pass filter kernels. These are formed by subtracting the corresponding lowpass filter kernels in Fig. 7-4 from a delta function. The distinguishing characteristic of high-pass filter kernels is a spike surrounded by many adjacent negative samples. 1.00 Amplitude Amplitude 1.0 -0.5 -20 129 0.50 0.00 -0.50 -20 -15 -10 Sample number The cutoff frequency of the filter is changed by making filter kernel wider or narrower. If a low-pass filter has a gain of one at DC (zero frequency), then the sum of all of the points in the impulse response must be equal to one. As illustrated in (a) and (c), some filter kernels theoretically extend to infinity without dropping to a value of zero. In actual practice, the tails are truncated after a certain number of samples, allowing it to be represented by a finite number of points. How else could it be stored in a computer? Figure 7-5 shows three common high-pass filter kernels, derived from the corresponding low-pass filter kernels in Fig. 7-4. This is a common strategy in filter design: first devise a low-pass filter and then transform it to what you need, high-pass, band-pass, band-reject, etc. To understand the low-pass to high-pass transform, remember that a delta function impulse response passes the entire signal, while a low-pass impulse response passes only the lowfrequency components. By superposition, a filter kernel consisting of a delta function minus the low-pass filter kernel will pass the entire signal minus the low-frequency components. A high-pass filter is born! As shown in Fig. 7-5, the delta function is usually added at the center of symmetry, or sample zero if the filter kernel is not symmetrical. High-pass filters have zero gain at DC (zero frequency), achieved by making the sum of all the points in the filter kernel equal to zero. 130 The Scientist and Engineer's Guide to Digital Signal Processing 0.4 0.4 a. Causal b. Causal Amplitude 0.3 0.2 0.1 0.0 -0.1 -20 0.2 0.1 0.0 -15 -10 -5 0 5 Sample number 10 15 20 -0.1 -20 -15 -10 -5 0 5 10 15 20 0 5 10 15 20 Sample number 0.4 c. Noncausal FIGURE 7-6 Examples of causal signals. An impulse response, or any signal, is said to be causal if all negative numbered samples have a value of zero. Three examples are shown here. Any noncausal signal with a finite number of points can be turned into a causal signal simply by shifting. 0.3 Amplitude Amplitude 0.3 0.2 0.1 0.0 -0.1 -20 -15 -10 -5 Sample number Causal and Noncausal Signals Imagine a simple analog electronic circuit. If you apply a short pulse to the input, you will see a response on the output. This is the kind of cause and effect that our universe is based on. One thing we definitely know: any effect must happen after the cause. This is a basic characteristic of what we call time. Now compare this to a DSP system that changes an input signal into an output signal, both stored in arrays in a computer. If this mimics a real world system, it must follow the same principle of causality as the real world does. For example, the value at sample number eight in the input signal can only affect sample number eight or greater in the output signal. Systems that operate in this manner are said to be causal. Of course, digital processing doesn't necessarily have to function this way. Since both the input and output signals are arrays of numbers stored in a computer, any of the input signal values can affect any of the output signal values. As shown by the examples in Fig. 7-6, the impulse response of a causal system must have a value of zero for all negative numbered samples. Think of this from the input side view of convolution. To be causal, an impulse in the input signal at sample number n must only affect those points in the output signal with a sample number of n or greater. In common usage, the term causal is applied to any signal where all the negative numbered samples have a value of zero, whether it is an impulse response or not. Chapter 7- Properties of Convolution 0.4 0.4 a. Zero phase b. Linear phase Amplitude 0.3 0.2 0.1 0.0 0.2 0.1 0.0 -15 -10 -5 0 5 Sample number 10 15 20 -0.1 -20 -15 -10 -5 0 5 10 15 20 0 5 10 15 20 Sample number 0.4 FIGURE 7-7 Examples of phase linearity. Signals that have a left-right symmetry are said to be linear phase. If the axis of symmetry occurs at sample number zero, they are additionally said to be zero phase. Any linear phase signal can be transformed into a zero phase signal simply by shifting. Signals that do not have a leftright symmetry are said to be nonlinear phase. Do not confuse these terms with the linear in linear systems. They are completely different concepts. c. Nonlinear phase 0.3 Amplitude Amplitude 0.3 -0.1 -20 131 0.2 0.1 0.0 -0.1 -20 -15 -10 -5 Sample number Zero Phase, Linear Phase, and Nonlinear Phase As shown in Fig. 7-7, a signal is said to be zero phase if it has left-right symmetry around sample number zero. A signal is said to be linear phase if it has left-right symmetry, but around some point other than zero. This means that any linear phase signal can be changed into a zero phase signal simply by shifting left or right. Lastly, a signal is said to be nonlinear phase if it does not have left-right symmetry. You are probably thinking that these names don't seem to follow from their definitions. What does phase have to do with symmetry? The answer lies in the frequency spectrum, and will be discussed in more detail in later chapters. Briefly, the frequency spectrum of any signal is composed of two parts, the magnitude and the phase. The frequency spectrum of a signal that is symmetrical around zero has a phase that is zero. Likewise, the frequency spectrum of a signal that is symmetrical around some nonzero point has a phase that is a straight line, i.e., a linear phase. Lastly, the frequency spectrum of a signal that is not symmetrical has a phase that is not a straight line, i.e., it has a nonlinear phase. A special note about the potentially confusing terms: linear and nonlinear phase. What does this have to do the concept of system linearity discussed in previous chapters? Absolutely nothing! System linearity is the broad concept 132 The Scientist and Engineer's Guide to Digital Signal Processing that nearly all of DSP is based on (superposition, homogeneity, additivity, etc). Linear and nonlinear phase mean that the phase is, or is not, a straight line. In fact, a system must be linear even to say that the phase is zero, linear, or nonlinear. Mathematical Properties Commutative Property The commutative property for convolution is expressed in mathematical form: EQUATION 7-6 The commutative property of convolution. This states that the order in which signals are convolved can be exchanged. a[n ] ( b[n ] ' b[n ] ( a[n ] In words, the order in which two signals are convolved makes no difference; the results are identical. As shown in Fig. 7-8, this has a strange meaning for system theory. In any linear system, the input signal and the system's impulse response can be exchanged without changing the output signal. This is interesting, but usually doesn't have any physical meaning. The input signal and the impulse response are very different things. Just because the mathematics allows you to do something, doesn't mean that it makes sense to do it. For example, suppose you make: $10/hour × 2,000 hours/year = $20,000/year. The commutative property for multiplication provides that you can make the same annual salary by only working 10 hours/year at $2000/hour. Let's see you convince your boss that this is meaningful! In spite of this, the commutative property sees great use in DSP for manipulating equations, just as in ordinary algebra. IF a[n] b[n] y[n] b[n] a[n] y[n] THEN FIGURE 7-8 The commutative property in system theory. The commutative property of convolution allows the input signal and the impulse response of a system to be exchanged without changing the output. While interesting, this usually has no physical significance. (A signal appearing inside of a box, such as b[n] and a[n] in this figure, represent the impulse response of the system). Chapter 7- Properties of Convolution 133 Associative Property Is it possible to convolve three or more signals? The answer is yes, and the associative property describes how: convolve two of the signals to produce an intermediate signal, then convolve the intermediate signal with the third signal. The associative property provides that the order of the convolutions doesn't matter. As an equation: EQUATION 7-7 The associative property of convolution describes how three or more signals are convolved. ( a [n] ( b [n] ) ( c[n ] ' a[n ] ( ( b[n ] ( c[n ] ) The associative property is used in system theory to describe how cascaded systems behave. As shown in Fig. 7-9, two or more systems are said to be in a cascade if the output of one system is used as the input for the next system. From the associative property, the order of the systems can be rearranged without changing the overall response of the cascade. Further, any number of cascaded systems can be replaced with a single system. The impulse response of the replacement system is found by convolving the impulse responses of all of the original systems. IF x[n] h1[n] h2[n] y[n] h2[n] h1[n] y[n] THEN x[n] ALSO x[n] h1[n] h2[n] y[n] FIGURE 7-9 The associative property in system theory. The associative property provides two important characteristics of cascaded linear systems. First, the order of the systems can be rearranged without changing the overall operation of the cascade. Second, two or more systems in a cascade can be replaced by a single system. The impulse response of the replacement system is found by convolving the impulse responses of the stages being replaced. 134 The Scientist and Engineer's Guide to Digital Signal Processing Distributive Property In equation form, the distributive property is written: EQUATION 7-8 The distributive property of convolution describes how parallel systems are analyzed. a[n ]( b[n ] % a [n ]( c[n ] ' a[n ] ( ( b [n] % c [n] ) The distributive property describes the operation of parallel systems with added outputs. As shown in Fig. 7-10, two or more systems can share the same input, x[n] , and have their outputs added to produce y[n] . The distributive property allows this combination of systems to be replaced with a single system, having an impulse response equal to the sum of the impulse responses of the original systems. IF h1[n] y[n] x[n] h2[n] THEN x[n] h1[n] + h2[n] y[n] FIGURE 7-10 The distributive property in system theory. The distributive property shows that parallel systems with added outputs can be replaced with a single system. The impulse response of the replacement system is equal to the sum of the impulse responses of all the original systems. Transference between the Input and Output Rather than being a formal mathematical property, this is a way of thinking about a common situation in signal processing. As illustrated in Fig. 7-11, imagine a linear system receiving an input signal, x[n] , and generating an output signal, y[n] . Now suppose that the input signal is changed in some linear way, resulting in a new input signal, which we will call x 3[n] . This results in a new output signal, y 3[n] . The question is, how does the change in Chapter 7- Properties of Convolution 135 the input signal relate to the change in the output signal? The answer is: the output signal is changed in exactly the same linear way that the input signal was changed. For example, if the input signal is amplified by a factor of two, the output signal will also be amplified by a factor of two. If the derivative is taken of the input signal, the derivative will also be taken of the output signal. If the input is filtered in some way, the output will be filtered in an identical manner. This can easily be proven by using the associative property. IF x[n] h[n] Same Linear Change Linear Change THEN x [n] y[n] h[n] y [n] FIGURE 7-11 Tranference between the input and output. This is a way of thinking about a common situation in signal processing. A linear change made to the input signal results in the same linear change being made to the output signal. The Central Limit Theorem The Central Limit Theorem is an important tool in probability theory because it mathematically explains why the Gaussian probability distribution is observed so commonly in nature. For example: the amplitude of thermal noise in electronic circuits follows a Gaussian distribution; the cross-sectional intensity of a laser beam is Gaussian; even the pattern of holes around a dart board bull's eye is Gaussian. In its simplest form, the Central Limit Theorem states that a Gaussian distribution results when the observed variable is the sum of many random processes. Even if the component processes do not have a Gaussian distribution, the sum of them will. The Central Limit Theorem has an interesting implication for convolution. If a pulse-like signal is convolved with itself many times, a Gaussian is produced. Figure 7-12 shows an example of this. The signal in (a) is an 136 The Scientist and Engineer's Guide to Digital Signal Processing 3.00 18.0 b. x[n] x[n] a. x[n] 12.0 Amplitude Amplitude 2.00 1.00 0.00 -1.00 -25 6.0 0.0 -6.0 -20 -15 -10 -5 0 5 Sample number 10 15 20 25 -25 -20 -15 -10 -5 0 5 Sample number 10 15 20 25 10 15 20 25 1500 c. x[n] x[n] x[n] x[n] -20 -5 1000 Amplitude FIGURE 7-12 Example of convolving a pulse waveform with itself. The Central Limit Theorem shows that a Gaussian waveform is produced when an arbitrary shaped pulse is convolved with itself many times. Figure (a) is an example pulse. In (b), the pulse is convolved with itself once, and begins to appear smooth and regular. In (c), the pulse is convolved with itself three times, and closely approximates a Gaussian. 500 0 -500 -25 -15 -10 0 5 Sample number irregular pulse, purposely chosen to be very unlike a Gaussian. Figure (b) shows the result of convolving this signal with itself one time. Figure (c) shows the result of convolving this signal with itself three times. Even with only three convolutions, the waveform looks very much like a Gaussian. In mathematics jargon, the procedure converges to a Gaussian very quickly. The width of the resulting Gaussian (i.e., F in Eq. 2-7 or 2-8) is equal to the width of the original pulse (expressed as F in Eq. 2-7) multiplied by the square root of the number of convolutions. Correlation The concept of correlation can best be presented with an example. Figure 7-13 shows the key elements of a radar system. A specially designed antenna transmits a short burst of radio wave energy in a selected direction. If the propagating wave strikes an object, such as the helicopter in this illustration, a small fraction of the energy is reflected back toward a radio receiver located near the transmitter. The transmitted pulse is a specific shape that we have selected, such as the triangle shown in this example. The received signal will consist of two parts: (1) a shifted and scaled version of the transmitted pulse, and (2) random noise, resulting from interfering radio waves, thermal noise in the electronics, etc. Since radio signals travel at a known rate, the speed of Chapter 7- Properties of Convolution 137 light, the shift between the transmitted and received pulse is a direct measure of the distance to the object being detected. This is the problem: given a signal of some known shape, what is the best way to determine where (or if) the signal occurs in another signal. Correlation is the answer. Correlation is a mathematical operation that is very similar to convolution. Just as with convolution, correlation uses two signals to produce a third signal. This third signal is called the cross-correlation of the two input signals. If a signal is correlated with itself, the resulting signal is instead called the autocorrelation. The convolution machine was presented in the last chapter to show how convolution is performed. Figure 7-14 is a similar FIGURE 7-13 Key elements of a radar system. Like other echo location systems, radar transmits a short pulse of energy that is reflected by objects being examined. This makes the received waveform a shifted version of the transmitted waveform, plus random noise. Detection of a known waveform in a noisy signal is the fundamental problem in echo location. The answer to this problem is correlation. TRANSMIT Transmitted amplitude 200 100 0 -100 -10 0 -10 0 10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80 Sample number (or time) Received amplitude 0.2 0.1 0 -0.1 Sample number (or time) RECEIVE 138 The Scientist and Engineer's Guide to Digital Signal Processing illustration of a correlation machine. The received signal, x[n] , and the cross-correlation signal, y[n] , are fixed on the page. The waveform we are looking for, t[n] , commonly called the target signal, is contained within the correlation machine. Each sample in y[n] is calculated by moving the correlation machine left or right until it points to the sample being worked on. Next, the indicated samples from the received signal fall into the correlation machine, and are multiplied by the corresponding points in the target signal. The sum of these products then moves into the proper sample in the crosscorrelation signal. The amplitude of each sample in the cross-correlation signal is a measure of how much the received signal resembles the target signal, at that location. This means that a peak will occur in the cross-correlation signal for every target signal that is present in the received signal. In other words, the value of the cross-correlation is maximized when the target signal is aligned with the same features in the received signal. What if the target signal contains samples with a negative value? Nothing changes. Imagine that the correlation machine is positioned such that the target signal is perfectly aligned with the matching waveform in the received signal. As samples from the received signal fall into the correlation machine, they are multiplied by their matching samples in the target signal. Neglecting noise, a positive sample will be multiplied by itself, resulting in a positive number. Likewise, a negative sample will be multiplied by itself, also resulting in a positive number. Even if the target signal is completely negative, the peak in the cross-correlation will still be positive. If there is noise on the received signal, there will also be noise on the crosscorrelation signal. It is an unavoidable fact that random noise looks a certain amount like any target signal you can choose. The noise on the cross-correlation signal is simply measuring this similarity. Except for this noise, the peak generated in the cross-correlation signal is symmetrical between its left and right. This is true even if the target signal isn't symmetrical. In addition, the width of the peak is twice the width of the target signal. Remember, the cross-correlation is trying to detect the target signal, not recreate it. There is no reason to expect that the peak will even look like the target signal. Correlation is the optimal technique for detecting a known waveform in random noise. That is, the peak is higher above the noise using correlation than can be produced by any other linear system. (To be perfectly correct, it is only optimal for random white noise). Using correlation to detect a known waveform is frequently called matched filtering. More on this in Chapter 17. The correlation machine and convolution machine are identical, except for one small difference. As discussed in the last chapter, the signal inside of the convolution machine is flipped left-for-right. This means that samples numbers: 1, 2, 3 þ run from the right to the left. In the correlation machine this flip doesn't take place, and the samples run in the normal direction. Chapter 7- Properties of Convolution 139 2 1 x[n] 0 -1 0 10 20 30 40 50 60 70 80 50 60 70 80 Sample number (or time) 2 1 t[n] 0 -1 0 10 20 3 2 y[n] 1 0 -1 0 10 20 30 40 Sample number (or time) FIGURE 7-14 The correlation machine. This is a flowchart showing how the cross-correlation of two signals is calculated. In this example, y [n] is the cross-correlation of x [n] and t [n] . The dashed box is moved left or right so that its output points at the sample being calculated in y [n] . The indicated samples from x [n] are multiplied by the corresponding samples in t [n] , and the products added. The correlation machine is identical to the convolution machine (Figs. 6-8 and 6-9), except that the signal inside of the dashed box is not reversed. In this illustration, the only samples calculated in y [n] are where t [n] is fully immersed in x [n] . Since this signal reversal is the only difference between the two operations, it is possible to represent correlation using the same mathematics as convolution. This requires preflipping one of the two signals being correlated, so that the left-for-right flip inherent in convolution is canceled. For instance, when a[n] a n d b[n] , are convolved to produce c[n] , the equation is written: a[n ]( b [n] ' c [n] . In comparison, the cross-correlation of a[n] and b[n] can 140 The Scientist and Engineer's Guide to Digital Signal Processing be written: a[n ]( b [& n] ' c [n] . That is, flipping b [n] left-for-right is accomplished by reversing the sign of the index, i.e., b [& n] . Don't let the mathematical similarity between convolution and correlation fool you; they represent very different DSP procedures. Convolution is the relationship between a system's input signal, output signal, and impulse response. Correlation is a way to detect a known waveform in a noisy background. The similar mathematics is only a convenient coincidence. Speed Writing a program to convolve one signal by another is a simple task, only requiring a few lines of code. Executing the program may be more painful. The problem is the large number of additions and multiplications required by the algorithm, resulting in long execution times. As shown by the programs in the last chapter, the time-consuming operation is composed of multiplying two numbers and adding the result to an accumulator. Other parts of the algorithm, such as indexing the arrays, are very quick. The multiply-accumulate is a basic building block in DSP, and we will see it repeated in several other important algorithms. In fact, the speed of DSP computers is often specified by how long it takes to preform a multiply-accumulate operation. If a signal composed of N samples is convolved with a signal composed of M samples, N ×M multiply-accumulations must be preformed. This can be seen from the programs of the last chapter. Personal computers of the mid 1990's requires about one microsecond per multiply-accumulation (100 MHz Pentium using single precision floating point, see Table 4-6). Therefore, convolving a 10,000 sample signal with a 100 sample signal requires about one second. To process a one million point signal with a 3000 point impulse response requires nearly an hour. A decade earlier (80286 at 12 MHz), this calculation would have required three days! The problem of excessive execution time is commonly handled in one of three ways. First, simply keep the signals as short as possible and use integers instead of floating point. If you only need to run the convolution a few times, this will probably be the best trade-off between execution time and programming effort. Second, use a computer designed for DSP. DSP microprocessors are available with multiply-accumulate times of only a few tens of nanoseconds. This is the route to go if you plan to perform the convolution many times, such as in the design of commercial products. The third solution is to use a better algorithm for implementing the convolution. Chapter 17 describes a very sophisticated algorithm called FFT convolution. FFT convolution produces exactly the same result as the convolution algorithms presented in the last chapter; however, the execution time is dramatically reduced. For signals with thousands of samples, FFT convolution can be hundreds of times faster. The disadvantage is program complexity. Even if you are familiar with the technique, expect to spend several hours getting the program to run. CHAPTER 8 The Discrete Fourier Transform Fourier analysis is a family of mathematical techniques, all based on decomposing signals into sinusoids. The discrete Fourier transform (DFT) is the family member used with digitized signals. This is the first of four chapters on the real DFT, a version of the discrete Fourier transform that uses real numbers to represent the input and output signals. The complex DFT, a more advanced technique that uses complex numbers, will be discussed in Chapter 31. In this chapter we look at the mathematics and algorithms of the Fourier decomposition, the heart of the DFT. The Family of Fourier Transform Fourier analysis is named after Jean Baptiste Joseph Fourier (1768-1830), a French mathematician and physicist. (Fourier is pronounced: for@ē@ā , and is always capitalized). While many contributed to the field, Fourier is honored for his mathematical discoveries and insight into the practical usefulness of the techniques. Fourier was interested in heat propagation, and presented a paper in 1807 to the Institut de France on the use of sinusoids to represent temperature distributions. The paper contained the controversial claim that any continuous periodic signal could be represented as the sum of properly chosen sinusoidal waves. Among the reviewers were two of history's most famous mathematicians, Joseph Louis Lagrange (1736-1813), and Pierre Simon de Laplace (1749-1827). While Laplace and the other reviewers voted to publish the paper, Lagrange adamantly protested. For nearly 50 years, Lagrange had insisted that such an approach could not be used to represent signals with corners, i.e., discontinuous slopes, such as in square waves. The Institut de France bowed to the prestige of Lagrange, and rejected Fourier's work. It was only after Lagrange died that the paper was finally published, some 15 years later. Luckily, Fourier had other things to keep him busy, political activities, expeditions to Egypt with Napoleon, and trying to avoid the guillotine after the French Revolution (literally!). 141 The Scientist and Engineer's Guide to Digital Signal Processing Who was right? It's a split decision. Lagrange was correct in his assertion that a summation of sinusoids cannot form a signal with a corner. However, you can get very close. So close that the difference between the two has zero energy. In this sense, Fourier was right, although 18th century science knew little about the concept of energy. This phenomenon now goes by the name: Gibbs Effect, and will be discussed in Chapter 11. Figure 8-1 illustrates how a signal can be decomposed into sine and cosine waves. Figure (a) shows an example signal, 16 points long, running from sample number 0 to 15. Figure (b) shows the Fourier decomposition of this signal, nine cosine waves and nine sine waves, each with a different frequency and amplitude. Although far from obvious, these 18 sinusoids 80 60 FIGURE 8-1a (see facing page) Amplitude 142 DECOMPOSE 40 20 0 SYNTHESIZE -20 -40 0 4 8 Sample number 12 16 add to produce the waveform in (a). It should be noted that the objection made by Lagrange only applies to continuous signals. For discrete signals, this decomposition is mathematically exact. There is no difference between the signal in (a) and the sum of the signals in (b), just as there is no difference between 7 and 3+4. Why are sinusoids used instead of, for instance, square or triangular waves? Remember, there are an infinite number of ways that a signal can be decomposed. The goal of decomposition is to end up with something easier to deal with than the original signal. For example, impulse decomposition allows signals to be examined one point at a time, leading to the powerful technique of convolution. The component sine and cosine waves are simpler than the original signal because they have a property that the original signal does not have: sinusoidal fidelity. As discussed in Chapter 5, a sinusoidal input to a system is guaranteed to produce a sinusoidal output. Only the amplitude and phase of the signal can change; the frequency and wave shape must remain the same. Sinusoids are the only waveform that have this useful property. While square and triangular decompositions are possible, there is no general reason for them to be useful. The general term: Fourier transform, can be broken into four categories, resulting from the four basic types of signals that can be encountered. Chapter 8- The Discrete Fourier Transform 143 Cosine Waves 8 8 8 4 4 4 0 0 0 -4 -4 -4 -8 -8 0 2 4 6 8 10 12 14 16 -8 0 2 4 6 8 10 12 14 16 8 8 8 4 4 4 0 0 0 -4 -4 -4 -8 -8 0 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 8 8 4 4 4 0 0 0 -4 -4 -4 -8 0 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 -8 0 8 -8 0 -8 0 2 4 6 8 10 12 14 16 Sine Waves 8 8 8 4 4 4 0 0 0 -4 -4 -4 -8 -8 0 2 4 6 8 10 12 14 16 -8 0 2 4 6 8 10 12 14 16 8 8 8 4 4 4 0 0 0 -4 -4 -4 -8 -8 0 2 4 6 8 10 12 14 16 -8 0 2 4 6 8 10 12 14 16 8 8 8 4 4 4 0 0 0 -4 -4 -4 -8 -8 0 2 4 6 8 10 12 14 16 -8 0 2 4 6 8 10 12 14 16 FIGURE 8-1b Example of Fourier decomposition. A 16 point signal (opposite page) is decomposed into 9 cosine waves and 9 sine waves. The frequency of each sinusoid is fixed; only the amplitude is changed depending on the shape of the waveform being decomposed. 144 The Scientist and Engineer's Guide to Digital Signal Processing A signal can be either continuous or discrete, and it can be either periodic or aperiodic. The combination of these two features generates the four categories, described below and illustrated in Fig. 8-2. Aperiodic-Continuous This includes, for example, decaying exponentials and the Gaussian curve. These signals extend to both positive and negative infinity without repeating in a periodic pattern. The Fourier Transform for this type of signal is simply called the Fourier Transform. Periodic-Continuous Here the examples include: sine waves, square waves, and any waveform that repeats itself in a regular pattern from negative to positive infinity. This version of the Fourier transform is called the Fourier Series. Aperiodic-Discrete These signals are only defined at discrete points between positive and negative infinity, and do not repeat themselves in a periodic fashion. This type of Fourier transform is called the Discrete Time Fourier Transform. Periodic-Discrete These are discrete signals that repeat themselves in a periodic fashion from negative to positive infinity. This class of Fourier Transform is sometimes called the Discrete Fourier Series, but is most often called the Discrete Fourier Transform. You might be thinking that the names given to these four types of Fourier transforms are confusing and poorly organized. You're right; the names have evolved rather haphazardly over 200 years. There is nothing you can do but memorize them and move on. These four classes of signals all extend to positive and negative infinity. Hold on, you say! What if you only have a finite number of samples stored in your computer, say a signal formed from 1024 points. Isn't there a version of the Fourier Transform that uses finite length signals? No, there isn't. Sine and cosine waves are defined as extending from negative infinity to positive infinity. You cannot use a group of infinitely long signals to synthesize something finite in length. The way around this dilemma is to make the finite data look like an infinite length signal. This is done by imagining that the signal has an infinite number of samples on the left and right of the actual points. If all these “imagined” samples have a value of zero, the signal looks discrete and aperiodic, and the Discrete Time Fourier Transform applies. As an alternative, the imagined samples can be a duplication of the actual 1024 points. In this case, the signal looks discrete and periodic, with a period of 1024 samples. This calls for the Discrete Fourier Transform to be used. As it turns out, an infinite number of sinusoids are required to synthesize a signal that is aperiodic. This makes it impossible to calculate the Discrete Time Fourier Transform in a computer algorithm. By elimination, the only Chapter 8- The Discrete Fourier Transform Type of Transform 145 Example Signal Fourier Transform signals that are continious and aperiodic Fourier Series signals that are continious and periodic Discrete Time Fourier Transform signals that are discrete and aperiodic Discrete Fourier Transform signals that are discrete and periodic FIGURE 8-2 Illustration of the four Fourier transforms. A signal may be continuous or discrete, and it may be periodic or aperiodic. Together these define four possible combinations, each having its own version of the Fourier transform. The names are not well organized; simply memorize them. type of Fourier transform that can be used in DSP is the DFT. In other words, digital computers can only work with information that is discrete and finite in length. When you struggle with theoretical issues, grapple with homework problems, and ponder mathematical mysteries, you may find yourself using the first three members of the Fourier transform family. When you sit down to your computer, you will only use the DFT. We will briefly look at these other Fourier transforms in future chapters. For now, concentrate on understanding the Discrete Fourier Transform. Look back at the example DFT decomposition in Fig. 8-1. On the face of it, it appears to be a 16 point signal being decomposed into 18 sinusoids, each consisting of 16 points. In more formal terms, the 16 point signal, shown in (a), must be viewed as a single period of an infinitely long periodic signal. Likewise, each of the 18 sinusoids, shown in (b), represents a 16 point segment from an infinitely long sinusoid. Does it really matter if we view this as a 16 point signal being synthesized from 16 point sinusoids, or as an infinitely long periodic signal being synthesized from infinitely long sinusoids? The answer is: usually no, but sometimes, yes. In upcoming chapters we will encounter properties of the DFT that seem baffling if the signals are viewed as finite, but become obvious when the periodic nature is considered. The key point to understand is that this periodicity is invoked in order to use a mathematical tool, i.e., the DFT. It is usually meaningless in terms of where the signal originated or how it was acquired. 146 The Scientist and Engineer's Guide to Digital Signal Processing Each of the four Fourier Transforms can be subdivided into real and complex versions. The real version is the simplest, using ordinary numbers and algebra for the synthesis and decomposition. For instance, Fig. 8-1 is an example of the real DFT. The complex versions of the four Fourier transforms are immensely more complicated, requiring the use of complex numbers. These are numbers such as: 3 %4 j , where j is equal to &1 (electrical engineers use the variable j, while mathematicians use the variable, i). Complex mathematics can quickly become overwhelming, even to those that specialize in DSP. In fact, a primary goal of this book is to present the fundamentals of DSP without the use of complex math, allowing the material to be understood by a wider range of scientists and engineers. The complex Fourier transforms are the realm of those that specialize in DSP, and are willing to sink to their necks in the swamp of mathematics. If you are so inclined, Chapters 30-33 will take you there. The mathematical term: transform, is extensively used in Digital Signal Processing, such as: Fourier transform, Laplace transform, Z transform, Hilbert transform, Discrete Cosine transform, etc. Just what is a transform? To answer this question, remember what a function is. A function is an algorithm or procedure that changes one value into another value. For example, y ' 2 x %1 is a function. You pick some value for x, plug it into the equation, and out pops a value for y. Functions can also change several values into a single value, such as: y ' 2 a % 3 b % 4 c , where a, b, and c are changed into y. Transforms are a direct extension of this, allowing both the input and output to have multiple values. Suppose you have a signal composed of 100 samples. If you devise some equation, algorithm, or procedure for changing these 100 samples into another 100 samples, you have yourself a transform. If you think it is useful enough, you have the perfect right to attach your last name to it and expound its merits to your colleagues. (This works best if you are an eminent 18th century French mathematician). Transforms are not limited to any specific type or number of data. For example, you might have 100 samples of discrete data for the input and 200 samples of discrete data for the output. Likewise, you might have a continuous signal for the input and a continuous signal for the output. Mixed signals are also allowed, discrete in and continuous out, and vice versa. In short, a transform is any fixed procedure that changes one chunk of data into another chunk of data. Let's see how this applies to the topic at hand: the Discrete Fourier transform. Notation and Format of the Real DFT As shown in Fig. 8-3, the discrete Fourier transform changes an N point input signal into two N/2 %1 point output signals. The input signal contains the signal being decomposed, while the two output signals contain the amplitudes of the component sine and cosine waves (scaled in a way we will discuss shortly). The input signal is said to be in the time domain. This is because the most common type of signal entering the DFT is composed of Chapter 8- The Discrete Fourier Transform Time Domain Frequency Domain Forward DFT x[ ] 0 N samples 147 N-1 0 Inverse DFT Im X[ ] Re X[ ] N/2 N/2+1 samples (cosine wave amplitudes) 0 N/2 N/2+1 samples (sine wave amplitudes) collectively referred to as X[ ] FIGURE 8-3 DFT terminology. In the time domain, x[ ] consists of N points running from 0 to N& 1 . In the frequency domain, the DFT produces two signals, the real part, written: Re X[ ] , and the imaginary part, written: Im X [ ] . Each of these frequency domain signals are N/2 % 1 points long, and run from 0 to N/2 . The Forward DFT transforms from the time domain to the frequency domain, while the Inverse DFT transforms from the frequency domain to the time domain. (Take note: this figure describes the real DFT. The complex DFT, discussed in Chapter 31, changes N complex points into another set of N complex points). samples taken at regular intervals of time. Of course, any kind of sampled data can be fed into the DFT, regardless of how it was acquired. When you see the term "time domain" in Fourier analysis, it may actually refer to samples taken over time, or it might be a general reference to any discrete signal that is being decomposed. The term frequency domain is used to describe the amplitudes of the sine and cosine waves (including the special scaling we promised to explain). The frequency domain contains exactly the same information as the time domain, just in a different form. If you know one domain, you can calculate the other. Given the time domain signal, the process of calculating the frequency domain is called decomposition, analysis, the forward DFT, or simply, the DFT. If you know the frequency domain, calculation of the time domain is called synthesis, or the inverse DFT. Both synthesis and analysis can be represented in equation form and computer algorithms. The number of samples in the time domain is usually represented by the variable N. While N can be any positive integer, a power of two is usually chosen, i.e., 128, 256, 512, 1024, etc. There are two reasons for this. First, digital data storage uses binary addressing, making powers of two a natural signal length. Second, the most efficient algorithm for calculating the DFT, the Fast Fourier Transform (FFT), usually operates with N that is a power of two. Typically, N is selected between 32 and 4096. In most cases, the samples run from 0 to N&1 , rather than 1 to N. Standard DSP notation uses lower case letters to represent time domain signals, such as x[ ] , y[ ] , and z[ ] . The corresponding upper case letters are 148 The Scientist and Engineer's Guide to Digital Signal Processing used to represent their frequency domains, that is, X [ ] , Y[ ] , and Z [ ] . For illustration, assume an N point time domain signal is contained in x[ ] . The frequency domain of this signal is called X [ ] , and consists of two parts, each an array of N/2 %1 samples. These are called the Real part of X [ ] , written as: Re X [ ] , and the Imaginary part of X [ ] , written as: Im X [ ] . The values in Re X [ ] are the amplitudes of the cosine waves, while the values in Im X [ ] are the amplitudes of the sine waves (not worrying about the scaling factors for the moment). Just as the time domain runs from x[0] to x[N&1] , the frequency domain signals run from Re X[0] to Re X[N/2] , and from Im X[0] to Im X [N/2] . Study these notations carefully; they are critical to understanding the equations in DSP. Unfortunately, some computer languages don't distinguish between lower and upper case, making the variable names up to the individual programmer. The programs in this book use the array XX[ ] to hold the time domain signal, and the arrays REX[ ] and IMX[ ] to hold the frequency domain signals. The names real part and imaginary part originate from the complex DFT, where they are used to distinguish between real and imaginary numbers. Nothing so complicated is required for the real DFT. Until you get to Chapter 31, simply think that "real part" means the cosine wave amplitudes, while "imaginary part" means the sine wave amplitudes. Don't let these suggestive names mislead you; everything here uses ordinary numbers. Likewise, don't be misled by the lengths of the frequency domain signals. It is common in the DSP literature to see statements such as: "The DFT changes an N point time domain signal into an N point frequency domain signal." This is referring to the complex DFT, where each "point" is a complex number (consisting of real and imaginary parts). For now, focus on learning the real DFT, the difficult math will come soon enough. The Frequency Domain's Independent Variable Figure 8-4 shows an example DFT with N ' 128 . The time domain signal is contained in the array: x[0] to x[127] . The frequency domain signals are contained in the two arrays: Re X[0] to Re X[64] , and Im X [0] to Im X [64] . Notice that 128 points in the time domain corresponds to 65 points in each of the frequency domain signals, with the frequency indexes running from 0 to 64. That is, N points in the time domain corresponds to N/2 %1 points in the frequency domain (not N/2 points). Forgetting about this extra point is a common bug in DFT programs. The horizontal axis of the frequency domain can be referred to in four different ways, all of which are common in DSP. In the first method, the horizontal axis is labeled from 0 to 64, corresponding to the 0 to N/2 samples in the arrays. When this labeling is used, the index for the frequency domain is an integer, for example, Re X [k] and Im X [k] , where k runs from 0 to N/2 in steps of one. Programmers like this method because it is how they write code, using an index to access array locations. This notation is used in Fig. 8-4b. Chapter 8- The Discrete Fourier Transform Time Domain Frequency Domain 2 8 a. x[ ] b. Re X[ ] 4 Amplitude 1 0 -1 0 -4 -2 -8 0 16 32 48 64 80 Sample number 96 112 128 127 FIGURE 8-4 Example of the DFT. The DFT converts the time domain signal, x[ ] , into the frequency domain signals, Re X[ ] a n d Im X [ ] . The horizontal axis of the frequency domain can be labeled in one of three ways: (1) as an array index that runs between 0 and N/2 , (2) as a fraction of the sampling frequency, running between 0 and 0.5, (3) as a natural frequency, running between 0 and B. In the example shown here, (b) uses the first method, while (c) use the second method. 0 16 32 48 64 Frequency (sample number) 8 c. Im X[ ] 4 Amplitude Amplitude 149 0 -4 -8 0 0.1 0.2 0.3 0.4 Frequency (fraction of sampling rate) 0.5 In the second method, used in (c), the horizontal axis is labeled as a fraction of the sampling rate. This means that the values along the horizonal axis always run between 0 and 0.5, since discrete data can only contain frequencies between DC and one-half the sampling rate. The index used with this notation is f, for frequency. The real and imaginary parts are written: Re X [f ] and Im X [f ] , where f takes on N/2 %1 equally spaced values between 0 and 0.5. To convert from the first notation, k, to the second notation, f , divide the horizontal axis by N. That is, f ' k/N . Most of the graphs in this book use this second method, reinforcing that discrete signals only contain frequencies between 0 and 0.5 of the sampling rate. The third style is similar to the second, except the horizontal axis is multiplied by 2 B . The index used with this labeling is T , a lower case Greek omega. In this notation, the real and imaginary parts are written: Re X [T] and Im X [T] , where T takes on N/2 %1 equally spaced values between 0 and B . The parameter, T , is called the natural frequency, and has the units of radians. This is based on the idea that there are 2 B radians in a circle. Mathematicians like this method because it makes the equations shorter. For instance, consider how a cosine wave is written in each of t h e s e f i r s t t h r e e n o t a t i o n s : u s i n g k : c [n ] ' cos (2Bkn /N) , u s i n g f : c [n ] ' cos (2Bfn ) , and using T : c [n ] ' cos (Tn ) . 150 The Scientist and Engineer's Guide to Digital Signal Processing The fourth method is to label the horizontal axis in terms of the analog frequencies used in a particular application. For instance, if the system being examined has a sampling rate of 10 kHz (i.e., 10,000 samples per second), graphs of the frequency domain would run from 0 to 5 kHz. This method has the advantage of presenting the frequency data in terms of a real world meaning. The disadvantage is that it is tied to a particular sampling rate, and is therefore not applicable to general DSP algorithm development, such as designing digital filters. All of these four notations are used in DSP, and you need to become comfortable with converting between them. This includes both graphs and mathematical equations. To find which notation is being used, look at the independent variable and its range of values. You should find one of four notations: k (or some other integer index), running from 0 to N/2 ; f, running from 0 to 0.5; T, running from 0 to B ; or a frequency expressed in hertz, running from DC to one-half of an actual sampling rate. DFT Basis Functions The sine and cosine waves used in the DFT are commonly called the DFT basis functions. In other words, the output of the DFT is a set of numbers that represent amplitudes. The basis functions are a set of sine and cosine waves with unity amplitude. If you assign each amplitude (the frequency domain) to the proper sine or cosine wave (the basis functions), the result is a set of scaled sine and cosine waves that can be added to form the time domain signal. The DFT basis functions are generated from the equations: EQUATION 8-1 Equations for the DFT basis functions. In these equations, ck[ i] and sk[ i] are the cosine and sine waves, each N points in length, running from i ' 0 to N& 1 . The parameter, k, determines the frequency of the wave. In an N point DFT, k takes on values between 0 and N/2 . ck [i ] ' cos ( 2B ki / N ) sk [i ] ' sin ( 2B ki / N ) where: ck[ ] is the cosine wave for the amplitude held in Re X[k] , and sk[ ] is the sine wave for the amplitude held in Im X [k] . For example, Fig. 8-5 shows some of the 17 sine and 17 cosine waves used in an N ' 32 point DFT. Since these sinusoids add to form the input signal, they must be the same length as the input signal. In this case, each has 32 points running from i ' 0 to 31. The parameter, k, sets the frequency of each sinusoid. In particular, c1[ ] is the cosine wave that makes one complete cycle in N points, c5[ ] is the cosine wave that makes five complete cycles in N points, etc. This is an important concept in understanding the basis functions; the frequency parameter, k, is equal to the number of complete cycles that occur over the N points of the signal. Chapter 8- The Discrete Fourier Transform 2 2 a. c0[ ] b. s0[ ] 1 Amplitude Amplitude 1 0 -1 0 -1 -2 -2 0 8 16 Sample number 24 32 0 24 32 16 24 32 16 24 32 16 24 32 d. s2[ ] 1 Amplitude 1 0 -1 0 -1 -2 -2 0 8 16 24 32 0 8 Sample number 2 Sample number 2 e. c10[ ] f. s10[ ] 1 Amplitude 1 Amplitude 16 2 c. c2[ ] Amplitude 8 Sample number 2 0 -1 0 -1 -2 -2 0 8 16 Sample number 24 32 0 2 8 Sample number 2 g. c16[ ] h. s16[ ] 1 Amplitude 1 Amplitude 151 0 -1 0 -1 -2 -2 0 8 16 Sample number 24 32 0 8 Sample number FIGURE 8-5 DFT basis functions. A 32 point DFT has 17 discrete cosine waves and 17 discrete sine waves for its basis functions. Eight of these are shown in this figure. These are discrete signals; the continuous lines are shown in these graphs only to help the reader's eye follow the waveforms. 152 The Scientist and Engineer's Guide to Digital Signal Processing Let's look at several of these basis functions in detail. Figure (a) shows the cosine wave c0[ ] . This is a cosine wave of zero frequency, which is a constant value of one. This means that Re X[0] holds the average value of all the points in the time domain signal. In electronics, it would be said that Re X[0] holds the DC offset. The sine wave of zero frequency, s0[ ] , is shown in (b), a signal composed of all zeros. Since this can not affect the time domain signal being synthesized, the value of Im X [0] is irrelevant, and always set to zero. More about this shortly. Figures (c) & (d) show c2[ ] & s2[ ] , the sinusoids that complete two cycles in the N points. These correspond to Re X [2] & Im X [2] , respectively. Likewise, (e) & (f) show c10[ ] & s10[ ] , the sinusoids that complete ten cycles in the N points. These sinusoids correspond to the amplitudes held in the arrays Re X[10] & Im X [10] . The problem is, the samples in (e) and (f) no longer look like sine and cosine waves. If the continuous curves were not present in these graphs, you would have a difficult time even detecting the pattern of the waveforms. This may make you a little uneasy, but don't worry about it. From a mathematical point of view, these samples do form discrete sinusoids, even if your eye cannot follow the pattern. The highest frequencies in the basis functions are shown in (g) and (h). These are cN/2[ ] & sN/2[ ] , or in this example, c16[ ] & s16[ ] . The discrete cosine wave alternates in value between 1 and -1, which can be interpreted as sampling a continuous sinusoid at the peaks. In contrast, the discrete sine wave contains all zeros, resulting from sampling at the zero crossings. This makes the value of Im X [N/2] the same as Im X [0] , always equal to zero, and not affecting the synthesis of the time domain signal. Here's a puzzle: If there are N samples entering the DFT, and N%2 samples exiting, where did the extra information come from? The answer: two of the output samples contain no information, allowing the other N samples to be fully independent. As you might have guessed, the points that carry no information are Im X [0] and Im X [N/2] , the samples that always have a value of zero. Synthesis, Calculating the Inverse DFT Pulling together everything said so far, we can write the synthesis equation: x[i ] ' j Re X̄ [k ] cos ( 2B ki / N) % j Im X̄ [k ] sin( 2B ki / N) N /2 N /2 k'0 k '0 EQUATION 8-2 The synthesis equation. In this relation, x [i ] is the signal being synthesized, with the index, i, running from 0 to N& 1 . Re X̄ [k ] and Im X̄ [k ] hold the amplitudes of the cosine and sine waves, respectively, with k running from 0 to N/2 . Equation 8-3 provides the normalization to change this equation into the inverse DFT. Chapter 8- The Discrete Fourier Transform 153 In words, any N point signal, x[i] , can be created by adding N/2 % 1 cosine waves and N/2 % 1 sine waves. The amplitudes of the cosine and sine waves are held in the arrays Im X̄ [k] and Re X̄ [k] , respectively. The synthesis equation multiplies these amplitudes by the basis functions to create a set of scaled sine and cosine waves. Adding the scaled sine and cosine waves produces the time domain signal, x[i] . In Eq. 8-2, the arrays are called Im X̄ [k] and Re X̄ [k] , rather than Im X [k] and Re X [k] . This is because the amplitudes needed for synthesis (called in this discussion: Im X̄ [k] and Re X̄ [k] ), are slightly different from the frequency domain of a signal (denoted by: Im X [k] and Re X [k] ). This is the scaling factor issue we referred to earlier. Although the conversion is only a simple normalization, it is a common bug in computer programs. Look out for it! In equation form, the conversion between the two is given by: EQUATIONS 8-3 Conversion between the sinusoidal amplitudes and the frequency domain values. In these equations, Re X̄ [k ] and Im X̄ [k ] hold the amplitudes of the cosine and sine waves needed for synthesis, while Re X [k ] and Im X [k ] hold the real and imaginary parts of the frequency domain. As usual, N is the number of points in the time domain signal, and k is an index that runs from 0 to N/2. Re X [k ] N/ 2 Re X̄ [k ] ' Im X̄ [k ] ' & Im X [k ] N/ 2 except for two special cases: ' Re X [ 0 ] N Re X̄ [N/2 ] ' Re X [N/2 ] N Re X̄ [ 0] Suppose you are given a frequency domain representation, and asked to synthesize the corresponding time domain signal. To start, you must find the amplitudes of the sine and cosine waves. In other words, given Im X [k] and Re X [k] , you must find Im X̄ [k] and Re X̄ [k] . Equation 8-3 shows this in a mathematical form. To do this in a computer program, three actions must be taken. First, divide all the values in the frequency domain by N/2 . Second, change the sign of all the imaginary values. Third, divide the first and last samples in the real part, Re X[0] and Re X[N/2 ] , by two. This provides the amplitudes needed for the synthesis described by Eq. 8-2. Taken together, Eqs. 8-2 and 8-3 define the inverse DFT. The entire Inverse DFT is shown in the computer program listed in Table 8-1. There are two ways that the synthesis (Eq. 8-2) can be programmed, and both are shown. In the first method, each of the scaled sinusoids are generated one at a time and added to an accumulation array, which ends up becoming the time domain signal. In the second method, each sample in the time domain signal is calculated one at a time, as the sum of all the 154 The Scientist and Engineer's Guide to Digital Signal Processing 100 'THE INVERSE DISCRETE FOURIER TRANSFORM 110 'The time domain signal, held in XX[ ], is calculated from the frequency domain signals, 120 'held in REX[ ] and IMX[ ]. 130 ' 140 DIM XX[511] 'XX[ ] holds the time domain signal 150 DIM REX[256] 'REX[ ] holds the real part of the frequency domain 160 DIM IMX[256] 'IMX[ ] holds the imaginary part of the frequency domain 170 ' 180 PI = 3.14159265 'Set the constant, PI 190 N% = 512 'N% is the number of points in XX[ ] 200 ' 210 GOSUB XXXX 'Mythical subroutine to load data into REX[ ] and IMX[ ] 220 ' 230 240 ' 'Find the cosine and sine wave amplitudes using Eq. 8-3 250 FOR K% = 0 TO 256 260 REX[K%] = REX[K%] / (N%/2) 270 IMX[K%] = -IMX[K%] / (N%/2) 280 NEXT K% 290 ' 300 REX[0] = REX[0] / 2 310 REX[256] = REX[256] / 2 320 ' 330 ' 340 FOR I% = 0 TO 511 'Zero XX[ ] so it can be used as an accumulator 350 XX[I%] = 0 360 NEXT I% 370 ' 380 ' Eq. 8-2 SYNTHESIS METHOD #1. Loop through each 390 ' frequency generating the entire length of the sine and cosine 400 ' waves, and add them to the accumulator signal, XX[ ] 410 ' 420 FOR K% = 0 TO 256 'K% loops through each sample in REX[ ] and IMX[ ] 430 FOR I% = 0 TO 511 'I% loops through each sample in XX[ ] 440 ' 450 XX[I%] = XX[I%] + REX[K%] * COS(2*PI*K%*I%/N%) 460 XX[I%] = XX[I%] + IMX[K%] * SIN(2*PI*K%*I%/N%) 470 ' 480 NEXT I% 490 NEXT K% 500 ' 510 END Alternate code for lines 380 to 510 380 ' Eq. 8-2 SYNTHESIS METHOD #2. Loop through each 390 ' sample in the time domain, and sum the corresponding 400 ' samples from each cosine and sine wave 410 ' 420 FOR I% = 0 TO 511 'I% loops through each sample in XX[ ] 430 FOR K% = 0 TO 256 'K% loops through each sample in REX[ ] and IMX[ ] 440 ' 450 XX[I%] = XX[I%] + REX[K%] * COS(2*PI*K%*I%/N%) 460 XX[I%] = XX[I%] + IMX[K%] * SIN(2*PI*K%*I%/N%) 470 ' 480 NEXT K% 490 NEXT I% 500 ' 510 END TABLE 8-1 Chapter 8- The Discrete Fourier Transform Time Domain Frequency Domain 50 50 a. The time domain signal b. Re X[ ] (the frequency domain) 40 30 I DFT 20 Amplitude 40 Amplitude 155 30 20 10 10 0 0 0 8 16 Sample number 24 32 0 Eq. 8.2 4 8 12 Frequency sample number 16 Eq. 8.3 3.0 Amplitude c. Re X[ ] (cosine wave amplitudes) 2.0 1.0 0.0 0 4 8 12 Frequency sample number 16 FIGURE 8-6 Example of the Inverse DFT. Figure (a) shows an example time domain signal, an impulse at sample zero with an amplitude of 32. Figure (b) shows the real part of the frequency domain of this signal, a constant value of 32. The imaginary part of the frequency domain (not shown) is composed of all zeros. Figure(c) shows the amplitudes of the cosine waves needed to reconstruct (a) using Eq. 8-2. The values in (c) are found from (b) by using Eq. 8-3. corresponding samples in the cosine and sine waves. Both methods produce the same result. The difference between these two programs is very minor; the inner and outer loops are swapped during the synthesis. Figure 8-6 illustrates the operation of the Inverse DFT, and the slight differences between the frequency domain and the amplitudes needed for synthesis. Figure 8-6a is an example signal we wish to synthesize, an impulse at sample zero with an amplitude of 32. Figure 8-6b shows the frequency domain representation of this signal. The real part of the frequency domain is a constant value of 32. The imaginary part (not shown) is composed of all zeros. As discussed in the next chapter, this is an important DFT pair: an impulse in the time domain corresponds to a constant value in the frequency domain. For now, the important point is that (b) is the DFT of (a), and (a) is the Inverse DFT of (b). The Scientist and Engineer's Guide to Digital Signal Processing Equation 8-3 is used to convert the frequency domain signal, (b), into the amplitudes of the cosine waves, (c). As shown, all of the cosine waves have an amplitude of two, except for samples 0 and 16, which have a value of one. The amplitudes of the sine waves are not shown in this example because they have a value of zero, and therefore provide no contribution. The synthesis equation, Eq. 8-2, is then used to convert the amplitudes of the cosine waves, (b), into the time domain signal, (a). This describes how the frequency domain is different from the sinusoidal amplitudes, but it doesn't explain why it is different. The difference occurs because the frequency domain is defined as a spectral density. Figure 8-7 shows how this works. The example in this figure is the real part of the frequency domain of a 32 point signal. As you should expect, the samples run from 0 to 16, representing 17 frequencies equally spaced between 0 and 1/2 of the sampling rate. Spectral density describes how much signal (amplitude) is present per unit of bandwidth. To convert the sinusoidal amplitudes into a spectral density, divide each amplitude by the bandwidth represented by each amplitude. This brings up the next issue: how do we determine the bandwidth of each of the discrete frequencies in the frequency domain? As shown in the figure, the bandwidth can be defined by drawing dividing lines between the samples. For instance, sample number 5 occurs in the band between 4.5 and 5.5; sample number 6 occurs in the band between 5.5 and 6.5, etc. Expressed as a fraction of the total bandwidth (i.e., N/2 ), the bandwidth of each sample is 2/N . An exception to this is the samples on each end, which have one-half of this bandwidth, 1/N . This accounts for the 2/N scaling factor between the sinusoidal amplitudes and frequency domain, as well as the additional factor of two needed for the first and last samples. Why the negation of the imaginary part? This is done solely to make the real DFT consistent with its big brother, the complex DFT. In Chapter 29 we will show that it is necessary to make the mathematics of the complex DFT work. When dealing only with the real DFT, many authors do not include this negation. For that matter, many authors do not even include 10 9 FIGURE 8-7 The bandwidth of frequency domain samples. Each sample in the frequency domain can be thought of as being contained in a frequency band of width 2/N, expressed as a fraction of the total bandwidth. An exception to this is the first and last samples, which have a bandwidth only one-half this wide, 1/N. 8 7 Amplitude 156 6 5 4 width: 1/N 3 width: 1/N width: 1/N 2/N 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Frequency sample number Chapter 8- The Discrete Fourier Transform 157 the 2/N scaling factor. Be prepared to find both of these missing in some discussions. They are included here for a tremendously important reason: The most efficient way to calculate the DFT is through the Fast Fourier Transform (FFT) algorithm, presented in Chapter 12. The FFT generates a frequency domain defined according to Eq. 8-2 and 8-3. If you start messing with these normalization factors, your programs containing the FFT are not going to work as expected. Analysis, Calculating the DFT The DFT can be calculated in three completely different ways. First, the problem can be approached as a set of simultaneous equations. This method is useful for understanding the DFT, but it is too inefficient to be of practical use. The second method brings in an idea from the last chapter: correlation. This is based on detecting a known waveform in another signal. The third method, called the Fast Fourier Transform (FFT), is an ingenious algorithm that decomposes a DFT with N points, into N DFTs each with a single point. The FFT is typically hundreds of times faster than the other methods. The first two methods are discussed here, while the FFT is the topic of Chapter 12. It is important to remember that all three of these methods produce an identical output. Which should you use? In actual practice, correlation is the preferred technique if the DFT has less than about 32 points, otherwise the FFT is used. DFT by Simultaneous Equations Think about the DFT calculation in the following way. You are given N values from the time domain, and asked to calculate the N values of the frequency domain (ignoring the two frequency domain values that you know must be zero). Basic algebra provides the answer: to solve for N unknowns, you must be able to write N linearly independent equations. To do this, take the first sample from each sinusoid and add them together. The sum must be equal to the first sample in the time domain signal, thus providing the first equation. Likewise, an equation can be written for each of the remaining points in the time domain signal, resulting in the required N equations. The solution can then be found by using established methods for solving simultaneous equations, such as Gauss Elimination. Unfortunately, this method requires a tremendous number of calculations, and is virtually never used in DSP. However, it is important for another reason, it shows why it is possible to decompose a signal into sinusoids, how many sinusoids are needed, and that the basis functions must be linearly independent (more about this shortly). DFT by Correlation Let's move on to a better way, the standard way of calculating the DFT. An example will show how this method works. Suppose we are trying to calculate the DFT of a 64 point signal. This means we need to calculate the 33 points in the real part, and the 33 points in the imaginary part of the frequency domain. In this example we will only show how to calculate a single sample, Im X [3] , i.e., the amplitude of the sine wave that makes three complete cycles 158 The Scientist and Engineer's Guide to Digital Signal Processing between point 0 and point 63. All of the other frequency domain values are calculated in a similar manner. Figure 8-8 illustrates using correlation to calculate Im X [3] . Figures (a) and (b) show two example time domain signals, called: x1[ ] and x2[ ] , respectively. The first signal, x1[ ] , is composed of nothing but a sine wave that makes three cycles between points 0 and 63. In contrast, x2[ ] is composed of several sine and cosine waves, none of which make three cycles between points 0 and 63. These two signals illustrate what the algorithm for calculating Im X [3] must do. When fed x1[ ] , the algorithm must produce a value of 32, the amplitude of the sine wave present in the signal (modified by the scaling factors of Eq. 8-3). In comparison, when the algorithm is fed the other signal, x2[ ] , a value of zero must be produced, indicating that this particular sine wave is not present in this signal. The concept of correlation was introduced in Chapter 7. As you recall, to detect a known waveform contained in another signal, multiply the two signals and add all the points in the resulting signal. The single number that results from this procedure is a measure of how similar the two signals are. Figure 8-8 illustrates this approach. Figures (c) and (d) both display the signal we are looking for, a sine wave that makes 3 cycles between samples 0 and 63. Figure (e) shows the result of multiplying (a) and (c). Likewise, (f) shows the result of multiplying (b) and (d). The sum of all the points in (e) is 32, while the sum of all the points in (f) is zero, showing we have found the desired algorithm. The other samples in the frequency domain are calculated in the same way. This procedure is formalized in the analysis equation, the mathematical way to calculate the frequency domain from the time domain: EQUATION 8-4 The analysis equations for calculating the DFT. In these equations, x[i] is the time domain signal being analyzed, and Re X [k] & Im X [k] are the frequency domain signals being calculated. The index i runs from 0 to N & 1 , while the index k runs from 0 to N/2 . j x [i ] cos (2B k i / N ) N&1 Re X [k] ' i'0 Im X [k] ' & j x [i ] sin(2B k i / N ) N&1 i'0 In words, each sample in the frequency domain is found by multiplying the time domain signal by the sine or cosine wave being looked for, and adding the resulting points. If someone asks you what you are doing, say with confidence: "I am correlating the input signal with each basis function." Table 8-2 shows a computer program for calculating the DFT in this way. The analysis equation does not require special handling of the first and last points, as did the synthesis equation. There is, however, a negative sign in the imaginary part in Eq. 8-4. Just as before, this negative sign makes the real DFT consistent with the complex DFT, and is not always included. Chapter 8- The Discrete Fourier Transform Example 1 Example 2 2 2 a. x1[ ], signal being analyzed b. x2[ ], signal being analyzed 1 Amplitude Amplitude 1 0 -1 0 -1 -2 -2 0 16 32 Sample number 48 64 0 2 32 Sample number 48 64 d. s3[ ], basis function being sought 1 Amplitude 1 Amplitude 16 2 c. s3[ ], basis function being sought 0 -1 0 -1 -2 -2 0 16 32 Sample number 48 64 0 16 32 48 64 32 48 64 Sample number 2 2 f. x2[ ] × s3[ ] e. x1[ ] × s3[ ] 1 Amplitude 1 Amplitude 159 0 0 -1 -1 -2 -2 0 16 32 Sample number 48 64 0 16 Sample number FIGURE 8-8 Two example signals, (a) and (b), are analyzed for containing the specific basis function shown in (c) and (d). Figures (e) and (f) show the result of multiplying each example signal by the basis function. Figure (e) has an average of 0.5, indicating that x1 [ ] contains the basis function with an amplitude of 1.0. Conversely, (f) has a zero average, indicating that x2 [ ] does not contain the basis function. In order for this correlation algorithm to work, the basis functions must have an interesting property: each of them must be completely uncorrelated with all of the others. This means that if you multiply any two of the basis functions, the sum of the resulting points will be equal to zero. Basis functions that have this property are called orthognal. Many other 160 The Scientist and Engineer's Guide to Digital Signal Processing 100 'THE DISCRETE FOURIER TRANSFORM 110 'The frequency domain signals, held in REX[ ] and IMX[ ], are calculated from 120 'the time domain signal, held in XX[ ]. 130 ' 140 DIM XX[511] 'XX[ ] holds the time domain signal 150 DIM REX[256] 'REX[ ] holds the real part of the frequency domain 160 DIM IMX[256] 'IMX[ ] holds the imaginary part of the frequency domain 170 ' 180 PI = 3.14159265 'Set the constant, PI 190 N% = 512 'N% is the number of points in XX[ ] 200 ' 210 GOSUB XXXX 'Mythical subroutine to load data into XX[ ] 220 ' 230 ' 240 FOR K% = 0 TO 256 'Zero REX[ ] & IMX[ ] so they can be used as accumulators 250 REX[K%] = 0 260 IMX[K%] = 0 270 NEXT K% 280 ' 290 ' 'Correlate XX[ ] with the cosine and sine waves, Eq. 8-4 300 ' 310 FOR K% = 0 TO 256 'K% loops through each sample in REX[ ] and IMX[ ] 320 FOR I% = 0 TO 511 'I% loops through each sample in XX[ ] 330 ' 340 REX[K%] = REX[K%] + XX[I%] * COS(2*PI*K%*I%/N%) 350 IMX[K%] = IMX[K%] - XX[I%] * SIN(2*PI*K%*I%/N%) 360 ' 370 NEXT I% 380 NEXT K% 390 ' 400 END TABLE 8-2 orthognal basis functions exist, including: square waves, triangle waves, impulses, etc. Signals can be decomposed into these other orthognal basis functions using correlation, just as done here with sinusoids. This is not to suggest that this is useful, only that it is possible. As previously shown in Table 8-1, the Inverse DFT has two ways to be implemented in a computer program. This difference involves swapping the inner and outer loops during the synthesis. While this does not change the output of the program, it makes a difference in how you view what is being done. The DFT program in Table 8-2 can also be changed in this fashion, by swapping the inner and outer loops in lines 310 to 380. Just as before, the output of the program is the same, but the way you think about the calculation is different. (These two different ways of viewing the DFT and inverse DFT could be described as "input side" and "output side" algorithms, just as for convolution). As the program in Table 8-2 is written, it describes how an individual sample in the frequency domain is affected by all of the samples in the time domain. That is, the program calculates each of the values in the frequency domain in succession, not as a group. When the inner and outer loops are exchanged, the program loops through each sample in the time domain, calculating the Chapter 8- The Discrete Fourier Transform 161 contribution of that point to the frequency domain. The overall frequency domain is found by adding the contributions from the individual time domain points. This brings up our next question: what kind of contribution does an individual sample in the time domain provide to the frequency domain? The answer is contained in an interesting aspect of Fourier analysis called duality. Duality The synthesis and analysis equations (Eqs. 8-2 and 8-4) are strikingly similar. To move from one domain to the other, the known values are multiplied by the basis functions, and the resulting products added. The fact that the DFT and the Inverse DFT use this same mathematical approach is really quite remarkable, considering the totally different way we arrived at the two procedures. In fact, the only significant difference between the two equations is a result of the time domain being one signal of N points, while the frequency domain is two signals of N/2 % 1 points. As discussed in later chapters, the complex DFT expresses both the time and the frequency domains as complex signals of N points each. This makes the two domains completely symmetrical, and the equations for moving between them virtually identical. This symmetry between the time and frequency domains is called duality, and gives rise to many interesting properties. For example, a single point in the frequency domain corresponds to a sinusoid in the time domain. By duality, the inverse is also true, a single point in the time domain corresponds to a sinusoid in the frequency domain. As another example, convolution in the time domain corresponds to multiplication in the frequency domain. By duality, the reverse is also true: convolution in the frequency domain corresponds to multiplication in the time domain. These and other duality relationships are discussed in more detail in Chapters 10 and 11. Polar Notation As it has been described so far, the frequency domain is a group of amplitudes of cosine and sine waves (with slight scaling modifications). This is called rectangular notation. Alternatively, the frequency domain can be expressed in polar form. In this notation, Re X[ ] & Im X [ ] are replaced with two other arrays, called the Magnitude of X [ ] , written in equations as: Mag X [ ] , and the Phase of X [ ] , written as: Phase X [ ] . The magnitude and phase are a pair-for-pair replacement for the real and imaginary parts. For example, Mag X [0] and Phase X [0] are calculated using only Re X[0] and Im X [0] . Likewise, Mag X [14] and Phase X [14] are calculated using only Re X[14] and Im X [14] , and so forth. To understand the conversion, consider what happens when you add a cosine wave and a sine wave of the same frequency. The result is a cosine wave of the same 162 The Scientist and Engineer's Guide to Digital Signal Processing frequency, but with a new amplitude and a new phase shift. In equation form, the two representations are related: EQUATION 8-5 The addition of a cosine and sine wave results in a cosine wave with a different amplitude and phase shift. The information contained in A & B is transferred to two other variables, M and 2. A cos (x) % B sin(x) ' M cos ( x % 2) The important point is that no information is lost in this process; given one representation you can calculate the other. In other words, the information contained in the amplitudes A and B, is also contained in the variables M and 2. Although this equation involves sine and cosine waves, it follows the same conversion equations as do simple vectors. Figure 8-9 shows the analogous vector representation of how the two variables, A and B, can be viewed in a rectangular coordinate system, while M and 2 are parameters in polar coordinates. FIGURE 8-9 Rectangular-to-polar conversion. The addition of a cosine wave and a sine wave (of the same frequency) follows the same mathematics as the addition of simple vectors. B M 2 A 2 2 1/2 M = (A + B ) 2 = arctan(B/A) In polar notation, Mag X [ ] holds the amplitude of the cosine wave (M in Eq. 8-4 and Fig. 8-9), while Phase X [ ] holds the phase angle of the cosine wave ( 2 in Eq. 8-4 and Fig. 8-9). The following equations convert the frequency domain from rectangular to polar notation, and vice versa: EQUATION 8-6 Rectangular-to-polar conversion. The rectangular representation of the frequency domain, Re X[k] and Im X [k] , is changed into the polar form, Mag X [k] and Phase X [k] . EQUATION 8-7 Polar-to-rectangular conversion. The two arrays, Mag X [k] and Phase X [k] , are converted into Re X[k] and Im X [k] . Mag X [k ] ' ( Re X [k ]2 % Im X [k ]2 ) 1/ 2 Phase X [k] ' arctan Im X [k ] Re X [k ] Re X [k ] ' Mag X [k ] cos ( Phase X [k] ) Im X [k ] ' Mag X [k ] sin( Phase X [k] ) Chapter 8- The Discrete Fourier Transform 163 Rectangular and polar notation allow you to think of the DFT in two different ways. With rectangular notation, the DFT decomposes an N point signal into N/2 % 1 cosine waves and N/2 % 1 sine waves, each with a specified amplitude. In polar notation, the DFT decomposes an N point signal into N/2 % 1 cosine waves, each with a specified amplitude (called the magnitude) and phase shift. Why does polar notation use cosine waves instead of sine waves? Sine waves cannot represent the DC component of a signal, since a sine wave of zero frequency is composed of all zeros (see Figs. 8-5 a&b). Even though the polar and rectangular representations contain exactly the same information, there are many instances where one is easier to use that the other. For example, Fig. 8-10 shows a frequency domain signal in both rectangular and polar form. Warning: Don't try to understand the shape of the real and imaginary parts; your head will explode! In comparison, the polar curves are straightforward: only frequencies below about 0.25 are present, and the phase shift is approximately proportional to the frequency. This is the frequency response of a low-pass filter. Rectangular Polar 2 2 a. Re X[ ] c. Mag X[ ] Amplitude Amplitude 1 0 1 0 -1 -2 -1 0 0.1 0.2 0.3 Frequency 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 0.4 0.5 Frequency 2 6 b. Im X[ ] d. Phase X[ ] 4 Phase (radians) Amplitude 1 0 2 0 -2 -1 -4 -2 -6 0 0.1 0.2 0.3 Frequency 0.4 0.5 0 0.1 0.2 0.3 Frequency FIGURE 8-10 Example of rectangular and polar frequency domains. This example shows a frequency domain expressed in both rectangular and polar notation. As in this case, polar notation usually provides human observers with a better understanding of the characteristics of the signal. In comparison, the rectangular form is almost always used when math computations are required. Pay special notice to the fact that the first and last samples in the phase must be zero, just as they are in the imaginary part. 164 The Scientist and Engineer's Guide to Digital Signal Processing When should you use rectangular notation and when should you use polar? Rectangular notation is usually the best choice for calculations, such as in equations and computer programs. In comparison, graphs are almost always in polar form. As shown by the previous example, it is nearly impossible for humans to understand the characteristics of a frequency domain signal by looking at the real and imaginary parts. In a typical program, the frequency domain signals are kept in rectangular notation until an observer needs to look at them, at which time a rectangular-to-polar conversion is done. Why is it easier to understand the frequency domain in polar notation? This question goes to the heart of why decomposing a signal into sinusoids is useful. Recall the property of sinusoidal fidelity from Chapter 5: if a sinusoid enters a linear system, the output will also be a sinusoid, and at exactly the same frequency as the input. Only the amplitude and phase can change. Polar notation directly represents signals in terms of the amplitude and phase of the component cosine waves. In turn, systems can be represented by how they modify the amplitude and phase of each of these cosine waves. Now consider what happens if rectangular notation is used with this scenario. A mixture of cosine and sine waves enter the linear system, resulting in a mixture of cosine and sine waves leaving the system. The problem is, a cosine wave on the input may result in both cosine and sine waves on the output. Likewise, a sine wave on the input can result in both cosine and sine waves on the output. While these cross-terms can be straightened out, the overall method doesn't match with why we wanted to use sinusoids in the first place. Polar Nuisances There are many nuisances associated with using polar notation. None of these are overwhelming, just really annoying! Table 8-3 shows a computer program for converting between rectangular and polar notation, and provides solutions for some of these pests. Nuisance 1: Radians vs. Degrees It is possible to express the phase in either degrees or radians. When expressed in degrees, the values in the phase signal are between -180 and 180. Using radians, each of the values will be between -B and B, that is, between -3.141592 to 3.141592. Most computer languages require the use radians for their trigonometric functions, such as cosine, sine, arctangent, etc. It can be irritating to work with these long decimal numbers, and difficult to interpret the data you receive. For example, if you want to introduce a 90 degree phase shift into a signal, you need to add 1.570796 to the phase. While it isn't going to kill you to type this into your program, it does become tiresome. The best way to handle this problem is to define the constant, PI = 3.141592, at the beginning of your program. A 90 degree phase shift can then be written as PI /2 . Degrees and radians are both widely used in DSP and you need to become comfortable with both. Chapter 8- The Discrete Fourier Transform 165 100 'RECTANGULAR-TO-POLAR & POLAR-TO-RECTANGULAR CONVERSION 110 ' 120 DIM REX[256] 'REX[ ] holds the real part 130 DIM IMX[256] 'IMX[ ] holds the imaginary part 140 DIM MAG[256] 'MAG[ ] holds the magnitude 150 DIM PHASE[256] 'PHASE[ ] holds the phase 160 ' 170 PI = 3.14159265 180 ' 190 GOSUB XXXX 'Mythical subroutine to load data into REX[ ] and IMX[ ] 200 ' 210 ' 220 ' 'Rectangular-to-polar conversion, Eq. 8-6 230 FOR K% = 0 TO 256 240 MAG[K%] = SQR( REX[K%]^2 + IMX[K%]^2 ) 'from Eq. 8-6 250 IF REX[K%] = 0 THEN REX[K%] = 1E-20 'prevent divide by 0 (nuisance 2) 260 PHASE[K%] = ATN( IMX[K%] / REX[K%] ) 'from Eq. 8-6 270 ' 'correct the arctan (nuisance 3) 280 IF REX[K%] < 0 AND IMX[K%] < 0 THEN PHASE[K%] = PHASE[K%] - PI 290 IF REX[K%] < 0 AND IMX[K%] >= 0 THEN PHASE[K%] = PHASE[K%] + PI 300 NEXT K% 310 ' 320 ' 330 ' 'Polar-to-rectangular conversion, Eq. 8-7 340 FOR K% = 0 TO 256 350 REX[K%] = MAG[K%] * COS( PHASE[K%] ) 360 IMX[K%] = MAG[K%] * SIN( PHASE[K%] ) 370 NEXT K% 380 ' 390 END TABLE 8-3 Nuisance 2: Divide by zero error When converting from rectangular to polar notation, it is very common to find frequencies where the real part is zero and the imaginary part is some nonzero value. This simply means that the phase is exactly 90 or -90 degrees. Try to tell your computer this! When your program tries to calculate the phase from: Phase X [k] ' arctan( Im X [k] / Re X [k]) , a divide by zero error occurs. Even if the program execution doesn't halt, the phase you obtain for this frequency won't be correct. To avoid this problem, the real part must be tested for being zero before the division. If it is zero, the imaginary part must be tested for being positive or negative, to determine whether to set the phase to B /2 or - B /2, respectively. Lastly, the division needs to be bypassed. Nothing difficult in all these steps, just the potential for aggravation. An alternative way to handle this problem is shown in line 250 of Table 8-3. If the real part is zero, change it to a negligibly small number to keep the math processor happy during the division. Nuisance 3: Incorrect arctan Consider a frequency domain sample where Re X[k] ' 1 and Im X [k] ' 1 . Equation 8-6 provides the corresponding polar values of Mag X[k] ' 1.414 and Phase X [k] ' 45E . Now consider another sample where Re X[k] ' & 1 and 166 The Scientist and Engineer's Guide to Digital Signal Processing Im X [k] ' & 1 . Again, Eq. 8-6 provides the values of Mag X [k] ' 1.414 and Phase X [k] ' 45E . The problem is, the phase is wrong! It should be & 135E . This error occurs whenever the real part is negative. This problem can be corrected by testing the real and imaginary parts after the phase has been calculated. If both the real and imaginary parts are negative, subtract 180E (or B radians) from the calculated phase. If the real part is negative and the imaginary part is positive, add 180E (or B radians). Lines 340 and 350 of the program in Table 8-3 show how this is done. If you fail to catch this problem, the calculated value of the phase will only run between -B/2 and B/2, rather than between -B and B. Drill this into your mind. If you see the phase only extending to ±1.5708, you have forgotten to correct the ambiguity in the arctangent calculation. Nuisance 4: Phase of very small magnitudes Imagine the following scenario. You are grinding away at some DSP task, and suddenly notice that part of the phase doesn't look right. It might be noisy, jumping all over, or just plain wrong. After spending the next hour looking through hundreds of lines of computer code, you find the answer. The corresponding values in the magnitude are so small that they are buried in round-off noise. If the magnitude is negligibly small, the phase doesn't have any meaning, and can assume unusual values. An example of this is shown in Fig. 8-11. It is usually obvious when an amplitude signal is lost in noise; the values are so small that you are forced to suspect that the values are meaningless. The phase is different. When a polar signal is contaminated with noise, the values in the phase are random numbers between -B and B. Unfortunately, this often looks like a real signal, rather than the nonsense it really is. Nuisance 5: 2 B ambiguity of the phase Look again at Fig. 8-10d, and notice the several discontinuities in the data. Every time a point looks as if it is going to dip below -3.14592, it snaps back to 3.141592. This is a result of the periodic nature of sinusoids. For 1.5 5 a. Mag X[ ] b. Phase X[ ] 4 1.0 Phase (radians) Amplitude 3 0.5 2 1 0 -1 -2 -3 -4 0.0 -5 0 0.1 0.2 0.3 Frequency 0.4 0.5 0 0.1 0.2 0.3 Frequency 0.4 0.5 FIGURE 8-11 The phase of small magnitude signals. At frequencies where the magnitude drops to a very low value, round-off noise can cause wild excursions of the phase. Don't make the mistake of thinking this is a meaningful signal. Chapter 8- The Discrete Fourier Transform 167 10 0 Phase (radians) FIGURE 8-12 Example of phase unwrapping. The top curve shows a typical phase signal obtained from a rectangular-to-polar conversion routine. Each value in the signal must be between -B and B (i.e., -3.14159 and 3.14159). As shown in the lower curve, the phase can be unwrapped by adding or subtracting integer multiplies of 2B from each sample, where the integer is chosen to minimize the discontinuities between points. wrapped -10 -20 unwrapped -30 -40 0 0.1 0.2 0.3 Frequency 0.4 0.5 example, a phase shift of θ is exactly the same as a phase shift of θ + 2π , θ + 4π , θ + 6π , etc. Any sinusoid is unchanged when you add an integer multiple of 2B to the phase. The apparent discontinuities in the signal are a result of the computer algorithm picking its favorite choice from an infinite number of equivalent possibilities. The smallest possible value is always chosen, keeping the phase between -B and B. It is often easier to understand the phase if it does not have these discontinuities, even if it means that the phase extends above B, or below -B. This is called unwrapping the phase, and an example is shown in Fig. 8-12. As shown by the program in Table 8-4, a multiple of 2B is added or subtracted from each value of the phase. The exact value is determined by an algorithm that minimizes the difference between adjacent samples. Nuisance 6: The magnitude is always positive (B ambiguity of the phase) Figure 8-13 shows a frequency domain signal in rectangular and polar form. The real part is smooth and quite easy to understand, while the imaginary part is entirely zero. In comparison, the polar signals contain abrupt 100 ' PHASE UNWRAPPING 110 ' 120 DIM PHASE[256] 'PHASE[ ] holds the original phase 130 DIM UWPHASE[256] 'UWPHASE[ ] holds the unwrapped phase 140 ' 150 PI = 3.14159265 160 ' 170 GOSUB XXXX 'Mythical subroutine to load data into PHASE[ ] 180 ' 190 UWPHASE[0] = 0 'The first point of all phase signals is zero 200 ' 210 ' 'Go through the unwrapping algorithm 220 FOR K% = 1 TO 256 230 C% = CINT( (UWPHASE[K%-1] - PHASE[K%]) / (2 * PI) ) 240 UWPHASE[K%] = PHASE[K%] + C%*2*PI 250 NEXT K% 260 ' 270 END TABLE 8-4 168 The Scientist and Engineer's Guide to Digital Signal Processing Rectangular Polar 3 3 a. Re X[ ] c. Mag X[ ] 2 Amplitude Amplitude 2 1 0 0 -1 -1 0 0.1 0.2 0.3 Frequency 0.4 0.5 0 3 0.1 0.2 0.3 0.4 0.5 0.2 0.3 0.4 0.5 Frequency 5 b. Im X[ ] d. Phase X[ ] 4 2 3 Phase (radians) Amplitude 1 1 0 -1 2 1 0 -1 -2 -3 -2 -4 -3 -5 0 0.1 0.2 0.3 Frequency 0.4 0.5 0 0.1 Frequency FIGURE 8-13 Example signals in rectangular and polar form. Since the magnitude must always be positive (by definition), the magnitude and phase may contain abrupt discontinuities and sharp corners. Figure (d) also shows another nuisance: random noise can cause the phase to rapidly oscillate between B or -B . discontinuities and sharp corners. This is because the magnitude must always be positive, by definition. Whenever the real part dips below zero, the magnitude remains positive by changing the phase by B (or -B, which is the same thing). While this is not a problem for the mathematics, the irregular curves can be difficult to interpret. One solution is to allow the magnitude to have negative values. In the example of Fig. 8-13, this would make the magnitude appear the same as the real part, while the phase would be entirely zero. There is nothing wrong with this if it helps your understanding. Just be careful not to call a signal with negative values the "magnitude" since this violates its formal definition. In this book we use the weasel words: unwrapped magnitude to indicate a "magnitude" that is allowed to have negative values. Nuisance 7: Spikes between B and - B Since B and - B represent the same phase shift, round-off noise can cause adjacent points in the phase to rapidly switch between the two values. As shown in Fig. 8-13d, this can produce sharp breaks and spikes in an otherwise smooth curve. Don't be fooled, the phase isn't really this discontinuous. CHAPTER 9 Applications of the DFT The Discrete Fourier Transform (DFT) is one of the most important tools in Digital Signal Processing. This chapter discusses three common ways it is used. First, the DFT can calculate a signal's frequency spectrum. This is a direct examination of information encoded in the frequency, phase, and amplitude of the component sinusoids. For example, human speech and hearing use signals with this type of encoding. Second, the DFT can find a system's frequency response from the system's impulse response, and vice versa. This allows systems to be analyzed in the frequency domain, just as convolution allows systems to be analyzed in the time domain. Third, the DFT can be used as an intermediate step in more elaborate signal processing techniques. The classic example of this is FFT convolution, an algorithm for convolving signals that is hundreds of times faster than conventional methods. Spectral Analysis of Signals It is very common for information to be encoded in the sinusoids that form a signal. This is true of naturally occurring signals, as well as those that have been created by humans. Many things oscillate in our universe. For example, speech is a result of vibration of the human vocal cords; stars and planets change their brightness as they rotate on their axes and revolve around each other; ship's propellers generate periodic displacement of the water, and so on. The shape of the time domain waveform is not important in these signals; the key information is in the frequency, phase and amplitude of the component sinusoids. The DFT is used to extract this information. An example will show how this works. Suppose we want to investigate the sounds that travel through the ocean. To begin, a microphone is placed in the water and the resulting electronic signal amplified to a reasonable level, say a few volts. An analog low-pass filter is then used to remove all frequencies above 80 hertz, so that the signal can be digitized at 160 samples per second. After acquiring and storing several thousand samples, what next? 169 170 The Scientist and Engineer's Guide to Digital Signal Processing The first thing is to simply look at the data. Figure 9-1a shows 256 samples from our imaginary experiment. All that can be seen is a noisy waveform that conveys little information to the human eye. For reasons explained shortly, the next step is to multiply this signal by a smooth curve called a Hamming window, shown in (b). (Chapter 16 provides the equations for the Hamming and other windows; see Eqs. 16-1 and 16-2, and Fig. 16-2a). This results in a 256 point signal where the samples near the ends have been reduced in amplitude, as shown in (c). Taking the DFT, and converting to polar notation, results in the 129 point frequency spectrum in (d). Unfortunately, this also looks like a noisy mess. This is because there is not enough information in the original 256 points to obtain a well behaved curve. Using a longer DFT does nothing to help this problem. For example, if a 2048 point DFT is used, the frequency spectrum becomes 1025 samples long. Even though the original 2048 points contain more information, the greater number of samples in the spectrum dilutes the information by the same factor. Longer DFTs provide better frequency resolution, but the same noise level. The answer is to use more of the original signal in a way that doesn't increase the number of points in the frequency spectrum. This can be done by breaking the input signal into many 256 point segments. Each of these segments is multiplied by the Hamming window, run through a 256 point DFT, and converted to polar notation. The resulting frequency spectra are then averaged to form a single 129 point frequency spectrum. Figure (e) shows an example of averaging 100 of the frequency spectra typified by (d). The improvement is obvious; the noise has been reduced to a level that allows interesting features of the signal to be observed. Only the magnitude of the frequency domain is averaged in this manner; the phase is usually discarded because it doesn't contain useful information. The random noise reduces in proportion to the square-root of the number of segments. While 100 segments is typical, some applications might average millions of segments to bring out weak features. There is also a second method for reducing spectral noise. Start by taking a very long DFT, say 16,384 points. The resulting frequency spectrum is high resolution (8193 samples), but very noisy. A low-pass digital filter is then used to smooth the spectrum, reducing the noise at the expense of the resolution. For example, the simplest digital filter might average 64 adjacent samples in the original spectrum to produce each sample in the filtered spectrum. Going through the calculations, this provides about the same noise and resolution as the first method, where the 16,384 points would be broken into 64 segments of 256 points each. Which method should you use? The first method is easier, because the digital filter isn't needed. The second method has the potential of better performance, because the digital filter can be tailored to optimize the tradeoff between noise and resolution. However, this improved performance is seldom worth the trouble. This is because both noise and resolution can be improved by using more data from the input signal. For example, Chapter 9- Applications of the DFT Time Domain 171 Frequency Domain 1.0 a. Measured signal Amplitude 0.5 0.0 -0.5 -1.0 0 32 64 96 128 160 192 224 255 256 160 192 224 255 256 Sample number 1.5 b. Hamming window Amplitude 1.0 0.5 0.0 -0.5 0 32 64 96 128 Sample number 1.0 10 c. Windowed signal 0.5 8 DFT 7 Amplitude Amplitude d. Single spectrum 9 0.0 6 5 4 3 -0.5 2 1 -1.0 0 0 32 64 96 128 160 Sample number 192 224 256 255 0 0.1 0.2 0.3 0.4 0.5 0.3 0.4 0.5 Frequency 10 e. Averaged spectrum 9 8 7 Amplitude FIGURE 9-1 An example of spectral analysis. Figure (a) shows 256 samples taken from a (simulated) undersea microphone at a rate of 160 samples per second. This signal is multiplied by the Hamming window shown in (b), resulting in the windowed signal in (c). The frequency spectrum of the windowed signal is found using the DFT, and is displayed in (d) (magnitude only). Averaging 100 of these spectra reduces the random noise, resulting in the averaged frequency spectrum shown in (e). 6 5 4 3 2 1 0 0 0.1 0.2 Frequency 172 The Scientist and Engineer's Guide to Digital Signal Processing imagine breaking the acquired data into 10,000 segments of 16,384 samples each. This resulting frequency spectrum is high resolution (8193 points) and low noise (10,000 averages). Problem solved! For this reason, we will only look at the averaged segment method in this discussion. Figure 9-2 shows an example spectrum from our undersea microphone, illustrating the features that commonly appear in the frequency spectra of acquired signals. Ignore the sharp peaks for a moment. Between 10 and 70 hertz, the signal consists of a relatively flat region. This is called white noise because it contains an equal amount of all frequencies, the same as white light. It results from the noise on the time domain waveform being uncorrelated from sample-to-sample. That is, knowing the noise value present on any one sample provides no information on the noise value present on any other sample. For example, the random motion of electrons in electronic circuits produces white noise. As a more familiar example, the sound of the water spray hitting the shower floor is white noise. The white noise shown in Fig. 9-2 could be originating from any of several sources, including the analog electronics, or the ocean itself. Above 70 hertz, the white noise rapidly decreases in amplitude. This is a result of the roll-off of the antialias filter. An ideal filter would pass all frequencies below 80 hertz, and block all frequencies above. In practice, a perfectly sharp cutoff isn't possible, and you should expect to see this gradual drop. If you don't, suspect that an aliasing problem is present. Below about 10 hertz, the noise rapidly increases due to a curiosity called 1/f noise (one-over-f noise). 1/f noise is a mystery. It has been measured in very diverse systems, such as traffic density on freeways and electronic noise in transistors. It probably could be measured in all systems, if you look low enough in frequency. In spite of its wide occurrence, a general theory and understanding of 1/f noise has eluded researchers. The cause of this noise can be identified in some specific systems; however, this doesn't answer the question of why 1/f noise is everywhere. For common analog electronics and most physical systems, the transition between white noise and 1/f noise occurs between about 1 and 100 hertz. Now we come to the sharp peaks in Fig. 9-2. The easiest to explain is at 60 hertz, a result of electromagnetic interference from commercial electrical power. Also expect to see smaller peaks at multiples of this frequency (120, 180, 240 hertz, etc.) since the power line waveform is not a perfect sinusoid. It is also common to find interfering peaks between 25-40 kHz, a favorite for designers of switching power supplies. Nearby radio and television stations produce interfering peaks in the megahertz range. Low frequency peaks can be caused by components in the system vibrating when shaken. This is called microphonics, and typically creates peaks at 10 to 100 hertz. Now we come to the actual signals. There is a strong peak at 13 hertz, with weaker peaks at 26 and 39 hertz. As discussed in the next chapter, this is the frequency spectrum of a nonsinusoidal periodic waveform. The peak at 13 hertz is called the fundamental frequency, while the peaks at 26 and 39 Chapter 9- Applications of the DFT 173 10 9 8 1/f noise 7 Amplitude FIGURE 9-2 Example frequency spectrum. Three types of features appear in the spectra of acquired signals: (1) random noise, such as white noise and 1/f noise, (2) interfering signals from power lines, switching power supplies, radio and TV stations, microphonics, etc., and (3) real signals, usually appearing as a fundamental plus harmonics. This example spectrum (magnitude only) shows several of these features. 6 13 Hz. 5 4 3 39 Hz. 26 Hz. 60 Hz. 2 1 white noise 0 0 10 antialias filter roll-off 20 30 40 50 60 70 80 Frequency (hertz) hertz are referred to as the second and third harmonic respectively. You would also expect to find peaks at other multiples of 13 hertz, such as 52, 65, 78 hertz, etc. You don't see these in Fig. 9-2 because they are buried in the white noise. This 13 hertz signal might be generated, for example, by a submarines's three bladed propeller turning at 4.33 revolutions per second. This is the basis of passive sonar, identifying undersea sounds by their frequency and harmonic content. Suppose there are peaks very close together, such as shown in Fig. 9-3. There are two factors that limit the frequency resolution that can be obtained, that is, how close the peaks can be without merging into a single entity. The first factor is the length of the DFT. The frequency spectrum produced by an N point DFT consists of N/2 %1 samples equally spaced between zero and onehalf of the sampling frequency. To separate two closely spaced frequencies, the sample spacing must be smaller than the distance between the two peaks. For example, a 512 point DFT is sufficient to separate the peaks in Fig. 9-3, while a 128 point DFT is not. 320 100 a. N = 128 b. N = 512 240 Amplitude Amplitude 80 60 40 160 80 20 0 0 0 0.1 0.2 0.3 Frequency 0.4 0.5 0 0.1 0.2 0.3 Frequency 0.4 0.5 FIGURE 9-3 Frequency spectrum resolution. The longer the DFT, the better the ability to separate closely spaced features. In these example magnitudes, a 128 point DFT cannot resolve the two peaks, while a 512 point DFT can. 174 The Scientist and Engineer's Guide to Digital Signal Processing The second factor limiting resolution is more subtle. Imagine a signal created by adding two sine waves with only a slight difference in their frequencies. Over a short segment of this signal, say a few periods, the waveform will look like a single sine wave. The closer the frequencies, the longer the segment must be to conclude that more than one frequency is present. In other words, the length of the signal limits the frequency resolution. This is distinct from the first factor, because the length of the input signal does not have to be the same as the length of the DFT. For example, a 256 point signal could be padded with zeros to make it 2048 points long. Taking a 2048 point DFT produces a frequency spectrum with 1025 samples. The added zeros don't change the shape of the spectrum, they only provide more samples in the frequency domain. In spite of this very close sampling, the ability to separate closely spaced peaks would be only slightly better than using a 256 point DFT. When the DFT is the same length as the input signal, the resolution is limited about equally by these two factors. We will come back to this issue shortly. Next question: What happens if the input signal contains a sinusoid with a frequency between two of the basis functions? Figure 9-4a shows the answer. This is the frequency spectrum of a signal composed of two sine waves, one having a frequency matching a basis function, and the other with a frequency between two of the basis functions. As you should expect, the first sine wave is represented as a single point. The other peak is more difficult to understand. Since it cannot be represented by a single sample, it becomes a peak with tails that extend a significant distance away. The solution? Multiply the signal by a Hamming window before taking the DFT, as was previously discussed. Figure (b) shows that the spectrum is changed in three ways by using the window. First, the two peaks are made to look more alike. This is good. Second, the tails are greatly reduced. 100 200 a. No window b. With Hamming window 80 on basis function 120 Amplitude Amplitude 160 between basis functions 80 40 on basis function 60 between basis functions 40 20 tails 0 0 0 0.1 0.2 0.3 Frequency 0.4 0.5 0 0.1 0.2 0.3 Frequency 0.4 0.5 FIGURE 9-4 Example of using a window in spectral analysis. Figure (a) shows the frequency spectrum (magnitude only) of a signal consisting of two sine waves. One sine wave has a frequency exactly equal to a basis function, allowing it to be represented by a single sample. The other sine wave has a frequency between two of the basis functions, resulting in tails on the peak. Figure (b) shows the frequency spectrum of the same signal, but with a Blackman window applied before taking the DFT. The window makes the peaks look the same and reduces the tails, but broadens the peaks. Chapter 9- Applications of the DFT 175 This is also good. Third, the window reduces the resolution in the spectrum by making the peaks wider. This is bad. In DSP jargon, windows provide a tradeoff between resolution (the width of the peak) and spectral leakage (the amplitude of the tails). To explore the theoretical aspects of this in more detail, imagine an infinitely long discrete sine wave at a frequency of 0.1 the sampling rate. The frequency spectrum of this signal is an infinitesimally narrow peak, with all other frequencies being zero. Of course, neither this signal nor its frequency spectrum can be brought into a digital computer, because of their infinite and infinitesimal nature. To get around this, we change the signal in two ways, both of which distort the true frequency spectrum. First, we truncate the information in the signal, by multiplying it by a window. For example, a 256 point rectangular window would allow 256 points to retain their correct value, while all the other samples in the infinitely long signal would be set to a value of zero. Likewise, the Hamming window would shape the retained samples, besides setting all points outside the window to zero. The signal is still infinitely long, but only a finite number of the samples have a nonzero value. How does this windowing affect the frequency domain? As discussed in Chapter 10, when two time domain signals are multiplied, the corresponding frequency domains are convolved. Since the original spectrum is an infinitesimally narrow peak (i.e., a delta function), the spectrum of the windowed signal is the spectrum of the window shifted to the location of the peak. Figure 9-5 shows how the spectral peak would appear using four different window options (If you need a refresher on dB, look ahead to Chapter 14). Figure 9-5a results from a rectangular window. Figures (b) and (c) result from using two popular windows, the Hamming and the Blackman (as previously mentioned, see Eqs. 16-1 and 16-2, and Fig. 16-2a for information on these windows). As shown in Fig. 9-5, all these windows have degraded the original spectrum by broadening the peak and adding tails composed of numerous side lobes. This is an unavoidable result of using only a portion of the original time domain signal. Here we can see the tradeoff between the three windows. The Blackman has the widest main lobe (bad), but the lowest amplitude tails (good). The rectangular window has the narrowest main lobe (good) but the largest tails (bad). The Hamming window sits between these two. Notice in Fig. 9-5 that the frequency spectra are continuous curves, not discrete samples. After windowing, the time domain signal is still infinitely long, even though most of the samples are zero. This means that the frequency spectrum consists of 4 /2 %1 samples between 0 and 0.5, the same as a continuous line. This brings in the second way we need to modify the time domain signal to allow it to be represented in a computer: select N points from the signal. These N points must contain all the nonzero points identified by the window, but may also include any number of the zeros. This has the effect 176 The Scientist and Engineer's Guide to Digital Signal Processing 20 20 b. Hamming window 0 0 -20 -20 Amplitude (dB) Amplitude (dB) a. Rectangular window -40 -60 -80 -40 -60 -80 -100 -100 -120 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 -120 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 Frequency Frequency 20 20 c. Blackman window d. Flat-top window 0 0 -40 tails -60 -80 Amplitude (dB) Amplitude (dB) main lobe -20 -20 -40 -60 -80 -100 -100 -120 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 -120 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 Frequency Frequency FIGURE 9-5 Detailed view of a spectral peak using various windows. Each peak in the frequency spectrum is a central lobe surrounded by tails formed from side lobes. By changing the window shape, the amplitude of the side lobes can be reduced at the expense of making the main lobe wider. The rectangular window, (a), has the narrowest main lobe but the largest amplitude side lobes. The Hamming window, (b), and the Blackman window, (c), have lower amplitude side lobes at the expense of a wider main lobe. The flat-top window, (d), is used when the amplitude of a peak must be accurately measured. These curves are for 255 point windows; longer windows produce proportionately narrower peaks. of sampling the frequency spectrum's continuous curve. For example, if N is chosen to be 1024, the spectrum's continuous curve will be sampled 513 times between 0 and 0.5. If N is chosen to be much larger than the window length, the samples in the frequency domain will be close enough that the peaks and valleys of the continuous curve will be preserved in the new spectrum. If N is made the same as the window length, the fewer number of samples in the spectrum results in the regular pattern of peaks and valleys turning into irregular tails, depending on where the samples happen to fall. This explains why the two peaks in Fig. 94a do not look alike. Each peak in Fig 9-4a is a sampling of the underlying curve in Fig. 9-5a. The presence or absence of the tails depends on where the samples are taken in relation to the peaks and valleys. If the sine wave exactly matches a basis function, the samples occur exactly at the valleys, eliminating the tails. If the sine wave is between two basis functions, the samples occur somewhere along the peaks and valleys, resulting in various patterns of tails. Chapter 9- Applications of the DFT 177 This leads us to the flat-top window, shown in Fig. 9-5d. In some applications the amplitude of a spectral peak must be measured very accurately. Since the DFT’s frequency spectrum is formed from samples, there is nothing to guarantee that a sample will occur exactly at the top of a peak. More than likely, the nearest sample will be slightly off-center, giving a value lower than the true amplitude. The solution is to use a window that produces a spectral peak with a flat top, insuring that one or more of the samples will always have the correct peak value. As shown in Fig. 9-5d, the penalty for this is a very broad main lobe, resulting in poor frequency resolution. As it turns out, the shape we want for a flat-top window is exactly the same shape as the filter kernel of a low-pass filter. We will discuss the theoretical reasons for this in later chapters; for now, here is a cookbook description of how the technique is used. Chapter 16 discusses a low-pass filter called the windowed-sinc. Equation 16-4 describes how to generate the filter kernel (which we want to use as a window), and Fig. 16-4a illustrates the typical shape of the curve. To use this equation, you will need to know the value of two parameters: M and fc . These are found from the relations: M ' N & 2 , and fc ' s/N , where N is the length of the DFT being used, and s is the number of samples you want on the flat portion of the peak (usually between 3 and 5). Table 16-1 shows a program for calculating the filter kernel (our window), including two subtle features: the normalization constant, K, and how to avoid a divide-by-zero error on the center sample. When using this method, remember that a DC value of one in the time domain will produce a peak of amplitude one in the frequency domain. However, a sinusoid of amplitude one in the time domain will only produce a spectral peak of amplitude one-half. (This is discussed in the last chapter: Synthesis, Calculating the Inverse DFT). Frequency Response of Systems Systems are analyzed in the time domain by using convolution. A similar analysis can be done in the frequency domain. Using the Fourier transform, every input signal can be represented as a group of cosine waves, each with a specified amplitude and phase shift. Likewise, the DFT can be used to represent every output signal in a similar form. This means that any linear system can be completely described by how it changes the amplitude and phase of cosine waves passing through it. This information is called the system's frequency response. Since both the impulse response and the frequency response contain complete information about the system, there must be a oneto-one correspondence between the two. Given one, you can calculate the other. The relationship between the impulse response and the frequency response is one of the foundations of signal processing: A system's frequency response is the Fourier Transform of its impulse response. Figure 9-6 illustrates these relationships. Keeping with standard DSP notation, impulse responses use lower case variables, while the corresponding frequency responses are upper case. Since h[ ] is the common symbol for the impulse response, H[ ] is used for the frequency response. Systems are described in the time domain by convolution, that is: The Scientist and Engineer's Guide to Digital Signal Processing x[n] t h[n] ' y[n] . In the frequency domain, the input spectrum is multiplied by the frequency response, resulting in the output spectrum. As an equation: X [f ] × H[ f ] ' Y[ f ] . That is, convolution in the time domain corresponds to multiplication in the frequency domain. Figure 9-7 shows an example of using the DFT to convert a system's impulse response into its frequency response. Figure (a) is the impulse response of the system. Looking at this curve isn't going to give you the slightest idea what the system does. Taking a 64 point DFT of this impulse response produces the frequency response of the system, shown in (b). Now the function of this system becomes obvious, it passes frequencies between 0.2 and 0.3, and rejects all others. It is a band-pass filter. The phase of the frequency response could also be examined; however, it is more difficult to interpret and less interesting. It will be discussed in upcoming chapters. Figure (b) is very jagged due to the low number of samples defining the curve. This situation can be improved by padding the impulse response with zeros before taking the DFT. For example, adding zeros to make the impulse response 512 samples long, as shown in (c), results in the higher resolution frequency response shown in (d). How much resolution can you obtain in the frequency response? The answer is: infinitely high, if you are willing to pad the impulse response with an infinite number of zeros. In other words, there is nothing limiting the frequency resolution except the length of the DFT. This leads to a very important concept. Even though the impulse response is a discrete signal, the corresponding frequency response is continuous. An N point DFT of the impulse response provides N/2 % 1 samples of this continuous curve. If you make the DFT longer, the resolution improves, and you obtain a better idea of X[f] IDFT y[n] H[f] DFT DFT FREQUENCY DOMAIN IDFT TIME DOMAIN h[n] IDFT x[n] DFT 178 Y[f] FIGURE 9-6 Comparing system operation in the time and frequency domains. In the time domain, an input signal is convolved with an impulse response, resulting in the output signal, that is, x[n] t h[n] ' y[n] . In the frequency domain, an input spectrum is multiplied by a frequency response, resulting in the output spectrum, that is, X[f] × H[f] ' Y[f] . The DFT and the Inverse DFT relate the signals in the two domain. Chapter 9- Applications of the DFT Time Domain Frequency Domain 0.4 2.0 a. Impulse response a. Impulse b. Frequency response response 1.5 Amplitude Amplitude 0.2 0 -0.2 1.0 0.5 -0.4 0.0 0 8 16 24 32 40 Sample number 48 56 64 63 0 0.1 0.2 0.3 0.4 0.5 Frequency 0.4 2.0 a. Impulse response padded with zeros c. d. Frequency response (high resolution) 0.2 1.5 Amplitude Amplitude 179 0 -0.2 1.0 0.5 -0.4 0.0 0 64 128 192 256 320 Sample number 384 448 511 512 0 0.1 0.2 0.3 0.4 0.5 Frequency FIGURE 9-7 Finding the frequency response from the impulse response. By using the DFT, a system's impulse response, (a), can be transformed into the system's frequency response, (b). By padding the impulse response with zeros (c), higher resolution can be obtained in the frequency response, (d). Only the magnitude of the frequency response is shown in this example; discussion of the phase is postponed until the next chapter. what the continuous curve looks like. Remember what the frequency response represents: amplitude and phase changes experienced by cosine waves as they pass through the system. Since the input signal can contain any frequency between 0 and 0.5, the system's frequency response must be a continuous curve over this range. This can be better understood by bringing in another member of the Fourier transform family, the Discrete Time Fourier Transform (DTFT). Consider an N sample signal being run through an N point DFT, producing an N/2 % 1 sample frequency domain. Remember from the last chapter that the DFT considers the time domain signal to be infinitely long and periodic. That is, the N points are repeated over and over from negative to positive infinity. Now consider what happens when we start to pad the time domain signal with an ever increasing number of zeros, to obtain a finer and finer sampling in the frequency domain. Adding zeros makes the period of the time domain longer, while simultaneously making the frequency domain samples closer together. 180 The Scientist and Engineer's Guide to Digital Signal Processing Now we will take this to the extreme, by adding an infinite number of zeros to the time domain signal. This produces a different situation in two respects. First, the time domain signal now has an infinitely long period. In other words, it has turned into an aperiodic signal. Second, the frequency domain has achieved an infinitesimally small spacing between samples. That is, it has become a continuous signal. This is the DTFT, the procedure that changes a discrete aperiodic signal in the time domain into a frequency domain that is a continuous curve. In mathematical terms, a system's frequency response is found by taking the DTFT of its impulse response. Since this cannot be done in a computer, the DFT is used to calculate a sampling of the true frequency response. This is the difference between what you do in a computer (the DFT) and what you do with mathematical equations (the DTFT). Convolution via the Frequency Domain Suppose that you despise convolution. What are you going to do if given an input signal and impulse response, and need to find the resulting output signal? Figure 9-8 provides an answer: transform the two signals into the frequency domain, multiply them, and then transform the result back into the time domain. This replaces one convolution with two DFTs, a multiplication, and an Inverse DFT. Even though the intermediate steps are very different, the output is identical to the standard convolution algorithm. Does anyone hate convolution enough to go to this trouble? The answer is yes. Convolution is avoided for two reasons. First, convolution is mathematically difficult to deal with. For instance, suppose you are given a system's impulse response, and its output signal. How do you calculate what the input signal is? This is called deconvolution, and is virtually impossible to understand in the time domain. However, deconvolution can be carried out in the frequency domain as a simple division, the inverse operation of multiplication. The frequency domain becomes attractive whenever the complexity of the Fourier Transform is less than the complexity of the convolution. This isn't a matter of which you like better; it is a matter of which you hate less. The second reason for avoiding convolution is computation speed. For example, suppose you design a digital filter with a kernel (impulse response) containing 512 samples. Using a 200 MHz personal computer with floating point numbers, each sample in the output signal requires about one millisecond to calculate, using the standard convolution algorithm. In other words, the throughput of the system is only about 1,000 samples per second. This is 40 times too slow for high-fidelity audio, and 10,000 times too slow for television quality video! The standard convolution algorithm is slow because of the large number of multiplications and additions that must be calculated. Unfortunately, simply bringing the problem into the frequency domain via the DFT doesn't help at all. Just as many calculations are required to calculate the DFTs, as are required to directly calculate the convolution. A breakthrough was made in the problem in the early 1960s when the Fast Fourier Transform (FFT) was developed. Chapter 9- Applications of the DFT 2 181 2 30 a. x[n] d. h[n] g. y[n] 20 1 = 0 Amplitude t 0 Amplitude Amplitude 1 10 0 -10 -1 -20 -1 128 256 384 Sample number -30 512 511 0 128 384 Sample number TIME DFT 256 512 511 0 128 256 384 511 512 Sample number DFT 0 IDFT -2 FREQUENCY 60 2000 60 b. Re X[f] 40 e. Re H[f] 40 h. Re Y[f] 0 -20 20 Amplitude 20 Amplitude Amplitude 1000 0 -20 0 -1000 -40 -40 -60 -2000 -60 0 64 128 Frequency 192 256 0 x 128 192 = 60 c. Im X[f] 0 256 Frequency 60 40 64 128 Frequency 192 256 192 256 2000 f. Im H[f] 40 64 i. Im Y[f] 0 -20 20 Amplitude 20 Amplitude Amplitude 1000 0 -20 0 -1000 -40 -40 -60 -60 0 64 128 Frequency 192 256 -2000 0 64 128 Frequency 192 256 0 64 128 Frequency FIGURE 9-8 Frequency domain convolution. In the time domain, x[n] is convolved with h[n] resulting in y[n] , as is shown in Figs. (a), (d), and (g). This same procedure to be accomplished in the frequency domain. The DFT is used to find the frequency spectrum of the input signal, (b) & (c), and the system's frequency response, (e) & (f). Multiplying these two frequency domain signals results in the frequency spectrum of the output signal, (h) & (i). The Inverse DFT is then used to find the output signal, (g). The FFT is a clever algorithm for rapidly calculating the DFT. Using the FFT, convolution by multiplication in the frequency domain can be hundreds of times faster than conventional convolution. Problems that take hours of calculation time are reduced to only minutes. This is why people get excited about the FFT, and processing signals in the frequency domain. The FFT will be presented in 182 The Scientist and Engineer's Guide to Digital Signal Processing Chapter 12, and the method of FFT convolution in Chapter 18. For now, focus on how signals are convolved by frequency domain multiplication. To start, we need to define how to multiply one frequency domain signal by another, i.e., what it means to write: X [f ] × H[ f ] ' Y[f ] . In polar form, the magnitudes are multiplied: Mag Y[ f ] ' Mag X [f ] × MagH [f ] , and the phases are added: Phase Y[ f ] ' PhaseX [f ] % Phase H[ f ] . To understand this, imagine a cosine wave entering a system with some amplitude and phase. Likewise, the output signal is also a cosine wave with some amplitude and phase. The polar form of the frequency response directly describes how the two amplitudes are related and how the two phases are related. When frequency domain multiplication is carried out in rectangular form there are cross terms between the real and imaginary parts. For example, a sine wave entering the system can produce both cosine and sine waves in the output. To multiply frequency domain signals in rectangular notation: EQUATION 9-1 Multiplication of frequency domain signals in rectangular form: Y[ f ] ' X[ f ] × H [ f ] . Re Y [ f ] ' Re X [ f ] Re H [ f ] & Im X [ f ] Im H [ f ] Im Y [ f ] ' Im X [ f ] Re H [ f ] % Re X [ f ] Im H [ f ] Focus on understanding multiplication using polar notation, and the idea of cosine waves passing through the system. Then simply accept that these more elaborate equations result when the same operations are carried out in rectangular form. For instance, let's look at the division of one frequency domain signal by another. In polar form, the division of frequency domain signals is achieved by the inverse operations we used for multiplication. To calculate: H [f ] ' Y[f ] / X [f ] , divide the magnitudes and subtract the phases, i.e., Mag H[ f ] ' Mag Y[f ] / MagX [f ] , Phase H[ f ] ' PhaseY[f ] & Phase X[ f ] . In rectangular form this becomes: EQUATION 9-2 Division of frequency domain signals in rectangular form, where: H [ f ] ' Y[ f ] / X[ f ] . Re H [ f ] ' Re Y [ f ] Re X [ f ] % Im Y [ f ] Im X [ f ] Re X [ f ]2 % Im X [ f ]2 Im H [ f ] ' Im Y [ f ] Re X [ f ] & Re Y [ f ] Im X [ f ] Re X [ f ]2 % Im X [ f ]2 Now back to frequency domain convolution. You may have noticed that we cheated slightly in Fig. 9-8. Remember, the convolution of an N point signal with an M point impulse response results in an N% M& 1 point output signal. We cheated by making the last part of the input signal all zeros to allow this expansion to occur. Specifically, (a) contains 453 nonzero samples, and (b) contains 60 nonzero samples. This means the convolution of the two, shown in (c), can fit comfortably in the 512 points provided. Chapter 9- Applications of the DFT 2 2 20 a. Input signal b. Impulse response 10 = 0 -1 -1 -2 -2 Amplitude t 0 c. Convolution of (a) and (b) 15 1 Amplitude 1 5 0 -5 -10 -15 0 64 128 192 Sample number 256 255 -20 0 64 128 192 Sample number 256 255 0 64 128 192 Sample number 256 255 305 20 d. Overlap of adjecent periods 15 10 Amplitude 5 0 -5 -10 -15 -20 0 256 Sample number 512 768 767 20 e. Circular convolution 15 10 Amplitude Amplitude 183 5 0 -5 overlap -10 -15 -20 0 64 128 192 Sample number 255 256 FIGURE 9-9 Circular convolution. A 256 sample signal, (a), convolved with a 51 sample impulse response, (b), results in a 306 sample signal, (c). If this convolution is performed in the frequency domain using 256 point DFTs, the 306 points in the correct convolution cannot fit into the 256 samples provided. As shown in (d), samples 256 through 305 of the output signal are pushed into the next period to the right, where they add to the beginning of the next period's signal. Figure (e) is a single period of the resulting signal. Now consider the more general case in Fig. 9-9. The input signal, (a), is 256 points long, while the impulse response, (b), contains 51 nonzero points. This makes the convolution of the two signals 306 samples long, as shown in (c). The problem is, if we use frequency domain multiplication to perform the convolution, there are only 256 samples allowed in the output signal. In other words, 256 point DFTs are used to move (a) and (b) into the frequency 184 The Scientist and Engineer's Guide to Digital Signal Processing domain. After the multiplication, a 256 point Inverse DFT is used to find the output signal. How do you squeeze 306 values of the correct signal into the 256 points provided by the frequency domain algorithm? The answer is, you can't! The 256 points end up being a distorted version of the correct signal. This process is called circular convolution. It is important because you want to avoid it. To understand circular convolution, remember that an N point DFT views the time domain as being an infinitely long periodic signal, with N samples per period. Figure (d) shows three periods of how the DFT views the output signal in this example. Since N ' 256 , each period consists of 256 points: 0-255, 256-511, and 512-767. Frequency domain convolution tries to place the 306 point correct output signal, shown in (c), into each of these 256 point periods. This results in 49 of the samples being pushed into the neighboring period to the right, where they overlap with the samples that are legitimately there. These overlapping sections add, resulting in each of the periods appearing as shown in (e), the circular convolution. Once the nature of circular convolution is understood, it is quite easy to avoid. Simply pad each of the signals being convolved with enough zerosto allow the output signal room to handle the N% M& 1 points in the correct convolution. For example, the signals in (a) and (b) could be padded with zeros to make them 512 points long, allowing the use of 512 point DFTs. After the frequency domain convolution, the output signal would consist of 306 nonzero samples, plus 206 samples with a value of zero. Chapter 18 explains this procedure in detail. Why is it called circular convolution? Look back at Fig. 9-9d and examine the center period, samples 256 to 511. Since all of the periods are the same, the portion of the signal that flows out of this period to the right, is the same that flows into this period from the left. If you only consider a single period, such as in (e), it appears that the right side of the signal is somehow connected to the left side. Imagine a snake biting its own tail; sample 255 is located next to sample 0, just as sample 100 is located next to sample 101. When a portion of the signal exits to the right, it magically reappears on the left. In other words, the N point time domain behaves as if it were circular. In the last chapter we posed the question: does it really matter if the DFT's time domain is viewed as being N points, rather than an infinitely long periodic signal of period N? Circular convolution is an example where it does matter. If the time domain signal is understood to be periodic, the distortion encountered in circular convolution can be simply explained as the signal expanding from one period to the next. In comparison, a rather bizarre conclusion is reached if only N points of the time domain are considered. That is, frequency domain convolution acts as if the time domain is somehow wrapping into a circular ring with sample 0 being positioned next to sample N-1. CHAPTER 10 Fourier Transform Properties The time and frequency domains are alternative ways of representing signals. The Fourier transform is the mathematical relationship between these two representations. If a signal is modified in one domain, it will also be changed in the other domain, although usually not in the same way. For example, it was shown in the last chapter that convolving time domain signals results in their frequency spectra being multiplied. Other mathematical operations, such as addition, scaling and shifting, also have a matching operation in the opposite domain. These relationships are called properties of the Fourier Transform, how a mathematical change in one domain results in a mathematical change in the other domain. Linearity of the Fourier Transform The Fourier Transform is linear, that is, it possesses the properties of homogeneity and additivity. This is true for all four members of the Fourier transform family (Fourier transform, Fourier Series, DFT, and DTFT). Figure 10-1 provides an example of how homogeneity is a property of the Fourier transform. Figure (a) shows an arbitrary time domain signal, with the corresponding frequency spectrum shown in (b). We will call these two signals: x[ ] and X[ ] , respectively. Homogeneity means that a change in amplitude in one domain produces an identical change in amplitude in the other domain. This should make intuitive sense: when the amplitude of a time domain waveform is changed, the amplitude of the sine and cosine waves making up that waveform must also change by an equal amount. In mathematical form, if x[ ] and X [ ] are a Fourier Transform pair, then k x[ ] and k X[ ] are also a Fourier Transform pair, for any constant k. If the frequency domain is represented in rectangular notation, k X[ ] means that both the real part and the imaginary part are multiplied by k. If the frequency domain is represented in polar notation, k X[ ] means that the magnitude is multiplied by k, while the phase remains unchanged. 185 186 The Scientist and Engineer's Guide to Digital Signal Processing Time Domain Frequency Domain 3 50 a. x[ ] b. X[ ] 40 1 0 F.T. -1 Amplitude Amplitude 2 30 20 10 -2 -3 0 0 64 128 Sample number 192 255 256 0 3 0.2 0.3 0.4 0.5 0.2 0.3 0.4 0.5 Frequency 50 c. k x[ ] d. k X[ ] 2 40 1 0 F.T. -1 Amplitude Amplitude 0.1 30 20 10 -2 -3 0 0 64 128 Sample number 192 255 256 0 0.1 Frequency FIGURE 10-1 Homogeneity of the Fourier transform. If the amplitude is changed in one domain, it is changed by the same amount in the other domain. In other words, scaling in one domain corresponds to scaling in the other domain. Additivity of the Fourier transform means that addition in one domain corresponds to addition in the other domain. An example of this is shown in Fig. 10-2. In this illustration, (a) and (b) are signals in the time domain called x1[ ] and x2[ ] , respectively. Adding these signals produces a third time domain signal called x3[ ] , shown in (c). Each of these three signals has a frequency spectrum consisting of a real and an imaginary part, shown in (d) through (i). Since the two time domain signals add to produce the third time domain signal, the two corresponding spectra add to produce the third spectrum. Frequency spectra are added in rectangular notation by adding the real parts to the real parts and the imaginary parts to the imaginary parts. If: x1[n] % x2[n] ' x3[n] , then: Re X1[f ] % ReX2[ f ] ' ReX3[ f ] and Im X1[f ] % ImX2[ f ] ' Im X3[f ] . Think of this in terms of cosine and sine waves. All the cosine waves add (the real parts) and all the sine waves add (the imaginary parts) with no interaction between the two. Frequency spectra in polar form cannot be directly added; they must be converted into rectangular notation, added, and then reconverted back to Chapter 10- Fourier Transform Properties Time Domain Frequency Domain 4 200 200 a. x1[ ] d. Re X1[ ] g. Im X1[ ] F.T. 0 -2 100 Amplitude 100 Amplitude 2 Amplitude 187 0 -100 -4 -100 -200 0 64 128 192 Sample number 256 255 -200 0 + 3 0.1 0.2 0.3 Frequency 0.4 0.5 0 + 200 b. x2[ ] 2 0.1 e. Re X2[ ] -1 0.3 0.4 0.5 h. Im X2[ ] 100 Amplitude F.T. 0 Amplitude 1 0.2 Frequency + 200 100 Amplitude 0 0 -100 0 -100 -2 -3 -200 0 64 128 192 Sample number 256 255 = 4 -200 0 0.1 0.3 Frequency 0.4 0.5 0 = 200 c. x3[ ] -2 0 128 192 Sample number 255 256 0.5 0 -100 -200 64 0.4 i. Im X3[ ] -100 -4 0.3 100 Amplitude Amplitude F.T. 0 0.2 Frequency = 200 100 0 0.1 f. Re X3[ ] 2 Amplitude 0.2 -200 0 0.1 0.2 0.3 Frequency 0.4 0.5 0 0.1 0.2 0.3 Frequency 0.4 0.5 FIGURE 10-2 Additivity of the Fourier transform. Adding two or more signals in one domain results in the corresponding signals being added in the other domain. In this illustration, the time domain signals in (a) and (b) are added to produce the signal in (c). This results in the corresponding real and imaginary parts of the frequency spectra being added. polar form. This can also be understood in terms of how sinusoids behave. Imagine adding two sinusoids having the same frequency, but with different amplitudes ( A1 and A2 ) and phases ( N1 and N2 ). If the two phases happen to be same ( N1 ' N2 ), the amplitudes will add ( A1 %A2 ) when the sinusoids are added. However, if the two phases happen to be exactly opposite ( N1 ' &N2 ), the amplitudes will subtract ( A1 &A2 ) when the sinusoids are added. The point is, when sinusoids (or spectra) are in polar form, they cannot be added by simply adding the magnitudes and phases. 188 The Scientist and Engineer's Guide to Digital Signal Processing In spite of being linear, the Fourier transform is not shift invariant. In other words, a shift in the time domain does not correspond to a shift in the frequency domain. This is the topic of the next section. Characteristics of the Phase In mathematical form: if x[n] : Mag X [f ] & Phase X [f ] , then a shift in the time domain results in: x[n%s] : Mag X [f ] & Phase X [f ] % 2B sf , (where f is expressed as a fraction of the sampling rate, running between 0 and 0.5). In words, a shift of s samples in the time domain leaves the magnitude unchanged, but adds a linear term to the phase, 2B sf . Let's look at an example of how this works. Figure 10-3 shows how the phase is affected when the time domain waveform is shifted to the left or right. The magnitude has not been included in this illustration because it isn't interesting; it is not changed by the time domain shift. In Figs. (a) through (d), the waveform is gradually shifted from having the peak centered on sample 128, to having it centered on sample 0. This sequence of graphs takes into account that the DFT views the time domain as circular; when portions of the waveform exit to the right, they reappear on the left. The time domain waveform in Fig. 10-3 is symmetrical around a vertical axis, that is, the left and right sides are mirror images of each other. As mentioned in Chapter 7, signals with this type of symmetry are called linear phase, because the phase of their frequency spectrum is a straight line. Likewise, signals that don't have this left-right symmetry are called nonlinear phase, and have phases that are something other than a straight line. Figures (e) through (h) show the phase of the signals in (a) through (d). As described in Chapter 7, these phase signals are unwrapped, allowing them to appear without the discontinuities associated with keeping the value between B and -B. When the time domain waveform is shifted to the right, the phase remains a straight line, but experiences a decrease in slope. When the time domain is shifted to the left, there is an increase in the slope. This is the main property you need to remember from this section; a shift in the time domain corresponds to changing the slope of the phase. Figures (b) and (f) display a unique case where the phase is entirely zero. This occurs when the time domain signal is symmetrical around sample zero. At first glance, this symmetry may not be obvious in (b); it may appear that the signal is symmetrical around sample 256 (i.e., N/2) instead. Remember that the DFT views the time domain as circular, with sample zero inherently connected to sample N-1. Any signal that is symmetrical around sample zero will also be symmetrical around sample N/2, and vice versa. When using members of the Fourier Transform family that do not view the time domain as periodic (such as the DTFT), the symmetry must be around sample zero to produces a zero phase. Chapter 10- Fourier Transform Properties Time Domain 189 Frequency Domain 2 900 a. e. Phase (radians) Amplitude 600 1 0 300 0 -300 -600 -1 -900 0 64 128 192 256 320 Sample number 384 448 512 511 0 2 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.4 0.5 Frequency 900 b. f. 600 Phase (radians) Amplitude 0.1 1 0 300 0 -300 -600 -1 -900 0 64 128 192 256 320 Sample number 384 448 512 511 0 2 Frequency 900 g. c. Phase (radians) Amplitude 600 1 0 300 0 -300 -600 -1 -900 0 64 128 192 256 320 Sample number 384 448 512 511 0 2 900 d. h. 600 Phase (radians) Amplitude Frequency 1 0 2 300 0 3 -300 -600 1 -1 -900 0 64 128 192 256 320 Sample number 384 448 512 511 0 0.1 FIGURE 10-3 Phase changes resulting from a time domain shift. 0.2 0.3 Frequency 190 The Scientist and Engineer's Guide to Digital Signal Processing Figures (d) and (h) shows something of a riddle. First imagine that (d) was formed by shifting the waveform in (c) slightly more to the right. This means that the phase in (h) would have a slightly more negative slope than in (g). This phase is shown as line 1. Next, imagine that (d) was formed by starting with (a) and shifting it to the left. In this case, the phase should have a slightly more positive slope than (e), as is illustrated by line 2. Lastly, notice that (d) is symmetrical around sample N/2, and should therefore have a zero phase, as illustrated by line 3. Which of these three phases is correct? They all are, depending on how the B and 2B phase ambiguities (discussed in Chapter 8) are arranged. For instance, every sample in line 2 differs from the corresponding sample in line 1 by an integer multiple of 2B , making them equal. To relate line 3 to lines 1 and 2, the B ambiguities must also be taken into account. To understand why the phase behaves as it does, imagine shifting a waveform by one sample to the right. This means that all of the sinusoids that compose the waveform must also be shifted by one sample to the right. Figure 10-4 shows two sinusoids that might be a part of the waveform. In (a), the sine wave has a very low frequency, and a one sample shift is only a small fraction of a full cycle. In (b), the sinusoid has a frequency of one-half of the sampling rate, the highest frequency that can exist in sampled data. A one sample shift at this frequency is equal to an entire 1/2 cycle, or B radians. That is, when a shift is expressed in terms of a phase change, it becomes proportional to the frequency of the sinusoid being shifted. For example, consider a waveform that is symmetrical around sample zero, and therefore has a zero phase. Figure 10-5a shows how the phase of this signal changes when it is shifted left or right. At the highest frequency, one-half of the sampling rate, the phase increases by B for each one sample shift to the left, and decreases by B for each one sample shift to the right. At zero frequency there is no phase shift, and all of the frequencies between follow in a straight line. 2 2 a. A low frequency b. 1/2 of sampling frequency 1 Amplitude Amplitude 1 0 0 -1 -1 1 sample shift = 1/32 cycle 1 sample shift = 1/2 cycle -2 -2 0 8 16 Sample number 24 32 0 8 16 Sample number 24 32 FIGURE 10-4 The relationship between samples and phase. Figures (a) and (b) show low and high frequency sinusoids, respectively. In (a), a one sample shift is equal to 1/32 of a cycle. In (b), a one sample shift is equal to 1/2 of a cycle. This is why a shift in the waveform changes the phase more at high frequencies than at low frequencies. Chapter 10- Fourier Transform Properties 30 3B 150 15B a. 20 2B b. -3 100 10B -2 -1 10 B Phase (radians) Phase (radians) 191 0 00 1 2 -10 -B 3 -20 -2B -14 50 5B -7 0 00 number of samples shifted in time domain number of samples shifted in time domain -30 -3B -5B -50 0 0.1 0.2 0.3 Frequency 0.4 0.5 0 0.1 0.2 0.3 Frequency 0.4 0.5 FIGURE 10-5 Phases resulting from time domain shifting. For each sample that a time domain signal is shifted in the positive direction (i.e., to the right), the phase at frequency 0.5 will decrease by B radians. For each sample shifted in the negative direction (i.e., to the left), the phase at frequency 0.5 will increase by B radians. Figure (a) shows this for a linear phase (a straight line), while (b) is an example using a nonlinear phase. All of the examples we have used so far are linear phase. Figure 10-5b shows that nonlinear phase signals react to shifting in the same way. In this example the nonlinear phase is a straight line with two rectangular pulses. When the time domain is shifted, these nonlinear features are simply superimposed on the changing slope. What happens in the real and imaginary parts when the time domain waveform is shifted? Recall that frequency domain signals in rectangular notation are nearly impossible for humans to understand. The real and imaginary parts typically look like random oscillations with no apparent pattern. When the time domain signal is shifted, the wiggly patterns of the real and imaginary parts become even more oscillatory and difficult to interpret. Don't waste your time trying to understand these signals, or how they are changed by time domain shifting. Figure 10-6 is an interesting demonstration of what information is contained in the phase, and what information is contained in the magnitude. The waveform in (a) has two very distinct features: a rising edge at sample number 55, and a falling edge at sample number 110. Edges are very important when information is encoded in the shape of a waveform. An edge indicates when something happens, dividing whatever is on the left from whatever is on the right. It is time domain encoded information in its purest form. To begin the demonstration, the DFT is taken of the signal in (a), and the frequency spectrum converted into polar notation. To find the signal in (b), the phase is replaced with random numbers between -B and B, and the inverse DFT used to reconstruct the time domain waveform. In other words, (b) is based only on the information contained in the magnitude. In a similar manner, (c) is found by replacing the magnitude with small random numbers before using the inverse DFT. This makes the reconstruction of (c) based solely on the information contained in the phase. 192 The Scientist and Engineer's Guide to Digital Signal Processing 3 3 a. Original signal b. Reconstructed from the magnitude Amplitude 2 1 0 -1 1 0 -1 -2 -2 0 64 128 Sample number 192 255 256 0 64 128 Sample number 192 255 256 192 255 256 3 FIGURE 10-6 Information contained in the phase. Figure (a) shows a pulse-like waveform. The signal in (b) is created by taking the DFT of (a), replacing the phase with random numbers, and taking the Inverse DFT. The signal in (c) is found by taking the DFT of (a), replacing the magnitude with random numbers, and taking the Inverse DFT. The location of the edges is retained in (c), but not in (b). This shows that the phase contains information on the location of events in the time domain signal. c. Reconstructed from the phase 2 Amplitude Amplitude 2 1 0 -1 -2 0 64 128 Sample number The result? The locations of the edges are clearly present in (c), but totally absent in (b). This is because an edge is formed when many sinusoids rise at the same location, possible only when their phases are coordinated. In short, much of the information about the shape of the time domain waveform is contained in the phase, rather than the magnitude. This can be contrasted with signals that have their information encoded in the frequency domain, such as audio signals. The magnitude is most important for these signals, with the phase playing only a minor role. In later chapters we will see that this type of understanding provides strategies for designing filters and other methods of processing signals. Understanding how information is represented in signals is always the first step in successful DSP. Why does left-right symmetry correspond to a zero (or linear) phase? Figure 10-7 provides the answer. Such a signal can be decomposed into a left half and a right half, as shown in (a), (b) and (c). The sample at the center of symmetry (zero in this case) is divided equally between the left and right halves, allowing the two sides to be perfect mirror images of each other. The magnitudes of these two halves will be identical, as shown in (e) and (f), while the phases will be opposite in sign, as in (h) and (i). Two important concepts fall out of this. First, every signal that is symmetrical between the left and right will have a linear phase because the nonlinear phase of the left half exactly cancels the nonlinear phase of the right half. Chapter 10- Fourier Transform Properties Time Domain 193 Frequency Domain 2 20 4 a. x[ ] d. Mag X[ ] 0 2 Amplitude Amplitude Amplitude 15 1 g. Phase X[ ] 3 10 5 1 0 -1 -2 -3 -1 0 -64 -32 0 32 Sample number 64 63 -4 0 0.1 0.2 0.3 Frequency 0.4 0.5 0 0.1 0.2 0.3 Frequency 0.4 0.5 0.4 0.5 0.4 0.5 Decompose 2 20 4 b. x1[ ] e. Mag X1[ ] 0 2 Amplitude Amplitude Amplitude 1 h. Phase X1[ ] 3 15 10 5 1 0 -1 -2 -3 -1 0 -64 -32 0 32 Sample number 64 63 -4 0 2 0.1 0.2 0.3 Frequency 0.4 0.5 0 20 f. Mag X 2[ ] 2 Amplitude Amplitude Amplitude 0.3 i. Phase X2[ ] 3 15 0 0.2 Frequency 4 c. x2[ ] 1 0.1 10 5 1 0 -1 -2 -3 0 -1 -64 -32 0 32 Sample number 64 63 -4 0 0.1 0.2 0.3 Frequency 0.4 0.5 0 0.1 0.2 0.3 Frequency FIGURE 10-7 Phase characteristics of left-right symmetry. A signal with left-right symmetry, shown in (a), can be decomposed into a right half, (b), and a left half, (c). The magnitudes of the two halves are identical, (e) and (f), while the phases are the negative of each other, (h) and (i). Second, imagine flipping (b) such that it becomes (c). This left-right flip in the time domain does nothing to the magnitude, but changes the sign of every point in the phase. Likewise, changing the sign of the phase flips the time domain signal left-for-right. If the signals are continuous, the flip is around zero. If the signals are discrete, the flip is around sample zero and sample N/2, simultaneously. Changing the sign of the phase is a common enough operation that it is given its own name and symbol. The name is complex conjugation, and it is 194 The Scientist and Engineer's Guide to Digital Signal Processing represented by placing a star to the upper-right of the variable. For example, if X [f ] consists of Mag X [f ] and Phase X [f ] , then X t[f ] is called the complex conjugate and is composed of Mag X [f ] and & Phase X [f ] . In rectangular notation, the complex conjugate is found by leaving the real part alone, and changing the sign of the imaginary part. In mathematical terms, if X [f ] is composed of Re X[ f ] and Im X[ f ] , then X t[f ] is made up of Re X[ f ] and & Im X[ f ] . Here are several examples of how the complex conjugate is used in DSP. If x[ n ] has a Fourier transform of X [f ] , then x [& n ] has a Fourier transform of X t[f ] . In words, flipping the time domain left-for-right corresponds to changing the sign of the phase. As another example, recall from Chapter 7 that correlation can be performed as a convolution. This is done by flipping one of the signals left-for-right. In mathematical form, a[ n ] t b[n ] is convolution, while a[ n ] t b[& n ] is correlation. In the frequency domain these operations correspond to A [f ] × B[ f ] and A [f ] × B t[f ] , respectively. As the last example, consider an arbitrary signal, x[ n ] , and its frequency spectrum, X [f ] . The frequency spectrum can be changed to zero phase by multiplying it by its complex conjugate, that is, X[ f ] × X t[ f ] . In words, whatever phase X [f ] happens to have will be canceled by adding its opposite (remember, when frequency spectra are multiplied, their phases are added). In the time domain, this means that x [n ] t x [& n ] (a signal convolved with a left-right flipped version of itself) will have left-right symmetry around sample zero, regardless of what x [n ] is. To many engineers and mathematicians, this kind of manipulation is DSP. If you want to be able to communicate with this group, get used to using their language. Periodic Nature of the DFT Unlike the other three Fourier Transforms, the DFT views both the time domain and the frequency domain as periodic. This can be confusing and inconvenient since most of the signals used in DSP are not periodic. Nevertheless, if you want to use the DFT, you must conform with the DFT's view of the world. Figure 10-8 shows two different interpretations of the time domain signal. First, look at the upper signal, the time domain viewed as N points. This represents how digital signals are typically acquired in scientific experiments and engineering applications. For instance, these 128 samples might have been acquired by sampling some parameter at regular intervals of time. Sample 0 is distinct and separate from sample 127 because they were acquired at different times. From the way this signal was formed, there is no reason to think that the samples on the left of the signal are even related to the samples on the right. Unfortunately, the DFT doesn't see things this way. As shown in the lower figure, the DFT views these 128 points to be a single period of an infinitely long periodic signal. This means that the left side of the acquired signal is Chapter 10- Fourier Transform Properties 195 connected to the right side of a duplicate signal. Likewise, the right side of the acquired signal is connected to the left side of an identical period. This can also be thought of as the right side of the acquired signal wrapping around and connecting to its left side. In this view, sample 127 occurs next to sample 0, just as sample 43 occurs next to sample 44. This is referred to as being circular, and is identical to viewing the signal as being periodic. The most serious consequence of time domain periodicity is time domain aliasing. To illustrate this, suppose we take a time domain signal and pass it through the DFT to find its frequency spectrum. We could immediately pass this frequency spectrum through an Inverse DFT to reconstruct the original time domain signal, but the entire procedure wouldn't be very interesting. Instead, we will modify the frequency spectrum in some manner before using the Inverse DFT. For instance, selected frequencies might be deleted, changed in amplitude or phase, shifted around, etc. These are the kinds of things routinely done in DSP. Unfortunately, these changes in the frequency domain can create a time domain signal that is too long to fit into 2 Amplitude 1 The time domain viewed as N points 0 -1 -2 0 The time domain viewed as periodic 32 64 Sample number 96 127 128 Amplitude 2 1 0 -1 -2 -384 -256 -128 0 128 256 384 Sample number FIGURE 10-8 Periodicity of the DFT's time domain signal. The time domain can be viewed as N samples in length, shown in the upper figure, or as an infinitely long periodic signal, shown in the lower figure. 196 The Scientist and Engineer's Guide to Digital Signal Processing a single period. This forces the signal to spill over from one period into the adjacent periods. When the time domain is viewed as circular, portions of the signal that overflow on the right suddenly seem to reappear on the left side of the signal, and vice versa. That is, the overflowing portions of the signal alias themselves to a new location in the time domain. If this new location happens to already contain an existing signal, the whole mess adds, resulting in a loss of information. Circular convolution resulting from frequency domain multiplication (discussed in Chapter 9), is an excellent example of this type of aliasing. Periodicity in the frequency domain behaves in much the same way, but is more complicated. Figure 10-9 shows an example. The upper figures show the magnitude and phase of the frequency spectrum, viewed as being composed of N /2 % 1 samples spread between 0 and 0.5 of the sampling rate. This is the simplest way of viewing the frequency spectrum, but it doesn't explain many of the DFT's properties. The lower two figures show how the DFT views this frequency spectrum as being periodic. The key feature is that the frequency spectrum between 0 and 0.5 appears to have a mirror image of frequencies that run between 0 and -0.5. This mirror image of negative frequencies is slightly different for the magnitude and the phase signals. In the magnitude, the signal is flipped leftfor-right. In the phase, the signal is flipped left-for-right, and changed in sign. As you recall, these two types of symmetry are given names: the magnitude is said to be an even signal (it has even symmetry), while the phase is said to be an odd signal (it has odd symmetry). If the frequency spectrum is converted into the real and imaginary parts, the real part will always be even, while the imaginary part will always be odd. Taking these negative frequencies into account, the DFT views the frequency domain as periodic, with a period of 1.0 times the sampling rate, such as -0.5 to 0.5, or 0 to 1.0. In terms of sample numbers, this makes the length of the frequency domain period equal to N, the same as in the time domain. The periodicity of the frequency domain makes it susceptible to frequency domain aliasing, completely analogous to the previously described time domain aliasing. Imagine a time domain signal that corresponds to some frequency spectrum. If the time domain signal is modified, it is obvious that the frequency spectrum will also be changed. If the modified frequency spectrum cannot fit in the space provided, it will push into the adjacent periods. Just as before, this aliasing causes two problems: frequencies aren't where they should be, and overlapping frequencies from different periods add, destroying information. Frequency domain aliasing is more difficult to understand than time domain aliasing, since the periodic pattern is more complicated in the frequency domain. Consider a single frequency that is being forced to move from 0.01 to 0.49 in the frequency domain. The corresponding negative frequency is therefore moving from -0.01 to -0.49. When the positive frequency moves Chapter 10- Fourier Transform Properties 197 3 Magnitude Amplitude 2 1 0 -1 0 The frequency domain viewed as 0 to 0.5 of the sampling rate 0.1 0.2 0.3 0.4 0.5 0.2 0.3 0.4 0.5 Frequency 4 Phase 3 Phase (radians) 2 1 0 -1 -2 -3 -4 The frequency domain viewed as periodic 0 W 0.1 Frequency W W Amplitude 3 2 1 0 -1 -1.5 -1 -0.5 0 0.5 1 1.5 0.5 1 1.5 Phase Frequency 4 3 2 1 0 -1 -2 -3 -4 -1.5 -1 -0.5 0 Frequency FIGURE 10-9 Periodicity of the DFT's frequency domain. The frequency domain can be viewed as running from 0 to 0.5 of the sampling rate (upper two figures), or an infinity long periodic signal with every other 0 to 0.5 segment flipped left-for-right (lower two figures). 198 The Scientist and Engineer's Guide to Digital Signal Processing across the 0.5 barrier, the negative frequency is pushed across the -0.5 barrier. Since the frequency domain is periodic, these same events are occurring in the other periods, such as between 0.5 and 1.5. A clone of the positive frequency is crossing frequency 1.5 from left to right, while a clone of the negative frequency is crossing 0.5 from right to left. Now imagine what this looks like if you can only see the frequency band of 0 to 0.5. It appears that a frequency leaving to the right, reappears on the right, but moving in the opposite direction. Figure 10-10 illustrates how aliasing appears in the time and frequency domains when only a single period is viewed. As shown in (a), if one end of a time domain signal is too long to fit inside a single period, the protruding end will be cut off and pasted onto the other side. In comparison, (b) shows that when a frequency domain signal overflows the period, the protruding end is folded over. Regardless of where the aliased segment ends up, it adds to whatever signal is already there, destroying information. Let's take a closer look at these strange things called negative frequencies. Are they just some bizarre artifact of the mathematics, or do they have a real world meaning? Figure 10-11 shows what they are about. Figure (a) is a discrete signal composed of 32 samples. Imagine that you are given the task of finding the frequency spectrum that corresponds to these 32 points. To make your job easier, you are told that these points represent a discrete cosine wave. In other words, you must find the frequency and phase shift ( f and 2 ) such that x[n] ' cos(2Bn f / N % 2) matches the given samples. It isn't long before you come up with the solution shown in (b), that is, f ' 3 and 2 ' & B/4 . If you stopped your analysis at this point, you only get 1/3 credit for the problem. This is because there are two other solutions that you have missed. As shown in (c), the second solution is f ' & 3 and 2 ' B/4 . Even if the idea of a negative frequency offends your sensibilities, it doesn't a. Time domain aliasing b. Frequency domain aliasing signal exiting to the right signal exiting to the right reappears on the left 0 Time reappears on the right N-1 0 Frequency 0.5 FIGURE 10-10 Examples of aliasing in the time and frequency domains, when only a single period is considered. In the time domain, shown in (a), portions of the signal that exits to the right, reappear on the left. In the frequency domain, (b), portions of the signal that exit to the right, reappear on the right as if they had been folded over. Chapter 10- Fourier Transform Properties 2 2 b. Solution #1 f = 3, 2 = -B/4 a. Samples 1 Amplitude Amplitude 1 0 -1 0 -1 -2 -2 0 8 16 24 32 0 8 Sample number 16 24 32 24 32 Sample number 2 2 c. Solution #2 f = -3, 2 = B/4 d. Solution #3 f = 35, 2 = -B/4 1 Amplitude 1 Amplitude 199 0 -1 0 -1 -2 -2 0 8 16 Sample number 24 32 0 8 16 Sample number FIGURE 10-11 The meaning of negative frequencies. The problem is to find the frequency spectrum of the discrete signal shown in (a). That is, we want to find the frequency and phase of the sinusoid that passed through all of the samples. Figure (b) is a solution using a positive frequency, while (c) is a solution using a negative frequency. Figure (d) represents a family of solutions to the problem. change the fact that it is a mathematically valid solution to the defined problem. Every positive frequency sinusoid can alternately be expressed as a negative frequency sinusoid. This applies to continuous as well as discrete signals The third solution is not a single answer, but an infinite family of solutions. As shown in (d), the sinusoid with f ' 35 and 2 ' & B/4 passes through all of the discrete points, and is therefore a correct solution. The fact that it shows oscillation between the samples may be confusing, but it doesn't disqualify it from being an authentic answer. Likewise, f ' ±29 , f ' ±35 , f ' ±61 , and f ' ±67 are all solutions with multiple oscillations between the points. Each of these three solutions corresponds to a different section of the frequency spectrum. For discrete signals, the first solution corresponds to frequencies between 0 and 0.5 of the sampling rate. The second solution 200 The Scientist and Engineer's Guide to Digital Signal Processing results in frequencies between 0 and -0.5. Lastly, the third solution makes up the infinite number of duplicated frequencies below -0.5 and above 0.5. If the signal we are analyzing is continuous, the first solution results in frequencies from zero to positive infinity, while the second solution results in frequencies from zero to negative infinity. The third group of solutions does not exist for continuous signals. Many DSP techniques do not require the use of negative frequencies, or an understanding of the DFT's periodicity. For example, two common ones were described in the last chapter, spectral analysis, and the frequency response of systems. For these applications, it is completely sufficient to view the time domain as extending from sample 0 to N-1, and the frequency domain from zero to one-half of the sampling frequency. These techniques can use a simpler view of the world because they never result in portions of one period moving into another period. In these cases, looking at a single period is just as good as looking at the entire periodic signal. However, certain procedures can only be analyzed by considering how signals overflow between periods. Two examples of this have already been presented, circular convolution and analog-to-digital conversion. In circular convolution, multiplication of the frequency spectra results in the time domain signals being convolved. If the resulting time domain signal is too long to fit inside a single period, it overflows into the adjacent periods, resulting in time domain aliasing. In contrast, analog-to-digital conversion is an example of frequency domain aliasing. A nonlinear action is taken in the time domain, that is, changing a continuous signal into a discrete signal by sampling. The problem is, the spectrum of the original analog signal may be too long to fit inside the discrete signal's spectrum. When we force the situation, the ends of the spectrum protrude into adjacent periods. Let's look at two more examples where the periodic nature of the DFT is important, compression & expansion of signals, and amplitude modulation. Compression and Expansion, Multirate methods As shown in Fig. 10-12, a compression of the signal in one domain results in an expansion in the other, and vice versa. For continuous signals, if X (f ) is the Fourier Transform of x( t) , then 1/k × X( f/ k) is the Fourier Transform of x(k t) , where k is the parameter controlling the expansion or contraction. If an event happens faster (it is compressed in time), it must be composed of higher frequencies. If an event happens slower (it is expanded in time), it must be composed of lower frequencies. This pattern holds if taken to either of the two extremes. That is, if the time domain signal is compressed so far that it becomes an impulse, the corresponding frequency spectrum is expanded so far that it becomes a constant value. Likewise, if the time domain is expanded until it becomes a constant value, the frequency domain becomes an impulse. Discrete signals behave in a similar fashion, but there are a few more details. The first issue with discrete signals is aliasing. Imagine that the Chapter 10- Fourier Transform Properties Time Domain Frequency Domain 3 20 a. Signal compressed b. Expanded frequency spectrum 16 Amplitude Amplitude 2 1 0 12 8 4 0 -1 0 16 32 48 64 80 Sample number 96 112 0 127 128 c. Signal 0.2 0.3 0.4 0.5 0.3 0.4 0.5 Frequency d. Frequency spectrum 24 2 samples 1 Amplitude Amplitude 0.1 30 3 underlying continious waveform 0 18 12 6 0 -1 0 16 32 48 64 80 Sample number 96 112 0 127 128 3 0.1 0.2 Frequency 50 e. Signal expanded f. Compressed frequency spectrum 40 Amplitude 2 Amplitude 201 1 0 30 20 10 0 -1 0 16 32 48 64 80 Sample number 96 112 128 127 0 0.1 0.2 0.3 Frequency 0.4 0.5 FIGURE 10-12 Compression and expansion. Compressing a signal in one domain results in the signal being expanded in the other domain, and vice versa. Figures (c) and (d) show a discrete signal and its spectrum, respectively. In (a) and (b), the time domain signal has been compressed, resulting in the frequency spectrum being expanded. Figures (e) and (f) show the opposite process. As shown in these figures, discrete signals are expanded or contracted by expanding or contracting the underlying continuous waveform. This underlying waveform is then resampled to find the new discrete signal. pulse in (a) is compressed several times more than is shown. The frequency spectrum is expanded by an equal factor, and several of the humps in (b) are pushed to frequencies beyond 0.5. The resulting aliasing breaks the simple expansion/contraction relationship. This type of aliasing can also happen in the 202 The Scientist and Engineer's Guide to Digital Signal Processing time domain. Imagine that the frequency spectrum in (f) is compressed much harder, resulting in the time domain signal in (e) expanding into neighboring periods. A second issue is to define exactly what it means to compress or expand a discrete signal. As shown in Fig. 10-12a, a discrete signal is compressed by compressing the underlying continuous curve that the samples lie on, and then resampling the new continuous curve to find the new discrete signal. Likewise, this same process for the expansion of discrete signals is shown in (e). When a discrete signal is compressed, events in the signal (such as the width of the pulse) happen over a fewer number of samples. Likewise, events in an expanded signal happen over a greater number of samples. An equivalent way of looking at this procedure is to keep the underlying continuous waveform the same, but resample it at a different sampling rate. For instance, look at Fig. 10-13a, a discrete Gaussian waveform composed of 50 samples. In (b), the same underlying curve is represented by 400 samples. The change between (a) and (b) can be viewed in two ways: (1) the sampling rate has been kept constant, but the underlying waveform has been expanded to be eight times wider, or (2) the underlying waveform has been kept constant, but the sampling rate has increased by a factor of eight. Methods for changing the sampling rate in this way are called multirate techniques. If more samples are added, it is called interpolation. If fewer samples are used to represent the signal, it is called decimation. Chapter 3 describes how multirate techniques are used in ADC and DAC. Here is the problem: if we are given an arbitrary discrete signal, how do we know what the underlying continuous curve is? It depends on if the signal's information is encoded in the time domain or in the frequency domain. For time domain encoded signals, we want the underlying continuous waveform to be a smooth curve that passes through all the samples. In the simplest case, we might draw straight lines between the points and then round the rough corners. The next level of sophistication is to use a curve fitting algorithm, such as a spline function or polynomial fit. There is not a single "correct" answer to this problem. This approach is based on minimizing irregularities in the time domain waveform, and completely ignores the frequency domain. When a signal has information encoded in the frequency domain, we ignore the time domain waveform and concentrate on the frequency spectrum. As discussed in the last chapter, a finer sampling of a frequency spectrum (more samples between frequency 0 and 0.5) can be obtained by padding the time domain signal with zeros before taking the DFT. Duality allows this to work in the opposite direction. If we want a finer sampling in the time domain (interpolation), pad the frequency spectrum with zeros before taking the Inverse DFT. Say we want to interpolate a 50 sample signal into a 400 sample signal. It's done like this: (1) Take the 50 samples and add zeros to make the signal 64 samples long. (2) Use a 64 point DFT to find the frequency spectrum, which will consist of a 33 point real part and a 33 point imaginary part. (3) Pad the right side of the frequency spectrum Chapter 10- Fourier Transform Properties 2 2 c. Sharp edges 1 Amplitude Amplitude a. Smooth waveform 0 -1 1 0 -1 0 10 20 30 Sample number 40 50 0 10 20 30 40 50 320 400 Sample number 2 2 b. Fig. (a) interpolated d. Fig. (c) interpolated 1 Amplitude Amplitude 203 0 -1 1 0 -1 0 80 160 240 Sample number 320 400 0 80 160 240 Sample number FIGURE 10-13 Interpolation by padding the frequency domain. Figures (a) and (c) each consist of 50 samples. These are interpolated to 400 samples by padding the frequency domain with zeros, resulting in (b) and (d), respectively. (Figures (b) and (d) are discrete signals, but are drawn as continuous lines because of the large number of samples). (both the real and imaginary parts) with 224 zeros to make the frequency spectrum 257 points long. (4) Use a 512 point Inverse DFT to transform the data back into the time domain. This will result in a 512 sample signal that is a high resolution version of the 64 sample signal. The first 400 samples of this signal are an interpolated version of the original 50 samples. The key feature of this technique is that the interpolated signal is composed of exactly the same frequencies as the original signal. This may or may not provide a well-behaved fit in the time domain. For example, Figs. 10-13 (a) and (b) show a 50 sample signal being interpolated into a 400 sample signal by this method. The interpolation is a smooth fit between the original points, much as if a curve fitting routine had been used. In comparison, (c) and (d) show another example where the time domain is a mess! The oscillatory behavior shown in (d) arises at edges or other discontinuities in the signal. This also includes any discontinuity between sample zero and N-1, since the time domain is viewed as being circular. This overshoot at discontinuities is called the Gibbs effect, and is discussed in Chapter 11. Another frequency domain interpolation technique is presented in Chapter 3, adding zeros between the time domain samples and low-pass filtering. 204 The Scientist and Engineer's Guide to Digital Signal Processing Multiplying Signals (Amplitude Modulation) An important Fourier transform property is that convolution in one domain corresponds to multiplication in the other domain. One side of this was discussed in the last chapter: time domain signals can be convolved by multiplying their frequency spectra. Amplitude modulation is an example of the reverse situation, multiplication in the time domain corresponds to convolution in the frequency domain. In addition, amplitude modulation provides an excellent example of how the elusive negative frequencies enter into everyday science and engineering problems. Audio signals are great for short distance communication; when you speak, someone across the room hears you. On the other hand, radio frequencies are very good at propagating long distances. For instance, if a 100 volt, 1 MHz sine wave is fed into an antenna, the resulting radio wave can be detected in the next room, the next country, and even on the next planet. Modulation is the process of merging two signals to form a third signal with desirable characteristics of both. This always involves nonlinear processes such as multiplication; you can't just add the two signals together. In radio communication, modulation results in radio signals that can propagate long distances and carry along audio or other information. Radio communication is an extremely well developed discipline, and many modulation schemes have been developed. One of the simplest is called amplitude modulation. Figure 10-14 shows an example of how amplitude modulation appears in both the time and frequency domains. Continuous signals will be used in this example, since modulation is usually carried out in analog electronics. However, the whole procedure could be carried out in discrete form if needed (the shape of the future!). Figure (a) shows an audio signal with a DC bias such that the signal always has a positive value. Figure (b) shows that its frequency spectrum is composed of frequencies from 300 Hz to 3 kHz, the range needed for voice communication, plus a spike for the DC component. All other frequencies have been removed by analog filtering. Figures (c) and (d) show the carrier wave, a pure sinusoid of much higher frequency than the audio signal. In the time domain, amplitude modulation consists of multiplying the audio signal by the carrier wave. As shown in (e), this results in an oscillatory waveform that has an instantaneous amplitude proportional to the original audio signal. In the jargon of the field, the envelope of the carrier wave is equal to the modulating signal. This signal can be routed to an antenna, converted into a radio wave, and then detected by a receiving antenna. This results in a signal identical to (e) being generated in the radio receiver's electronics. A detector or demodulator circuit is then used to convert the waveform in (e) back into the waveform in (a). Since the time domain signals are multiplied, the corresponding frequency spectra are convolved. That is, (f) is found by convolving (b) & (d). Since the spectrum of the carrier is a shifted delta function, the spectrum of the Chapter 10- Fourier Transform Properties Time Domain Frequency Domain 2 4 b. Audio signal a. Audio signal 3 Amplitude Voltage 1 0 -1 2 1 -2 0 0.0 0.2 0.4 0.6 Time (milliseconds) 0.8 1.0 0 2 10 15 20 25 30 35 40 20 25 30 35 40 Frequency (kHz) d. Carrier signal 3 Amplitude 1 Voltage 5 4 c. Carrier signal 0 -1 2 1 -2 0 0.0 0.2 0.4 0.6 Time (milliseconds) 0.8 1.0 0 2 5 10 15 Frequency (kHz) 4 f. Modulated signal e. Modulated signal 0 -1 upper sideband carrier 3 Amplitude 1 Voltage 205 lower sideband 2 1 -2 0 0.0 0.2 0.4 0.6 Time (milliseconds) 0.8 1.0 0 5 10 15 20 25 Frequency (kHz) 30 35 40 FIGURE 10-14 Amplitude modulation. In the time domain, amplitude modulation is achieved by multiplying the audio signal, (a), by the carrier signal, (c), to produce the modulated signal, (e). Since multiplication in the time domain corresponds to convolution in the frequency domain, the spectrum of the modulated signal is the spectrum of the audio signal shifted to the frequency of the carrier. modulated signal is equal to the audio spectrum shifted to the frequency of the carrier. This results in a modulated spectrum composed of three components: a carrier wave, an upper sideband, and a lower sideband. These correspond to the three parts of the original audio signal: the DC component, the positive frequencies between 0.3 and 3 kHz, and the negative 206 The Scientist and Engineer's Guide to Digital Signal Processing frequencies between -0.3 and -3 kHz, respectively. Even though the negative frequencies in the original audio signal are somewhat elusive and abstract, the resulting frequencies in the lower sideband are as real as you could want them to be. The ghosts have taken human form! Communication engineers live and die by this type of frequency domain analysis. For example, consider the frequency spectrum for television transmission. A standard TV signal has a frequency spectrum from DC to 6 MHz. By using these frequency shifting techniques, 82 of these 6 MHz wide channels are stacked end-to-end. For instance, channel 3 is from 60 to 66 MHz, channel 4 is from 66 to 72 MHz, channel 83 is from 884 to 890 MHz, etc. The television receiver moves the desired channel back to the DC to 6 MHz band for display on the screen. This scheme is called frequency domain multiplexing. The Discrete Time Fourier Transform The Discrete Time Fourier Transform (DTFT) is the member of the Fourier transform family that operates on aperiodic, discrete signals. The best way to understand the DTFT is how it relates to the DFT. To start, imagine that you acquire an N sample signal, and want to find its frequency spectrum. By using the DFT, the signal can be decomposed into N/2 % 1 sine and cosine waves, with frequencies equally spaced between zero and one-half of the sampling rate. As discussed in the last chapter, padding the time domain signal with zeros makes the period of the time domain longer, as well as making the spacing between samples in the frequency domain narrower. As N approaches infinity, the time domain becomes aperiodic, and the frequency domain becomes a continuous signal. This is the DTFT, the Fourier transform that relates an aperiodic, discrete signal, with a periodic, continuous frequency spectrum. The mathematics of the DTFT can be understood by starting with the synthesis and analysis equations for the DFT (Eqs. 8-2, 8-3 and 8-4), and taking N to infinity: j x [n] cos (T n ) %4 EQUATION 10-1 The DTFT analysis equation. In this relation, x[n] is the time domain signal with n running from 0 to N& 1 . The frequency spectrum is held in: Re X (T) and Im X ( T ), with T taking on values Re X (T) ' EQUATION 10-2 The DTFT synthesis equation. 1 Re X (T) cos ( T n ) & Im X (T) sin( T n) dT B m n ' &4 Im X (T) ' & j x [n] sin(T n ) %4 n ' &4 B x[n ] ' 0 Chapter 10- Fourier Transform Properties 207 There are many subtle details in these relations. First, the time domain signal, x[n] , is still discrete, and therefore is represented by brackets. In comparison, the frequency domain signals, Re X(T) & Im X (T) , are continuous, and are thus written with parentheses. Since the frequency domain is continuous, the synthesis equation must be written as an integral, rather than a summation. As discussed in Chapter 8, frequency is represented in the DFT's frequency domain by one of three variables: k, an index that runs from 0 to N/2; f, the fraction of the sampling rate, running from 0 to 0.5; or T, the fraction of the sampling rate expressed as a natural frequency, running from 0 to B. The spectrum of the DTFT is continuous, so either f or T can be used. The common choice is T, because it makes the equations shorter by eliminating the always present factor of 2B. Remember, when T is used, the frequency spectrum extends from 0 to B, which corresponds to DC to one-half of the sampling rate. To make things even more complicated, many authors use S (an upper case omega) to represent this frequency in the DTFT, rather than T (a lower case omega). When calculating the inverse DFT, samples 0 and N/2 must be divided by two (Eq. 8-3) before the synthesis can be carried out (Eq. 8-2). This is not necessary with the DTFT. As you recall, this action in the DFT is related to the frequency spectrum being defined as a spectral density, i.e., amplitude per unit of bandwidth. When the spectrum becomes continuous, the special treatment of the end points disappear. However, there is still a normalization factor that must be included, the 2/N in the DFT (Eq. 8-3) becomes 1/ B in the DTFT (Eq. 10-2). Some authors place these terms in front of the synthesis equation, while others place them in front of the analysis equation. Suppose you start with some time domain signal. After taking the Fourier transform, and then the Inverse Fourier transform, you want to end up with what you started. That is, the 1/ B term (or the 2/N term) must be encountered somewhere along the way, either in the forward or in the inverse transform. Some authors even split the term between the two transforms by placing 1/ B in front of both. Since the DTFT involves infinite summations and integrals, it cannot be calculated with a digital computer. Its main use is in theoretical problems as an alternative to the DFT. For instance, suppose you want to find the frequency response of a system from its impulse response. If the impulse response is known as an array of numbers, such as might be obtained from an experimental measurement or computer simulation, a DFT program is run on a computer. This provides the frequency spectrum as another array of numbers, equally spaced between 0 and 0.5 of the sampling rate. In other cases, the impulse response might be know as an equation, such as a sinc function (described in the next chapter) or an exponentially decaying sinusoid. The DTFT is used here to mathematically calculate the frequency domain as another equation, specifying the entire continuous curve between 0 and 0.5. While the DFT could also be used for this calculation, it would only provide an equation for samples of the frequency response, not the entire curve. 208 The Scientist and Engineer's Guide to Digital Signal Processing Parseval's Relation Since the time and frequency domains are equivalent representations of the same signal, they must have the same energy. This is called Parseval's relation, and holds for all members of the Fourier transform family. For the DFT, Parseval's relation is expressed: EQUATION 10-3 Parseval's relation. In this equation, x[i] is a time domain signal with i running from 0 to N-1, and X[k] is its modified frequency spectrum, with k running from 0 to N/2. The modified frequency spectrum is found by taking the DFT of the signal, and dividing the first and last frequencies (sample 0 and N/2) by the square-root of two. j x [i ] i' 0 2 2 j Mag X [k ] N k' 0 N /2 N& 1 2 ' The left side of this equation is the total energy contained in the time domain signal, found by summing the energies of the N individual samples. Likewise, the right side is the energy contained in the frequency domain, found by summing the energies of the N /2 % 1 sinusoids. Remember from physics that energy is proportional to the amplitude squared. For example, the energy in a spring is proportional to the displacement squared, and the energy stored in a capacitor is proportional to the voltage squared. In Eq. 10-3, X [f ] is the frequency spectrum of x[n] , with one slight modification: the first and last frequency components, X [0 ] & X [N/2 ] , have been divided by 2 . This modification, along with the 2/N factor on the right side of the equation, accounts for several subtle details of calculating and summing energies. To understand these corrections, start by finding the frequency domain representation of the signal by using the DFT. Next, convert the frequency domain into the amplitudes of the sinusoids needed to reconstruct the signal, as previously defined in Eq. 8-3. This is done by dividing the first and last points (sample 0 and N/2) by 2, and then dividing all of the points by N/2. While this provides the amplitudes of the sinusoids, they are expressed as a peak amplitude, not the root-mean-square (rms) amplitude needed for energy calculations. In a sinusoid, the peak amplitude is converted to rms by dividing by 2 . This correction must be made to all of the frequency domain values, except sample 0 and N/2. This is because these two sinusoids are unique; one is a constant value, while the other alternates between two constant values. For these two special cases, the peak amplitude is already equal to the rms value. All of the values in the frequency domain are squared and then summed. The last step is to divide the summed value by N, to account for each sample in the frequency domain being converted into a sinusoid that covers N values in the time domain. Working through all of these details produces Eq. 10-3. While Parseval's relation is interesting from the physics it describes (conservation of energy), it has few practical uses in DSP. CHAPTER Fourier Transform Pairs 11 For every time domain waveform there is a corresponding frequency domain waveform, and vice versa. For example, a rectangular pulse in the time domain coincides with a sinc function [i.e., sin(x)/x] in the frequency domain. Duality provides that the reverse is also true; a rectangular pulse in the frequency domain matches a sinc function in the time domain. Waveforms that correspond to each other in this manner are called Fourier transform pairs. Several common pairs are presented in this chapter. Delta Function Pairs For discrete signals, the delta function is a simple waveform, and has an equally simple Fourier transform pair. Figure 11-1a shows a delta function in the time domain, with its frequency spectrum in (b) and (c). The magnitude is a constant value, while the phase is entirely zero. As discussed in the last chapter, this can be understood by using the expansion/compression property. When the time domain is compressed until it becomes an impulse, the frequency domain is expanded until it becomes a constant value. In (d) and (g), the time domain waveform is shifted four and eight samples to the right, respectively. As expected from the properties in the last chapter, shifting the time domain waveform does not affect the magnitude, but adds a linear component to the phase. The phase signals in this figure have not been unwrapped, and thus extend only from -B to B. Also notice that the horizontal axes in the frequency domain run from -0.5 to 0.5. That is, they show the negative frequencies in the spectrum, as well as the positive ones. The negative frequencies are redundant information, but they are often included in DSP graphs and you should become accustomed to seeing them. Figure 11-2 presents the same information as Fig. 11-1, but with the frequency domain in rectangular form. There are two lessons to be learned here. First, compare the polar and rectangular representations of the 209 210 The Scientist and Engineer's Guide to Digital Signal Processing Time Domain Frequency Domain 2 2 6 b. Magnitude a. Impulse at x[0] 4 0 Phase (radians) Amplitude Amplitude 1 1 0 -1 c. Phase 2 0 -2 -4 -2 -0.5 -1 0 16 32 48 Sample number 64 63 2 0 Frequency 2 4 Amplitude 0 Phase (radians) 1 1 0 0.5 0 0.5 0 0.5 Frequency 6 e. Magnitude d. Impulse at x[4] Amplitude -6 -0.5 0.5 0 -1 f. Phase 2 0 -2 -4 -2 -0.5 -1 0 16 32 48 Sample number 64 63 2 0 Frequency 2 4 Amplitude 0 Phase (radians) 1 1 Frequency 6 h. Magnitude g. Impulse at x[8] Amplitude -6 -0.5 0.5 0 -1 i. Phase 2 0 -2 -4 -1 0 16 32 48 Sample number 64 63 -2 -0.5 0 Frequency 0.5 -6 -0.5 Frequency FIGURE 11-1 Delta function pairs in polar form. An impulse in the time domain corresponds to a constant magnitude and a linear phase in the frequency domain. frequency domains. As is usually the case, the polar form is much easier to understand; the magnitude is nothing more than a constant, while the phase is a straight line. In comparison, the real and imaginary parts are sinusoidal oscillations that are difficult to attach a meaning to. The second interesting feature in Fig. 11-2 is the duality of the DFT. In the conventional view, each sample in the DFT's frequency domain corresponds to a sinusoid in the time domain. However, the reverse of this is also true, each sample in the time domain corresponds to sinusoids in the frequency domain. Including the negative frequencies in these graphs allows the duality property to be more symmetrical. For instance, Figs. (d), (e), and Chapter 11- Fourier Transform Pairs Time Domain 211 Frequency Domain 2 2 2 b. Real Part a. Impulse at x[0] c. Imaginary part 0 1 Amplitude Amplitude Amplitude 1 1 0 -1 -2 -0.5 -1 0 16 32 48 Sample number 64 63 2 -1 0 Frequency -2 -0.5 0.5 2 1 Amplitude Amplitude Amplitude 0 0 -1 -2 -0.5 -1 16 32 48 Sample number 64 63 2 0 Frequency -2 -0.5 0.5 0 i. Imaginary part 0 -1 -1 16 32 48 Sample number 64 63 -2 -0.5 0.5 1 Amplitude Amplitude Amplitude 1 0 Frequency 2 h. Real Part 1 0 0 -1 2 g. Impulse at x[8] 0.5 f. Imaginary part 1 1 0 Frequency 2 e. Real Part d. Impulse at x[4] 0 0 0 -1 0 Frequency 0.5 -2 -0.5 0 Frequency 0.5 FIGURE 11-2 Delta function pairs in rectangular form. Each sample in the time domain results in a cosine wave in the real part, and a negative sine wave in the imaginary part of the frequency domain. (f) show that an impulse at sample number four in the time domain results in four cycles of a cosine wave in the real part of the frequency spectrum, and four cycles of a negative sine wave in the imaginary part. As you recall, an impulse at sample number four in the real part of the frequency spectrum results in four cycles of a cosine wave in the time domain. Likewise, an impulse at sample number four in the imaginary part of the frequency spectrum results in four cycles of a negative sine wave being added to the time domain wave. As mentioned in Chapter 8, this can be used as another way to calculate the DFT (besides correlating the time domain with sinusoids). Each sample in the time domain results in a cosine wave being added to the real part of the 212 The Scientist and Engineer's Guide to Digital Signal Processing frequency domain, and a negative sine wave being added to the imaginary part. The amplitude of each sinusoid is given by the amplitude of the time domain sample. The frequency of each sinusoid is provided by the sample number of the time domain point. The algorithm involves: (1) stepping through each time domain sample, (2) calculating the sine and cosine waves that correspond to each sample, and (3) adding up all of the contributing sinusoids. The resulting program is nearly identical to the correlation method (Table 8-2), except that the outer and inner loops are exchanged. The Sinc Function Figure 11-4 illustrates a common transform pair: the rectangular pulse and the sinc function (pronounced “sink”). The sinc function is defined as: sinc(a) ' sin(Ba)/ (Ba) , however, it is common to see the vague statement: "the sinc function is of the general form: sin(x)/x ." In other words, the sinc is a sine wave that decays in amplitude as 1/x. In (a), the rectangular pulse is symmetrically centered on sample zero, making one-half of the pulse on the right of the graph and the other one-half on the left. This appears to the DFT as a single pulse because of the time domain periodicity. The DFT of this signal is shown in (b) and (c), with the unwrapped version in (d) and (e). First look at the unwrapped spectrum, (d) and (e). The unwrapped magnitude is an oscillation that decreases in amplitude with increasing frequency. The phase is composed of all zeros, as you should expect for a time domain signal that is symmetrical around sample number zero. We are using the term unwrapped magnitude to indicate that it can have both positive and negative values. By definition, the magnitude must always be positive. This is shown in (b) and (c) where the magnitude is made all positive by introducing a phase shift of B at all frequencies where the unwrapped magnitude is negative in (d). In (f), the signal is shifted so that it appears as one contiguous pulse, but is no longer centered on sample number zero. While this doesn't change the magnitude of the frequency domain, it does add a linear component to the phase, making it a jumbled mess. What does the frequency spectrum look like as real and imaginary parts ? Too confusing to even worry about. An N point time domain signal that contains a unity amplitude rectangular pulse M points wide, has a DFT frequency spectrum given by: EQUATION 11-1 DFT spectrum of a rectangular pulse. In this equation, N is the number of points in the time domain signal, all of which have a value of zero, except M adjacent points that have a value of one. The frequency spectrum is contained in X[k] , where k runs from 0 to N/2. To avoid the division by zero, use X[0] ' M . The sine function uses radians, not degrees. This equation takes into account that the signal is aliased. sin(BkM / N ) Mag X [ k] ' /0 /0 00 sin(Bk /N ) 00 Chapter 11- Fourier Transform Pairs Time Domain 213 Frequency Domain 2 20 6 a. Rectangular pulse b. Magnitude 0 Phase (radians) Amplitude Amplitude 1 10 5 0 -1 32 64 96 Sample number 128 127 0.1 0.2 0.3 Frequency 0.4 0 Phase (radians) Amplitude 10 5 0.1 0.2 0.3 Frequency 0.4 0.2 0.5 0.3 0.4 0.5 Frequency h. Phase 4 Phase (radians) Amplitude Amplitude 0.1 6 15 10 5 0 2 0 -2 -4 -6 -5 Sample number 0.4 0 -2 0 g. Magnitude 128 127 0.3 2 0.5 20 f. Rectangular pulse 96 0.5 -6 0 64 0.4 -4 -5 32 0.3 e. Phase 4 0 0 0.2 Frequency 6 15 -1 0.1 or d. Unwrapped Magnitude 0 -2 0.5 20 1 0 -6 0 2 2 -4 -5 0 c. Phase 4 15 0 0.1 0.2 0.3 Frequency 0.4 0.5 0 0.1 0.2 Frequency FIGURE 11-3 DFT of a rectangular pulse. A rectangular pulse in one domain corresponds to a sinc function in the other domain. Alternatively, the DTFT can be used to express the frequency spectrum as a fraction of the sampling rate, f: EQUATION 11-2 Equation 11-1 rewritten in terms of the sampling frequency. The parameter, f , is the fraction of the sampling rate, running continiously from 0 to 0.5. To avoid the division by zero, use Mag X(0) ' M . sin(B f M ) Mag X ( f ) ' /0 / 00 sin(B f ) 000 In other words, Eq. 11-1 provides N/2 % 1 samples in the frequency spectrum, while Eq. 11-2 provides the continuous curve that the samples lie on. These The Scientist and Engineer's Guide to Digital Signal Processing equations only provide the magnitude. The phase is determined solely by the left-right positioning of the time domain waveform, as discussed in the last chapter. Notice in Fig. 11-3b that the amplitude of the oscillation does not decay to zero before a frequency of 0.5 is reached. As you should suspect, the waveform continues into the next period where it is aliased. This changes the shape of the frequency domain, an effect that is included in Eqs. 11-1 and 11-2. It is often important to understand what the frequency spectrum looks like when aliasing isn't present. This is because discrete signals are often used to represent or model continuous signals, and continuous signals don't alias. To remove the aliasing in Eqs. 11-1 and 11-2, change the denominators from sin (Bk M/ N ) to BkM /N , and sin (Bf ) to Bf , respectively. Figure 11-4 shows the significance of this. The quantity Bf can only run from 0 to 1.5708, since f can only run from 0 to 0.5. Over this range there isn't much difference between sin (Bf ) and Bf . At zero frequency they have the same value, and at a frequency of 0.5 there is only about a 36% difference. Without aliasing, the curve in Fig. 11-3b would show a slightly lower amplitude near the right side of the graph, and no change near the left side. When the frequency spectrum of the rectangular pulse is not aliased (because the time domain signal is continuous, or because you are ignoring the aliasing), it is of the general form: sin (x)/ x , i.e., a sinc function. For continuous signals, the rectangular pulse and the sinc function are Fourier transform pairs. For discrete signals this is only an approximation, with the error being due to aliasing. The sinc function has an annoying problem at x ' 0 , where sin (x)/x becomes zero divided by zero. This is not a difficult mathematical problem; as x becomes very small, sin (x) approaches the value of x (see Fig. 11-4). 1.6 1.4 FIGURE 11-4 Comparing x and sin(x). The functions: y(x) ' x , and y(x) ' sin(x) are similar for small values of x, and only differ by about 36% at 1.57 (B /2). This describes how aliasing distorts the frequency spectrum of the rectangular pulse from a pure sinc function. y(x) = x 1.2 y(x) 214 1 0.8 y(x) = sin(x) 0.6 0.4 0.2 0 0.0 0.2 0.4 0.6 0.8 x 1.0 1.2 1.4 1.6 Chapter 11- Fourier Transform Pairs 215 This turns the sinc function into x/x , which has a value of one. In other words, as x becomes smaller and smaller, the value of sinc (x) approaches one, which includes sinc (0) ' 1 . Now try to tell your computer this! All it sees is a division by zero, causing it to complain and stop your program. The important point to remember is that your program must include special handling at x ' 0 when calculating the sinc function. A key trait of the sinc function is the location of the zero crossings. These occur at frequencies where an integer number of the sinusoid's cycles fit evenly into the rectangular pulse. For example, if the rectangular pulse is 20 points wide, the first zero in the frequency domain is at the frequency that makes one complete cycle in 20 points. The second zero is at the frequency that makes two complete cycles in 20 points, etc. This can be understood by remembering how the DFT is calculated by correlation. The amplitude of a frequency component is found by multiplying the time domain signal by a sinusoid and adding up the resulting samples. If the time domain waveform is a rectangular pulse of unity amplitude, this is the same as adding the sinusoid's samples that are within the rectangular pulse. If this summation occurs over an integral number of the sinusoid's cycles, the result will be zero. The sinc function is widely used in DSP because it is the Fourier transform pair of a very simple waveform, the rectangular pulse. For example, the sinc function is used in spectral analysis, as discussed in Chapter 9. Consider the analysis of an infinitely long discrete signal. Since the DFT can only work with finite length signals, N samples are selected to represent the longer signal. The key here is that "selecting N samples from a longer signal" is the same as multiplying the longer signal by a rectangular pulse. The ones in the rectangular pulse retain the corresponding samples, while the zeros eliminate them. How does this affect the frequency spectrum of the signal? Multiplying the time domain by a rectangular pulse results in the frequency domain being convolved with a sinc function. This reduces the frequency spectrum's resolution, as previously shown in Fig. 9-5a. Other Transform Pairs Figure 11-5 (a) and (b) show the duality of the above: a rectangular pulse in the frequency domain corresponds to a sinc function (plus aliasing) in the time domain. Including the effects of aliasing, the time domain signal is given by: EQUATION 11-3 Inverse DFT of the rectangular pulse. In the frequency domain, the pulse has an amplitude of one, and runs from sample number 0 through sample number M-1. The parameter N is the length of the DFT, and x[i] is the time domain signal with i running from 0 to N-1. To avoid the division by zero, use x[0] ' (2M& 1)/ N . x [i ] ' 1 sin(2B i (M & 1/2) / N ) N sin(B i / N ) 216 The Scientist and Engineer's Guide to Digital Signal Processing To eliminate the effects of aliasing from this equation, imagine that the frequency domain is so finely sampled that it turns into a continuous curve. This makes the time domain infinitely long with no periodicity. The DTFT is the Fourier transform to use here, resulting in the time domain signal being given by the relation: EQUATION 11-4 Inverse DTFT of the rectangular pulse. In the frequency domain, the pulse has an amplitude of one, and runs from zero frequency to the cutoff frequency, fc , a value between 0 and 0.5. The time domain signal is held in x [i ] with i running from 0 to N-1. To avoid the division by zero, use x[0] ' 2fc . x[i ] ' sin(2B f c i ) iB This equation is very important in DSP, because the rectangular pulse in the frequency domain is the perfect low-pass filter. Therefore, the sinc function described by this equation is the filter kernel for the perfect low-pass filter. This is the basis for a very useful class of digital filters called the windowedsinc filters, described in Chapter 15. Figures (c) & (d) show that a triangular pulse in the time domain coincides with a sinc function squared (plus aliasing) in the frequency domain. This transform pair isn't as important as the reason it is true. A 2M & 1 point triangle in the time domain can be formed by convolving an M point rectangular pulse with itself. Since convolution in the time domain results in multiplication in the frequency domain, convolving a waveform with itself will square the frequency spectrum. Is there a waveform that is its own Fourier Transform? The answer is yes, and there is only one: the Gaussian. Figure (e) shows a Gaussian curve, and (f) shows the corresponding frequency spectrum, also a Gaussian curve. This relationship is only true if you ignore aliasing. The relationship between the standard deviation of the time domain and frequency domain is given by: 2BFf ' 1/Ft . While only one side of a Gaussian is shown in (f), the negative frequencies in the spectrum complete the full curve, with the center of symmetry at zero frequency. Figure (g) shows what can be called a Gaussian burst. It is formed by multiplying a sine wave by a Gaussian. For example, (g) is a sine wave multiplied by the same Gaussian shown in (e). The corresponding frequency domain is a Gaussian centered somewhere other than zero frequency. As before, this transform pair is not as important as the reason it is true. Since the time domain signal is the multiplication of two signals, the frequency domain will be the convolution of the two frequency spectra. The frequency spectrum of the sine wave is a delta function centered at the frequency of the sine wave. The frequency spectrum of a Gaussian is a Gaussian centered at zero frequency. Convolving the two produces a Gaussian centered at the frequency of the sine wave. This should look familiar; it is identical to the procedure of amplitude modulation described in the last chapter. Chapter 11- Fourier Transform Pairs Time Domain Frequency Domain 15 3 a. Sinc b. Rectangular pulse 10 Amplitude 2 Amplitude 217 1 0 5 0 -1 -5 0 16 32 48 64 80 Sample number 96 112 128 127 0 2 0.1 0.2 0.3 0.4 0.5 0.2 0.3 0.4 0.5 0.2 0.3 0.4 0.5 0.2 0.3 0.4 0.5 Frequency 15 c. Triangle d. Sinc squared 1 Amplitude Amplitude 10 0 5 0 -1 -5 0 16 32 48 64 80 Sample number 96 112 128 127 0 2 0.1 Frequency 15 e. Gaussian f. Gaussian 1 Amplitude Amplitude 10 0 5 0 -1 -5 0 16 32 48 64 80 Sample number 96 112 128 127 0 3 Frequency 15 g. Gaussian burst h. Gaussian 2 10 Amplitude Amplitude 0.1 1 0 5 0 -1 -5 -2 0 16 32 48 64 80 Sample number 96 112 128 127 0 FIGURE 11-5 Common transform pairs. 0.1 Frequency 218 The Scientist and Engineer's Guide to Digital Signal Processing Gibbs Effect Figure 11-6 shows a time domain signal being synthesized from sinusoids. The signal being reconstructed is shown in the last graph, (h). Since this signal is 1024 points long, there will be 513 individual frequencies needed for a complete reconstruction. Figures (a) through (g) show what the reconstructed signal looks like if only some of these frequencies are used. For example, (f) shows a reconstructed signal using frequencies 0 through 100. This signal was created by taking the DFT of the signal in (h), setting frequencies 101 through 512 to a value of zero, and then using the Inverse DFT to find the resulting time domain signal. As more frequencies are added to the reconstruction, the signal becomes closer to the final solution. The interesting thing is how the final solution is approached at the edges in the signal. There are three sharp edges in (h). Two are the edges of the rectangular pulse. The third is between sample numbers 1023 and 0, since the DFT views the time domain as periodic. When only some of the frequencies are used in the reconstruction, each edge shows overshoot and ringing (decaying oscillations). This overshoot and ringing is known as the Gibbs effect, after the mathematical physicist Josiah Gibbs, who explained the phenomenon in 1899. Look closely at the overshoot in (e), (f), and (g). As more sinusoids are added, the width of the overshoot decreases; however, the amplitude of the overshoot remains about the same, roughly 9 percent. With discrete signals this is not a problem; the overshoot is eliminated when the last frequency is added. However, the reconstruction of continuous signals cannot be explained so easily. An infinite number of sinusoids must be added to synthesize a continuous signal. The problem is, the amplitude of the overshoot does not decrease as the number of sinusoids approaches infinity, it stays about the same 9%. Given this situation (and other arguments), it is reasonable to question if a summation of continuous sinusoids can reconstruct an edge. Remember the squabble between Lagrange and Fourier? The critical factor in resolving this puzzle is that the width of the overshoot becomes smaller as more sinusoids are included. The overshoot is still present with an infinite number of sinusoids, but it has zero width. Exactly at the discontinuity the value of the reconstructed signal converges to the midpoint of the step. As shown by Gibbs, the summation converges to the signal in the sense that the error between the two has zero energy. Problems related to the Gibbs effect are frequently encountered in DSP. For example, a low-pass filter is a truncation of the higher frequencies, resulting in overshoot and ringing at the edges in the time domain. Another common procedure is to truncate the ends of a time domain signal to prevent them from extending into neighboring periods. By duality, this distorts the edges in the frequency domain. These issues will resurface in future chapters on filter design. Chapter 11- Fourier Transform Pairs 2 2 b. Frequencies: 0 & 1 1 Amplitude Amplitude a. Frequencies: 0 0 -1 1 0 -1 0 256 512 Sample number 768 1024 1023 0 2 1 Amplitude Amplitude 512 Sample number 768 1024 1023 768 1024 1023 768 1024 1023 768 1024 1023 d. Frequencies: 0 to 10 0 -1 1 0 -1 0 256 512 Sample number 768 1024 1023 0 2 256 512 Sample number 2 e. Frequencies: 0 to 30 f. Frequencies: 0 to 100 1 Amplitude Amplitude 256 2 c. Frequencies: 0 to 3 0 -1 1 0 -1 0 256 512 Sample number 768 1024 1023 0 2 256 512 Sample number 2 g. Frequencies: 0 to 300 h. Frequencies: 0 to 512 1 Amplitude Amplitude 219 0 -1 1 0 -1 0 256 512 Sample number 768 1024 1023 0 FIGURE 11-6. The Gibbs effect. 256 512 Sample number 220 The Scientist and Engineer's Guide to Digital Signal Processing Harmonics If a signal is periodic with frequency f, the only frequencies composing the signal are integer multiples of f, i.e., f, 2f, 3f, 4f, etc. These frequencies are called harmonics. The first harmonic is f, the second harmonic is 2f, the third harmonic is 3f, and so forth. The first harmonic (i.e., f) is also given a special name, the fundamental frequency. Figure 11-7 shows an Time Domain Frequency Domain 2 700 a. Sine wave 1 500 Amplitude Amplitude b. Fundamental 600 0 400 300 200 -1 100 -2 0 0 128 256 384 512 640 Sample number 768 896 1024 1023 0 0.04 0.06 0.08 0.1 0.06 0.08 0.1 0.06 0.08 0.1 Frequency 700 2 c. Asymmetrical distortion d. Fundamental plus even and odd harmonics 600 1 500 Amplitude Amplitude 0.02 0 400 300 200 -1 100 0 -2 0 128 256 384 512 640 Sample number 768 896 1024 1023 0 0.04 Frequency 700 2 e. Symmetrical distortion f. Fundamental plus odd harmonics 600 1 500 Amplitude Amplitude 0.02 0 400 300 200 -1 100 0 -2 0 128 256 384 512 640 Sample number 768 896 1024 1023 0 0.02 0.04 Frequency FIGURE 11-7 Example of harmonics. Asymmetrical distortion, shown in (c), results in even and odd harmonics, (d), while symmetrical distortion, shown in (e), produces only even harmonics, (f). Chapter 11- Fourier Transform Pairs Time Domain Frequency Domain 2 700 a. Distorted sine wave b. Frequency spectrum 600 500 Amplitude 1 0 -1 400 300 200 100 0 -100 -2 0 20 40 60 Sample number 80 100 0 0.1 0.2 0.3 Frequency 0.4 0.5 100 105 FIGURE 11-8 Harmonic aliasing. Figures (a) and (b) show a distorted sine wave and its frequency spectrum, respectively. Harmonics with a frequency greater than 0.5 will become aliased to a frequency between 0 and 0.5. Figure (c) displays the same frequency spectrum on a logarithmic scale, revealing many aliased peaks with very low amplitude. c. Frequency spectrum (Log scale) 804 10 603 10 402 10 Amplitude Amplitude 221 201 10 1000 -20-1 10 -40-2 10 -60-3 10 -80-4 10 -100 10-5 0 0.1 0.2 0.3 Frequency 0.4 0.5 example. Figure (a) is a pure sine wave, and (b) is its DFT, a single peak. In (c), the sine wave has been distorted by poking in the tops of the peaks. Figure (d) shows the result of this distortion in the frequency domain. Because the distorted signal is periodic with the same frequency as the original sine wave, the frequency domain is composed of the original peak plus harmonics. Harmonics can be of any amplitude; however, they usually become smaller as they increase in frequency. As with any signal, sharp edges result in higher frequencies. For example, consider a common TTL logic gate generating a 1 kHz square wave. The edges rise in a few nanoseconds, resulting in harmonics being generated to nearly 100 MHz, the ten-thousandth harmonic! Figure (e) demonstrates a subtlety of harmonic analysis. If the signal is symmetrical around a horizontal axis, i.e., the top lobes are mirror images of the bottom lobes, all of the even harmonics will have a value of zero. As shown in (f), the only frequencies contained in the signal are the fundamental, the third harmonic, the fifth harmonic, etc. All continuous periodic signals can be represented as a summation of harmonics, just as described. Discrete periodic signals have a problem that disrupts this simple relation. As you might have guessed, the problem is aliasing. Figure 11-8a shows a sine wave distorted in the same manner as before, by poking in the tops of the peaks. This waveform looks much less 222 The Scientist and Engineer's Guide to Digital Signal Processing regular and smooth than in the previous example because the sine wave is at a much higher frequency, resulting in fewer samples per cycle. Figure (b) shows the frequency spectrum of this signal. As you would expect, you can identify the fundamental and harmonics. This example shows that harmonics can extend to frequencies greater than 0.5 of the sampling frequency, and will be aliased to frequencies somewhere between 0 and 0.5. You don't notice them in (b) because their amplitudes are too low. Figure (c) shows the frequency spectrum plotted on a logarithmic scale to reveal these low amplitude aliased peaks. At first glance, this spectrum looks like random noise. It isn't; this is a result of the many harmonics overlapping as they are aliased. It is important to understand that this example involves distorting a signal after it has been digitally represented. If this distortion occurred in an analog signal, you would remove the offending harmonics with an antialias filter before digitization. Harmonic aliasing is only a problem when nonlinear operations are performed directly on a discrete signal. Even then, the amplitude of these aliased harmonics is often low enough that they can be ignored. The concept of harmonics is also useful for another reason: it explains why the DFT views the time and frequency domains as periodic. In the frequency domain, an N point DFT consists of N/2+1 equally spaced frequencies. You can view the frequencies between these samples as (1) having a value of zero, or (2) not existing. Either way they don't contribute to the synthesis of the time domain signal. In other words, a discrete frequency spectrum consists of harmonics, rather than a continuous range of frequencies. This requires the time domain to be periodic with a frequency equal to the lowest sinusoid in the frequency domain, i.e., the fundamental frequency. Neglecting the DC value, the lowest frequency represented in the frequency domain makes one complete cycle every N samples, resulting in the time domain being periodic with a period of N. In other words, if one domain is discrete, the other domain must be periodic, and vice versa. This holds for all four members of the Fourier transform family. Since the DFT views both domains as discrete, it must also view both domains as periodic. The samples in each domain represent harmonics of the periodicity of the opposite domain. Chirp Signals Chirp signals are an ingenious way of handling a practical problem in echo location systems, such as radar and sonar. Figure 11-9 shows the frequency response of the chirp system. The magnitude has a constant value of one, while the phase is a parabola: EQUATION 11-7 Phase of the chirp system. Phase X [k ] ' "k % $k 2 Chapter 11- Fourier Transform Pairs 2 223 50 a. Chirp magnitude b. Chirp phase Phase (radians) Amplitude 0 1 0 -50 -100 -150 -200 -1 0 0.1 0.2 0.3 0.4 Frequency 0.5 0 0.1 0.2 0.3 Frequency 0.4 0.5 FIGURE 11-9 Frequency response of the chirp system. The magnitude is a constant, while the phase is a parabola. The parameter " introduces a linear slope in the phase, that is, it simply shifts the impulse response left or right as desired. The parameter $ controls the curvature of the phase. These two parameters must be chosen such that the phase at frequency 0.5 (i.e. k = N/2) is a multiple of 2B. Remember, whenever the phase is directly manipulated, frequency 0 and 0.5 must both have a phase of zero (or a multiple of 2B, which is the same thing). Figure 11-10 shows an impulse entering a chirp system, and the impulse response exiting the system. The impulse response is an oscillatory burst that starts at a low frequency and changes to a high frequency as time progresses. This is called a chirp signal for a very simple reason: it sounds like the chirp of a bird when played through a speaker. The key feature of the chirp system is that it is completely reversible. If you run the chirp signal through an antichirp system, the signal is again made into an impulse. This requires the antichirp system to have a magnitude of one, and the opposite phase of the chirp system. As discussed in the last 1.5 1.5 b. Impulse Response (chirp signal) a. Impulse 1 Chirp System 0.5 0 Amplitude Amplitude 1 0.5 0 -0.5 -0.5 0 20 40 60 80 Sample number 100 120 0 20 40 60 80 Sample number FIGURE 11-10 The chirp system. The impulse response of a chirp system is a chirp signal. 100 120 224 The Scientist and Engineer's Guide to Digital Signal Processing chapter, this means that the impulse response of the antichirp system is found by preforming a left-for-right flip of the chirp system's impulse response. Interesting, but what is it good for? Consider how a radar system operates. A short burst of radio frequency energy is emitted from a directional antenna. Aircraft and other objects reflect some of this energy back to a radio receiver located next to the transmitter. Since radio waves travel at a constant rate, the elapsed time between the transmitted and received signals provides the distance to the target. This brings up the first requirement for the pulse: it needs to be as short as possible. For example, a 1 microsecond pulse provides a radio burst about 300 meters long. This means that the distance information we obtain with the system will have a resolution of about this same length. If we want better distance resolution, we need a shorter pulse. The second requirement is obvious: if we want to detect objects farther away, you need more energy in your pulse. Unfortunately, more energy and shorter pulse are conflicting requirements. The electrical power needed to provide a pulse is equal to the energy of the pulse divided by the pulse length. Requiring both more energy and a shorter pulse makes electrical power handling a limiting factor in the system. The output stage of a radio transmitter can only handle so much power without destroying itself. Chirp signals provide a way of breaking this limitation. Before the impulse reaches the final stage of the radio transmitter, it is passed through a chirp system. Instead of bouncing an impulse off the target aircraft, a chirp signal is used. After the chirp echo is received, the signal is passed through an antichirp system, restoring the signal to an impulse. This allows the portions of the system that measure distance to see short pulses, while the power handling circuits see long duration signals. This type of waveshaping is a fundamental part of modern radar systems. CHAPTER 12 The Fast Fourier Transform There are several ways to calculate the Discrete Fourier Transform (DFT), such as solving simultaneous linear equations or the correlation method described in Chapter 8. The Fast Fourier Transform (FFT) is another method for calculating the DFT. While it produces the same result as the other approaches, it is incredibly more efficient, often reducing the computation time by hundreds. This is the same improvement as flying in a jet aircraft versus walking! If the FFT were not available, many of the techniques described in this book would not be practical. While the FFT only requires a few dozen lines of code, it is one of the most complicated algorithms in DSP. But don't despair! You can easily use published FFT routines without fully understanding the internal workings. Real DFT Using the Complex DFT J.W. Cooley and J.W. Tukey are given credit for bringing the FFT to the world in their paper: "An algorithm for the machine calculation of complex Fourier Series," Mathematics Computation, Vol. 19, 1965, pp 297-301. In retrospect, others had discovered the technique many years before. For instance, the great German mathematician Karl Friedrich Gauss (1777-1855) had used the method more than a century earlier. This early work was largely forgotten because it lacked the tool to make it practical: the digital computer. Cooley and Tukey are honored because they discovered the FFT at the right time, the beginning of the computer revolution. The FFT is based on the complex DFT, a more sophisticated version of the real DFT discussed in the last four chapters. These transforms are named for the way each represents data, that is, using complex numbers or using real numbers. The term complex does not mean that this representation is difficult or complicated, but that a specific type of mathematics is used. Complex mathematics often is difficult and complicated, but that isn't where the name comes from. Chapter 29 discusses the complex DFT and provides the background needed to understand the details of the FFT algorithm. The 225 226 The Scientist and Engineer's Guide to Digital Signal Processing Real DFT Time Domain Frequency Domain Time Domain Signal 0 Real Part N-1 0 N/2 Imaginary Part 0 N/2 Complex DFT Time Domain Frequency Domain Real Part 0 Real Part N-1 Imaginary Part 0 0 N/2 N-1 N/2 N-1 Imaginary Part N-1 0 FIGURE 12-1 Comparing the real and complex DFTs. The real DFT takes an N point time domain signal and creates two N/2 % 1 point frequency domain signals. The complex DFT takes two N point time domain signals and creates two N point frequency domain signals. The crosshatched regions shows the values common to the two transforms. topic of this chapter is simpler: how to use the FFT to calculate the real DFT, without drowning in a mire of advanced mathematics. Since the FFT is an algorithm for calculating the complex DFT, it is important to understand how to transfer real DFT data into and out of the complex DFT format. Figure 12-1 compares how the real DFT and the complex DFT store data. The real DFT transforms an N point time domain signal into two N /2 % 1 point frequency domain signals. The time domain signal is called just that: the time domain signal. The two signals in the frequency domain are called the real part and the imaginary part, holding the amplitudes of the cosine waves and sine waves, respectively. This should be very familiar from past chapters. In comparison, the complex DFT transforms two N point time domain signals into two N point frequency domain signals. The two time domain signals are called the real part and the imaginary part, just as are the frequency domain signals. In spite of their names, all of the values in these arrays are just ordinary numbers. (If you are familiar with complex numbers: the j's are not included in the array values; they are a part of the mathematics. Recall that the operator, Im( ), returns a real number). Chapter 12- The Fast Fourier Transform 227 Suppose you have an N point signal, and need to calculate the real DFT by means of the Complex DFT (such as by using the FFT algorithm). First, move the N point signal into the real part of the complex DFT's time domain, and then set all of the samples in the imaginary part to zero. Calculation of the complex DFT results in a real and an imaginary signal in the frequency domain, each composed of N points. Samples 0 through N/2 of these signals correspond to the real DFT's spectrum. As discussed in Chapter 10, the DFT's frequency domain is periodic when the negative frequencies are included (see Fig. 10-9). The choice of a single period is arbitrary; it can be chosen between -1.0 and 0, -0.5 and 0.5, 0 and 1.0, or any other one unit interval referenced to the sampling rate. The complex DFT's frequency spectrum includes the negative frequencies in the 0 to 1.0 arrangement. In other words, one full period stretches from sample 0 to sample N& 1 , corresponding with 0 to 1.0 times the sampling rate. The positive frequencies sit between sample 0 and N/2 , corresponding with 0 to 0.5. The other samples, between N/2 % 1 and N& 1 , contain the negative frequency values (which are usually ignored). Calculating a real Inverse DFT using a complex Inverse DFT is slightly harder. This is because you need to insure that the negative frequencies are loaded in the proper format. Remember, points 0 through N/2 in the complex DFT are the same as in the real DFT, for both the real and the imaginary parts. For the real part, point N/2 % 1 is the same as point N/2 & 1 , point N/2 % 2 is the same as point N/2 & 2 , etc. This continues to point N& 1 being the same as point 1. The same basic pattern is used for the imaginary part, except the sign is changed. That is, point N/2 % 1 is the negative of point N/2 & 1 , point N/2 % 2 is the negative of point N/2 & 2 , etc. Notice that samples 0 and N/2 do not have a matching point in this duplication scheme. Use Fig. 10-9 as a guide to understanding this symmetry. In practice, you load the real DFT's frequency spectrum into samples 0 to N/2 of the complex DFT's arrays, and then use a subroutine to generate the negative frequencies between samples N/2 % 1 and N& 1 . Table 12-1 shows such a program. To check that the proper symmetry is present, after taking the inverse FFT, look at the imaginary part of the time domain. It will contain all zeros if everything is correct (except for a few parts-permillion of noise, using single precision calculations). 6000 'NEGATIVE FREQUENCY GENERATION 6010 'This subroutine creates the complex frequency domain from the real frequency domain. 6020 'Upon entry to this subroutine, N% contains the number of points in the signals, and 6030 'REX[ ] and IMX[ ] contain the real frequency domain in samples 0 to N%/2. 6040 'On return, REX[ ] and IMX[ ] contain the complex frequency domain in samples 0 to N%-1. 6050 ' 6060 FOR K% = (N%/2+1) TO (N%-1) 6070 REX[K%] = REX[N%-K%] 6080 IMX[K%] = -IMX[N%-K%] 6090 NEXT K% 6100 ' 6110 RETURN TABLE 12-1 228 The Scientist and Engineer's Guide to Digital Signal Processing How the FFT works The FFT is a complicated algorithm, and its details are usually left to those that specialize in such things. This section describes the general operation of the FFT, but skirts a key issue: the use of complex numbers. If you have a background in complex mathematics, you can read between the lines to understand the true nature of the algorithm. Don't worry if the details elude you; few scientists and engineers that use the FFT could write the program from scratch. In complex notation, the time and frequency domains each contain one signal made up of N complex points. Each of these complex points is composed of two numbers, the real part and the imaginary part. For example, when we talk about complex sample X[42] , it refers to the combination of Re X[42] and Im X[42] . In other words, each complex variable holds two numbers. When two complex variables are multiplied, the four individual components must be combined to form the two components of the product (such as in Eq. 9-1). The following discussion on "How the FFT works" uses this jargon of complex notation. That is, the singular terms: signal, point, sample, and value, refer to the combination of the real part and the imaginary part. The FFT operates by decomposing an N point time domain signal into N time domain signals each composed of a single point. The second step is to calculate the N frequency spectra corresponding to these N time domain signals. Lastly, the N spectra are synthesized into a single frequency spectrum. Figure 12-2 shows an example of the time domain decomposition used in the FFT. In this example, a 16 point signal is decomposed through four 1 signal of 16 points 0 1 2 2 signals of 8 points 0 2 4 6 8 10 12 14 3 4 5 6 7 8 9 10 11 12 13 14 15 1 3 5 7 9 11 13 15 4 signals of 4 points 0 4 8 12 2 6 10 14 1 5 9 13 3 7 11 15 8 signals of 2 points 0 8 4 12 2 10 1 9 3 11 16 signals of 1 point 0 4 12 2 10 6 14 1 8 6 14 9 5 13 7 15 5 13 3 11 7 15 FIGURE 12-2 The FFT decomposition. An N point signal is decomposed into N signals each containing a single point. Each stage uses an interlace decomposition, separating the even and odd numbered samples. Chapter 12- The Fast Fourier Transform Sample numbers in normal order Sample numbers after bit reversal Decimal Binary Decimal Binary 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 0 8 4 12 2 10 6 14 1 9 5 13 3 11 7 15 229 0000 1000 0100 1100 0010 1010 0100 1110 0001 1001 0101 1101 0011 1011 0111 1111 FIGURE 12-3 The FFT bit reversal sorting. The FFT time domain decomposition can be implemented by sorting the samples according to bit reversed order. separate stages. The first stage breaks the 16 point signal into two signals each consisting of 8 points. The second stage decomposes the data into four signals of 4 points. This pattern continues until there are N signals composed of a single point. An interlaced decomposition is used each time a signal is broken in two, that is, the signal is separated into its even and odd numbered samples. The best way to understand this is by inspecting Fig. 12-2 until you grasp the pattern. There are Log2 N stages required in this decomposition, i.e., a 16 point signal (24) requires 4 stages, a 512 point signal (27) requires 7 stages, a 4096 point signal (212) requires 12 stages, etc. Remember this value, Log2 N ; it will be referenced many times in this chapter. Now that you understand the structure of the decomposition, it can be greatly simplified. The decomposition is nothing more than a reordering of the samples in the signal. Figure 12-3 shows the rearrangement pattern required. On the left, the sample numbers of the original signal are listed along with their binary equivalents. On the right, the rearranged sample numbers are listed, also along with their binary equivalents. The important idea is that the binary numbers are the reversals of each other. For example, sample 3 (0011) is exchanged with sample number 12 (1100). Likewise, sample number 14 (1110) is swapped with sample number 7 (0111), and so forth. The FFT time domain decomposition is usually carried out by a bit reversal sorting algorithm. This involves rearranging the order of the N time domain samples by counting in binary with the bits flipped left-for-right (such as in the far right column in Fig. 12-3). 230 The Scientist and Engineer's Guide to Digital Signal Processing The next step in the FFT algorithm is to find the frequency spectra of the 1 point time domain signals. Nothing could be easier; the frequency spectrum of a 1 point signal is equal to itself. This means that nothing is required to do this step. Although there is no work involved, don't forget that each of the 1 point signals is now a frequency spectrum, and not a time domain signal. The last step in the FFT is to combine the N frequency spectra in the exact reverse order that the time domain decomposition took place. This is where the algorithm gets messy. Unfortunately, the bit reversal shortcut is not applicable, and we must go back one stage at a time. In the first stage, 16 frequency spectra (1 point each) are synthesized into 8 frequency spectra (2 points each). In the second stage, the 8 frequency spectra (2 points each) are synthesized into 4 frequency spectra (4 points each), and so on. The last stage results in the output of the FFT, a 16 point frequency spectrum. Figure 12-4 shows how two frequency spectra, each composed of 4 points, are combined into a single frequency spectrum of 8 points. This synthesis must undo the interlaced decomposition done in the time domain. In other words, the frequency domain operation must correspond to the time domain procedure of combining two 4 point signals by interlacing. Consider two time domain signals, abcd and efgh. An 8 point time domain signal can be formed by two steps: dilute each 4 point signal with zeros to make it an Time Domain Frequency Domain a b c d A B C D a 0 b 0 c 0 d 0 AB C DA B C D e f g h E F G H × sinusoid 0 e 0 f 0 g 0 h E F G H E F G H FIGURE 12-4 The FFT synthesis. When a time domain signal is diluted with zeros, the frequency domain is duplicated. If the time domain signal is also shifted by one sample during the dilution, the spectrum will additionally be multiplied by a sinusoid. Chapter 12- The Fast Fourier Transform Odd- Four Point Frequency Spectrum xS xS xS xS + + + + 231 Even- Four Point Frequency Spectrum FIGURE 12-5 FFT synthesis flow diagram. This shows the method of combining two 4 point frequency spectra into a single 8 point frequency spectrum. The ×S operation means that the signal is multiplied by a sinusoid with an appropriately selected frequency. + + + + Eight Point Frequency Spectrum 8 point signal, and then add the signals together. That is, abcd becomes a0b0c0d0, and efgh becomes 0e0f0g0h. Adding these two 8 point signals produces aebfcgdh. As shown in Fig. 12-4, diluting the time domain with zeros corresponds to a duplication of the frequency spectrum. Therefore, the frequency spectra are combined in the FFT by duplicating them, and then adding the duplicated spectra together. In order to match up when added, the two time domain signals are diluted with zeros in a slightly different way. In one signal, the odd points are zero, while in the other signal, the even points are zero. In other words, one of the time domain signals (0e0f0g0h in Fig. 12-4) is shifted to the right by one sample. This time domain shift corresponds to multiplying the spectrum by a sinusoid. To see this, recall that a shift in the time domain is equivalent to convolving the signal with a shifted delta function. This multiplies the signal's spectrum with the spectrum of the shifted delta function. The spectrum of a shifted delta function is a sinusoid (see Fig 11-2). Figure 12-5 shows a flow diagram for combining two 4 point spectra into a single 8 point spectrum. To reduce the situation even more, notice that Fig. 125 is formed from the basic pattern in Fig 12-6 repeated over and over. 2 point input FIGURE 12-6 The FFT butterfly. This is the basic calculation element in the FFT, taking two complex points and converting them into two other complex points. xS 2 point output 232 The Scientist and Engineer's Guide to Digital Signal Processing This simple flow diagram is called a butterfly due to its winged appearance. The butterfly is the basic computational element of the FFT, transforming two complex points into two other complex points. Figure 12-7 shows the structure of the entire FFT. The time domain decomposition is accomplished with a bit reversal sorting algorithm. Transforming the decomposed data into the frequency domain involves nothing and therefore does not appear in the figure. The frequency domain synthesis requires three loops. The outer loop runs through the Log2 N stages (i.e., each level in Fig. 12-2, starting from the bottom and moving to the top). The middle loop moves through each of the individual frequency spectra in the stage being worked on (i.e., each of the boxes on any one level in Fig. 12-2). The innermost loop uses the butterfly to calculate the points in each frequency spectra (i.e., looping through the samples inside any one box in Fig. 12-2). The overhead boxes in Fig. 12-7 determine the beginning and ending indexes for the loops, as well as calculating the sinusoids needed in the butterflies. Now we come to the heart of this chapter, the actual FFT programs. Time Domain Data FIGURE 12-7 Flow diagram of the FFT. This is based on three steps: (1) decompose an N point time domain signal into N signals each containing a single point, (2) find the spectrum of each of the N point signals (nothing required), and (3) synthesize the N frequency spectra into a single frequency spectrum. Bit Reversal Data Sorting Time Domain Decomposition Loop for each Butterfly Loop for Leach sub-DFT Loop for Log 2N stages Overhead Overhead Butterfly Calculation Frequency Domain Data Frequency Domain Synthesis Chapter 12- The Fast Fourier Transform 233 FFT Programs As discussed in Chapter 8, the real DFT can be calculated by correlating the time domain signal with sine and cosine waves (see Table 8-2). Table 12-2 shows a program to calculate the complex DFT by the same method. In an apples-to-apples comparison, this is the program that the FFT improves upon. Tables 12-3 and 12-4 show two different FFT programs, one in FORTRAN and one in BASIC. First we will look at the BASIC routine in Table 12-4. This subroutine produces exactly the same output as the correlation technique in Table 12-2, except it does it much faster. The block diagram in Fig. 12-7 can be used to identify the different sections of this program. Data are passed to this FFT subroutine in the arrays: REX[ ] and IMX[ ], each running from sample 0 to N& 1 . Upon return from the subroutine, REX[ ] and IMX[ ] are overwritten with the frequency domain data. This is another way that the FFT is highly optimized; the same arrays are used for the input, intermediate storage, and output. This efficient use of memory is important for designing fast hardware to calculate the FFT. The term in-place computation is used to describe this memory usage. While all FFT programs produce the same numerical result, there are subtle variations in programming that you need to look out for. Several of these 5000 'COMPLEX DFT BY CORRELATION 5010 'Upon entry, N% contains the number of points in the DFT, and 5020 'XR[ ] and XI[ ] contain the real and imaginary parts of the time domain. 5030 'Upon return, REX[ ] and IMX[ ] contain the frequency domain data. 5040 'All signals run from 0 to N%-1. 5050 ' 5060 PI = 3.14159265 'Set constants 5070 ' 5080 FOR K% = 0 TO N%-1 'Zero REX[ ] and IMX[ ], so they can be used 5090 REX[K%] = 0 'as accumulators during the correlation 5100 IMX[K%] = 0 5110 NEXT K% 5120 ' 5130 FOR K% = 0 TO N%-1 'Loop for each value in frequency domain 5140 FOR I% = 0 TO N%-1 'Correlate with the complex sinusoid, SR & SI 5150 ' 5160 SR = COS(2*PI*K%*I%/N%) 'Calculate complex sinusoid 5170 SI = -SIN(2*PI*K%*I%/N%) 5180 REX[K%] = REX[K%] + XR[I%]*SR - XI[I%]*SI 5190 IMX[K%] = IMX[K%] + XR[I%]*SI + XI[I%]*SR 5200 ' 5210 NEXT I% 5220 NEXT K% 5230 ' 5240 RETURN TABLE 12-2 234 The Scientist and Engineer's Guide to Digital Signal Processing of these differences are illustrated by the FORTRAN program listed in Table 12-3. This program uses an algorithm called decimation in frequency, while the previously described algorithm is called decimation in time. In a decimation in frequency algorithm, the bit reversal sorting is done after the three nested loops. There are also FFT routines that completely eliminate the bit reversal sorting. None of these variations significantly improve the performance of the FFT, and you shouldn't worry about which one you are using. The important differences between FFT algorithms concern how data are passed to and from the subroutines. In the BASIC program, data enter and leave the subroutine in the arrays REX[ ] and IMX[ ], with the samples running from index 0 to N& 1 . In the FORTRAN program, data are passed in the complex array X( ) , with the samples running from 1 to N. Since this is an array of complex variables, each sample in X( ) consists of two numbers, a real part and an imaginary part. The length of the DFT must also be passed to these subroutines. In the BASIC program, the variable N% is used for this purpose. In comparison, the FORTRAN program uses the variable M, which is defined to equal Log2 N . For instance, M will be 10 20 30 40 50 SUBROUTINE FFT(X,M) COMPLEX X(4096),U,S,T PI=3.14159265 N=2**M DO 20 L=1,M LE=2**(M+1-L) LE2=LE/2 U=(1.0,0.0) S=CMPLX(COS(PI/FLOAT(LE2)),-SIN(PI/FLOAT(LE2))) DO 20 J=1,LE2 DO 10 I=J,N,LE IP=I+LE2 T=X(I)+X(IP) X(IP)=(X(I)-X(IP))*U X(I)=T U=U*S TABLE 12-3 ND2=N/2 The Fast Fourier Transform in FORTRAN. NM1=N-1 Data are passed to this subroutine in the J=1 variables X( ) and M. The integer, M, is the DO 50 I=1,NM1 base two logarithm of the length of the DFT, IF(I.GE.J) GO TO 30 i.e., M = 8 for a 256 point DFT, M = 12 for a T=X(J) 4096 point DFT, etc. The complex array, X( ), X(J)=X(I) holds the time domain data upon entering the DFT. Upon return from this subroutine, X( ) is X(I)=T overwritten with the frequency domain data. K=ND2 Take note: this subroutine requires that the IF(K.GE.J) GO TO 50 input and output signals run from X(1) through J=J-K X(N), rather than the customary X(0) through K=K/2 X(N-1). GO TO 40 J=J+K RETURN END Chapter 12- The Fast Fourier Transform 1000 'THE FAST FOURIER TRANSFORM 1010 'Upon entry, N% contains the number of points in the DFT, REX[ ] and 1020 'IMX[ ] contain the real and imaginary parts of the input. Upon return, 1030 'REX[ ] and IMX[ ] contain the DFT output. All signals run from 0 to N%-1. 1040 ' 1050 PI = 3.14159265 'Set constants 1060 NM1% = N%-1 1070 ND2% = N%/2 1080 M% = CINT(LOG(N%)/LOG(2)) 1090 J% = ND2% 1100 ' 1110 FOR I% = 1 TO N%-2 'Bit reversal sorting 1120 IF I% >= J% THEN GOTO 1190 1130 TR = REX[J%] 1140 TI = IMX[J%] 1150 REX[J%] = REX[I%] 1160 IMX[J%] = IMX[I%] 1170 REX[I%] = TR 1180 IMX[I%] = TI 1190 K% = ND2% 1200 IF K% > J% THEN GOTO 1240 1210 J% = J%-K% 1220 K% = K%/2 1230 GOTO 1200 1240 J% = J%+K% 1250 NEXT I% 1260 ' 1270 FOR L% = 1 TO M% 'Loop for each stage 1280 LE% = CINT(2^L%) 1290 LE2% = LE%/2 1300 UR = 1 1310 UI = 0 1320 SR = COS(PI/LE2%) 'Calculate sine & cosine values 1330 SI = -SIN(PI/LE2%) 1340 FOR J% = 1 TO LE2% 'Loop for each sub DFT 1350 JM1% = J%-1 1360 FOR I% = JM1% TO NM1% STEP LE% 'Loop for each butterfly 1370 IP% = I%+LE2% 1380 TR = REX[IP%]*UR - IMX[IP%]*UI 'Butterfly calculation 1390 TI = REX[IP%]*UI + IMX[IP%]*UR 1400 REX[IP%] = REX[I%]-TR 1410 IMX[IP%] = IMX[I%]-TI 1420 REX[I%] = REX[I%]+TR 1430 IMX[I%] = IMX[I%]+TI 1440 NEXT I% 1450 TR = UR 1460 UR = TR*SR - UI*SI 1470 UI = TR*SI + UI*SR 1480 NEXT J% 1490 NEXT L% 1500 ' 1510 RETURN TABLE 12-4 The Fast Fourier Transform in BASIC. 235 236 The Scientist and Engineer's Guide to Digital Signal Processing 8 for a 256 point DFT, 12 for a 4096 point DFT, etc. The point is, the programmer who writes an FFT subroutine has many options for interfacing with the host program. Arrays that run from 1 to N, such as in the FORTRAN program, are especially aggravating. Most of the DSP literature (including this book) explains algorithms assuming the arrays run from sample 0 to N& 1 . For instance, if the arrays run from 1 to N, the symmetry in the frequency domain is around points 1 and N/2 % 1 , rather than points 0 and N/2 , Using the complex DFT to calculate the real DFT has another interesting advantage. The complex DFT is more symmetrical between the time and frequency domains than the real DFT. That is, the duality is stronger. Among other things, this means that the Inverse DFT is nearly identical to the Forward DFT. In fact, the easiest way to calculate an Inverse FFT is to calculate a Forward FFT, and then adjust the data. Table 12-5 shows a subroutine for calculating the Inverse FFT in this manner. Suppose you copy one of these FFT algorithms into your computer program and start it running. How do you know if it is operating properly? Two tricks are commonly used for debugging. First, start with some arbitrary time domain signal, such as from a random number generator, and run it through the FFT. Next, run the resultant frequency spectrum through the Inverse FFT and compare the result with the original signal. They should be identical, except round-off noise (a few parts-per-million for single precision). The second test of proper operation is that the signals have the correct symmetry. When the imaginary part of the time domain signal is composed of all zeros (the normal case), the frequency domain of the complex DFT will be symmetrical around samples 0 and N/2 , as previously described. 2000 'INVERSE FAST FOURIER TRANSFORM SUBROUTINE 2010 'Upon entry, N% contains the number of points in the IDFT, REX[ ] and 2020 'IMX[ ] contain the real and imaginary parts of the complex frequency domain. 2030 'Upon return, REX[ ] and IMX[ ] contain the complex time domain. 2040 'All signals run from 0 to N%-1. 2050 ' 2060 FOR K% = 0 TO N%-1 'Change the sign of IMX[ ] 2070 IMX[K%] = -IMX[K%] 2080 NEXT K% 2090 ' 2100 GOSUB 1000 'Calculate forward FFT (Table 12-3) 2110 ' 2120 FOR I% = 0 TO N%-1 'Divide the time domain by N% and 2130 REX[I%] = REX[I%]/N% 'change the sign of IMX[ ] 2140 IMX[I%] = -IMX[I%]/N% 2150 NEXT I% 2160 ' 2170 RETURN TABLE 12-5 Chapter 12- The Fast Fourier Transform 237 Likewise, when this correct symmetry is present in the frequency domain, the Inverse DFT will produce a time domain that has an imaginary part composes of all zeros (plus round-off noise). These debugging techniques are essential for using the FFT; become familiar with them. Speed and Precision Comparisons When the DFT is calculated by correlation (as in Table 12-2), the program uses two nested loops, each running through N points. This means that the total number of operations is proportional to N times N. The time to complete the program is thus given by: EQUATION 12-1 DFT execution time. The time required to calculate a DFT by correlation is proportional to the length of the DFT squared. ExecutionTime ' kDFT N 2 where N is the number of points in the DFT and k D F T is a constant of proportionality. If the sine and cosine values are calculated within the nested loops, k DFT is equal to about 25 microseconds on a Pentium at 100 MHz. If you precalculate the sine and cosine values and store them in a look-up-table, k DFT drops to about 7 microseconds. For example, a 1024 point DFT will require about 25 seconds, or nearly 25 milliseconds per point. That's slow! Using this same strategy we can derive the execution time for the FFT. The time required for the bit reversal is negligible. In each of the Log2 N stages there are N/2 butterfly computations. This means the execution time for the program is approximated by: EQUATION 12-2 FFT execution time. The time required to calculate a DFT using the FFT is proportional to N multiplied by the logarithm of N. ExecutionTime ' kFFT N log2N The value of k FFT is about 10 microseconds on a 100 MHz Pentium system. A 1024 point FFT requires about 70 milliseconds to execute, or 70 microseconds per point. This is more than 300 times faster than the DFT calculated by correlation! Not only is N Log2 N less than N 2 , it increases much more slowly as N becomes larger. For example, a 32 point FFT is about ten times faster than the correlation method. However, a 4096 point FFT is one-thousand times faster. For small values of N (say, 32 to 128), the FFT is important. For large values of N (1024 and above), the FFT is absolutely critical. Figure 12-8 compares the execution times of the two algorithms in a graphical form. 238 The Scientist and Engineer's Guide to Digital Signal Processing FIGURE 12-8 Execution times for calculating the DFT. The correlation method refers to the algorithm described in Table 12-2. This method can be made faster by precalculating the sine and cosine values and storing them in a look-up table (LUT). The FFT (Table 12-3) is the fastest algorithm when the DFT is greater than 16 points long. The times shown are for a Pentium processor at 100 MHz. Execution time (seconds) 1000 correlation 100 10 correlation w/LUT 1 0.1 FFT 0.01 0.001 8 16 32 64 128 256 512 1024 2048 4096 Number points in DFT The FFT has another advantage besides raw speed. The FFT is calculated more precisely because the fewer number of calculations results in less round-off error. This can be demonstrated by taking the FFT of an arbitrary signal, and then running the frequency spectrum through an Inverse FFT. This reconstructs the original time domain signal, except for the addition of roundoff noise from the calculations. A single number characterizing this noise can be obtained by calculating the standard deviation of the difference between the two signals. For comparison, this same procedure can be repeated using a DFT calculated by correlation, and a corresponding Inverse DFT. How does the round-off noise of the FFT compare to the DFT by correlation? See for yourself in Fig. 12-9. Further Speed Increases There are several techniques for making the FFT even faster; however, the improvements are only about 20-40%. In one of these methods, the time 70 FIGURE 12-9 DFT precision. Since the FFT calculates the DFT faster than the correlation method, it also calculates it with less round-off error. Error (parts per million) 60 50 40 30 20 correlation 10 FFT 0 16 32 64 128 256 Number of points in DFT 512 1024 Chapter 12- The Fast Fourier Transform 239 domain decomposition is stopped two stages early, when each signals is composed of only four points. Instead of calculating the last two stages, highly optimized code is used to jump directly into the frequency domain, using the simplicity of four point sine and cosine waves. Another popular algorithm eliminates the wasted calculations associated with the imaginary part of the time domain being zero, and the frequency spectrum being symmetrical. In other words, the FFT is modified to calculate the real DFT, instead of the complex DFT. These algorithms are called the real FFT and the real Inverse FFT (or similar names). Expect them to be about 30% faster than the conventional FFT routines. Tables 12-6 and 12-7 show programs for these algorithms. There are two small disadvantages in using the real FFT. First, the code is about twice as long. While your computer doesn't care, you must take the time to convert someone else's program to run on your computer. Second, debugging these programs is slightly harder because you cannot use symmetry as a check for proper operation. These algorithms force the imaginary part of the time domain to be zero, and the frequency domain to have left-right symmetry. For debugging, check that these programs produce the same output as the conventional FFT algorithms. Figures 12-10 and 12-11 illustrate how the real FFT works. In Fig. 12-10, (a) and (b) show a time domain signal that consists of a pulse in the real part, and all zeros in the imaginary part. Figures (c) and (d) show the corresponding frequency spectrum. As previously described, the frequency domain's real part has an even symmetry around sample 0 and sample N/2 , while the imaginary part has an odd symmetry around these same points. 4000 'INVERSE FFT FOR REAL SIGNALS 4010 'Upon entry, N% contains the number of points in the IDFT, REX[ ] and 4020 'IMX[ ] contain the real and imaginary parts of the frequency domain running from 4030 'index 0 to N%/2. The remaining samples in REX[ ] and IMX[ ] are ignored. 4040 'Upon return, REX[ ] contains the real time domain, IMX[ ] contains zeros. 4050 ' 4060 ' 4070 FOR K% = (N%/2+1) TO (N%-1) 'Make frequency domain symmetrical 4080 REX[K%] = REX[N%-K%] '(as in Table 12-1) 4090 IMX[K%] = -IMX[N%-K%] 4100 NEXT K% 4110 ' 4120 FOR K% = 0 TO N%-1 'Add real and imaginary parts together 4130 REX[K%] = REX[K%]+IMX[K%] 4140 NEXT K% 4150 ' 4160 GOSUB 3000 'Calculate forward real DFT (TABLE 12-6) 4170 ' 4180 FOR I% = 0 TO N%-1 'Add real and imaginary parts together 4190 REX[I%] = (REX[I%]+IMX[I%])/N% 'and divide the time domain by N% 4200 IMX[I%] = 0 4210 NEXT I% 4220 ' 4230 RETURN TABLE 12-6 240 The Scientist and Engineer's Guide to Digital Signal Processing Time Domain Frequency Domain 8 2 a. Real part c. Real part (even symmetry) Amplitude Amplitude 4 1 0 0 -4 -1 -8 0 16 32 Sample number 48 64 63 0 2 16 32 Freqeuncy 48 63 8 b. Imaginary part d. Imaginary part (odd symmetry) Amplitude Amplitude 4 1 0 0 -4 -1 -8 0 16 32 Sample number 48 64 63 0 16 32 Frequency 48 63 FIGURE 12-10 Real part symmetry of the DFT. Now consider Fig. 12-11, where the pulse is in the imaginary part of the time domain, and the real part is all zeros. The symmetry in the frequency domain is reversed; the real part is odd, while the imaginary part is even. This situation will be discussed in Chapter 29. For now, take it for granted that this is how the complex DFT behaves. What if there is a signal in both parts of the time domain? By additivity, the frequency domain will be the sum of the two frequency spectra. Now the key element: a frequency spectrum composed of these two types of symmetry can be perfectly separated into the two component signals. This is achieved by the even/odd decomposition discussed in Chapter 6. In other words, two real DFT's can be calculated for the price of single FFT. One of the signals is placed in the real part of the time domain, and the other signal is placed in the imaginary part. After calculating the complex DFT (via the FFT, of course), the spectra are separated using the even/odd decomposition. When two or more signals need to be passed through the FFT, this technique reduces the execution time by about 40%. The improvement isn't a full factor of two because of the calculation time required for the even/odd decomposition. This is a relatively simple technique with few pitfalls, nothing like writing an FFT routine from scratch. Chapter 12- The Fast Fourier Transform Time Domain 241 Frequency Domain 2 8 a. Real part c. Real part (odd symmetry) Amplitude Amplitude 4 1 0 0 -4 -1 -8 0 16 32 Sample number 48 64 63 0 2 16 32 Frequency 48 63 8 b. Imaginary part d. Imaginary part (even symmetry) Amplitude Amplitude 4 1 0 0 -4 -1 -8 0 16 32 Sample number 48 64 63 0 16 32 Frequency 48 63 FIGURE 12-11 Imaginary part symmetry of the DFT. The next step is to modify the algorithm to calculate a single DFT faster. It's ugly, but here is how it is done. The input signal is broken in half by using an interlaced decomposition. The N/2 even points are placed into the real part of the time domain signal, while the N/2 odd points go into the imaginary part. An N/2 point FFT is then calculated, requiring about one-half the time as an N point FFT. The resulting frequency domain is then separated by the even/odd decomposition, resulting in the frequency spectra of the two interlaced time domain signals. These two frequency spectra are then combined into a single spectrum, just as in the last synthesis stage of the FFT. To close this chapter, consider that the FFT is to Digital Signal Processing what the transistor is to electronics. It is a foundation of the technology; everyone in the field knows its characteristics and how to use it. However, only a small number of specialists really understand the details of the internal workings. 242 The Scientist and Engineer's Guide to Digital Signal Processing 3000 'FFT FOR REAL SIGNALS 3010 'Upon entry, N% contains the number of points in the DFT, REX[ ] contains 3020 'the real input signal, while values in IMX[ ] are ignored. Upon return, 3030 'REX[ ] and IMX[ ] contain the DFT output. All signals run from 0 to N%-1. 3040 ' 3050 NH% = N%/2-1 'Separate even and odd points 3060 FOR I% = 0 TO NH% 3070 REX(I%) = REX(2*I%) 3080 IMX(I%) = REX(2*I%+1) 3090 NEXT I% 3100 ' 3110 N% = N%/2 'Calculate N%/2 point FFT 3120 GOSUB 1000 '(GOSUB 1000 is the FFT in Table 12-3) 3130 N% = N%*2 3140 ' 3150 NM1% = N%-1 'Even/odd frequency domain decomposition 3160 ND2% = N%/2 3170 N4% = N%/4-1 3180 FOR I% = 1 TO N4% 3190 IM% = ND2%-I% 3200 IP2% = I%+ND2% 3210 IPM% = IM%+ND2% 3220 REX(IP2%) = (IMX(I%) + IMX(IM%))/2 3230 REX(IPM%) = REX(IP2%) 3240 IMX(IP2%) = -(REX(I%) - REX(IM%))/2 3250 IMX(IPM%) = -IMX(IP2%) 3260 REX(I%) = (REX(I%) + REX(IM%))/2 3270 REX(IM%) = REX(I%) 3280 IMX(I%) = (IMX(I%) - IMX(IM%))/2 3290 IMX(IM%) = -IMX(I%) 3300 NEXT I% 3310 REX(N%*3/4) = IMX(N%/4) 3320 REX(ND2%) = IMX(0) 3330 IMX(N%*3/4) = 0 3340 IMX(ND2%) = 0 3350 IMX(N%/4) = 0 3360 IMX(0) = 0 3370 ' 3380 PI = 3.14159265 'Complete the last FFT stage 3390 L% = CINT(LOG(N%)/LOG(2)) 3400 LE% = CINT(2^L%) 3410 LE2% = LE%/2 3420 UR = 1 3430 UI = 0 3440 SR = COS(PI/LE2%) 3450 SI = -SIN(PI/LE2%) 3460 FOR J% = 1 TO LE2% 3470 JM1% = J%-1 3480 FOR I% = JM1% TO NM1% STEP LE% 3490 IP% = I%+LE2% 3500 TR = REX[IP%]*UR - IMX[IP%]*UI 3510 TI = REX[IP%]*UI + IMX[IP%]*UR 3520 REX[IP%] = REX[I%]-TR 3530 IMX[IP%] = IMX[I%]-TI 3540 REX[I%] = REX[I%]+TR 3550 IMX[I%] = IMX[I%]+TI 3560 NEXT I% 3570 TR = UR 3580 UR = TR*SR - UI*SI 3590 UI = TR*SI + UI*SR 3600 NEXT J% 3610 RETURN TABLE 12-7 CHAPTER 13 Continuous Signal Processing Continuous signal processing is a parallel field to DSP, and most of the techniques are nearly identical. For example, both DSP and continuous signal processing are based on linearity, decomposition, convolution and Fourier analysis. Since continuous signals cannot be directly represented in digital computers, don't expect to find computer programs in this chapter. Continuous signal processing is based on mathematics; signals are represented as equations, and systems change one equation into another. Just as the digital computer is the primary tool used in DSP, calculus is the primary tool used in continuous signal processing. These techniques have been used for centuries, long before computers were developed. The Delta Function Continuous signals can be decomposed into scaled and shifted delta functions, just as done with discrete signals. The difference is that the continuous delta function is much more complicated and mathematically abstract than its discrete counterpart. Instead of defining the continuous delta function by what it is, we will define it by the characteristics it has. A thought experiment will show how this works. Imagine an electronic circuit composed of linear components, such as resistors, capacitors and inductors. Connected to the input is a signal generator that produces various shapes of short pulses. The output of the circuit is connected to an oscilloscope, displaying the waveform produced by the circuit in response to each input pulse. The question we want to answer is: how is the shape of the output pulse related to the characteristics of the input pulse? To simplify the investigation, we will only use input pulses that are much shorter than the output. For instance, if the system responds in milliseconds, we might use input pulses only a few microseconds in length. After taking many measurement, we come to three conclusions: First, the shape of the input pulse does not affect the shape of the output signal. This 243 244 The Scientist and Engineer's Guide to Digital Signal Processing is illustrated in Fig. 13-1, where various shapes of short input pulses produce exactly the same shape of output pulse. Second, the shape of the output waveform is totally determined by the characteristics of the system, i.e., the value and configuration of the resistors, capacitors and inductors. Third, the amplitude of the output pulse is directly proportional to the area of the input pulse. For example, the output will have the same amplitude for inputs of: 1 volt for 1 microsecond, 10 volts for 0.1 microseconds, 1,000 volts for 1 nanosecond, etc. This relationship also allows for input pulses with negative areas. For instance, imagine the combination of a 2 volt pulse lasting 2 microseconds being quickly followed by a -1 volt pulse lasting 4 microseconds. The total area of the input signal is zero, resulting in the output doing nothing. Input signals that are brief enough to have these three properties are called impulses. In other words, an impulse is any signal that is entirely zero except for a short blip of arbitrary shape. For example, an impulse to a microwave transmitter may have to be in the picosecond range because the electronics responds in nanoseconds. In comparison, a volcano that erupts for years may be a perfectly good impulse to geological changes that take millennia. Mathematicians don't like to be limited by any particular system, and commonly use the term impulse to mean a signal that is short enough to be an impulse to any possible system. That is, a signal that is infinitesimally narrow. The continuous delta function is a normalized version of this type of impulse. Specifically, the continuous delta function is mathematically defined by three idealized characteristics: (1) the signal must be infinitesimally brief, (2) the pulse must occur at time zero, and (3) the pulse must have an area of one. Since the delta function is defined to be infinitesimally narrow and have a fixed area, the amplitude is implied to be infinite. Don't let this bother you; it is completely unimportant. Since the amplitude is part of the shape of the impulse, you will never encounter a problem where the amplitude makes any difference, infinite or not. The delta function is a mathematical construct, not a real world signal. Signals in the real world that act as delta functions will always have a finite duration and amplitude. Just as in the discrete case, the continuous delta function is given the mathematical symbol: * ( ) . Likewise, the output of a continuous system in response to a delta function is called the impulse response, and is often denoted by: h ( ) . Notice that parentheses, ( ), are used to denote continuous signals, as compared to brackets, [ ], for discrete signals. This notation is used in this book and elsewhere in DSP, but isn't universal. Impulses are displayed in graphs as vertical arrows (see Fig. 13-1d), with the length of the arrow indicating the area of the impulse. To better understand real world impulses, look into the night sky at a planet and a star, for instance, Mars and Sirius. Both appear about the same brightness and size to the unaided eye. The reason for this similarity is not Chapter 13- Continuous Signal Processing a. Linear System b. Linear System c Linear System d. * (t) 245 Linear System FIGURE 13-1 The continuous delta function. If the input to a linear system is brief compared to the resulting output, the shape of the output depends only on the characteristics of the system, and not the shape of the input. Such short input signals are called impulses. Figures a,b & c illustrate example input signals that are impulses for this particular system. The term delta function is used to describe a normalized impulse, i.e., one that occurs at t ' 0 and has an area of one. The mathematical symbols for the delta function are shown in (d), a vertical arrow and *(t) . obvious, since the viewing geometry is drastically different. Mars is about 6000 kilometers in diameter and 60 million kilometers from earth. In comparison, Sirius is about 300 times larger and over one-million times farther away. These dimensions should make Mars appear more than three-thousand times larger than Sirius. How is it possible that they look alike? These objects look the same because they are small enough to be impulses to the human visual system. The perceived shape is the impulse response of the eye, not the actual image of the star or planet. This becomes obvious when the two objects are viewed through a small telescope; Mars appears as a dim disk, while Sirius still appears as a bright impulse. This is also the reason that stars twinkle while planets do not. The image of a star is small enough that it can be briefly blocked by particles or turbulence in the atmosphere, whereas the larger image of the planet is much less affected. 246 The Scientist and Engineer's Guide to Digital Signal Processing Convolution Just as with discrete signals, the convolution of continuous signals can be viewed from the input signal, or the output signal. The input side viewpoint is the best conceptual description of how convolution operates. In comparison, the output side viewpoint describes the mathematics that must be used. These descriptions are virtually identical to those presented in Chapter 6 for discrete signals. Figure 13-2 shows how convolution is viewed from the input side. An input signal, x (t) , is passed through a system characterized by an impulse response, h (t) , to produce an output signal, y (t) . This can be written in the familiar mathematical equation, y (t) ' x(t) th (t) . The input signal is divided into narrow columns, each short enough to act as an impulse to the system. In other words, the input signal is decomposed into an infinite number of scaled and shifted delta functions. Each of these impulses produces a scaled and shifted version of the impulse response in the output signal. The final output signal is then equal to the combined effect, i.e., the sum of all of the individual responses. For this scheme to work, the width of the columns must be much shorter than the response of the system. Of course, mathematicians take this to the extreme by making the input segments infinitesimally narrow, turning the situation into a calculus problem. In this manner, the input viewpoint describes how a single point (or narrow region) in the input signal affects a larger portion of output signal. In comparison, the output viewpoint examines how a single point in the output signal is determined by the various values from the input signal. Just as with discrete signals, each instantaneous value in the output signal is affected by a section of the input signal, weighted by the impulse response flipped left-for-right. In the discrete case, the signals are multiplied and summed. In the continuous case, the signals are multiplied and integrated. In equation form: EQUATION 13-1 The convolution integral. This equation defines the meaning of: y (t ) ' x (t )t h (t ) . %4 y(t ) ' m x(J) h (t & J) dJ &4 This equation is called the convolution integral, and is the twin of the convolution sum (Eq. 6-1) used with discrete signals. Figure 13-3 shows how this equation can be understood. The goal is to find an expression for calculating the value of the output signal at an arbitrary time, t. The first step is to change the independent variable used to move through the input signal and the impulse response. That is, we replace t with J (a lower case Chapter 13- Continuous Signal Processing a 247 a x(t) b Linear System c y(t) time (t) c b time(t) FIGURE 13-2 Convolution viewed from the input side. The input signal, x(t) , is divided into narrow segments, each acting as an impulse to the system. The output signal, y (t) , is the sum of the resulting scaled and shifted impulse responses. This illustration shows how three points in the input signal contribute to the output signal. Greek tau). This makes x (t) and h (t) become x (J) and h (J) , respectively. This change of variable names is needed because t is already being used to represent the point in the output signal being calculated. The next step is to flip the impulse response left-for-right, turning it into h (& J) . Shifting the flipped impulse response to the location t, results in the expression becoming h(t& J) . The input signal is then weighted by the flipped and shifted impulse response by multiplying the two, i.e., x (J) h (t& J) . The value of the output signal is then found by integrating this weighted input signal from negative to positive infinity, as described by Eq. 13-1. If you have trouble understanding how this works, go back and review the same concepts for discrete signals in Chapter 6. Figure 13-3 is just another way of describing the convolution machine in Fig. 6-8. The only difference is that integrals are being used instead of summations. Treat this as an extension of what you already know, not something new. An example will illustrate how continuous convolution is used in real world problems and the mathematics required. Figure 13-4 shows a simple continuous linear system: an electronic low-pass filter composed of a single resistor and a single capacitor. As shown in the figure, an impulse entering this system produces an output that quickly jumps to some value, and then exponentially decays toward zero. In other words, the impulse response of this simple electronic circuit is a one-sided exponential. Mathematically, the h(t-J) t t Linear System x(J) x(J) time (J) ? y(t) time(t) FIGURE 13-3 Convolution viewed from the output side. Each value in the output signal is influenced by many points from the input signal. In this figure, the output signal at time t is being calculated. The input signal, x (J) , is weighted (multiplied) by the flipped and shifted impulse response, given by h ( t&J) . Integrating the weighted input signal produces the value of the output point, y (t) 248 The Scientist and Engineer's Guide to Digital Signal Processing *(t) h(t) 2 R Amplitude Amplitude 2 1 C 0 1 0 -1 0 1 2 Time 3 -1 0 1 Time 2 3 FIGURE 13-4 Example of a continuous linear system. This electronic circuit is a low-pass filter composed of a single resistor and capacitor. The impulse response of this system is a one-sided exponential. impulse response of this system is broken into two sections, each represented by an equation: h(t ) ' 0 for t < 0 h(t ) ' "e &" t for t $ 0 where " ' 1/RC (R is in ohms, C is in farads, and t is in seconds). Just as in the discrete case, the continuous impulse response contains complete information about the system, that is, how it will react to all possible signals. To pursue this example further, Fig. 13-5 shows a square pulse entering the system, mathematically expressed by: x(t ) ' 1 for 0 # t # 1 x(t ) ' 0 otherwise Since both the input signal and the impulse response are completely known as mathematical expressions, the output signal, y (t) , can be calculated by evaluating the convolution integral of Eq. 13-1. This is complicated by the fact that both signals are defined by regions rather than a single x(t) h(t) y(t) 2 1 Amplitude 2 Amplitude Amplitude 2 1 0 0 -1 0 1 Time 2 3 1 0 -1 0 1 Time 2 3 -1 0 1 Time 2 3 FIGURE 13-5 Example of continuous convolution. This figure illustrates a square pulse entering an RC low-pass filter (Fig. 13-4). The square pulse is convolved with the system's impulse response to produce the output. Chapter 13- Continuous Signal Processing a. No overlap (t < 0) c. Full overlap b. Partial overlap (0 # t # 1) t 0 J 1 (t > 1) t 0 249 0 1 J t J 1 FIGURE 13-6 Calculating a convolution by segments. Since many continuous signals are defined by regions, the convolution calculation must be performed region-by-region. In this example, calculation of the output signal is broken into three sections: (a) no overlap, (b) partial overlap, and (c) total overlap, of the input signal and the shiftedflipped impulse response. mathematical expression. This is very common in continuous signal processing. It is usually essential to draw a picture of how the two signals shift over each other for various values of t. In this example, Fig. 13-6a shows that the two signals do not overlap at all for t < 0 . This means that the product of the two signals is zero at all locations along the J axis, and the resulting output signal is: y(t ) ' 0 for t < 0 A second case is illustrated in (b), where t is between 0 and 1. Here the two signals partially overlap, resulting in their product having nonzero values between J ' 0 and J ' t . Since this is the only nonzero region, it is the only section where the integral needs to be evaluated. This provides the output signal for 0 # t # 1 , given by: 4 y(t ) ' m x(J) h (t & J) dJ (start with Eq. 13-1) &4 t y(t ) ' m 1@ "e & "( t& J) d J (plug in the signals) 0 t y(t ) ' e & "t [e "J ] / (evaluate the integral) 0 y(t ) ' e & "t [ e "t & 1 ] (reduce) y(t ) ' 1 & e & "t for 0 # t # 1 250 The Scientist and Engineer's Guide to Digital Signal Processing Figure (c) shows the calculation for the third section of the output signal, where t > 1. Here the overlap occurs between J ' 0 and J ' 1 , making the calculation the same as for the second segment, except a change to the limits of integration: 1 y(t ) ' m 1@ "e & "( t& J) d J (plug into Eq. 13-1) 0 1 y(t ) ' e & " t [ e "J ] / (evaluate the integral) 0 y(t ) ' [ e " & 1 ] e & " t for t > 1 The waveform in each of these three segments should agree with your knowledge of electronics: (1) The output signal must be zero until the input signal becomes nonzero. That is, the first segment is given by y (t) ' 0 for t < 0 . (2) When the step occurs, the RC circuit exponentially increases to match the input, according to the equation: y (t) ' 1 & e & " t . (3) When the input is returned to zero, the output exponentially decays toward zero, given by the equation: y (t) ' ke & " t (where k ' e " & 1 , the voltage on the capacitor just before the discharge was started). More intricate waveforms can be handled in the same way, although the mathematical complexity can rapidly become unmanageable. When faced with a nasty continuous convolution problem, you need to spend significant time evaluating strategies for solving the problem. If you start blindly evaluating integrals you are likely to end up with a mathematical mess. A common strategy is to break one of the signals into simpler additive components that can be individually convolved. Using the principles of linearity, the resulting waveforms can be added to find the answer to the original problem. Figure 13-7 shows another strategy: modify one of the signals in some linear way, perform the convolution, and then undo the original modification. In this example the modification is the derivative, and it is undone by taking the integral. The derivative of a unit amplitude square pulse is two impulses, the first with an area of one, and the second with an area of negative one. To understand this, think about the opposite process of taking the integral of the two impulses. As you integrate past the first impulse, the integral rapidly increases from zero to one, i.e., a step function. After passing the negative impulse, the integral of the signal rapidly returns from one back to zero, completing the square pulse. Taking the derivative simplifies this problem because convolution is easy when one of the signals is composed of impulses. Each of the two impulses in x ) (t) contributes a scaled and shifted version of the impulse response to Chapter 13- Continuous Signal Processing x(t) h(t) 2 y(t) 1 0 2 Amplitude 2 Amplitude Amplitude 251 1 0 -1 0 1 2 Time 3 1 0 -1 0 1 Time 2 3 -1 0 1 h(t) yN(t) 2 2 1 1 1 0 -1 Amplitude 2 Amplitude Amplitude xN(t) 0 -1 -2 1 Time 2 3 0 -1 -2 0 3 I d/dt -1 2 Time -2 -1 0 1 Time 2 3 -1 0 1 Time 2 3 FIGURE 13-7 A strategy for convolving signals. Convolution problems can often be simplified by clever use of the rules governing linear systems. In this example, the convolution of two signals is simplified by taking the derivative of one of them. After performing the convolution, the derivative is undone by taking the integral. the derivative of the output signal, y ) (t) . That is, by inspection it is known that: y ) (t) ' h (t ) & h (t& 1) . The output signal, y (t) , can then be found by plugging in the exact equation for h (t) , and integrating the expression. A slight nuisance in this procedure is that the DC value of the input signal is lost when the derivative is taken. This can result in an error in the DC value of the calculated output signal. The mathematics reflects this as the arbitrary constant that can be added during the integration. There is no systematic way of identifying this error, but it can usually be corrected by inspection of the problem. For instance, there is no DC error in the example of Fig. 13-7. This is known because the calculated output signal has the correct DC value when t becomes very large. If an error is present in a particular problem, an appropriate DC term is manually added to the output signal to complete the calculation. This method also works for signals that can be reduced to impulses by taking the derivative multiple times. In the jargon of the field, these signals are called piecewise polynomials. After the convolution, the initial operation of multiple derivatives is undone by taking multiple integrals. The only catch is that the lost DC value must be found at each stage by finding the correct constant of integration. 252 The Scientist and Engineer's Guide to Digital Signal Processing Before starting a difficult continuous convolution problem, there is another approach that you should consider. Ask yourself the question: Is a mathematical expression really needed for the output signal, or is a graph of the waveform sufficient? If a graph is adequate, you may be better off to handle the problem with discrete techniques. That is, approximate the continuous signals by samples that can be directly convolved by a computer program. While not as mathematically pure, it can be much easier. The Fourier Transform The Fourier Transform for continuous signals is divided into two categories, one for signals that are periodic, and one for signals that are aperiodic. Periodic signals use a version of the Fourier Transform called the Fourier Series, and are discussed in the next section. The Fourier Transform used with aperiodic signals is simply called the Fourier Transform. This chapter describes these Fourier techniques using only real mathematics, just as the last several chapters have done for discrete signals. The more powerful use of complex mathematics will be reserved for Chapter 31. Figure 13-8 shows an example of a continuous aperiodic signal and its frequency spectrum. The time domain signal extends from negative infinity to positive infinity, while each of the frequency domain signals extends from zero to positive infinity. This frequency spectrum is shown in rectangular form (real and imaginary parts); however, the polar form (magnitude and phase) is also used with continuous signals. Just as in the discrete case, the synthesis equation describes a recipe for constructing the time domain signal using the data in the frequency domain. In mathematical form: %4 x (t ) ' 1 Re X (T) cos (T t ) & Im X (T) sin(T t ) dT Bm 0 EQUATION 13-2 The Fourier transform synthesis equation. In this equation, x(t) is the time domain signal being synthesized, and Re X(T) & Im X(T) are the real and imaginary parts of the frequency spectrum, respectively. In words, the time domain signal is formed by adding (with the use of an integral) an infinite number of scaled sine and cosine waves. The real part of the frequency domain consists of the scaling factors for the cosine waves, while the imaginary part consists of the scaling factors for the sine waves. Just as with discrete signals, the synthesis equation is usually written with negative sine waves. Although the negative sign has no significance in this discussion, it is necessary to make the notation compatible with the complex mathematics described in Chapter 29. The key point to remember is that some authors put this negative sign in the equation, while others do not. Also notice that frequency is represented by the symbol, T , a lower case Chapter 13- Continuous Signal Processing Time Domain 253 Frequency Domain 100 b. Re X(T) Amplitude 80 8 60 40 a. x(t) 20 Amplitude 4 0 0 0 -4 100 -8 80 20 40 60 80 100 120 140 60 80 100 120 140 Frequency (hertz) c. Im X(T) 0 10 20 Time (milliseconds) 30 40 50 Amplitude -50 -40 -30 -20 -10 60 40 20 0 0 20 40 Frequency (hertz) FIGURE 13-8 Example of the Fourier Transform. The time domain signal, x(t ) , extends from negative to positive infinity. The frequency domain is composed of a real part, Re X(T) , and an imaginary part, Im X(T) , each extending from zero to positive infinity. The frequency axis in this illustration is labeled in cycles per second (hertz). To convert to natural frequency, multiply the numbers on the frequency axis by 2B . Greek omega. As you recall, this notation is called the natural frequency, and has the units of radians per second. That is, T' 2Bf , where f is the frequency in cycles per second (hertz). The natural frequency notation is favored by mathematicians and others doing signal processing by solving equations, because there are usually fewer symbols to write. The analysis equations for continuous signals follow the same strategy as the discrete case: correlation with sine and cosine waves. The equations are: %4 EQUATION 13-3 The Fourier transform analysis equations. In this equation, Re X(T) & Im X(T) are the real and imaginary parts of the frequency spectrum, respectively, and x(t) is the time domain signal being analyzed. Re X (T) ' m x(t ) cos (T t ) d t &4 %4 Im X (T) ' & m &4 x(t ) sin(T t ) d t 254 The Scientist and Engineer's Guide to Digital Signal Processing As an example of using the analysis equations, we will find the frequency response of the RC low-pass filter. This is done by taking the Fourier transform of its impulse response, previously shown in Fig. 13-4, and described by: h(t ) ' 0 for t < 0 h(t ) ' "e & " t for t $ 0 The frequency response is found by plugging the impulse response into the analysis equations. First, the real part: %4 m Re H(T) ' h (t ) cos (T t ) dt (start with Eq. 13-3) &4 %4 Re H(T) ' m "e & " t cos (T t ) dt (plug in the signal) 0 %4 "e & " t Re H (T) ' [ & " cos (T t ) % T sin(T t ) ] / "2 % T2 0 Re H (T) ' (evaluate) "2 "2 % T2 Using this same approach, the imaginary part of the frequency response is calculated to be: Im H (T) ' & T" "2 % T2 Just as with discrete signals, the rectangular representation of the frequency domain is great for mathematical manipulation, but difficult for human understanding. The situation can be remedied by converting into polar notation with the standard relations: Mag H (T) ' [ Re H (T)2 % Im H(T)2 ]½ and Phase H(T) ' arctan [ Re H(T) / Im H(T)] . Working through the algebra Chapter 13- Continuous Signal Processing 255 provides the frequency response of the RC low-pass filter as magnitude and phase (i.e., polar form): " [ " % T2 ] 1/2 Mag H (T) ' 2 T " Phase H (T) ' arctan & Figure 13-9 shows graphs of these curves for a cutoff frequency of 1000 hertz (i.e., " ' 2B1000 ). 1.2 1.6 a. Magnitude 1.0 b. Phase 1.2 Phase (radians) Amplitude 0.8 0.8 0.6 0.4 0.0 -0.4 0.4 -0.8 0.2 -1.2 0.0 -1.6 0 1000 2000 3000 4000 Frequency (hertz) 5000 6000 0 1000 2000 3000 4000 Frequency (hertz) 5000 6000 FIGURE 13-9 Frequency response of an RC low-pass filter. These curves were derived by calculating the Fourier transform of the impulse response, and then converting to polar form. The Fourier Series This brings us to the last member of the Fourier transform family: the Fourier series. The time domain signal used in the Fourier series is periodic and continuous. Figure 13-10 shows several examples of continuous waveforms that repeat themselves from negative to positive infinity. Chapter 11 showed that periodic signals have a frequency spectrum consisting of harmonics. For instance, if the time domain repeats at 1000 hertz (a period of 1 millisecond), the frequency spectrum will contain a first harmonic at 1000 hertz, a second harmonic at 2000 hertz, a third harmonic at 3000 hertz, and so forth. The first harmonic, i.e., the frequency that the time domain repeats itself, is also called the fundamental frequency. This means that the frequency spectrum can be viewed in two ways: (1) the frequency spectrum is continuous, but zero at all frequencies except the harmonics, or (2) the frequency spectrum is discrete, and only defined at the harmonic frequencies. In other words, the frequencies between the harmonics can be thought of as having a value of zero, or simply 256 The Scientist and Engineer's Guide to Digital Signal Processing not existing. The important point is that they do not contribute to forming the time domain signal. The Fourier series synthesis equation creates a continuous periodic signal with a fundamental frequency, f, by adding scaled cosine and sine waves with frequencies: f, 2f, 3f, 4f, etc. The amplitudes of the cosine waves are held in the variables: a1, a2, a3, a4, etc., while the amplitudes of the sine waves are held in: b1, b2, b3, b4, and so on. In other words, the "a" and "b" coefficients are the real and imaginary parts of the frequency spectrum, respectively. In addition, the coefficient a0 is used to hold the DC value of the time domain waveform. This can be viewed as the amplitude of a cosine wave with zero frequency (a constant value). Sometimes a0 is grouped with the other "a" coefficients, but it is often handled separately because it requires special calculations. There is no b0 coefficient since a sine wave of zero frequency has a constant value of zero, and would be quite useless. The synthesis equation is written: x(t ) ' a0 % j an cos (2B f t n) & j bn sin(2B f t n) 4 4 n'1 n'1 EQUATION 13-4 The Fourier series synthesis equation. Any periodic signal, x(t) , can be reconstructed from sine and cosine waves with frequencies that are multiples of the fundamental, f. The a n and b n coefficients hold the amplitudes of the cosine and sine waves, respectively. The corresponding analysis equations for the Fourier series are usually written in terms of the period of the waveform, denoted by T, rather than the fundamental frequency, f (where f ' 1/T ). Since the time domain signal is periodic, the sine and cosine wave correlation only needs to be evaluated over a single period, i.e., & T /2 to T /2 , 0 to T, -T to 0, etc. Selecting different limits makes the mathematics different, but the final answer is always the same. The Fourier series analysis equations are: T /2 a0 ' 1 x (t ) d t T m T /2 an ' & T /2 EQUATION 13-5 Fourier series analysis equations. In these equations, x(t) is the time domain signal being decomposed, a0 is the DC component, an & bn hold the amplitudes of the cosine and sine waves, respectively, and T is the period of the signal, i.e., the reciprocal of the fundamental frequency. 2 2Btn x(t ) cos dt T m T & T /2 T /2 bn &2 2Btn ' x(t ) sin dt T m T & T /2 Chapter 13- Continuous Signal Processing Time Domain a. Pulse k Frequency Domain A a0 ' A d 2A an ' sin (nBd ) nB bn ' 0 A d = k/T T 0 0 t=0 b. Square 257 f 2f 3f 4f 5f 6f ( d ' 0.27 in this example) a0 ' 0 2A nB an ' sin nB 2 bn ' 0 A A 0 0 t=0 c. Triangle f 2f 3f 4f 5f 6f (all even harmonics are zero) a0 ' 0 A 4A (nB)2 bn ' 0 an ' A 0 0 t=0 d. Sawtooth f 2f 3f 4f 5f 6f A a0 ' 0 an ' 0 A bn ' nB A 0 0 t=0 e. Rectified f 2f 3f 4f 5f 6f A a0 ' 2 A /B &4A an ' B(4n 2& 1) A bn ' 0 0 0 t=0 f. Cosine wave (all even harmonics are zero) f 2f 3f 4f 5f 6f A a1 ' A A (all other coefficients are zero) 0 t=0 0 f 2f 3f 4f 5f 6f FIGURE 13-10 Examples of the Fourier series. Six common time domain waveforms are shown, along with the equations to calculate their "a" and "b" coefficients. 258 The Scientist and Engineer's Guide to Digital Signal Processing -T/2 -k/2 k/2 T/2 Amplitude A 0 -3T -2T -T 0 T 2T 3T Time FIGURE 13-11 Example of calculating a Fourier series. This is a pulse train with a duty cycle of d = k/T. The Fourier series coefficients are calculated by correlating the waveform with cosine and sine waves over any full period. In this example, the period from -T/2 to T/2 is used. Figure 13-11 shows an example of calculating a Fourier series using these equations. The time domain signal being analyzed is a pulse train, a square wave with unequal high and low durations. Over a single period from & T /2 to T /2 , the waveform is given by: x(t ) ' A for -k/2 # t # k/2 x(t ) ' 0 otherwise The duty cycle of the waveform (the fraction of time that the pulse is "high") is thus given by d ' k/T . The Fourier series coefficients can be found by evaluating Eq. 13-5. First, we will find the DC component, a0 : T/2 a0 ' 1 x (t ) d t T m (start with Eq. 13-5) & T/2 k/2 a0 ' 1 A dt T m (plug in the signal) & k/2 a0 ' Ak T a0 ' A d (evaluate the integral) (substitute: d = k/T) This result should make intuitive sense; the DC component is simply the average value of the signal. A similar analysis provides the "a" coefficients: Chapter 13- Continuous Signal Processing 259 T /2 an ' 2 2Bt n x(t ) cos dt T m T (start with Eq. 13-4) & T /2 k /2 an ' 2 2Bt n A cos m T T dt (plug in the signal) & k /2 k /2 an ' 2A T 2Bt n sin T 2Bn T an ' 2A sin(Bn d ) nB / (evaluate the integral) & k /2 (reduce) The "b" coefficients are calculated in this same way; however, they all turn out to be zero. In other words, this waveform can be constructed using only cosine waves, with no sine waves being needed. The "a" and "b" coefficients will change if the time domain waveform is shifted left or right. For instance, the "b" coefficients in this example will be zero only if one of the pulses is centered on t ' 0 . Think about it this way. If the waveform is even (i.e., symmetrical around t ' 0 ), it will be composed solely of even sinusoids, that is, cosine waves. This makes all of the "b" coefficients equal to zero. If the waveform if odd (i.e., symmetrical but opposite in sign around t ' 0 ), it will be composed of odd sinusoids, i.e., sine waves. This results in the "a" coefficients being zero. If the coefficients are converted to polar notation (say, M n and 2n coefficients), a shift in the time domain leaves the magnitude unchanged, but adds a linear component to the phase. To complete this example, imagine a pulse train existing in an electronic circuit, with a frequency of 1 kHz, an amplitude of one volt, and a duty cycle of 0.2. The table in Fig. 13-12 provides the amplitude of each harmonic contained in this waveform. Figure 13-12 also shows the synthesis of the waveform using only the first fourteen of these harmonics. Even with this number of harmonics, the reconstruction is not very good. In mathematical jargon, the Fourier series converges very slowly. This is just another way of saying that sharp edges in the time domain waveform results in very high frequencies in the spectrum. Lastly, be sure and notice the overshoot at the sharp edges, i.e., the Gibbs effect discussed in Chapter 11. An important application of the Fourier series is electronic frequency multiplication. Suppose you want to construct a very stable sine wave oscillator at 150 MHz. This might be needed, for example, in a radio 260 The Scientist and Engineer's Guide to Digital Signal Processing Amplitude (volts) transmitter operating at this frequency. High stability calls for the circuit to be crystal controlled. That is, the frequency of the oscillator is determined by a resonating quartz crystal that is a part of the circuit. The problem is, quartz crystals only work to about 10 MHz. The solution is to build a crystal controlled oscillator operating somewhere between 1 and 10 MHz, and then multiply the frequency to whatever you need. This is accomplished by distorting the sine wave, such as by clipping the peaks with a diode, or running the waveform through a squaring circuit. The harmonics in the distorted waveform are then isolated with band-pass filters. This allows the frequency to be doubled, tripled, or multiplied by even higher integers numbers. The most common technique is to use sequential stages of doublers and triplers to generate the required frequency multiplication, rather than just a single stage. The Fourier series is important to this type of design because it describes the amplitude of the multiplied signal, depending on the type of distortion and harmonic selected. 1.5 frequency amplitude (volts) 1.0 DC 1 kHz 2 kHz 3 kHz 4 kHz 5 kHz 6 kHz 7 kHz 8 kHz 9 kHz 10 kHz 11 kHz 12 kHz 0.20000 0.37420 0.30273 0.20182 0.09355 0.00000 -0.06237 -0.08649 -0.07568 -0.04158 0.00000 0.03402 0.05046 123 kHz 124 kHz 125 kHz 126 kHz 0.00492 0.00302 0.00000 -0.00297 803 kHz 804 kHz 805 kHz 806 kHz 0.00075 0.00046 0.00000 -0.00046 0.5 0.0 -0.5 0 1 2 Time (milliseconds) 3 FIGURE 13-12 Example of Fourier series synthesis. The waveform being constructed is a pulse train at 1 kHz, an amplitude of one volt, and a duty cycle of 0.2 (as illustrated in Fig. 13-11). This table shows the amplitude of the harmonics, while the graph shows the reconstructed waveform using only the first fourteen harmonics. 4 ! ! CHAPTER 14 Introduction to Digital Filters Digital filters are used for two general purposes: (1) separation of signals that have been combined, and (2) restoration of signals that have been distorted in some way. Analog (electronic) filters can be used for these same tasks; however, digital filters can achieve far superior results. The most popular digital filters are described and compared in the next seven chapters. This introductory chapter describes the parameters you want to look for when learning about each of these filters. Filter Basics Digital filters are a very important part of DSP. In fact, their extraordinary performance is one of the key reasons that DSP has become so popular. As mentioned in the introduction, filters have two uses: signal separation and signal restoration. Signal separation is needed when a signal has been contaminated with interference, noise, or other signals. For example, imagine a device for measuring the electrical activity of a baby's heart (EKG) while still in the womb. The raw signal will likely be corrupted by the breathing and heartbeat of the mother. A filter might be used to separate these signals so that they can be individually analyzed. Signal restoration is used when a signal has been distorted in some way. For example, an audio recording made with poor equipment may be filtered to better represent the sound as it actually occurred. Another example is the deblurring of an image acquired with an improperly focused lens, or a shaky camera. These problems can be attacked with either analog or digital filters. Which is better? Analog filters are cheap, fast, and have a large dynamic range in both amplitude and frequency. Digital filters, in comparison, are vastly superior in the level of performance that can be achieved. For example, a low-pass digital filter presented in Chapter 16 has a gain of 1 +/- 0.0002 from DC to 1000 hertz, and a gain of less than 0.0002 for frequencies above 261 262 The Scientist and Engineer's Guide to Digital Signal Processing 1001 hertz. The entire transition occurs within only 1 hertz. Don't expect this from an op amp circuit! Digital filters can achieve thousands of times better performance than analog filters. This makes a dramatic difference in how filtering problems are approached. With analog filters, the emphasis is on handling limitations of the electronics, such as the accuracy and stability of the resistors and capacitors. In comparison, digital filters are so good that the performance of the filter is frequently ignored. The emphasis shifts to the limitations of the signals, and the theoretical issues regarding their processing. It is common in DSP to say that a filter's input and output signals are in the time domain. This is because signals are usually created by sampling at regular intervals of time. But this is not the only way sampling can take place. The second most common way of sampling is at equal intervals in space. For example, imagine taking simultaneous readings from an array of strain sensors mounted at one centimeter increments along the length of an aircraft wing. Many other domains are possible; however, time and space are by far the most common. When you see the term time domain in DSP, remember that it may actually refer to samples taken over time, or it may be a general reference to any domain that the samples are taken in. As shown in Fig. 14-1, every linear filter has an impulse response, a step response and a frequency response. Each of these responses contains complete information about the filter, but in a different form. If one of the three is specified, the other two are fixed and can be directly calculated. All three of these representations are important, because they describe how the filter will react under different circumstances. The most straightforward way to implement a digital filter is by convolving the input signal with the digital filter's impulse response. All possible linear filters can be made in this manner. (This should be obvious. If it isn't, you probably don't have the background to understand this section on filter design. Try reviewing the previous section on DSP fundamentals). When the impulse response is used in this way, filter designers give it a special name: the filter kernel. There is also another way to make digital filters, called recursion. When a filter is implemented by convolution, each sample in the output is calculated by weighting the samples in the input, and adding them together. Recursive filters are an extension of this, using previously calculated values from the output, besides points from the input. Instead of using a filter kernel, recursive filters are defined by a set of recursion coefficients. This method will be discussed in detail in Chapter 19. For now, the important point is that all linear filters have an impulse response, even if you don't use it to implement the filter. To find the impulse response of a recursive filter, simply feed in an impulse, and see what comes out. The impulse responses of recursive filters are composed of sinusoids that exponentially decay in amplitude. In principle, this makes their impulse responses infinitely long. However, the amplitude eventually drops below the round-off noise of the system, and the remaining samples can be ignored. Because Chapter 14- Introduction to Digital Filters 1.5 0.3 a. Impulse response c. Frequency response 1.0 FFT 0.1 Amplitude Amplitude 0.2 0.0 0.5 0.0 -0.1 -0.5 0 32 64 Sample number 96 127 128 0 0.1 Integrate 0.2 0.3 Frequency 0.4 0.5 20 Log( ) 1.5 40 b. Step response d. Frequency response (in dB) 20 Amplitude (dB) 1.0 Amplitude 263 0.5 0.0 0 -20 -40 -0.5 -60 0 32 64 Sample number 96 127 128 0 0.1 0.2 0.3 Frequency 0.4 0.5 FIGURE 14-1 Filter parameters. Every linear filter has an impulse response, a step response, and a frequency response. The step response, (b), can be found by discrete integration of the impulse response, (a). The frequency response can be found from the impulse response by using the Fast Fourier Transform (FFT), and can be displayed either on a linear scale, (c), or in decibels, (d). of this characteristic, recursive filters are also called Infinite Impulse Response or IIR filters. In comparison, filters carried out by convolution are called Finite Impulse Response or FIR filters. As you know, the impulse response is the output of a system when the input is an impulse. In this same manner, the step response is the output when the input is a step (also called an edge, and an edge response). Since the step is the integral of the impulse, the step response is the integral of the impulse response. This provides two ways to find the step response: (1) feed a step waveform into the filter and see what comes out, or (2) integrate the impulse response. (To be mathematically correct: integration is used with continuous signals, while discrete integration, i.e., a running sum, is used with discrete signals). The frequency response can be found by taking the DFT (using the FFT algorithm) of the impulse response. This will be reviewed later in this 264 The Scientist and Engineer's Guide to Digital Signal Processing chapter. The frequency response can be plotted on a linear vertical axis, such as in (c), or on a logarithmic scale (decibels), as shown in (d). The linear scale is best at showing the passband ripple and roll-off, while the decibel scale is needed to show the stopband attenuation. Don't remember decibels? Here is a quick review. A bel (in honor of Alexander Graham Bell) means that the power is changed by a factor of ten. For example, an electronic circuit that has 3 bels of amplification produces an output signal with 10 ×10 ×10 ' 1000 times the power of the input. A decibel (dB) is one-tenth of a bel. Therefore, the decibel values of: -20dB, -10dB, 0dB, 10dB & 20dB, mean the power ratios: 0.01, 0.1, 1, 10, & 100, respectively. In other words, every ten decibels mean that the power has changed by a factor of ten. Here's the catch: you usually want to work with a signal's amplitude, not its power. For example, imagine an amplifier with 20dB of gain. By definition, this means that the power in the signal has increased by a factor of 100. Since amplitude is proportional to the square-root of power, the amplitude of the output is 10 times the amplitude of the input. While 20dB means a factor of 100 in power, it only means a factor of 10 in amplitude. Every twenty decibels mean that the amplitude has changed by a factor of ten. In equation form: EQUATION 14-1 Definition of decibels. Decibels are a way of expressing a ratio between two signals. Ratios of power (P1 & P2) use a different equation from ratios of amplitude (A1 & A2). dB ' 10 log10 dB ' 20 log10 P2 P1 A2 A1 The above equations use the base 10 logarithm; however, many computer languages only provide a function for the base e logarithm (the natural log, written loge x or ln x ). The natural log can be use by modifying the above equations: dB ' 4.342945 loge (P2 / P1) and dB ' 8.685890 loge (A2 / A1) . Since decibels are a way of expressing the ratio between two signals, they are ideal for describing the gain of a system, i.e., the ratio between the output and the input signal. However, engineers also use decibels to specify the amplitude (or power) of a single signal, by referencing it to some standard. For example, the term: dBV means that the signal is being referenced to a 1 volt rms signal. Likewise, dBm indicates a reference signal producing 1 mW into a 600 ohms load (about 0.78 volts rms). If you understand nothing else about decibels, remember two things: First, -3dB means that the amplitude is reduced to 0.707 (and the power is Chapter 14- Introduction to Digital Filters 265 therefore reduced to 0.5). Second, memorize the following conversions between decibels and amplitude ratios: 60dB 40dB 20dB 0dB -20dB -40dB -60dB = = = = = = = 1000 100 10 1 0.1 0.01 0.001 How Information is Represented in Signals The most important part of any DSP task is understanding how information is contained in the signals you are working with. There are many ways that information can be contained in a signal. This is especially true if the signal is manmade. For instance, consider all of the modulation schemes that have been devised: AM, FM, single-sideband, pulse-code modulation, pulse-width modulation, etc. The list goes on and on. Fortunately, there are only two ways that are common for information to be represented in naturally occurring signals. We will call these: information represented in the time domain, and information represented in the frequency domain. Information represented in the time domain describes when something occurs and what the amplitude of the occurrence is. For example, imagine an experiment to study the light output from the sun. The light output is measured and recorded once each second. Each sample in the signal indicates what is happening at that instant, and the level of the event. If a solar flare occurs, the signal directly provides information on the time it occurred, the duration, the development over time, etc. Each sample contains information that is interpretable without reference to any other sample. Even if you have only one sample from this signal, you still know something about what you are measuring. This is the simplest way for information to be contained in a signal. In contrast, information represented in the frequency domain is more indirect. Many things in our universe show periodic motion. For example, a wine glass struck with a fingernail will vibrate, producing a ringing sound; the pendulum of a grandfather clock swings back and forth; stars and planets rotate on their axis and revolve around each other, and so forth. By measuring the frequency, phase, and amplitude of this periodic motion, information can often be obtained about the system producing the motion. Suppose we sample the sound produced by the ringing wine glass. The fundamental frequency and harmonics of the periodic vibration relate to the mass and elasticity of the material. A single sample, in itself, contains no information about the periodic motion, and therefore no information about the wine glass. The information is contained in the relationship between many points in the signal. 266 The Scientist and Engineer's Guide to Digital Signal Processing This brings us to the importance of the step and frequency responses. The step response describes how information represented in the time domain is being modified by the system. In contrast, the frequency response shows how information represented in the frequency domain is being changed. This distinction is absolutely critical in filter design because it is not possible to optimize a filter for both applications. Good performance in the time domain results in poor performance in the frequency domain, and vice versa. If you are designing a filter to remove noise from an EKG signal (information represented in the time domain), the step response is the important parameter, and the frequency response is of little concern. If your task is to design a digital filter for a hearing aid (with the information in the frequency domain), the frequency response is all important, while the step response doesn't matter. Now let's look at what makes a filter optimal for time domain or frequency domain applications. Time Domain Parameters It may not be obvious why the step response is of such concern in time domain filters. You may be wondering why the impulse response isn't the important parameter. The answer lies in the way that the human mind understands and processes information. Remember that the step, impulse and frequency responses all contain identical information, just in different arrangements. The step response is useful in time domain analysis because it matches the way humans view the information contained in the signals. For example, suppose you are given a signal of some unknown origin and asked to analyze it. The first thing you will do is divide the signal into regions of similar characteristics. You can't stop from doing this; your mind will do it automatically. Some of the regions may be smooth; others may have large amplitude peaks; others may be noisy. This segmentation is accomplished by identifying the points that separate the regions. This is where the step function comes in. The step function is the purest way of representing a division between two dissimilar regions. It can mark when an event starts, or when an event ends. It tells you that whatever is on the left is somehow different from whatever is on the right. This is how the human mind views time domain information: a group of step functions dividing the information into regions of similar characteristics. The step response, in turn, is important because it describes how the dividing lines are being modified by the filter. The step response parameters that are important in filter design are shown in Fig. 14-2. To distinguish events in a signal, the duration of the step response must be shorter than the spacing of the events. This dictates that the step response should be as fast (the DSP jargon) as possible. This is shown in Figs. (a) & (b). The most common way to specify the risetime (more jargon) is to quote the number of samples between the 10% and 90% amplitude levels. Why isn't a very fast risetime always possible? There are many reasons, noise reduction, inherent limitations of the data acquisition system, avoiding aliasing, etc. Chapter 14- Introduction to Digital Filters POOR GOOD 1.5 1.5 a. Slow step response b. Fast step response 1.0 Amplitude Amplitude 1.0 0.5 0.0 0.5 0.0 -0.5 -0.5 0 16 32 Sample number 48 64 0 1.5 32 48 64 32 48 64 32 48 64 Sample number d. No overshoot 1.0 Amplitude 1.0 Amplitude 16 1.5 c. Overshoot 0.5 0.0 0.5 0.0 -0.5 -0.5 0 16 32 Sample number 48 64 0 1.5 16 Sample number 1.5 e. Nonlinear phase f. Linear phase 1.0 Amplitude 1.0 Amplitude 267 0.5 0.0 0.5 0.0 -0.5 -0.5 0 16 32 Sample number 48 64 0 16 Sample number FIGURE 14-2 Parameters for evaluating time domain performance. The step response is used to measure how well a filter performs in the time domain. Three parameters are important: (1) transition speed (risetime), shown in (a) and (b), (2) overshoot, shown in (c) and (d), and (3) phase linearity (symmetry between the top and bottom halves of the step), shown in (e) and (f). Figures (c) and (d) shows the next parameter that is important: overshoot in the step response. Overshoot must generally be eliminated because it changes the amplitude of samples in the signal; this is a basic distortion of the information contained in the time domain. This can be summed up in 268 The Scientist and Engineer's Guide to Digital Signal Processing one question: Is the overshoot you observe in a signal coming from the thing you are trying to measure, or from the filter you have used? Finally, it is often desired that the upper half of the step response be symmetrical with the lower half, as illustrated in (e) and (f). This symmetry is needed to make the rising edges look the same as the falling edges. This symmetry is called linear phase, because the frequency response has a phase that is a straight line (discussed in Chapter 19). Make sure you understand these three parameters; they are the key to evaluating time domain filters. Frequency Domain Parameters Figure 14-3 shows the four basic frequency responses. The purpose of these filters is to allow some frequencies to pass unaltered, while completely blocking other frequencies. The passband refers to those frequencies that are passed, while the stopband contains those frequencies that are blocked. The transition band is between. A fast roll-off means that the transition band is very narrow. The division between the passband and transition band is called the cutoff frequency. In analog filter design, the cutoff frequency is usually defined to be where the amplitude is reduced to 0.707 (i.e., -3dB). Digital filters are less standardized, and it is common to see 99%, 90%, 70.7%, and 50% amplitude levels defined to be the cutoff frequency. Figure 14-4 shows three parameters that measure how well a filter performs in the frequency domain. To separate closely spaced frequencies, the filter must have a fast roll-off, as illustrated in (a) and (b). For the passband frequencies to move through the filter unaltered, there must be no passband ripple, as shown in (c) and (d). Lastly, to adequately block the stopband frequencies, it is necessary to have good stopband attenuation, displayed in (e) and (f). c. Band-pass Amplitude transition band stopband Frequency Frequency d. Band-reject Amplitude b. High-pass Amplitude FIGURE 14-3 The four common frequency responses. Frequency domain filters are generally used to pass certain frequencies (the passband), while blocking others (the stopband). Four responses are the most common: low-pass, high-pass, band-pass, and band-reject. Amplitude a. Low-pass passband Frequency Frequency Chapter 14- Introduction to Digital Filters POOR GOOD 1.5 1.5 a. Slow roll-off b. Fast roll-off 1.0 Amplitude Amplitude 1.0 0.5 0.0 0.5 0.0 -0.5 -0.5 0 0.1 0.2 0.3 Frequency 0.4 0.5 0 1.5 0.2 0.3 0.4 0.5 0.3 0.4 0.5 0.4 0.5 Frequency d. Flat passband 1.0 Amplitude 1.0 Amplitude 0.1 1.5 c. Ripple in passband 0.5 0.0 0.5 0.0 -0.5 -0.5 0 0.1 0.2 0.3 Frequency 0.4 0.5 0 40 0.1 0.2 Frequency 40 e. Poor stopband attenuation 20 f. Good stopband attenuation 20 0 Amplitude (dB) 0 Amplitude (dB) 269 -20 -40 -60 -80 -100 -20 -40 -60 -80 -100 -120 -120 0 0.1 0.2 0.3 Frequency 0.4 0.5 0 0.1 0.2 0.3 Frequency FIGURE 14-4 Parameters for evaluating frequency domain performance. The frequency responses shown are for low-pass filters. Three parameters are important: (1) roll-off sharpness, shown in (a) and (b), (2) passband ripple, shown in (c) and (d), and (3) stopband attenuation, shown in (e) and (f). Why is there nothing about the phase in these parameters? First, the phase isn't important in most frequency domain applications. For example, the phase of an audio signal is almost completely random, and contains little useful information. Second, if the phase is important, it is very easy to make digital 270 The Scientist and Engineer's Guide to Digital Signal Processing filters with a perfect phase response, i.e., all frequencies pass through the filter with a zero phase shift (also discussed in Chapter 19). In comparison, analog filters are ghastly in this respect. Previous chapters have described how the DFT converts a system's impulse response into its frequency response. Here is a brief review. The quickest way to calculate the DFT is by means of the FFT algorithm presented in Chapter 12. Starting with a filter kernel N samples long, the FFT calculates the frequency spectrum consisting of an N point real part and an N point imaginary part. Only samples 0 to N/2 of the FFT's real and imaginary parts contain useful information; the remaining points are duplicates (negative frequencies) and can be ignored. Since the real and imaginary parts are difficult for humans to understand, they are usually converted into polar notation as described in Chapter 8. This provides the magnitude and phase signals, each running from sample 0 to sample N/2 (i.e., N/2 %1 samples in each signal). For example, an impulse response of 256 points will result in a frequency response running from point 0 to 128. Sample 0 represents DC, i.e., zero frequency. Sample 128 represents one-half of the sampling rate. Remember, no frequencies higher than one-half of the sampling rate can appear in sampled data. The number of samples used to represent the impulse response can be arbitrarily large. For instance, suppose you want to find the frequency response of a filter kernel that consists of 80 points. Since the FFT only works with signals that are a power of two, you need to add 48 zeros to the signal to bring it to a length of 128 samples. This padding with zeros does not change the impulse response. To understand why this is so, think about what happens to these added zeros when the input signal is convolved with the system's impulse response. The added zeros simply vanish in the convolution, and do not affect the outcome. Taking this a step further, you could add many zeros to the impulse response to make it, say, 256, 512, or 1024 points long. The important idea is that longer impulse responses result in a closer spacing of the data points in the frequency response. That is, there are more samples spread between DC and one-half of the sampling rate. Taking this to the extreme, if the impulse response is padded with an infinite number of zeros, the data points in the frequency response are infinitesimally close together, i.e., a continuous line. In other words, the frequency response of a filter is really a continuous signal between DC and one-half of the sampling rate. The output of the DFT is a sampling of this continuous line. What length of impulse response should you use when calculating a filter's frequency response? As a first thought, try N '1024 , but don't be afraid to change it if needed (such as insufficient resolution or excessive computation time). Keep in mind that the "good" and "bad" parameters discussed in this chapter are only generalizations. Many signals don't fall neatly into categories. For example, consider an EKG signal contaminated with 60 hertz interference. The information is encoded in the time domain, but the interference is best dealt with in the frequency domain. The best design for this application is Chapter 14- Introduction to Digital Filters Time Domain Frequency Domain 1.5 1.0 a. Original filter kernel 0.8 b. Original frequency response Amplitude 0.6 Amplitude 271 0.4 0.2 0.0 1.0 0.5 -0.2 -0.4 0.0 0 10 20 30 Sample number 40 50 0 1.0 0.2 0.3 Frequency 0.4 0.5 1.5 c. Filter kernel with spectral inversion 0.8 d. Inverted frequency response Amplitude 0.6 Amplitude 0.1 0.4 0.2 0.0 1.0 Flipped top-for-bottom 0.5 -0.2 -0.4 0.0 0 10 20 30 Sample number 40 50 0 0.1 0.2 0.3 0.4 0.5 Frequency FIGURE 14-5 Example of spectral inversion. The low-pass filter kernel in (a) has the frequency response shown in (b). A high-pass filter kernel, (c), is formed by changing the sign of each sample in (a), and adding one to the sample at the center of symmetry. This action in the time domain inverts the frequency spectrum (i.e., flips it top-forbottom), as shown by the high-pass frequency response in (d). bound to have trade-offs, and might go against the conventional wisdom of this chapter. Remember the number one rule of education: A paragraph in a book doesn't give you a license to stop thinking. High-Pass, Band-Pass and Band-Reject Filters High-pass, band-pass and band-reject filters are designed by starting with a low-pass filter, and then converting it into the desired response. For this reason, most discussions on filter design only give examples of low-pass filters. There are two methods for the low-pass to high-pass conversion: spectral inversion and spectral reversal. Both are equally useful. An example of spectral inversion is shown in 14-5. Figure (a) shows a lowpass filter kernel called a windowed-sinc (the topic of Chapter 16). This filter kernel is 51 points in length, although many of samples have a value so small that they appear to be zero in this graph. The corresponding 272 The Scientist and Engineer's Guide to Digital Signal Processing a. High-pass by adding parallel stages Low-pass h[n] FIGURE 14-6 Block diagram of spectral inversion. In (a), the input signal, x[n] , is applied to two systems in parallel, having impulse responses of h[n] and *[n] . As shown in (b), the combined system has an impulse response of *[n] & h[n] . This means that the frequency response of the combined system is the inversion of the frequency response of h[n] . x[n] y[n] *[n] All-pass b. High-pass in a single stage x[n] High-pass *[n] - h[n] y[n] frequency response is shown in (b), found by adding 13 zeros to the filter kernel and taking a 64 point FFT. Two things must be done to change the low-pass filter kernel into a high-pass filter kernel. First, change the sign of each sample in the filter kernel. Second, add one to the sample at the center of symmetry. This results in the high-pass filter kernel shown in (c), with the frequency response shown in (d). Spectral inversion flips the frequency response top-for-bottom, changing the passbands into stopbands, and the stopbands into passbands. In other words, it changes a filter from low-pass to high-pass, high-pass to low-pass, band-pass to band-reject, or band-reject to band-pass. Figure 14-6 shows why this two step modification to the time domain results in an inverted frequency spectrum. In (a), the input signal, x[n] , is applied to two systems in parallel. One of these systems is a low-pass filter, with an impulse response given by h[n] . The other system does nothing to the signal, and therefore has an impulse response that is a delta function, *[n] . The overall output, y[n] , is equal to the output of the all-pass system minus the output of the low-pass system. Since the low frequency components are subtracted from the original signal, only the high frequency components appear in the output. Thus, a high-pass filter is formed. This could be performed as a two step operation in a computer program: run the signal through a low-pass filter, and then subtract the filtered signal from the original. However, the entire operation can be performed in a signal stage by combining the two filter kernels. As described in Chapter Chapter 14- Introduction to Digital Filters Time Domain Frequency Domain 1.5 1.0 a. Original filter kernel 0.8 b. Original frequency response Amplitude 0.6 Amplitude 273 0.4 0.2 0.0 1.0 0.5 -0.2 -0.4 0.0 0 10 20 30 Sample number 40 50 0 1.0 0.2 0.3 Frequency 0.4 0.5 0.4 0.5 1.5 c. Filter kernel with spectral reversal 0.8 d. Reversed frequency response Amplitude 0.6 Amplitude 0.1 0.4 0.2 0.0 1.0 Flipped left-for-right 0.5 -0.2 -0.4 0.0 0 10 20 30 Sample number 40 50 0 0.1 0.2 0.3 Frequency FIGURE 14-7 Example of spectral reversal. The low-pass filter kernel in (a) has the frequency response shown in (b). A high-pass filter kernel, (c), is formed by changing the sign of every other sample in (a). This action in the time domain results in the frequency domain being flipped left-for-right, resulting in the high-pass frequency response shown in (d). 7, parallel systems with added outputs can be combined into a single stage by adding their impulse responses. As shown in (b), the filter kernel for the highpass filter is given by: *[n] & h[n] . That is, change the sign of all the samples, and then add one to the sample at the center of symmetry. For this technique to work, the low-frequency components exiting the low-pass filter must have the same phase as the low-frequency components exiting the all-pass system. Otherwise a complete subtraction cannot take place. This places two restrictions on the method: (1) the original filter kernel must have left-right symmetry (i.e., a zero or linear phase), and (2) the impulse must be added at the center of symmetry. The second method for low-pass to high-pass conversion, spectral reversal, is illustrated in Fig. 14-7. Just as before, the low-pass filter kernel in (a) corresponds to the frequency response in (b). The high-pass filter kernel, (c), is formed by changing the sign of every other sample in (a). As shown in (d), this flips the frequency domain left-for-right: 0 becomes 0.5 and 0.5 274 The Scientist and Engineer's Guide to Digital Signal Processing a. Band-pass by cascading stages FIGURE 14-8 Designing a band-pass filter. As shown in (a), a band-pass filter can be formed by cascading a low-pass filter and a high-pass filter. This can be reduced to a single stage, shown in (b). The filter kernel of the single stage is equal to the convolution of the low-pass and highpass filter kernels. x[n] b. Band-pass in a single stage x[n] Low-pass High-pass h1[n] h2[n] y[n] Band-pass h1[n] h2[n] y[n] becomes 0. The cutoff frequency of the example low-pass filter is 0.15, resulting in the cutoff frequency of the high-pass filter being 0.35. Changing the sign of every other sample is equivalent to multiplying the filter kernel by a sinusoid with a frequency of 0.5. As discussed in Chapter 10, this has the effect of shifting the frequency domain by 0.5. Look at (b) and imagine the negative frequencies between -0.5 and 0 that are of mirror image of the frequencies between 0 and 0.5. The frequencies that appear in (d) are the negative frequencies from (b) shifted by 0.5. Lastly, Figs. 14-8 and 14-9 show how low-pass and high-pass filter kernels can be combined to form band-pass and band-reject filters. In short, adding the filter kernels produces a band-reject filter, while convolving the filter kernels produces a band-pass filter. These are based on the way cascaded and parallel systems are be combined, as discussed in Chapter 7. Multiple combination of these techniques can also be used. For instance, a band-pass filter can be designed by adding the two filter kernels to form a stop-pass filter, and then use spectral inversion or spectral reversal as previously described. All these techniques work very well with few surprises. Filter Classification Table 14-1 summarizes how digital filters are classified by their use and by their implementation. The use of a digital filter can be broken into three categories: time domain, frequency domain and custom. As previously described, time domain filters are used when the information is encoded in the shape of the signal's waveform. Time domain filtering is used for such actions as: smoothing, DC removal, waveform shaping, etc. In contrast, frequency domain filters are used when the information is contained in the Chapter 14- Introduction to Digital Filters a. Band-reject by adding parallel stages 275 Low-pass h1[n] FIGURE 14-9 Designing a band-reject filter. As shown in (a), a band-reject filter is formed by the parallel combination of a low-pass filter and a high-pass filter with their outputs added. Figure (b) shows this reduced to a single stage, with the filter kernel found by adding the low-pass and high-pass filter kernels. x[n] y[n] h2[n] High-pass b. Band-reject in a single stage Band-reject x[n] h1[n] + h2[n] y[n] amplitude, frequency, and phase of the component sinusoids. The goal of these filters is to separate one band of frequencies from another. Custom filters are used when a special action is required by the filter, something more elaborate than the four basic responses (high-pass, low-pass, band-pass and band-reject). For instance, Chapter 17 describes how custom filters can be used for deconvolution, a way of counteracting an unwanted convolution. FILTER USED FOR: FILTER IMPLEMENTED BY: Convolution Recursion Finite Impulse Response (FIR) Infinite Impulse Response (IIR) Time Domain Moving average (Ch. 15) Single pole (Ch. 19) Frequency Domain Windowed-sinc (Ch. 16) Chebyshev (Ch. 20) FIR custom (Ch. 17) Iterative design (Ch. 26) (smoothing, DC removal) (separating frequencies) Custom (Deconvolution) TABLE 14-1 Filter classification. Filters can be divided by their use, and how they are implemented. 276 The Scientist and Engineer's Guide to Digital Signal Processing Digital filters can be implemented in two ways, by convolution (also called finite impulse response or FIR) and by recursion (also called infinite impulse response or IIR). Filters carried out by convolution can have far better performance than filters using recursion, but execute much more slowly. The next six chapters describe digital filters according to the classifications in Table 14-1. First, we will look at filters carried out by convolution. The moving average (Chapter 15) is used in the time domain, the windowed-sinc (Chapter 16) is used in the frequency domain, and FIR custom (Chapter 17) is used when something special is needed. To finish the discussion of FIR filters, Chapter 18 presents a technique called FFT convolution. This is an algorithm for increasing the speed of convolution, allowing FIR filters to execute faster. Next, we look at recursive filters. The single pole recursive filter (Chapter 19) is used in the time domain, while the Chebyshev (Chapter 20) is used in the frequency domain. Recursive filters having a custom response are designed by iterative techniques. For this reason, we will delay their discussion until Chapter 26, where they will be presented with another type of iterative procedure: the neural network. As shown in Table 14-1, convolution and recursion are rival techniques; you must use one or the other for a particular application. How do you choose? Chapter 21 presents a head-to-head comparison of the two, in both the time and frequency domains. CHAPTER 15 Moving Average Filters The moving average is the most common filter in DSP, mainly because it is the easiest digital filter to understand and use. In spite of its simplicity, the moving average filter is optimal for a common task: reducing random noise while retaining a sharp step response. This makes it the premier filter for time domain encoded signals. However, the moving average is the worst filter for frequency domain encoded signals, with little ability to separate one band of frequencies from another. Relatives of the moving average filter include the Gaussian, Blackman, and multiplepass moving average. These have slightly better performance in the frequency domain, at the expense of increased computation time. Implementation by Convolution As the name implies, the moving average filter operates by averaging a number of points from the input signal to produce each point in the output signal. In equation form, this is written: EQUATION 15-1 Equation of the moving average filter. In this equation, x[ ] is the input signal, y[ ] is the output signal, and M is the number of points used in the moving average. This equation only uses points on one side of the output sample being calculated. y [i ] ' 1 M j x [i %j ] M &1 j' 0 Where x [ ] is the input signal, y [ ] is the output signal, and M is the number of points in the average. For example, in a 5 point moving average filter, point 80 in the output signal is given by: y [80] ' x [80] % x [81] % x [82] % x [83] % x [84] 5 277 278 The Scientist and Engineer's Guide to Digital Signal Processing As an alternative, the group of points from the input signal can be chosen symmetrically around the output point: y [80] ' x [78] % x [79] % x [80] % x [81] % x [82] 5 This corresponds to changing the summation in Eq. 15-1 from: j ' 0 to M& 1 , to: j ' & (M& 1) /2 to (M& 1) /2 . For instance, in an 11 point moving average filter, the index, j, can run from 0 to 11 (one side averaging) or -5 to 5 (symmetrical averaging). Symmetrical averaging requires that M be an odd number. Programming is slightly easier with the points on only one side; however, this produces a relative shift between the input and output signals. You should recognize that the moving average filter is a convolution using a very simple filter kernel. For example, a 5 point filter has the filter kernel: þ 0, 0, 1/5, 1/5, 1/5, 1/5, 1/5, 0, 0 þ . That is, the moving average filter is a convolution of the input signal with a rectangular pulse having an area of one. Table 15-1 shows a program to implement the moving average filter. 100 'MOVING AVERAGE FILTER 110 'This program filters 5000 samples with a 101 point moving 120 'average filter, resulting in 4900 samples of filtered data. 130 ' 140 DIM X[4999] 'X[ ] holds the input signal 150 DIM Y[4999] 'Y[ ] holds the output signal 160 ' 170 GOSUB XXXX 'Mythical subroutine to load X[ ] 180 ' 190 FOR I% = 50 TO 4949 'Loop for each point in the output signal 200 Y[I%] = 0 'Zero, so it can be used as an accumulator 210 FOR J% = -50 TO 50 'Calculate the summation 220 Y[I%] = Y[I%] + X(I%+J%] 230 NEXT J% 240 Y[I%] = Y[I%]/101 'Complete the average by dividing 250 NEXT I% 260 ' 270 END TABLE 15-1 Noise Reduction vs. Step Response Many scientists and engineers feel guilty about using the moving average filter. Because it is so very simple, the moving average filter is often the first thing tried when faced with a problem. Even if the problem is completely solved, there is still the feeling that something more should be done. This situation is truly ironic. Not only is the moving average filter very good for many applications, it is optimal for a common problem, reducing random white noise while keeping the sharpest step response. Chapter 15- Moving Average Filters 2 2 a. Original signal b. 11 point moving average Amplitude 1 0 1 0 -1 -1 0 100 200 300 Sample number 400 0 500 100 200 300 Sample number 400 500 400 500 2 FIGURE 15-1 Example of a moving average filter. In (a), a rectangular pulse is buried in random noise. In (b) and (c), this signal is filtered with 11 and 51 point moving average filters, respectively. As the number of points in the filter increases, the noise becomes lower; however, the edges becoming less sharp. The moving average filter is the optimal solution for this problem, providing the lowest noise possible for a given edge sharpness. c. 51 point moving average Amplitude Amplitude 279 1 0 -1 0 100 200 300 Sample number Figure 15-1 shows an example of how this works. The signal in (a) is a pulse buried in random noise. In (b) and (c), the smoothing action of the moving average filter decreases the amplitude of the random noise (good), but also reduces the sharpness of the edges (bad). Of all the possible linear filters that could be used, the moving average produces the lowest noise for a given edge sharpness. The amount of noise reduction is equal to the square-root of the number of points in the average. For example, a 100 point moving average filter reduces the noise by a factor of 10. To understand why the moving average if the best solution, imagine we want to design a filter with a fixed edge sharpness. For example, let's assume we fix the edge sharpness by specifying that there are eleven points in the rise of the step response. This requires that the filter kernel have eleven points. The optimization question is: how do we choose the eleven values in the filter kernel to minimize the noise on the output signal? Since the noise we are trying to reduce is random, none of the input points is special; each is just as noisy as its neighbor. Therefore, it is useless to give preferential treatment to any one of the input points by assigning it a larger coefficient in the filter kernel. The lowest noise is obtained when all the input samples are treated equally, i.e., the moving average filter. (Later in this chapter we show that other filters are essentially as good. The point is, no filter is better than the simple moving average). 280 The Scientist and Engineer's Guide to Digital Signal Processing Frequency Response Figure 15-2 shows the frequency response of the moving average filter. It is mathematically described by the Fourier transform of the rectangular pulse, as discussed in Chapter 11: EQUATION 15-2 Frequency response of an M point moving average filter. The frequency, f, runs between 0 and 0.5. For f ' 0 , use: H [ f ] ' 1 H [f ] ' sin(B f M ) M sin(B f ) The roll-off is very slow and the stopband attenuation is ghastly. Clearly, the moving average filter cannot separate one band of frequencies from another. Remember, good performance in the time domain results in poor performance in the frequency domain, and vice versa. In short, the moving average is an exceptionally good smoothing filter (the action in the time domain), but an exceptionally bad low-pass filter (the action in the frequency domain). 1.2 1.0 3 point 0.8 Amplitude FIGURE 15-2 Frequency response of the moving average filter. The moving average is a very poor low-pass filter, due to its slow roll-off and poor stopband attenuation. These curves are generated by Eq. 15-2. 11 point 0.6 31 point 0.4 0.2 0.0 0 0.1 0.2 0.3 Frequency 0.4 0.5 Relatives of the Moving Average Filter In a perfect world, filter designers would only have to deal with time domain or frequency domain encoded information, but never a mixture of the two in the same signal. Unfortunately, there are some applications where both domains are simultaneously important. For instance, television signals fall into this nasty category. Video information is encoded in the time domain, that is, the shape of the waveform corresponds to the patterns of brightness in the image. However, during transmission the video signal is treated according to its frequency composition, such as its total bandwidth, how the carrier waves for sound & color are added, elimination & restoration of the DC component, etc. As another example, electromagnetic interference is best understood in the frequency domain, even if Chapter 15- Moving Average Filters 0.2 1.25 a. Filter kernel c. Frequency response 1.00 2 pass FFT 0.1 Amplitude 1 pass Amplitude 281 0.75 1 pass 2 pass 0.50 4 pass 0.25 4 pass 0.00 0.0 0 6 12 Sample number 18 0 24 0.1 0.2 Integrate 0.4 0.5 0.4 0.5 20 Log( ) 40 1.2 b. Step response Amplitude (dB) 1 pass 4 pass 0.8 0.6 0.4 0.2 0 6 12 Sample number 0 1 pass -20 -40 -60 -80 2 pass -100 2 pass 0.0 d. Frequency response (dB) 20 1.0 Amplitude 0.3 Frequency 4 pass -120 18 24 0 0.1 0.2 0.3 Frequency FIGURE 15-3 Characteristics of multiple-pass moving average filters. Figure (a) shows the filter kernels resulting from passing a seven point moving average filter over the data once, twice and four times. Figure (b) shows the corresponding step responses, while (c) and (d) show the corresponding frequency responses. the signal's information is encoded in the time domain. For instance, the temperature monitor in a scientific experiment might be contaminated with 60 hertz from the power lines, 30 kHz from a switching power supply, or 1320 kHz from a local AM radio station. Relatives of the moving average filter have better frequency domain performance, and can be useful in these mixed domain applications. Multiple-pass moving average filters involve passing the input signal through a moving average filter two or more times. Figure 15-3a shows the overall filter kernel resulting from one, two and four passes. Two passes are equivalent to using a triangular filter kernel (a rectangular filter kernel convolved with itself). After four or more passes, the equivalent filter kernel looks like a Gaussian (recall the Central Limit Theorem). As shown in (b), multiple passes produce an "s" shaped step response, as compared to the straight line of the single pass. The frequency responses in (c) and (d) are given by Eq. 15-2 multiplied by itself for each pass. That is, each time domain convolution results in a multiplication of the frequency spectra. 282 The Scientist and Engineer's Guide to Digital Signal Processing Figure 15-4 shows the frequency response of two other relatives of the moving average filter. When a pure Gaussian is used as a filter kernel, the frequency response is also a Gaussian, as discussed in Chapter 11. The Gaussian is important because it is the impulse response of many natural and manmade systems. For example, a brief pulse of light entering a long fiber optic transmission line will exit as a Gaussian pulse, due to the different paths taken by the photons within the fiber. The Gaussian filter kernel is also used extensively in image processing because it has unique properties that allow fast two-dimensional convolutions (see Chapter 24). The second frequency response in Fig. 15-4 corresponds to using a Blackman window as a filter kernel. (The term window has no meaning here; it is simply part of the accepted name of this curve). The exact shape of the Blackman window is given in Chapter 16 (Eq. 16-2, Fig. 16-2); however, it looks much like a Gaussian. How are these relatives of the moving average filter better than the moving average filter itself? Three ways: First, and most important, these filters have better stopband attenuation than the moving average filter. Second, the filter kernels taper to a smaller amplitude near the ends. Recall that each point in the output signal is a weighted sum of a group of samples from the input. If the filter kernel tapers, samples in the input signal that are farther away are given less weight than those close by. Third, the step responses are smooth curves, rather than the abrupt straight line of the moving average. These last two are usually of limited benefit, although you might find applications where they are genuine advantages. The moving average filter and its relatives are all about the same at reducing random noise while maintaining a sharp step response. The ambiguity lies in how the risetime of the step response is measured. If the risetime is measured from 0% to 100% of the step, the moving average filter is the best you can do, as previously shown. In comparison, measuring the risetime from 10% to 90% makes the Blackman window better than the moving average filter. The point is, this is just theoretical squabbling; consider these filters equal in this parameter. The biggest difference in these filters is execution speed. Using a recursive algorithm (described next), the moving average filter will run like lightning in your computer. In fact, it is the fastest digital filter available. Multiple passes of the moving average will be correspondingly slower, but still very quick. In comparison, the Gaussian and Blackman filters are excruciatingly slow, because they must use convolution. Think a factor of ten times the number of points in the filter kernel (based on multiplication being about 10 times slower than addition). For example, expect a 100 point Gaussian to be 1000 times slower than a moving average using recursion. Recursive Implementation A tremendous advantage of the moving average filter is that it can be implemented with an algorithm that is very fast. To understand this Chapter 15- Moving Average Filters 283 20 0 -20 Amplitude (dB) FIGURE 15-4 Frequency response of the Blackman window and Gaussian filter kernels. Both these filters provide better stopband attenuation than the moving average filter. This has no advantage in removing random noise from time domain encoded signals, but it can be useful in mixed domain problems. The disadvantage of these filters is that they must use convolution, a terribly slow algorithm. Gaussian -40 -60 Blackman -80 -100 -120 -140 0 0.1 0.2 0.3 Frequency 0.4 0.5 algorithm, imagine passing an input signal, x [ ] , through a seven point moving average filter to form an output signal, y [ ] . Now look at how two adjacent output points, y [50] and y [51] , are calculated: y [50] ' x [47] % x [48] % x [49] % x [50] % x [51] % x [52] % x [53] y [51] ' x [48] % x [49] % x [50] % x [51] % x [52] % x [53] % x [54] These are nearly the same calculation; points x [48] through x [53] must be added for y [50] , and again for y [51] . If y [50] has already been calculated, the most efficient way to calculate y [51] is: y [51] ' y [50] % x [54] & x [47] Once y [51] has been found using y [50] , then y [52] can be calculated from sample y [51] , and so on. After the first point is calculated in y [ ] , all of the other points can be found with only a single addition and subtraction per point. This can be expressed in the equation: EQUATION 15-3 Recursive implementation of the moving average filter. In this equation, x[ ] is the input signal, y[ ] is the output signal, M is the number of points in the moving average (an odd number). Before this equation can be used, the first point in the signal must be calculated using a standard summation. y [i ] ' y [i & 1] % x [i % p ] & x [i & q ] where: p ' (M& 1) /2 q' p% 1 Notice that this equation use two sources of data to calculate each point in the output: points from the input and previously calculated points from the output. This is called a recursive equation, meaning that the result of one calculation 284 The Scientist and Engineer's Guide to Digital Signal Processing is used in future calculations. (The term "recursive" also has other meanings, especially in computer science). Chapter 19 discusses a variety of recursive filters in more detail. Be aware that the moving average recursive filter is very different from typical recursive filters. In particular, most recursive filters have an infinitely long impulse response (IIR), composed of sinusoids and exponentials. The impulse response of the moving average is a rectangular pulse (finite impulse response, or FIR). This algorithm is faster than other digital filters for several reasons. First, there are only two computations per point, regardless of the length of the filter kernel. Second, addition and subtraction are the only math operations needed, while most digital filters require time-consuming multiplication. Third, the indexing scheme is very simple. Each index in Eq. 15-3 is found by adding or subtracting integer constants that can be calculated before the filtering starts (i.e., p and q). Forth, the entire algorithm can be carried out with integer representation. Depending on the hardware used, integers can be more than an order of magnitude faster than floating point. Surprisingly, integer representation works better than floating point with this algorithm, in addition to being faster. The round-off error from floating point arithmetic can produce unexpected results if you are not careful. For example, imagine a 10,000 sample signal being filtered with this method. The last sample in the filtered signal contains the accumulated error of 10,000 additions and 10,000 subtractions. This appears in the output signal as a drifting offset. Integers don't have this problem because there is no round-off error in the arithmetic. If you must use floating point with this algorithm, the program in Table 15-2 shows how to use a double precision accumulator to eliminate this drift. 100 'MOVING AVERAGE FILTER IMPLEMENTED BY RECURSION 110 'This program filters 5000 samples with a 101 point moving 120 'average filter, resulting in 4900 samples of filtered data. 130 'A double precision accumulator is used to prevent round-off drift. 140 ' 150 DIM X[4999] 'X[ ] holds the input signal 160 DIM Y[4999] 'Y[ ] holds the output signal 170 DEFDBL ACC 'Define the variable ACC to be double precision 180 ' 190 GOSUB XXXX 'Mythical subroutine to load X[ ] 200 ' 210 ACC = 0 'Find Y[50] by averaging points X[0] to X[100] 220 FOR I% = 0 TO 100 230 ACC = ACC + X[I%] 240 NEXT I% 250 Y[[50] = ACC/101 260 ' 'Recursive moving average filter (Eq. 15-3) 270 FOR I% = 51 TO 4949 280 ACC = ACC + X[I%+50] - X[I%-51] 290 Y[I%] = ACC 300 NEXT I% 310 ' 320 END TABLE 15-2 CHAPTER 16 Windowed-Sinc Filters Windowed-sinc filters are used to separate one band of frequencies from another. They are very stable, produce few surprises, and can be pushed to incredible performance levels. These exceptional frequency domain characteristics are obtained at the expense of poor performance in the time domain, including excessive ripple and overshoot in the step response. When carried out by standard convolution, windowed-sinc filters are easy to program, but slow to execute. Chapter 18 shows how the FFT can be used to dramatically improve the computational speed of these filters. Strategy of the Windowed-Sinc Figure 16-1 illustrates the idea behind the windowed-sinc filter. In (a), the frequency response of the ideal low-pass filter is shown. All frequencies below the cutoff frequency, fC , are passed with unity amplitude, while all higher frequencies are blocked. The passband is perfectly flat, the attenuation in the stopband is infinite, and the transition between the two is infinitesimally small. Taking the Inverse Fourier Transform of this ideal frequency response produces the ideal filter kernel (impulse response) shown in (b). As previously discussed (see Chapter 11, Eq. 11-4), this curve is of the general form: sin(x)/x , called the sinc function, given by: h [i ] ' sin(2B fC i ) iB Convolving an input signal with this filter kernel provides a perfect low-pass filter. The problem is, the sinc function continues to both negative and positive infinity without dropping to zero amplitude. While this infinite length is not a problem for mathematics, it is a show stopper for computers. 285 286 The Scientist and Engineer's Guide to Digital Signal Processing To get around this problem, we will make two modifications to the sinc function in (b), resulting in the waveform shown in (c). First, it is truncated to M % 1 points, symmetrically chosen around the main lobe, where M is an even number. All samples outside these M % 1 points are set to zero, or simply ignored. Second, the entire sequence is shifted to the right so that it runs from 0 to M. This allows the filter kernel to be represented using only positive indexes. While many programming languages allow negative indexes, they are a nuisance to use. The sole effect of this M /2 shift in the filter kernel is to shift the output signal by the same amount. Since the modified filter kernel is only an approximation to the ideal filter kernel, it will not have an ideal frequency response. To find the frequency response that is obtained, the Fourier transform can be taken of the signal in (c), resulting in the curve in (d). It's a mess! There is excessive ripple in the passband and poor attenuation in the stopband (recall the Gibbs effect discussed in Chapter 11). These problems result from the abrupt discontinuity at the ends of the truncated sinc function. Increasing the length of the filter kernel does not reduce these problems; the discontinuity is significant no matter how long M is made. Fortunately, there is a simple method of improving this situation. Figure (e) shows a smoothly tapered curve called a Blackman window. Multiplying the truncated-sinc, (c), by the Blackman window, (e), results in the windowedsinc filter kernel shown in (f). The idea is to reduce the abruptness of the truncated ends and thereby improve the frequency response. Figure (g) shows this improvement. The passband is now flat, and the stopband attenuation is so good it cannot be seen in this graph. Several different windows are available, most of them named after their original developers in the 1950s. Only two are worth using, the Hamming window and the Blackman window These are given by: EQUATION 16-1 The Hamming window. These windows run from i ' 0 to M, for a total of M % 1 points. EQUATION 16-2 The Blackman window. w[i ] ' 0.54 & 0.46 cos (2B i / M ) w[i ] ' 0.42 & 0.5 cos (2B i / M ) % 0.08 cos (4B i / M ) Figure 16-2a shows the shape of these two windows for M ' 50 (i.e., 51 total points in the curves). Which of these two windows should you use? It's a trade-off between parameters. As shown in Fig. 16-2b, the Hamming window has about a 20% faster roll-off than the Blackman. However, FIGURE 16-1 (facing page) Derivation of the windowed-sinc filter kernel. The frequency response of the ideal low-pass filter is shown in (a), with the corresponding filter kernel in (b), a sinc function. Since the sinc is infinitely long, it must be truncated to be used in a computer, as shown in (c). However, this truncation results in undesirable changes in the frequency response, (d). The solution is to multiply the truncated-sinc with a smooth window, (e), resulting in the windowed-sinc filter kernel, (f). The frequency response of the windowed-sinc, (g), is smooth and well behaved. These figures are not to scale. Chapter 16- Windowed-Sinc Filters Time Domain Frequency Domain 1.5 1.5 b. Ideal filter kernel a. Ideal frequency response 1.0 Amplitude Amplitude 1.0 0.5 0.0 0.5 0.0 -0.5 -50 -0.5 -25 0 Sample number 25 50 0 1.5 fc Frequency 0.5 1.5 c. Truncated-sinc filter kernel d. Truncated-sinc frequency response 1.0 Amplitude 1.0 Amplitude 287 0.5 0.0 0.5 0.0 abrupt end -0.5 0 M Sample number -0.5 1 0 fc Frequency 0.5 1.5 e. Blackman or Hamming window Amplitude 1.0 0.5 0.0 -0.5 0 M Sample number 1 1.5 1.5 f. Windowed-sinc filter kernel g. Windowed-sinc frequency response 1.0 Amplitude Amplitude 1.0 0.5 0.0 0.5 0.0 -0.5 0 M Sample number -0.5 1 FIGURE 16-1 0 fc Frequency 0.5 288 The Scientist and Engineer's Guide to Digital Signal Processing 1.5 1.5 b. Frequency response 1.0 Amplitude Amplitude a. Blackman and Hamming window Hamming 0.5 1.0 Blackman Hamming 0.5 Blackman 0.0 0.0 0 10 20 30 Sample number 40 50 0 0.1 0.2 0.3 0.4 0.5 Frequency 40 c. Frequency response (dB) 20 Amplitude (dB) FIGURE 16-2 Characteristics of the Blackman and Hamming windows. The shapes of these two windows are shown in (a), and given by Eqs. 16-1 and 16-2. As shown in (b), the Hamming window results in about 20% faster roll-off than the Blackman window. However, the Blackman window has better stopband attenuation (Blackman: 0.02%, Hamming: 0.2%), and a lower passband ripple (Blackman: 0.02% Hamming: 0.2%). 0 -20 -40 Hamming -60 -80 -100 Blackman -120 0 0.1 0.2 0.3 Frequency 0.4 0.5 (c) shows that the Blackman has a better stopband attenuation. To be exact, the stopband attenuation for the Blackman is -74dB ( - 0.02%), while the Hamming is only -53dB (-0.2%). Although it cannot be seen in these graphs, the Blackman has a passband ripple of only about 0.02%, while the Hamming is typically 0.2%. In general, the Blackman should be your first choice; a slow roll-off is easier to handle than poor stopband attenuation. There are other windows you might hear about, although they fall short of the Blackman and Hamming. The Bartlett window is a triangle, using straight lines for the taper. The Hanning window, also called the raised cosine window, is given by: w[i] ' 0.5 & 0.5 cos(2Bi /M) . These two windows have about the same roll-off speed as the Hamming, but worse stopband attenuation (Bartlett: -25dB or 5.6%, Hanning -44dB or 0.63%). You might also hear of a rectangular window. This is the same as no window, just a truncation of the tails (such as in Fig. 16-1c). While the roll-off is -2.5 times faster than the Blackman, the stopband attenuation is only -21dB (8.9%). Designing the Filter To design a windowed-sinc, two parameters must be selected: the cutoff frequency, fC , and the length of the filter kernel, M. The cutoff frequency Chapter 16- Windowed-Sinc Filters 1.5 1.5 a. Roll-off vs. kernel length b. Roll-off vs. cutoff frequency 1.0 1.0 Amplitude Amplitude 289 M=200 M=40 M=20 0.5 fC =0.05 fC=0.25 0.5 f C=0.45 0.0 0.0 0 0.1 0.2 0.3 Frequency 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 Frequency FIGURE 16-3 Filter length vs. roll-off of the windowed-sinc filter. As shown in (a), for M = 20, 40, and 200, the transition bandwidths are BW = 0.2, 0.1, and 0.02 of the sampling rate, respectively. As shown in (b), the shape of the frequency response does not change with different cutoff frequencies. In (b), M = 60. is expressed as a fraction of the sampling rate, and therefore must be between 0 and 0.5. The value for M sets the roll-off according to the approximation: EQUATION 16-3 Filter length vs. roll-off. The length of the filter kernel, M, determines the transition bandwidth of the filter, BW. This is only an approximation since roll-off depends on the particular window being used. M . 4 BW where BW is the width of the transition band, measured from where the curve just barely leaves one, to where it almost reaches zero (say, 99% to 1% of the curve). The transition bandwidth is also expressed as a fraction of the sampling frequency, and must between 0 and 0.5. Figure 16-3a shows an example of how this approximation is used. The three curves shown are generated from filter kernels with: M ' 20, 40, and 200 . From Eq. 16-3, the transition bandwidths are: BW ' 0.2, 0.1, and 0.02 , respectively. Figure (b) shows that the shape of the frequency response does not depend on the cutoff frequency selected. Since the time required for a convolution is proportional to the length of the signals, Eq. 16-3 expresses a trade-off between computation time (depends on the value of M) and filter sharpness (the value of BW). For instance, the 20% slower roll-off of the Blackman window (as compared with the Hamming) can be compensated for by using a filter kernel 20% longer. In other words, it could be said that the Blackman window is 20% slower to execute that an equivalent roll-off Hamming window. This is important because the execution speed of windowed-sinc filters is already terribly slow. As also shown in Fig. 16-3b, the cutoff frequency of the windowed-sinc filter is measured at the one-half amplitude point. Why use 0.5 instead of the 290 The Scientist and Engineer's Guide to Digital Signal Processing standard 0.707 (-3dB) used in analog electronics and other digital filters? This is because the windowed-sinc's frequency response is symmetrical between the passband and the stopband. For instance, the Hamming window results in a passband ripple of 0.2%, and an identical stopband attenuation (i.e., ripple in the stopband) of 0.2%. Other filters do not show this symmetry, and therefore have no advantage in using the one-half amplitude point to mark the cutoff frequency. As shown later in this chapter, this symmetry makes the windowedsinc ideal for spectral inversion. After fC and M have been selected, the filter kernel is calculated from the relation: h [i ] ' K sin(2BfC (i & M /2)) i & M /2 0.42 & 0.5 cos 2Bi M % 0.08 cos 4Bi M EQUATION 16-4 The windowed-sinc filter kernel. The cutoff frequency, fC , is expressed as a fraction of the sampling rate, a value between 0 and 0.5. The length of the filter kernel is determined by M, which must be an even integer. The sample number i, is an integer that runs from 0 to M, resulting in M%1 total points in the filter kernel. The constant, K, is chosen to provide unity gain at zero frequency. To avoid a divide-by-zero error, for i ' M/2 , use h[i ] ' 2B fC K . Don't be intimidated by this equation! Based on the previous discussion, you should be able to identify three components: the sinc function, the M/2 shift, and the Blackman window. For the filter to have unity gain at DC, the constant K must be chosen such that the sum of all the samples is equal to one. In practice, ignore K during the calculation of the filter kernel, and then normalize all of the samples as needed. The program listed in Table 16-1 shows how this is done. Also notice how the calculation is handled at the center of the sinc, i ' M/2 , which involves a division by zero. This equation may be long, but it is easy to use; simply type it into your computer program and forget it. Let the computer handle the calculations. If you find yourself trying to evaluate this equation by hand, you are doing something very very wrong. Let's be specific about where the filter kernel described by Eq. 16-4 is located in your computer array. As an example, M will be chosen to be 100. Remember, M must be an even number. The first point in the filter kernel is in array location 0, while the last point is in array location 100. This means that the entire signal is 101 points long. The center of symmetry is at point 50, i.e., M/2 . The 50 points to the left of point 50 are symmetrical with the 50 points to the right. Point 0 is the same value as point 100, and point 49 is the same as point 51. If you must have a specific number of samples in the filter kernel, such as to use the FFT, simply add zeros to one end or the other. For example, with M ' 100 , you could make samples 101 through 127 equal to zero, resulting in a filter kernel 128 points long. Chapter 16- Windowed-Sinc Filters Filter kernel Step response 1.2 0.10 a. fC = 0.015 M = 500 0.8 0.06 0.04 0.02 0.6 0.4 0.2 0.00 0.0 -0.2 -0.02 0 100 200 300 Sample number 400 0 500 0.10 100 200 300 400 500 200 300 400 500 200 300 400 500 Sample number 1.2 c. fC = 0.04 M = 500 0.08 d. fC = 0.04 M = 500 1.0 0.8 0.06 Amplitude Amplitude b. fC = 0.015 M = 500 1.0 Amplitude Amplitude 0.08 0.04 0.02 0.6 0.4 0.2 0.00 0.0 -0.02 -0.2 0 100 200 300 Sample number 400 500 0 0.10 0.08 Sample number f. fC = 0.04 M = 150 1.0 0.8 0.06 0.04 0.02 0.6 0.4 0.2 0.00 -0.02 -175 100 1.2 e. fC = 0.04 M = 150 Amplitude Amplitude 291 0.0 -0.2 0 150175 Sample number 0 100 Sample number FIGURE 16-4 Example filter kernels and the corresponding step responses. The frequency of the sinusoidal oscillation is approximately equal to the cutoff frequency, fC , while M determines the kernel length. Figure 16-4 shows examples of windowed-sinc filter kernels, and their corresponding step responses. The samples at the beginning and end of the filter kernels are so small that they can't even be seen in the graphs. Don't make the mistake of thinking they are unimportant! These samples may be small in value; however, they collectively have a large effect on the 292 The Scientist and Engineer's Guide to Digital Signal Processing performance of the filter. This is also why floating point representation is typically used to implement windowed-sinc filters. Integers usually don't have enough dynamic range to capture the large variation of values contained in the filter kernel. How does the windowed-sinc filter perform in the time domain? Terrible! The step response has overshoot and ringing; this is not a filter for signals with information encoded in the time domain. Examples of Windowed-Sinc Filters An electroencephalogram, or EEG, is a measurement of the electrical activity of the brain. It can be detected as millivolt level signals appearing on electrodes attached to the surface of the head. Each nerve cell in the brain generates small electrical pulses. The EEG is the combined result of an enormous number of these electrical pulses being generated in a (hopefully) coordinated manner. Although the relationship between thought and this electrical coordination is very poorly understood, different frequencies in the EEG can be identified with specific mental states. If you close your eyes and relax, the predominant EEG pattern will be a slow oscillation between about 7 and 12 hertz. This waveform is called the alpha rhythm, and is associated with contentment and a decreased level of attention. Opening your eyes and looking around causes the EEG to change to the beta rhythm, occurring between about 17 and 20 hertz. Other frequencies and waveforms are seen in children, different depths of sleep, and various brain disorders such as epilepsy. In this example, we will assume that the EEG signal has been amplified by analog electronics, and then digitized at a sampling rate of 100 samples per second. Acquiring data for 50 seconds produces a signal of 5,000 points. Our goal is to separate the alpha from the beta rhythms. To do this, we will design a digital low-pass filter with a cutoff frequency of 14 hertz, or 0.14 1.5 1.5 b. High-pass filter 0.0 0.5 Beta wave Beta wave 0.5 Alpha wave 1.0 Amplitude 1.0 Alpha wave Amplitude a. Low-pass filter 0.0 0 0.1 0.2 0.3 Frequency 0.4 0.5 0 0.1 0.2 0.3 Frequency 0.4 0.5 FIGURE 16-5 Example of windowed-sinc filters. The alpha and beta rhythms in an EEG are separated by low-pass and highpass filters with M ' 100 . The program to implement the low-pass filter is shown in Table 16-1. The program for the high-pass filter is identical, except for a spectral inversion of the low-pass filter kernel. Chapter 16- Windowed-Sinc Filters 1.50 293 40 a. Frequency response 20 b. Frequency response (dB) Amplitude (dB) Amplitude 0 1.00 0.50 -20 -40 -60 -80 -100 0.00 0.15 0.2 Frequency (discrete) 0.25 -120 1500 2000 Frequency (hertz) 2500 FIGURE 16-6 Example of a windowed-sinc band-pass filter. This filter was designed for a sampling rate of 10 kHz. When referenced to the analog signal, the center frequency of the passband is at 2 kHz, the passband is 80 hertz, and the transition bands are 50 hertz. The windowed-sinc uses 801 points in the filter kernel to achieve this roll-off, and a Blackman window for good stopband attenuation. Figure (a) shows the resulting frequency response on a linear scale, while (b) shows it in decibels. The frequency axis in (a) is expressed as a fraction of the sampling frequency, while (b) is expressed in terms of the analog signal before digitization. of the sampling rate. The transition bandwidth will be set at 4 hertz, or 0.04 of the sampling rate. From Eq. 16-3, the filter kernel needs to be about 101 points long, and we will arbitrarily choose to use a Hamming window. The program in Table 16-1 shows how the filter is carried out. The frequency response of the filter, obtained by taking the Fourier Transform of the filter kernel, is shown in Fig. 16-5. In a second example, we will design a band-pass filter to isolate a signaling tone in an audio signal, such as when a button on a telephone is pressed. We will assume that the signal has been digitized at 10 kHz, and the goal is to isolate an 80 hertz band of frequencies centered on 2 kHz. In terms of the sampling rate, we want to block all frequencies below 0.196 and above 0.204 (corresponding to 1960 hertz and 2040 hertz, respectively). To achieve a transition bandwidth of 50 hertz (0.005 of the sampling rate), we will make the filter kernel 801 points long, and use a Blackman window. Table 16-2 contains a program for calculating the filter kernel, while Fig. 16-6 shows the frequency response. The design involves several steps. First, two low-pass filters are designed, one with a cutoff at 0.196, and the other with a cutoff at 0.204. This second filter is then spectrally inverted, making it a high-pass filter (see Chapter 14, Fig. 14-6). Next, the two filter kernels are added, resulting in a band-reject filter (see Fig. 14-8). Finally, another spectral inversion makes this into the desired band-pass filter. Pushing it to the Limit The windowed-sinc filter can be pushed to incredible performance levels without nasty surprises. For instance, suppose you need to isolate a 1 millivolt signal riding on a 120 volt power line. The low-pass filter will need 294 The Scientist and Engineer's Guide to Digital Signal Processing 100 'LOW-PASS WINDOWED-SINC FILTER 110 'This program filters 5000 samples with a 101 point windowed-sinc filter, 120 'resulting in 4900 samples of filtered data. 130 ' 140 DIM X[4999] 'X[ ] holds the input signal 150 DIM Y[4999] 'Y[ ] holds the output signal 160 DIM H[100] 'H[ ] holds the filter kernel 170 ' 180 PI = 3.14159265 190 FC = .14 'Set the cutoff frequency (between 0 and 0.5) 200 M% = 100 'Set filter length (101 points) 210 ' 220 GOSUB XXXX 'Mythical subroutine to load X[ ] 230 ' 240 ' 'Calculate the low-pass filter kernel via Eq. 16-4 250 FOR I% = 0 TO 100 260 IF (I%-M%/2) = 0 THEN H[I%] = 2*PI*FC 270 IF (I%-M%/2) <> 0 THEN H[I%] = SIN(2*PI*FC * (I%-M%/2)) / (I%-M%/2) 280 H[I%] = H[I%] * (0.54 - 0.46*COS(2*PI*I%/M%) ) 290 NEXT I% 300 ' 310 SUM = 0 'Normalize the low-pass filter kernel for 320 FOR I% = 0 TO 100 'unity gain at DC 330 SUM = SUM + H[I%] 340 NEXT I% 350 ' 360 FOR I% = 0 TO 100 370 H[I%] = H[I%] / SUM 380 NEXT I% 390 ' 400 FOR J% = 100 TO 4999 'Convolve the input signal & filter kernel 410 Y[J%] = 0 420 FOR I% = 0 TO 100 430 Y[J%] = Y[J%] + X[J%-I%] * H[I%] 440 NEXT I% 450 NEXT J% 460 ' 470 END TABLE 16-1 a stopband attenuation of at least -120dB (one part in one-million for those that refuse to learn decibels). As previously shown, the Blackman window only provides -74dB (one part in five-thousand). Fortunately, greater stopband attenuation is easy to obtain. The input signal can be filtered using a conventional windowed-sinc filter kernel, providing an intermediate signal. The intermediate signal can then be passed through the filter a second time, further increasing the stopband attenuation to -148dB (1 part in 30 million, wow!). It is also possible to combine the two stages into a single filter. The kernel of the combined filter is equal to the convolution of the filter kernels of the two stages. This also means that convolving any filter kernel with itself results in a filter kernel with a much improved stopband attenuation. The price you pay is a longer filter kernel and a slower roll-off. Figure 16-7a shows the frequency response of a 201 point lowpass filter, formed by convolving a 101 point Blackman windowed-sinc with itself. Amazing performance! (If you really need more than -100dB of stopband attenuation, you should use double precision. Single precision Chapter 16- Windowed-Sinc Filters 100 'BAND-PASS WINDOWED-SINC FILTER 110 'This program calculates an 801 point band-pass filter kernel 120 ' 130 DIM A[800] 'A[ ] workspace for the lower cutoff 140 DIM B[800] 'B[ ] workspace for the upper cutoff 150 DIM H[800] 'H[ ] holds the final filter kernel 160 ' 170 PI = 3.1415926 180 M% = 800 'Set filter kernel length (801 points) 190 ' 200 ' 'Calculate the first low-pass filter kernel via Eq. 16-4, 210 FC = 0.196 'with a cutoff frequency of 0.196, store in A[ ] 220 FOR I% = 0 TO 800 230 IF (I%-M%/2) = 0 THEN A[I%] = 2*PI*FC 240 IF (I%-M%/2) <> 0 THEN A[I%] = SIN(2*PI*FC * (I%-M%/2)) / (I%-M%/2) 250 A[I%] = A[I%] * (0.42 - 0.5*COS(2*PI*I%/M%) + 0.08*COS(4*PI*I%/M%)) 260 NEXT I% 270 ' 280 SUM = 0 'Normalize the first low-pass filter kernel for 290 FOR I% = 0 TO 800 'unity gain at DC 300 SUM = SUM + A[I%] 310 NEXT I% 320 ' 330 FOR I% = 0 TO 800 340 A[I%] = A[I%] / SUM 350 NEXT I% 360 ' 'Calculate the second low-pass filter kernel via Eq. 16-4, 370 FC = 0.204 'with a cutoff frequency of 0.204, store in B[ ] 380 FOR I% = 0 TO 800 390 IF (I%-M%/2) = 0 THEN B[I%] = 2*PI*FC 400 IF (I%-M%/2) <> 0 THEN B[I%] = SIN(2*PI*FC * (I%-M%/2)) / (I%-M%/2) 410 B[I%] = B[I%] * (0.42 - 0.5*COS(2*PI*I%/M%) + 0.08*COS(4*PI*I%/M%)) 420 NEXT I% 430 ' 440 SUM = 0 'Normalize the second low-pass filter kernel for 450 FOR I% = 0 TO 800 'unity gain at DC 460 SUM = SUM + B[I%] 470 NEXT I% 480 ' 490 FOR I% = 0 TO 800 500 B[I%] = B[I%] / SUM 510 NEXT I% 520 ' 530 FOR I% = 0 TO 800 'Change the low-pass filter kernel in B[ ] into a high-pass 540 B[I%] = - B[I%] 'filter kernel using spectral inversion (as in Fig. 14-5) 550 NEXT I% 560 B[400] = B[400] + 1 570 ' 580 ' 590 FOR I% = 0 TO 800 'Add the low-pass filter kernel in A[ ], to the high-pass 600 H[I%] = A[I%] + B[I%] 'filter kernel in B[ ], to form a band-reject filter kernel 610 NEXT I% 'stored in H[ ] (as in Fig. 14-8) 620 ' 630 FOR I% = 0 TO 800 'Change the band-reject filter kernel into a band-pass 640 H[I%] = -H[I%] 'filter kernel by using spectral inversion 650 NEXT I% 660 H[400] = H[400] + 1 670 ' 'The band-pass filter kernel now resides in H[ ] 680 END TABLE 16-2 295 The Scientist and Engineer's Guide to Digital Signal Processing 1.5 60 40 20 0 -20 -40 -60 -80 -100 -120 -140 -160 -180 -200 b. Incredible roll-off ! a. Incredible stopband attenuation ! 1.0 Amplitude Amplitude (dB) 296 single precision round-off noise 0 0.1 0.2 0.3 Frequency 0.4 0.5 0.5 0.0 0.1995 0.2 Frequency 0.2005 FIGURE 16-7 The incredible performance of the windowed-sinc filter. Figure (a) shows the frequency response of a windowed-sinc filter with increased stopband attenuation. This is achieved by convolving a windowed-sinc filter kernel with itself. Figure (b) shows the very rapid roll-off a 32,001 point windowed-sinc filter. round-off noise on signals in the passband can erratically appear in the stopband with amplitudes in the -100dB to -120dB range). Figure 16-7b shows another example of the windowed-sinc's incredible performance: a low-pass filter with 32,001 points in the kernel. The frequency response appears as expected, with a roll-off of 0.000125 of the sampling rate. How good is this filter? Try building an analog electronic filter that passes signals from DC to 1000 hertz with less than a 0.02% variation, and blocks all frequencies above 1001 hertz with less than 0.02% residue. Now that's a filter! If you really want to be impressed, remember that both the filters in Fig. 16-7 use single precision. Using double precision allows these performance levels to be extended by a million times. The strongest limitation of the windowed-sinc filter is the execution time; it can be unacceptably long if there are many points in the filter kernel and standard convolution is used. A high-speed algorithm for this filter (FFT convolution) is presented in Chapter 18. Recursive filters (Chapter 19) also provide good frequency separation and are a reasonable alternative to the windowed-sinc filter. Is the windowed-sinc the optimal filter kernel for separating frequencies? No, filter kernels resulting from more sophisticated techniques can be better. But beware! Before you jump into this very mathematical field, you should consider exactly what you hope to gain. The windowed-sinc will provide any level of performance that you could possibly need. What the advanced filter design methods may provide is a slightly shorter filter kernel for a given level of performance. This, in turn, may mean a slightly faster execution speed. Be warned that you may get little return for the effort expended. CHAPTER 17 Custom Filters Most filters have one of the four standard frequency responses: low-pass, high-pass, band-pass or band-reject. This chapter presents a general method of designing digital filters with an arbitrary frequency response, tailored to the needs of your particular application. DSP excels in this area, solving problems that are far above the capabilities of analog electronics. Two important uses of custom filters are discussed in this chapter: deconvolution, a way of restoring signals that have undergone an unwanted convolution, and optimal filtering, the problem of separating signals with overlapping frequency spectra. This is DSP at its best. Arbitrary Frequency Response The approach used to derive the windowed-sinc filter in the last chapter can also be used to design filters with virtually any frequency response. The only difference is how the desired response is moved from the frequency domain into the time domain. In the windowed-sinc filter, the frequency response and the filter kernel are both represented by equations, and the conversion between them is made by evaluating the mathematics of the Fourier transform. In the method presented here, both signals are represented by arrays of numbers, with a computer program (the FFT) being used to find one from the other. Figure 17-1 shows an example of how this works. The frequency response we want the filter to produce is shown in (a). To say the least, it is very irregular and would be virtually impossible to obtain with analog electronics. This ideal frequency response is defined by an array of numbers that have been selected, not some mathematical equation. In this example, there are 513 samples spread between 0 and 0.5 of the sampling rate. More points could be used to better represent the desired frequency response, while a smaller number may be needed to reduce the computation time during the filter design. However, these concerns are usually small, and 513 is a good length for most applications. 297 298 The Scientist and Engineer's Guide to Digital Signal Processing Besides the desired magnitude array shown in (a), there must be a corresponding phase array of the same length. In this example, the phase of the desired frequency response is entirely zero (this array is not shown in Fig. 17-1). Just as with the magnitude array, the phase array can be loaded with any arbitrary curve you would like the filter to produce. However, remember that the first and last samples (i.e., 0 and 512) of the phase array must have a value of zero (or a multiple of 2 B , which is the same thing). The frequency response can also be specified in rectangular form by defining the array entries for the real and imaginary parts, instead of using the magnitude and phase. The next step is to take the Inverse DFT to move the filter into the time domain. The quickest way to do this is to convert the frequency domain to rectangular form, and then use the Inverse FFT. This results in a 1024 sample signal running from 0 to 1023, as shown in (b). This is the impulse response that corresponds to the frequency response we want; however, it is not suitable for use as a filter kernel (more about this shortly). Just as in the last chapter, it needs to be shifted, truncated, and windowed. In this example, we will design the filter kernel with M ' 40 , i.e., 41 points running from sample 0 to sample 40. Table 17-1 shows a computer program that converts the signal in (b) into the filter kernel shown in (c). As with the windowed-sinc filter, the points near the ends of the filter kernel are so small that they appear to be zero when plotted. Don't make the mistake of thinking they can be deleted! 100 'CUSTOM FILTER DESIGN 110 'This program converts an aliased 1024 point impulse response into an M+1 point 120 'filter kernel (such as Fig. 17-1b being converted into Fig. 17-1c) 130 ' 140 DIM REX[1023] 'REX[ ] holds the signal being converted 150 DIM T[1023] 'T[ ] is a temporary storage buffer 160 ' 170 PI = 3.14159265 180 M% = 40 'Set filter kernel length (41 total points) 190 ' 200 GOSUB XXXX 'Mythical subroutine to load REX[ ] with impulse response 210 ' 220 FOR I% = 0 TO 1023 'Shift (rotate) the signal M/2 points to the right 230 INDEX% = I% + M%/2 240 IF INDEX% > 1023 THEN INDEX% = INDEX%-1024 250 T[INDEX%] = REX[I%] 260 NEXT I% 270 ' 280 FOR I% = 0 TO 1023 290 REX[I%] = T[I%] 300 NEXT I% 310 ' 'Truncate and window the signal 320 FOR I% = 0 TO 1023 330 IF I% <= M% THEN REX[I%] = REX[I%] * (0.54 - 0.46 * COS(2*PI*I%/M%)) 340 IF I% > M% THEN REX[I%] = 0 350 NEXT I% 360 ' 'The filter kernel now resides in REX[0] to REX[40] 370 END TABLE 17-1 Chapter 17- Custom Filters Time Domain 299 Frequency Domain 1.5 3 b. Impulse response (aliased) a. Desired frequency response Amplitude Amplitude 1.0 0.5 2 1 0.0 -0.5 0 0 256 512 Sample number 768 1024 1023 0 1.5 0.1 0.2 0.3 Frequency 0.4 0.5 0.4 0.5 3 c. Filter kernel d. Actual frequency response Amplitude Amplitude 1.0 0.5 added zeros 2 1 0.0 -0.5 0 0 10 20 30 Sample number 40 1023 50 0 0.1 0.2 0.3 Frequency FIGURE 17-1 Example of FIR filter design. Figure (a) shows the desired frequency response, with 513 samples running between 0 to 0.5 of the sampling rate. Taking the Inverse DFT results in (b), an aliased impulse response composed of 1024 samples. To form the filter kernel, (c), the aliased impulse response is truncated to M% 1 samples, shifted to the right by M/2 samples, and multiplied by a Hamming or Blackman window. In this example, M is 40. The program in Table 17-1 shows how this is done. The filter kernel is tested by padding it with zeros and taking the DFT, providing the actual frequency response of the filter, (d). The last step is to test the filter kernel. This is done by taking the DFT (using the FFT) to find the actual frequency response, as shown in (d). To obtain better resolution in the frequency domain, pad the filter kernel with zeros before the FFT. For instance, using 1024 total samples (41 in the filter kernel, plus 983 zeros), results in 513 samples between 0 and 0.5. As shown in Fig. 17-2, the length of the filter kernel determines how well the actual frequency response matches the desired frequency response. The exceptional performance of FIR digital filters is apparent; virtually any frequency response can be obtained if a long enough filter kernel is used. This is the entire design method; however, there is a subtle theoretical issue that needs to be clarified. Why isn't it possible to directly use the impulse response shown in 17-1b as the filter kernel? After all, if (a) is the Fourier transform of (b), wouldn't convolving an input signal with (b) produce the exact frequency response we want? The answer is no, and here's why. 300 The Scientist and Engineer's Guide to Digital Signal Processing When designing a custom filter, the desired frequency response is defined by the values in an array. Now consider this: what does the frequency response do between the specified points? For simplicity, two cases can be imagined, one "good" and one "bad." In the "good" case, the frequency response is a smooth curve between the defined samples. In the "bad" case, there are wild fluctuations between. As luck would have it, the impulse response in (b) corresponds to the "bad" frequency response. This can be shown by padding it with a large number of zeros, and then taking the DFT. The frequency response obtained by this method will show the erratic behavior between the originally defined samples, and look just awful. To understand this, imagine that we force the frequency response to be what we want by defining it at an infinite number of points between 0 and 0.5. That is, we create a continuous curve. The inverse DTFT is then used to find the impulse response, which will be infinite in length. In other words, the "good" frequency response corresponds to something that cannot be represented in a computer, an infinitely long impulse response. When we represent the frequency spectrum with N/2 % 1 samples, only N points are provided in the time domain, making it unable to correctly contain the signal. The result is that the infinitely long impulse response wraps up (aliases) into the N points. When this aliasing occurs, the frequency response changes from "good" to "bad." Fortunately, windowing the N point impulse response greatly reduces this aliasing, providing a smooth curve between the frequency domain samples. Designing a digital filter to produce a given frequency response is quite simple. The hard part is finding what frequency response to use. Let's look at some strategies used in DSP to design custom filters. Deconvolution Unwanted convolution is an inherent problem in transferring analog information. For instance, all of the following can be modeled as a convolution: image blurring in a shaky camera, echoes in long distance telephone calls, the finite bandwidth of analog sensors and electronics, etc. Deconvolution is the process of filtering a signal to compensate for an undesired convolution. The goal of deconvolution is to recreate the signal as it existed before the convolution took place. This usually requires the characteristics of the convolution (i.e., the impulse or frequency response) to be known. This can be distinguished from blind deconvolution, where the characteristics of the parasitic convolution are not known. Blind deconvolution is a much more difficult problem that has no general solution, and the approach must be tailored to the particular application. Deconvolution is nearly impossible to understand in the time domain, but quite straightforward in the frequency domain. Each sinusoid that composes the original signal can be changed in amplitude and/or phase as it passes through the undesired convolution. To extract the original signal, the deconvolution filter must undo these amplitude and phase changes. For Chapter 17- Custom Filters 3 3 a. M = 10 c. M = 100 2 Amplitude Amplitude 2 1 1 0 0 0 0.1 0.2 0.3 Frequency 0.4 0.5 0 3 0.1 0.2 0.3 0.4 0.5 0.2 0.3 0.4 0.5 0.2 0.3 0.4 0.5 Frequency 3 b. M = 30 d. M = 300 2 Amplitude 2 1 0 1 0 0 0.1 0.2 0.3 Frequency 0.4 0.5 0 0.1 Frequency 3 FIGURE 17-2 Frequency response vs. filter kernel length. These figures show the frequency responses obtained with various lengths of filter kernels. The number of points in each filter kernel is equal to M% 1 , running from 0 to M. As more points are used in the filter kernel, the resulting frequency response more closely matches the desired frequency response. Figure 17-1a shows the desired frequency response for this example. e. M = 1000 2 Amplitude Amplitude 301 1 0 0 0.1 Frequency example, if the convolution changes a sinusoid's amplitude by 0.5 with a 30 degree phase shift, the deconvolution filter must amplify the sinusoid by 2.0 with a -30 degree phase change. The example we will use to illustrate deconvolution is a gamma ray detector. As illustrated in Fig. 17-3, this device is composed of two parts, a scintillator and a light detector. A scintillator is a special type of transparent material, such as sodium iodide or bismuth germanate. These compounds change the energy in each gamma ray into a brief burst of visible light. This light Voltage The Scientist and Engineer's Guide to Digital Signal Processing Energy 302 Time Time scintillator gamma ray amplifier light light detector FIGURE 17-3 Example of an unavoidable convolution. A gamma ray detector can be formed by mounting a scintillator on a light detector. When a gamma ray strikes the scintillator, its energy is converted into a pulse of light. This pulse of light is then converted into an electronic signal by the light detector. The gamma ray is an impulse, while the output of the detector (i.e., the impulse response) resembles a one-sided exponential. is then converted into an electronic signal by a light detector, such as a photodiode or photomultiplier tube. Each pulse produced by the detector resembles a one-sided exponential, with some rounding of the corners. This shape is determined by the characteristics of the scintillator used. When a gamma ray deposits its energy into the scintillator, nearby atoms are excited to a higher energy level. These atoms randomly deexcite, each producing a single photon of visible light. The net result is a light pulse whose amplitude decays over a few hundred nanoseconds (for sodium iodide). Since the arrival of each gamma ray is an impulse, the output pulse from the detector (i.e., the one-sided exponential) is the impulse response of the system. Figure 17-4a shows pulses generated by the detector in response to randomly arriving gamma rays. The information we would like to extract from this output signal is the amplitude of each pulse, which is proportional to the energy of the gamma ray that generated it. This is useful information because the energy can tell interesting things about where the gamma ray has been. For example, it may provide medical information on a patient, tell the age of a distant galaxy, detect a bomb in airline luggage, etc. Everything would be fine if only an occasional gamma ray were detected, but this is usually not the case. As shown in (a), two or more pulses may overlap, shifting the measured amplitude. One answer to this problem is to deconvolve the detector's output signal, making the pulses narrower so that less pile-up occurs. Ideally, we would like each pulse to resemble the original impulse. As you may suspect, this isn't possible and we must settle for a pulse that is finite in length, but significantly shorter than the detected pulse. This goal is illustrated in Fig. 17-4b. Chapter 17- Custom Filters 2 2 a. Detected pulses b. Filtered pulses 1 Amplitude Amplitude 303 0 -1 1 0 -1 0 100 200 300 Sample number 400 500 0 100 200 300 Sample number 400 500 FIGURE 17-4 Example of deconvolution. Figure (a) shows the output signal from a gamma ray detector in response to a series of randomly arriving gamma rays. The deconvolution filter is designed to convert (a) into (b), by reducing the width of the pulses. This minimizes the amplitude shift when pulses land on top of each other. Even though the detector signal has its information encoded in the time domain, much of our analysis must be done in the frequency domain, where the problem is easier to understand. Figure 17-5a is the signal produced by the detector (something we know). Figure (c) is the signal we wish to have (also something we know). This desired pulse was arbitrarily selected to be the same shape as a Blackman window, with a length about one-third that of the original pulse. Our goal is to find a filter kernel, (e), that when convolved with the signal in (a), produces the signal in (c). In equation form: if a t e ' c , and given a and c, find e. If these signals were combined by addition or multiplication instead of convolution, the solution would be easy: subtraction is used to "de-add" and division is used to "de-multiply." Convolution is different; there is not a simple inverse operation that can be called "deconvolution." Convolution is too messy to be undone by directly manipulating the time domain signals. Fortunately, this problem is simpler in the frequency domain. Remember, convolution in one domain corresponds with multiplication in the other domain. Again referring to the signals in Fig. 17-5: if b ×f ' d , and given b and d, find f. This is an easy problem to solve: the frequency response of the filter, (f), is the frequency spectrum of the desired pulse, (d), divided by the frequency spectrum of the detected pulse, (b). Since the detected pulse is asymmetrical, it will have a nonzero phase. This means that a complex division must be used (that is, a magnitude & phase divided by another magnitude & phase). In case you have forgotten, Chapter 9 defines how to perform a complex division of one spectrum by another. The required filter kernel, (e), is then found from the frequency response by the custom filter method (IDFT, shift, truncate, & multiply by a window). There are limits to the improvement that deconvolution can provide. In other words, if you get greedy, things will fall apart. Getting greedy in this 304 The Scientist and Engineer's Guide to Digital Signal Processing example means trying to make the desired pulse excessively narrow. Let's look at what happens. If the desired pulse is made narrower, its frequency spectrum must contain more high frequency components. Since these high frequency components are at a very low amplitude in the detected pulse, the filter must have a very high gain at these frequencies. For instance, (f) shows that some frequencies must be multiplied by a factor of three to achieve the desired pulse in (c). If the desired pulse is made narrower, the gain of the deconvolution filter will be even greater at high frequencies. The problem is, small errors are very unforgiving in this situation. For instance, if some frequency is amplified by 30, when only 28 is required, the deconvolved signal will probably be a mess. When the deconvolution is pushed to greater levels of performance, the characteristics of the unwanted convolution must be understood with greater accuracy and precision. There are always unknowns in real world applications, caused by such villains as: electronic noise, temperature drift, variation between devices, etc. These unknowns set a limit on how well deconvolution will work. Even if the unwanted convolution is perfectly understood, there is still a factor that limits the performance of deconvolution: noise. For instance, most unwanted convolutions take the form of a low-pass filter, reducing the amplitude of the high frequency components in the signal. Deconvolution corrects this by amplifying these frequencies. However, if the amplitude of these components falls below the inherent noise of the system, the information contained in these frequencies is lost. No amount of signal processing can retrieve it. It's gone forever. Adios! Goodbye! Sayonara! Trying to reclaim this data will only amplify the noise. As an extreme case, the amplitude of some frequencies may be completely reduced to zero. This not only obliterates the information, it will try to make the deconvolution filter have infinite gain at these frequencies. The solution: design a less aggressive deconvolution filter and/or place limits on how much gain is allowed at any of the frequencies. How far can you go? How greedy is too greedy? This depends totally on the problem you are attacking. If the signal is well behaved and has low noise, a significant improvement can probably be made (think a factor of 5-10). If the signal changes over time, isn't especially well understood, or is noisy, you won't do nearly as well (think a factor of 1-2). Successful deconvolution involves a great deal of testing. If it works at some level, try going farther; you will know when it falls apart. No amount of theoretical work will allow you to bypass this iterative process. Deconvolution can also be applied to frequency domain encoded signals. A classic example is the restoration of old recordings of the famous opera singer, Enrico Caruso (1873-1921). These recordings were made with very primitive equipment by modern standards. The most significant problem is the resonances of the long tubular recording horn used to gather the sound. Whenever the singer happens to hit one of these resonance frequencies, the loudness of the recording abruptly increases. Digital deconvolution has improved the subjective quality of these recordings by Chapter 17- Custom Filters Time Domain 305 Frequency Domain 1.5 1.5 a. Detected pulse b. Detected frequency spectrum Amplitude Amplitude 1.0 0.5 1.0 0.5 0.0 Gamma ray strikes -0.5 0.0 0 10 20 30 Sample number 40 50 0 1.5 0.1 0.2 0.3 Frequency 0.4 0.5 1.5 c. Desired pulse d. Desired frequency spectrum Amplitude Amplitude 1.0 0.5 1.0 0.5 0.0 -0.5 0.0 0 10 20 30 Sample number 40 50 0 0.4 0.2 0.3 Frequency 0.4 0.5 4.0 e. Required filter kernel f. Required Frequency response 0.2 3.0 Amplitude Amplitude 0.1 0.0 -0.2 2.0 1.0 -0.4 0.0 0 10 20 30 Sample number 40 50 0 0.1 0.2 0.3 0.4 0.5 Frequency FIGURE 17-5 Example of deconvolution in the time and frequency domains. The impulse response of the example gamma ray detector is shown in (a), while the desired impulse response is shown in (c). The frequency spectra of these two signals are shown in (b) and (d), respectively. The filter that changes (a) into (c) has a frequency response, (f), equal to (b) divided by (d). The filter kernel of this filter, (e), is then found from the frequency response using the custom filter design method (inverse DFT, truncation, windowing). Only the magnitudes of the frequency domain signals are shown in this illustration; however, the phases are nonzero and must also be used. reducing the loud spots in the music. We will only describe the general method; for a detailed description, see the original paper: T. Stockham, T. Cannon, and R. Ingebretsen, "Blind Deconvolution Through Digital Signal Processing", Proc. IEEE, vol. 63, Apr. 1975, pp. 678-692. The Scientist and Engineer's Guide to Digital Signal Processing Frequency e. Deconvolved spectrum Amplitude c. Recorded spectrum Amplitude a. Original spectrum Amplitude Frequency Frequency Undesired Convolution b. Frequency response d. Frequency response Amplitude Deconvolution Amplitude 306 Frequency Frequency FIGURE 17-6 Deconvolution of old phonograph recordings. The frequency spectrum produced by the original singer is illustrated in (a). Resonance peaks in the primitive equipment, (b), produce distortion in the recorded frequency spectrum, (c). The frequency response of the deconvolution filter, (d), is designed to counteracts the undesired convolution, restoring the original spectrum, (e). These graphs are for illustrative purposes only; they are not actual signals. Figure 17-6 shows the general approach. The frequency spectrum of the original audio signal is illustrated in (a). Figure (b) shows the frequency response of the recording equipment, a relatively smooth curve except for several sharp resonance peaks. The spectrum of the recorded signal, shown in (c), is equal to the true spectrum, (a), multiplied by the uneven frequency response, (b). The goal of the deconvolution is to counteract the undesired convolution. In other words, the frequency response of the deconvolution filter, (d), must be the inverse of (b). That is, each peak in (b) is cancelled by a corresponding dip in (d). If this filter were perfectly designed, the resulting signal would have a spectrum, (e), identical to that of the original. Here's the catch: the original recording equipment has long been discarded, and its frequency response, (b), is a mystery. In other words, this is a blind deconvolution problem; given only (c), how can we determine (d)? Blind deconvolution problems are usually attacked by making an estimate or assumption about the unknown parameters. To deal with this example, the average spectrum of the original music is assumed to match the average spectrum of the same music performed by a present day singer using modern equipment. The average spectrum is found by the techniques of Chapter 9: Chapter 17- Custom Filters 307 break the signal into a large number of segments, take the DFT of each segment, convert into polar form, and then average the magnitudes together. In the simplest case, the unknown frequency response is taken as the average spectrum of the old recording, divided by the average spectrum of the modern recording. (The method used by Stockham et al. is based on a more sophisticated technique called homomorphic processing, providing a better estimate of the characteristics of the recording system). Optimal Filters Figure 17-7a illustrates a common filtering problem: trying to extract a waveform (in this example, an exponential pulse) buried in random noise. As shown in (b), this problem is no easier in the frequency domain. The signal has a spectrum composed mainly of low frequency components. In comparison, the spectrum of the noise is white (the same amplitude at all frequencies). Since the spectra of the signal and noise overlap, it is not clear how the two can best be separated. In fact, the real question is how to define what "best" means. We will look at three filters, each of which is "best" (optimal) in a different way. Figure 17-8 shows the filter kernel and frequency response for each of these filters. Figure 17-9 shows the result of using these filters on the example waveform of Fig. 17-7a. The moving average filter is the topic of Chapter 15. As you recall, each output point produced by the moving average filter is the average of a certain number of points from the input signal. This makes the filter kernel a rectangular pulse with an amplitude equal to the reciprocal of the number of points in the average. The moving average filter is optimal in the sense that it provides the fastest step response for a given noise reduction. The matched filter was previously discussed in Chapter 7. As shown in Fig. 17-8a, the filter kernel of the matched filter is the same as the target signal 1.5 15 a. Signal + noise (time domain) b. Signal + noise (frequency spectrum) signal 10 Amplitude Amplitude 1.0 noise 0.5 signal noise 5 0.0 -0.5 0 0 100 200 300 Sample number 400 500 0 0.1 0.2 0.3 Frequency 0.4 0.5 FIGURE 17-7 Example of optimal filtering. In (a), an exponential pulse buried in random noise. The frequency spectra of the pulse and noise are shown in (b). Since the signal and noise overlap in both the time and frequency domains, the best way to separate them isn't obvious. 308 The Scientist and Engineer's Guide to Digital Signal Processing 1.5 0.25 b. Frequency response a. Filter kernel Wiener moving average 0.15 0.10 Amplitude Amplitude 0.20 matched moving average 1.0 Wiener matched 0.5 0.05 0.0 0.00 0 10 20 30 40 50 0 0.1 Sample number 0.2 0.3 Frequency 0.4 0.5 FIGURE 17-8 Example of optimal filters. In (a), three filter kernels are shown, each of which is optimal in some sense. The corresponding frequency responses are shown in (b). The moving average filter is designed to have a rectangular pulse for a filter kernel. In comparison, the filter kernel of the matched filter looks like the signal being detected. The Wiener filter is designed in the frequency domain, based on the relative amounts of signal and noise present at each frequency. being detected, except it has been flipped left-for-right. The idea behind the matched filter is correlation, and this flip is required to perform correlation using convolution. The amplitude of each point in the output signal is a measure of how well the filter kernel matches the corresponding section of the input signal. Recall that the output of a matched filter does not necessarily look like the signal being detected. This doesn't really matter; if a matched filter is used, the shape of the target signal must already be known. The matched filter is optimal in the sense that the top of the peak is farther above the noise than can be achieved with any other linear filter (see Fig. 17-9b). The Wiener filter (named after the optimal estimation theory of Norbert Wiener) separates signals based on their frequency spectra. As shown in Fig. 17-7b, at some frequencies there is mostly signal, while at others there is mostly noise. It seems logical that the "mostly signal" frequencies should be passed through the filter, while the "mostly noise" frequencies should be blocked. The Wiener filter takes this idea a step further; the gain of the filter at each frequency is determined by the relative amount of signal and noise at that frequency: EQUATION 17-1 The Wiener filter. The frequency response, represented by H [ f ] , is determined by the frequency spectra of the noise, N [ f ] , and the signal, S [ f ] . Only the magnitudes are important; all of the phases are zero. H[ f ] ' S[ f ]2 S [ f ]2 % N [ f ]2 This relation is used to convert the spectra in Fig. 17-7b into the Wiener filter's frequency response in Fig. 17-8b. The Wiener filter is optimal in the sense that it maximizes the ratio of the signal power to the noise power Chapter 17- Custom Filters 1.5 1.5 a. Moving average filter b. Matched filter 1.0 Amplitude 1.0 0.5 0.0 0.5 0.0 -0.5 -0.5 0 100 200 300 Sample number 400 500 0 100 200 300 400 500 300 400 500 Sample number 1.5 FIGURE 17-9 Example of using three optimal filters. These signals result from filtering the waveform in Fig. 17-7 with the filters in Fig. 17-8. Each of these three filters is optimal in some sense. In (a), the moving average filter results in the sharpest edge response for a given level of random noise reduction. In (b), the matched filter produces a peak that is farther above the residue noise than provided by any other filter. In (c), the Wiener filter optimizes the signal-to-noise ratio. c. Weiner filter 1.0 Amplitude Amplitude 309 0.5 0.0 -0.5 0 100 200 Sample number (over the length of the signal, not at each individual point). An appropriate filter kernel is designed from the Wiener frequency response using the custom method. While the ideas behind these optimal filters are mathematically elegant, they often fail in practicality. This isn't to say they should never be used. The point is, don't hear the word "optimal" and stop thinking. Let's look at several reasons why you might not want to use them. First, the difference between the signals in Fig. 17-9 is very unimpressive. In fact, if you weren't told what parameters were being optimized, you probably couldn't tell by looking at the signals. This is usually the case for problems involving overlapping frequency spectra. The small amount of extra performance obtained from an optimal filter may not be worth the the increased program complexity, the extra design effort, or the longer execution time. Second: The Wiener and matched filters are completely determined by the characteristics of the problem. Other filters, such as the windowed-sinc and moving average, can be tailored to your liking. Optimal filter advocates would claim that this diddling can only reduce the effectiveness of the filter. This is 310 The Scientist and Engineer's Guide to Digital Signal Processing very arguable. Remember, each of these filters is optimal in one specific way (i.e., "in some sense"). This is seldom sufficient to claim that the entire problem has been optimized, especially if the resulting signals are interpreted by a human observer. For instance, a biomedical engineer might use a Wiener filter to maximize the signal-to-noise ratio in an electro-cardiogram. However, it is not obvious that this also optimizes a physician's ability to detect irregular heart activity by looking at the signal. Third: The Wiener and matched filter must be carried out by convolution, making them extremely slow to execute. Even with the speed improvements discussed in the next chapter (FFT convolution), the computation time can be excessively long. In comparison, recursive filters (such as the moving average or others presented in Chapter 19) are much faster, and may provide an acceptable level of performance. CHAPTER FFT Convolution 18 This chapter presents two important DSP techniques, the overlap-add method, and FFT convolution. The overlap-add method is used to break long signals into smaller segments for easier processing. FFT convolution uses the overlap-add method together with the Fast Fourier Transform, allowing signals to be convolved by multiplying their frequency spectra. For filter kernels longer than about 64 points, FFT convolution is faster than standard convolution, while producing exactly the same result. The Overlap-Add Method There are many DSP applications where a long signal must be filtered in segments. For instance, high fidelity digital audio requires a data rate of about 5 Mbytes/min, while digital video requires about 500 Mbytes/min. With data rates this high, it is common for computers to have insufficient memory to simultaneously hold the entire signal to be processed. There are also systems that process segment-by-segment because they operate in real time. For example, telephone signals cannot be delayed by more than a few hundred milliseconds, limiting the amount of data that are available for processing at any one instant. In still other applications, the processing may require that the signal be segmented. An example is FFT convolution, the main topic of this chapter. The overlap-add method is based on the fundamental technique in DSP: (1) decompose the signal into simple components, (2) process each of the components in some useful way, and (3) recombine the processed components into the final signal. Figure 18-1 shows an example of how this is done for the overlap-add method. Figure (a) is the signal to be filtered, while (b) shows the filter kernel to be used, a windowed-sinc low-pass filter. Jumping to the bottom of the figure, (i) shows the filtered signal, a smoothed version of (a). The key to this method is how the lengths of these signals are affected by the convolution. When an N sample signal is convolved with an M sample 311 312 The Scientist and Engineer's Guide to Digital Signal Processing filter kernel, the output signal is N% M& 1 samples long. For instance, the input signal, (a), is 300 samples (running from 0 to 299), the filter kernel, (b), is 101 samples (running from 0 to 100), and the output signal, (i), is 400 samples (running from 0 to 399). In other words, when an N sample signal is filtered, it will be expanded by M& 1 points to the right. (This is assuming that the filter kernel runs from index 0 to M. If negative indexes are used in the filter kernel, the expansion will also be to the left). In (a), zeros have been added to the signal between sample 300 and 399 to illustrate where this expansion will occur. Don't be confused by the small values at the ends of the output signal, (i). This is simply a result of the windowed-sinc filter kernel having small values near its ends. All 400 samples in (i) are nonzero, even though some of them are too small to be seen in the graph. Figures (c), (d) and (e) show the decomposition used in the overlap-add method. The signal is broken into segments, with each segment having 100 samples from the original signal. In addition, 100 zeros are added to the right of each segment. In the next step, each segment is individually filtered by convolving it with the filter kernel. This produces the output segments shown in (f), (g), and (h). Since each input segment is 100 samples long, and the filter kernel is 101 samples long, each output segment will be 200 samples long. The important point to understand is that the 100 zeros were added to each input segment to allow for the expansion during the convolution. Notice that the expansion results in the output segments overlapping each other. These overlapping output segments are added to give the output signal, (i). For instance, samples 200 to 299 in (i) are found by adding the corresponding samples in (g) and (h). The overlap-add method produces exactly the same output signal as direct convolution. The disadvantage is a much greater program complexity to keep track of the overlapping samples. FFT Convolution FFT convolution uses the principle that multiplication in the frequency domain corresponds to convolution in the time domain. The input signal is transformed into the frequency domain using the DFT, multiplied by the frequency response of the filter, and then transformed back into the time domain using the Inverse DFT. This basic technique was known since the days of Fourier; however, no one really cared. This is because the time required to calculate the DFT was longer than the time to directly calculate the convolution. This changed in 1965 with the development of the Fast Fourier Transform (FFT). By using the FFT algorithm to calculate the DFT, convolution via the frequency domain can be faster than directly convolving the time domain signals. The final result is the same; only the number of calculations has been changed by a more efficient algorithm. For this reason, FFT convolution is also called high-speed convolution. Chapter 18- FFT Convolution 0.180 4 2 Amplitude Amplitude a. Input signal 0 -2 ? b. Filter kernel 0.120 0.060 0.000 -0.060 -4 0 100 200 300 Sample number 400 -50 0 50 100 150 Sample 4 4 f. Output segment 1 2 Amplitude Amplitude c. Input segment 1 0 -2 added zeros -4 2 0 -2 -4 0 100 200 300 Sample number 400 0 4 200 300 400 300 400 300 400 300 400 Sample number g. Output segment 2 2 Amplitude Amplitude 100 4 d. Input segment 2 0 -2 -4 2 0 -2 -4 0 100 200 300 Sample number 400 0 4 100 200 Sample number 4 e. Input segment 3 h. Output segment 3 2 Amplitude Amplitude 313 0 -2 -4 2 0 -2 -4 100 200 300 Sample number 400 FIGURE 18-1 The overlap-add method. The goal is to convolve the input signal, (a), with the filter kernel, (b). This is done by breaking the input signal into a number of segments, such as (c), (d) and (e), each padded with enough zeros to allow for the expansion during the convolution. Convolving each of the input segments with the filter kernel produces the output segments, (f), (g), and (h). The output signal, (i), is then found by adding the overlapping output segments. 0 100 200 Sample number 4 i. Output signal 2 Amplitude 0 0 -2 -4 0 100 200 Sample number 314 The Scientist and Engineer's Guide to Digital Signal Processing FFT convolution uses the overlap-add method shown in Fig. 18-1; only the way that the input segments are converted into the output segments is changed. Figure 18-2 shows an example of how an input segment is converted into an output segment by FFT convolution. To start, the frequency response of the filter is found by taking the DFT of the filter kernel, using the FFT. For instance, (a) shows an example filter kernel, a windowed-sinc band-pass filter. The FFT converts this into the real and imaginary parts of the frequency response, shown in (b) & (c). These frequency domain signals may not look like a band-pass filter because they are in rectangular form. Remember, polar form is usually best for humans to understand the frequency domain, while rectangular form is normally best for mathematical calculations. These real and imaginary parts are stored in the computer for use when each segment is being calculated. Figure (d) shows the input segment to being processed. The FFT is used to find its frequency spectrum, shown in (e) & (f). The frequency spectrum of the output segment, (h) & (i) is then found by multiplying the filter's frequency response, (b) & (c), by the spectrum of the input segment, (e) & (f). Since these spectra consist of real and imaginary parts, they are multiplied according to Eq. 9-1 in Chapter 9. The Inverse FFT is then used to find the output segment, (g), from its frequency spectrum, (h) & (i). It is important to recognize that this output segment is exactly the same as would be obtained by the direct convolution of the input segment, (d), and the filter kernel, (a). The FFTs must be long enough that circular convolution does not take place (also described in Chapter 9). This means that the FFT should be the same length as the output segment, (g). For instance, in the example of Fig. 18-2, the filter kernel contains 129 points and each segment contains 128 points, making output segment 256 points long. This calls for 256 point FFTs to be used. This means that the filter kernel, (a), must be padded with 127 zeros to bring it to a total length of 256 points. Likewise, each of the input segments, (d), must be padded with 128 zeros. As another example, imagine you need to convolve a very long signal with a filter kernel having 600 samples. One alternative would be to use segments of 425 points, and 1024 point FFTs. Another alternative would be to use segments of 1449 points, and 2048 point FFTs. Table 18-1 shows an example program to carry out FFT convolution. This program filters a 10 million point signal by convolving it with a 400 point filter kernel. This is done by breaking the input signal into 16000 segments, with each segment having 625 points. When each of these segments is convolved with the filter kernel, an output segment of 625 % 400 & 1 ' 1024 points is produced. Thus, 1024 point FFTs are used. After defining and initializing all the arrays (lines 130 to 230), the first step is to calculate and store the frequency response of the filter (lines 250 to 310). Line 260 calls a mythical subroutine that loads the filter kernel into XX[0] through XX[399], and sets XX[400] through XX[1023] to a value of zero. The subroutine in line 270 is the FFT, transforming the 1024 samples held in XX[ ] into the 513 samples held in REX[ ] & IMX[ ], the real and Chapter 18- FFT Convolution Time Domain Frequency Domain 0.3 2.0 2.0 b. Real a. Filter kernel 0.2 c. Imaginary 0.1 FFT 0.0 1.0 Amplitude 1.0 Amplitude Amplitude 315 0.0 -1.0 -0.1 -1.0 signal in 0 to 128 zeros in 129 to 255 -2.0 -0.2 0 64 128 Sample number 192 -2.0 0 255 256 64 128 0 Frequency 6.0 100 e. Real 50 Amplitude -2.0 Amplitude Amplitude FFT 0.0 128 f. Imaginary 50 2.0 64 Frequency 100 d. Input segment 4.0 0 -50 0 -50 signal in 0 to 127 zeros in 128 to 255 -4.0 -6.0 -100 0 64 128 Sample number 192 255 256 -100 0 6.0 64 Frequency 128 0 100 h. Real 50 Amplitude -2.0 Amplitude IFFT 0.0 0 -50 -4.0 128 i. Imaginary 50 2.0 64 Frequency 100 g. Output segment 4.0 Amplitude 0.0 0 -50 signal in 0 to 255 -6.0 -100 0 64 128 Sample number 192 255 256 -100 0 64 Frequency 128 0 64 Frequency 128 FIGURE 18-2 FFT convolution. The filter kernel, (a), and the signal segment, (d), are converted into their respective spectra, (b) & (c) and (d) & (e), via the FFT. These spectra are multiplied, resulting in the spectrum of the output segment, (h) & (i). The Inverse FFT then finds the output segment, (g). imaginary parts of the frequency response. These values are transferred into the arrays REFR[ ] & IMFR[ ] (for: REal and IMaginary Frequency Response), to be used later in the program. 316 The Scientist and Engineer's Guide to Digital Signal Processing The FOR-NEXT loop between lines 340 and 580 controls how the 16000 segments are processed. In line 360, a subroutine loads the next segment to be processed into XX[0] through XX[624], and sets XX[625] through XX[1023] to a value of zero. In line 370, the FFT subroutine is used to find this segment's frequency spectrum, with the real part being placed in the 513 points of REX[ ], and the imaginary part being placed in the 513 points of IMX[ ]. Lines 390 to 430 show the multiplication of the segment's frequency spectrum, held in REX[ ] & IMX[ ], by the filter's frequency response, held in REFR[ ] and IMFR[ ]. The result of the multiplication is stored in REX[ ] & IMX[ ], overwriting the data previously there. Since this is now the frequency spectrum of the output segment, the IFFT can be used to find the output segment. This is done by the mythical IFFT subroutine in line 450, which transforms the 513 points held in REX[ ] & IMX[ ] into the 1024 points held in XX[ ], the output segment. Lines 470 to 550 handle the overlapping of the segments. Each output segment is divided into two sections. The first 625 points (0 to 624) need to be combined with the overlap from the previous output segment, and then written to the output signal. The last 399 points (625 to 1023) need to be saved so that they can overlap with the next output segment. To understand this, look back at Fig 18-1. Samples 100 to 199 in (g) need to be combined with the overlap from the previous output segment, (f), and can then be moved to the output signal (i). In comparison, samples 200 to 299 in (g) need to be saved so that they can be combined with the next output segment, (h). Now back to the program. The array OLAP[ ] is used to hold the 399 samples that overlap from one segment to the next. In lines 470 to 490 the 399 values in this array (from the previous output segment) are added to the output segment currently being worked on, held in XX[ ]. The mythical subroutine in line 550 then outputs the 625 samples in XX[0] to XX[624] to the file holding the output signal. The 399 samples of the current output segment that need to be held over to the next output segment are then stored in OLAP[ ] in lines 510 to 530. After all 0 to 15999 segments have been processed, the array, OLAP[ ], will contain the 399 samples from segment 15999 that should overlap segment 16000. Since segment 16000 doesn't exist (or can be viewed as containing all zeros), the 399 samples are written to the output signal in line 600. This makes the length of the output signal 16000 ×625 % 399 ' 10,000,399 points. This matches the length of input signal, plus the length of the filter kernel, minus 1. Speed Improvements When is FFT convolution faster than standard convolution? The answer depends on the length of the filter kernel, as shown in Fig. 18-3. The time Chapter 18- FFT Convolution 317 100 'FFT CONVOLUTION 110 'This program convolves a 10 million point signal with a 400 point filter kernel. The input 120 'signal is broken into 16000 segments, each with 625 points. 1024 point FFTs are used. 130 ' 130 ' 'INITIALIZE THE ARRAYS 140 DIM XX[1023] 'the time domain signal (for the FFT) 150 DIM REX[512] 'real part of the frequency domain (for the FFT) 160 DIM IMX[512] 'imaginary part of the frequency domain (for the FFT) 170 DIM REFR[512] 'real part of the filter's frequency response 180 DIM IMFR[512] 'imaginary part of the filter's frequency response 190 DIM OLAP[398] 'holds the overlapping samples from segment to segment 200 ' 210 FOR I% = 0 TO 398 'zero the array holding the overlapping samples 220 OLAP[I%] = 0 230 NEXT I% 240 ' 250 ' 'FIND & STORE THE FILTER'S FREQUENCY RESPONSE 260 GOSUB XXXX 'Mythical subroutine to load the filter kernel into XX[ ] 270 GOSUB XXXX 'Mythical FFT subroutine: XX[ ] --> REX[ ] & IMX[ ] 280 FOR F% = 0 TO 512 'Save the frequency response in REFR[ ] & IMFR[ ] 290 REFR[F%] = REX[F%] 300 IMFR[F%] = IMX[F%] 310 NEXT F% 320 ' 330 ' 'PROCESS EACH OF THE 16000 SEGMENTS 340 FOR SEGMENT% = 0 TO 15999 350 ' 360 GOSUB XXXX 'Mythical subroutine to load the next input segment into XX[ ] 370 GOSUB XXXX 'Mythical FFT subroutine: XX[ ] --> REX[ ] & IMX[ ] 380 ' 390 FOR F% = 0 TO 512 'Multiply the frequency spectrum by the frequency response 400 TEMP = REX[F%]*REFR[F%] - IMX[F%]*IMFR[F%] 410 IMX[F%] = REX[F%]*IMFR[F%] + IMX[F%]*REFR[F%] 420 REX[F%] = TEMP 430 NEXT F% 440 ' 450 GOSUB XXXX 'Mythical IFFT subroutine: REX[ ] & IMX[ ] --> XX[ ] 460 ' 470 FOR I% = 0 TO 398 'Add the last segment's overlap to this segment 480 XX[I%] = XX[I%] + OLAP[I%] 490 NEXT I% 500 ' 510 FOR I% = 625 TO 1023 'Save the samples that will overlap the next segment 520 OLAP[I%-625] = XX[I%] 530 NEXT I% 540 ' 550 GOSUB XXXX 'Mythical subroutine to output the 625 samples stored 560 ' 'in XX[0] to XX[624] 570 ' 580 NEXT SEGMENT% 590 ' 600 GOSUB XXXX 'Mythical subroutine to output all 399 samples in OLAP[ ] 610 END TABLE 18-1 for standard convolution is directly proportional to the number of points in the filter kernel. In comparison, the time required for FFT convolution increases very slowly, only as the logarithm of the number of points in the 318 The Scientist and Engineer's Guide to Digital Signal Processing FIGURE 18-3 Execution times for FFT convolution. FFT convolution is faster than the standard method when the filter kernel is longer than about 60 points. These execution times are for a 100 MHz Pentium, using single precision floating point. Execution Time (msec/point) 1.5 1 Standard 0.5 FFT 0 8 16 32 64 128 256 Impulse Response Length 512 1024 filter kernel. The crossover occurs when the filter kernel has about 40 to 80 samples (depending on the particular hardware used). The important idea to remember: filter kernels shorter than about 60 points can be implemented faster with standard convolution, and the execution time is proportional to the kernel length. Longer filter kernels can be implemented faster with FFT convolution. With FFT convolution, the filter kernel can be made as long as you like, with very little penalty in execution time. For instance, a 16,000 point filter kernel only requires about twice as long to execute as one with only 64 points. The speed of the convolution also dictates the precision of the calculation (just as described for the FFT in Chapter 12). This is because the round-off error in the output signal depends on the total number of calculations, which is directly proportional to the computation time. If the output signal is calculated faster, it will also be calculated more precisely. For instance, imagine convolving a signal with a 1000 point filter kernel, with single precision floating point. Using standard convolution, the typical round-off noise can be expected to be about 1 part in 20,000 (from the guidelines in Chapter 4). In comparison, FFT convolution can be expected to be an order of magnitude faster, and an order of magnitude more precise (i.e., 1 part in 200,000). Keep FFT convolution tucked away for when you have a large amount of data to process and need an extremely long filter kernel. Think in terms of a million sample signal and a thousand point filter kernel. Anything less won't justify the extra programming effort. Don't want to write your own FFT convolution routine? Look in software libraries and packages for prewritten code. Start with this book's web site (see the copyright page). CHAPTER Recursive Filters 19 Recursive filters are an efficient way of achieving a long impulse response, without having to perform a long convolution. They execute very rapidly, but have less performance and flexibility than other digital filters. Recursive filters are also called Infinite Impulse Response (IIR) filters, since their impulse responses are composed of decaying exponentials. This distinguishes them from digital filters carried out by convolution, called Finite Impulse Response (FIR) filters. This chapter is an introduction to how recursive filters operate, and how simple members of the family can be designed. Chapters 20, 26 and 33 present more sophisticated design methods. The Recursive Method To start the discussion of recursive filters, imagine that you need to extract information from some signal, x[ ] . Your need is so great that you hire an old mathematics professor to process the data for you. The professor's task is to filter x[ ] to produce y[ ] , which hopefully contains the information you are interested in. The professor begins his work of calculating each point in y[ ] according to some algorithm that is locked tightly in his over-developed brain. Part way through the task, a most unfortunate event occurs. The professor begins to babble about analytic singularities and fractional transforms, and other demons from a mathematician's nightmare. It is clear that the professor has lost his mind. You watch with anxiety as the professor, and your algorithm, are taken away by several men in white coats. You frantically review the professor's notes to find the algorithm he was using. You find that he had completed the calculation of points y[0] through y[27] , and was about to start on point y[28] . As shown in Fig. 19-1, we will let the variable, n, represent the point that is currently being calculated. This means that y[n] is sample 28 in the output signal, y[n& 1] is sample 27, y[n& 2] is sample 26, etc. Likewise, x[n] is point 28 in the input signal, 319 320 The Scientist and Engineer's Guide to Digital Signal Processing x[n& 1] is point 27, etc. To understand the algorithm being used, we ask ourselves: "What information was available to the professor to calculate y[n] , the sample currently being worked on?" The most obvious source of information is the input signal, that is, the values: x[n], x[n& 1], x[n& 2], þ. The professor could have been multiplying each point in the input signal by a coefficient, and adding the products together: y [n ] ' a0 x [n ] % a1 x [n & 1] % a2 x [n & 2] % a3 x [n & 3] % þ You should recognize that this is nothing more than simple convolution, with the coefficients: a0, a1, a2, þ, forming the convolution kernel. If this was all the professor was doing, there wouldn't be much need for this story, or this chapter. However, there is another source of information that the professor had access to: the previously calculated values of the output signal, held in: y[n& 1], y[n& 2], y[n& 3], þ. Using this additional information, the algorithm would be in the form: y [n ] ' a0 x [n ] % a1 x [n & 1] % a2 x [n & 2] % a3 x [n & 3] % % b1 y [n & 1] % b2 y [n & 2] % b3 y [n & 3] % þ þ EQUATION 19-1 The recursion equation. In this equation, x[ ] is the input signal, y[ ] is the output signal, and the a's and b's are coefficients. In words, each point in the output signal is found by multiplying the values from the input signal by the "a" coefficients, multiplying the previously calculated values from the output signal by the "b" coefficients, and adding the products together. Notice that there isn't a value for b0 , because this corresponds to the sample being calculated. Equation 19-1 is called the recursion equation, and filters that use it are called recursive filters. The "a" and "b" values that define the filter are called the recursion coefficients. In actual practice, no more than about a dozen recursion coefficients can be used or the filter becomes unstable (i.e., the output continually increases or oscillates). Table 19-1 shows an example recursive filter program. Recursive filters are useful because they bypass a longer convolution. For instance, consider what happens when a delta function is passed through a recursive filter. The output is the filter's impulse response, and will typically be a sinusoidal oscillation that exponentially decays. Since this impulse response in infinitely long, recursive filters are often called infinite impulse response (IIR) filters. In effect, recursive filters convolve the input signal with a very long filter kernel, although only a few coefficients are involved. Chapter 19- Recursive Filters 2 2 a. The input signal, x[ ] b. The output signal, y[ ] 1 Amplitude 1 Amplitude 321 0 x[n-3] x[n-2] x[n-1] x[n] -1 0 y[n-3] y[n-2] y[n-1] y[n] -1 -2 -2 0 10 Sample number 20 30 0 10 Sample number 20 30 FIGURE 19-1 Recursive filter notation. The output sample being calculated, y[n] , is determined by the values from the input signal, x[n], x[n& 1], x[n& 2], þ, as well as the previously calculated values in the output signal, y[n& 1], y[n& 2], y[n& 3], þ. These figures are shown for n ' 28 . The relationship between the recursion coefficients and the filter's response is given by a mathematical technique called the z-transform, the topic of Chapter 33. For example, the z-transform can be used for such tasks as: converting between the recursion coefficients and the frequency response, combining cascaded and parallel stages into a single filter, designing recursive systems that mimic analog filters, etc. Unfortunately, the z-transform is very mathematical, and more complicated than most DSP users are willing to deal with. This is the realm of those that specialize in DSP. There are three ways to find the recursion coefficients without having to understand the z-transform. First, this chapter provides design equations for several types of simple recursive filters. Second, Chapter 20 provides a "cookbook" computer program for designing the more sophisticated Chebyshev low-pass and high-pass filters. Third, Chapter 26 describes an iterative method for designing recursive filters with an arbitrary frequency response. 100 'RECURSIVE FILTER 110 ' 120 DIM X[499] 'holds the input signal 130 DIM Y[499] 'holds the filtered output signal 140 ' 150 GOSUB XXXX 'mythical subroutine to calculate the recursion 160 ' 'coefficients: A0, A1, A2, B1, B2 170 ' 180 GOSUB XXXX 'mythical subroutine to load X[ ] with the input data 190 ' 200 FOR I% = 2 TO 499 210 Y[I%] = A0*X[I%] + A1*X[I%-1] + A2*X[I%-2] + B1*Y[I%-1] + B2*Y[I%-2] 220 NEXT I% 230 ' 240 END TABLE 19-1 322 The Scientist and Engineer's Guide to Digital Signal Processing Digital Filter 1.5 1.0 1.0 Recursive Filter 0.5 Amplitude Amplitude 1.5 a0 = 0.15 b1 = 0.85 0.0 0.5 0.0 -0.5 -0.5 0 10 20 Sample number 30 40 0 10 0 10 20 30 40 20 30 40 Sample number Analog Filter 1.5 R 1.0 0.5 C Amplitude Amplitude 1.5 1.0 0.5 0.0 0.0 -0.5 -0.5 0 10 Time 20 30 Time FIGURE 19-2 Single pole low-pass filter. Digital recursive filters can mimic analog filters composed of resistors and capacitors. As shown in this example, a single pole low-pass recursive filter smoothes the edge of a step input, just as an electronic RC filter. Single Pole Recursive Filters Figure 19-2 shows an example of what is called a single pole low-pass filter. This recursive filter uses just two coefficients, a0 ' 0.15 and b1 ' 0.85 . For this example, the input signal is a step function. As you should expect for a low-pass filter, the output is a smooth rise to the steady state level. This figure also shows something that ties into your knowledge of electronics. This lowpass recursive filter is completely analogous to an electronic low-pass filter composed of a single resistor and capacitor. The beauty of the recursive method is in its ability to create a wide variety of responses by changing only a few parameters. For example, Fig. 19-3 shows a filter with three coefficients: a0 ' 0.93 , a1 ' & 0.93 and b1 ' 0.86 . As shown by the similar step responses, this digital filter mimics an electronic RC high-pass filter. These single pole recursive filters are definitely something you want to keep in your DSP toolbox. You can use them to process digital signals just as you would use RC networks to process analog electronic signals. This includes everything you would expect: DC removal, high-frequency noise suppression, wave shaping, smoothing, etc. They are easy to program, fast Chapter 19- Recursive Filters 323 Digital Filter 1.5 1.0 Recursive Filter 0.5 a0 = 0.93 a1 = -0.93 b1 = 0.86 0.0 1.0 Amplitude Amplitude 1.5 0.5 0.0 -0.5 -0.5 0 10 20 Sample number 30 40 0 10 0 10 20 30 40 20 30 40 Sample number Analog Filter 1.5 C 1.0 0.5 R Amplitude Amplitude 1.5 1.0 0.5 0.0 0.0 -0.5 -0.5 0 10 Time 20 30 Time FIGURE 19-3 Single pole high-pass filter. Proper coefficient selection can also make the recursive filter mimic an electronic RC high-pass filter. These single pole recursive filters can be used in DSP just as you would use RC circuits in analog electronics. to execute, and produce few surprises. The coefficients are found from these simple equations: EQUATION 19-2 Single pole low-pass filter. The filter's response is controlled by the parameter, x, a value between zero and one. a0 ' 1& x b1 ' x EQUATION 19-3 Single pole high-pass filter. a0 ' (1% x ) / 2 a1 ' & (1% x) / 2 b1 ' x The characteristics of these filters are controlled by the parameter, x, a value between zero and one. Physically, x is the amount of decay between adjacent samples. For instance, x is 0.86 in Fig. 19-3, meaning that the value of each sample in the output signal is 0.86 the value of the sample before it. The higher the value of x, the slower the decay. Notice that the 324 The Scientist and Engineer's Guide to Digital Signal Processing 1.5 1.5 a. Original signal b. Filtered signals 1.0 Amplitude Amplitude 1.0 0.5 0.0 low-pass 0.5 0.0 high-pass -0.5 -0.5 0 100 200 300 Sample number 400 500 0 100 200 300 Sample number 400 500 FIGURE 19-4 Example of single pole recursive filters. In (a), a high frequency burst rides on a slowly varying signal. In (b), single pole low-pass and high-pass filters are used to separate the two components. The low-pass filter uses x = 0.95, while the high-pass filter is for x = 0.86. filter becomes unstable if x is made greater than one. That is, any nonzero value on the input will make the output increase until an overflow occurs. The value for x can be directly specified, or found from the desired time constant of the filter. Just as R×C is the number of seconds it takes an RC circuit to decay to 36.8% of its final value, d is the number of samples it takes for a recursive filter to decay to this same level: EQUATION 19-4 Time constant of single pole filters. This equation relates the amount of decay between samples, x, with the filter's time constant, d, the number of samples for the filter to decay to 36.8%. x ' e & 1 /d For instance, a sample-to-sample decay of x ' 0.86 corresponds to a time constant of d ' 6.63 samples (as shown in Fig 19-3). There is also a fixed relationship between x and the -3dB cutoff frequency, fC , of the digital filter: EQUATION 19-5 Cutoff frequency of single pole filters. The amount of decay between samples, x, is related to the cutoff frequency of the filter, fC , a value between 0 and 0.5. x ' e & 2 BfC This provides three ways to find the "a" and "b" coefficients, starting with the time constant, the cutoff frequency, or just directly picking x. Figure 19-4 shows an example of using single pole recursive filters. In (a), the original signal is a smooth curve, except a burst of a high frequency sine wave. Figure (b) shows the signal after passing through low-pass and high-pass filters. The signals have been separated fairly well, but not perfectly, just as if simple RC circuits were used on an analog signal. Chapter 19- Recursive Filters 1.5 325 1.5 a. High-pass filter b. Low-pass filter Amplitude 0.05 fc = 0.25 0.5 1.0 fc = 0.25 0.05 0.5 0.01 0.0 0.0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 Frequency 0.3 0.4 0.5 0.4 0.5 Frequency 1.5 FIGURE 19-5 Single pole frequency responses. Figures (a) and (b) show the frequency responses of highpass and low-pass single pole recursive filters, respectively. Figure (c) shows the frequency response of a cascade of four low-pass filters. The frequency response of recursive filters is not always what you expect, especially if the filter is pushed to extreme limits. For example, the fC ' 0.25 curve in (c) is quite useless. Many factors are to blame, including: aliasing, roundoff noise, and the nonlinear phase response. c. Low-pass filter (4 stage) Amplitude Amplitude 0.01 1.0 fc = 0.25 1.0 0.5 0.05 0.01 0.0 0 0.1 0.2 0.3 Frequency Figure 19-5 shows the frequency responses of various single pole recursive filters. These curves are obtained by passing a delta function through the filter to find the filter's impulse response. The FFT is then used to convert the impulse response into the frequency response. In principle, the impulse response is infinitely long; however, it decays below the single precision roundoff noise after about 15 to 20 time constants. For example, when the time constant of the filter is d ' 6.63 samples, the impulse response can be contained in about 128 samples. The key feature in Fig. 19-5 is that single pole recursive filters have little ability to separate one band of frequencies from another. In other words, they perform well in the time domain, and poorly in the frequency domain. The frequency response can be improved slightly by cascading several stages. This can be accomplished in two ways. First, the signal can be passed through the filter several times. Second, the z-transform can be used to find the recursion coefficients that combine the cascade into a single stage. Both ways work and are commonly used. Figure (c) shows the frequency response of a cascade of four low-pass filters. Although the stopband attenuation is significantly improved, the roll-off is still terrible. If you need better performance in the frequency domain, look at the Chebyshev filters of the next chapter. 326 The Scientist and Engineer's Guide to Digital Signal Processing The four stage low-pass filter is comparable to the Blackman and Gaussian filters (relatives of the moving average, Chapter 15), but with a much faster execution speed. The design equations for a four stage low-pass filter are: a0 ' b1 ' EQUATION 19-6 Four stage low-pass filter. These equations provide the "a" and "b" coefficients for a cascade of four single pole low-pass filters. The relationship between x and the cutoff frequency of this filter is given by Eq. 19-5, with the 2B replaced by 14.445. (1& x )4 4x b2 ' & 6x 2 b3 ' 4x 3 b4 ' & x 4 Narrow-band Filters A common need in electronics and DSP is to isolate a narrow band of frequencies from a wider bandwidth signal. For example, you may want to eliminate 60 hertz interference in an instrumentation system, or isolate the signaling tones in a telephone network. Two types of frequency responses are available: the band-pass and the band-reject (also called a notch filter). Figure 19-6 shows the frequency response of these filters, with the recursion coefficients provided by the following equations: EQUATION 19-7 Band-pass filter. An example frequency response is shown in Fig. 19-6a. To use these equations, first select the center frequency, f, and the bandwidth, BW. Both of these are expressed as a fraction of the sampling rate, and therefore in the range of 0 to 0.5. Next, calculate R, and then K, and then the recursion coefficients. EQUATION 19-8 Band-reject filter. This filter is commonly called a notch filter. Example frequency responses are shown in Fig. 19-6b. a0 a1 a2 b1 ' 1& K ' 2(K& R ) cos( 2Bf ) ' R 2& K ' 2R cos( 2Bf ) b2 ' & R 2 a0 a1 a2 b1 b2 ' ' ' ' ' K & 2K cos( 2Bf ) K 2R cos( 2Bf ) &R2 where: K ' 1 & 2R cos(2Bf ) % R 2 2 & 2 cos(2Bf ) R ' 1 & 3BW Chapter 19- Recursive Filters 327 1.5 1.5 a. Band-pass frequency response b. Band-reject frequency response Amplitude 0.5 single stage BW=0.0066 1.0 BW=0.033 0.5 cascade of three stages 0.0 0.0 0 0.1 0.2 0.3 0.4 0 0.5 0.1 0.2 0.3 Frequency Frequency 0.4 0.5 1.5 c. Band-reject step response FIGURE 19-6 Characteristics of narrow-band filters. Figure (a) and (b) shows the frequency responses of various band-pass and band-reject filters. The step response of the band-reject filter is shown in (c). The band-reject (notch) filter is useful for removing 60 Hz and similar interference from time domain encoded waveforms. Amplitude Amplitude BW=0.0066 1.0 1.0 BW=0.0066 0.5 0.0 0 50 100 150 200 Sample number Two parameters must be selected before using these equations: f, the center frequency, and BW, the bandwidth (measured at an amplitude of 0.707). Both of these are expressed as a fraction of the sampling frequency, and therefore must be between 0 and 0.5. From these two specified values, calculate the intermediate variables: R and K, and then the recursion coefficients. As shown in (a), the band-pass filter has relatively large tails extending from the main peak. This can be improved by cascading several stages. Since the design equations are quite long, it is simpler to implement this cascade by filtering the signal several times, rather than trying to find the coefficients needed for a single filter. Figure (b) shows examples of the band-reject filter. The narrowest bandwidth that can be obtain with single precision is about 0.0003 of the sampling frequency. When pushed beyond this limit, the attenuation of the notch will degrade. Figure (c) shows the step response of the band-reject filter. There is noticeable overshoot and ringing, but its amplitude is quite small. This allows the filter to remove narrowband interference (60 Hz and the like) with only a minor distortion to the time domain waveform. 328 The Scientist and Engineer's Guide to Digital Signal Processing Phase Response There are three types of phase response that a filter can have: zero phase, linear phase, and nonlinear phase. An example of each of these is shown in Figure 19-7. As shown in (a), the zero phase filter is characterized by an impulse response that is symmetrical around sample zero. The actual shape doesn't matter, only that the negative numbered samples are a mirror image of the positive numbered samples. When the Fourier transform is taken of this symmetrical waveform, the phase will be entirely zero, as shown in (b). The disadvantage of the zero phase filter is that it requires the use of negative indexes, which can be inconvenient to work with. The linear phase filter is a way around this. The impulse response in (d) is identical to that shown in (a), except it has been shifted to use only positive numbered samples. The impulse response is still symmetrical between the left and right; however, the location of symmetry has been shifted from zero. This shift results in the phase, (e), being a straight line, accounting for the name: linear phase. The slope of this straight line is directly proportional to the amount of the shift. Since the shift in the impulse response does nothing but produce an identical shift in the output signal, the linear phase filter is equivalent to the zero phase filter for most purposes. Figure (g) shows an impulse response that is not symmetrical between the left and right. Correspondingly, the phase, (h), is not a straight line. In other words, it has a nonlinear phase. Don't confuse the terms: nonlinear and linear phase with the concept of system linearity discussed in Chapter 5. Although both use the word linear, they are not related. Why does anyone care if the phase is linear or not? Figures (c), (f), and (i) show the answer. These are the pulse responses of each of the three filters. The pulse response is nothing more than a positive going step response followed by a negative going step response. The pulse response is used here because it displays what happens to both the rising and falling edges in a signal. Here is the important part: zero and linear phase filters have left and right edges that look the same, while nonlinear phase filters have left and right edges that look different. Many applications cannot tolerate the left and right edges looking different. One example is the display of an oscilloscope, where this difference could be misinterpreted as a feature of the signal being measured. Another example is in video processing. Can you imagine turning on your TV to find the left ear of your favorite actor looking different from his right ear? It is easy to make an FIR (finite impulse response) filter have a linear phase. This is because the impulse response (filter kernel) is directly specified in the design process. Making the filter kernel have left-right symmetry is all that is required. This is not the case with IIR (recursive) filters, since the recursion coefficients are what is specified, not the impulse response. The impulse response of a recursive filter is not symmetrical between the left and right, and therefore has a nonlinear phase. Chapter 19- Recursive Filters 329 Zero Phase Filter 12 0.25 a. Impulse response 8 Phase (radians) 0.15 Amplitude 2.0 b. Phase 0.10 0.05 0.00 c. Pulse response 1.5 4 Amplitude 0.20 0 -4 -0.10 -25 -12 0 25 Sample number 0 50 0.5 0.0 -8 -0.05 1.0 0.1 0.2 0.3 Frequency 0.4 0.5 -0.5 -25 0 25 50 Sample number 75 100 75 100 75 100 Linear Phase Filter 0.25 96 d. Impulse response 64 Phase (radians) Amplitude 0.15 0.10 0.05 f. Pulse response 1.5 32 Amplitude 0.20 2.0 e. Phase 0 -32 0.00 -0.10 -25 -96 0 25 Sample number 50 0.5 0.0 -64 -0.05 1.0 0 0.1 0.2 0.3 Frequency 0.4 0.5 -0.5 -25 0 25 50 Sample number Nonlinear Phase Filter 0.25 8 Phase (radians) Amplitude 0.15 0.10 0.05 0.00 i. Pulse response 1.5 4 0 -4 -12 0 25 Sample number 50 1.0 0.5 0.0 -8 -0.05 -0.10 -25 2.0 h. Phase Amplitude 0.20 12 g. Impulse response -0.5 0 0.1 0.2 0.3 Frequency 0.4 0.5 -25 0 25 50 Sample number FIGURE 19-7 Zero, linear, and nonlinear phase filters. A zero phase filter has an impulse response that has left-right symmetry around sample number zero, as in (a). This results in a frequency response that has a phase composed entirely of zeros, as in (b). Zero phase impulse responses are desirable because their step responses are symmetrical between the top and bottom, making the left and right edges of pulses look the same, as is shown in (c). Linear phase filters have left-right symmetry, but not around sample zero, as illustrated in (d). This results in a phase that is linear, that is, a straight line, as shown in (e). The linear phase pulse response, shown in (f), has all the advantages of the zero phase pulse response. In comparison, the impulse responses of nonlinear phase filters are not symmetrical between the left and right, as in (g), and the phases are not a straight line, as in (h). The worst part is that the left and right edges of the pulse response are not the same, as shown in (i). 330 The Scientist and Engineer's Guide to Digital Signal Processing Analog electronic circuits have this same problem with the phase response. Imagine a circuit composed of resistors and capacitors sitting on your desk. If the input has always been zero, the output will also have always been zero. When an impulse is applied to the input, the capacitors quickly charge to some value and then begin to exponentially decay through the resistors. The impulse response (i.e., the output signal) is a combination of these various decaying exponentials. The impulse response cannot be symmetrical, because the output was zero before the impulse, and the exponential decay never quite reaches a value of zero again. Analog filter designers attack this problem with the Bessel filter, presented in Chapter 3. The Bessel filter is designed to have as linear phase as possible; however, it is far below the performance of digital filters. The ability to provide an exact linear phase is a clear advantage of digital filters. Fortunately, there is a simple way to modify recursive filters to obtain a zero phase. Figure 19-8 shows an example of how this works. The input signal to be filtered is shown in (a). Figure (b) shows the signal after it has been filtered by a single pole low-pass filter. Since this is a nonlinear phase filter, the left and right edges do not look the same; they are inverted versions of each other. As previously described, this recursive filter is implemented by starting at sample 0 and working toward sample 150, calculating each sample along the way. Now, suppose that instead of moving from sample 0 toward sample 150, we start at sample 150 and move toward sample 0. In other words, each sample in the output signal is calculated from input and output samples to the right of the sample being worked on. This means that the recursion equation, Eq. 19-1, is changed to: y [n ] ' a0 x [n ] % a1 x [n % 1] % a2 x [n % 2] % a3 x [n % 3] % % b1 y [n % 1] % b2 y [n % 2] % b3 y [n % 3] % þ þ EQUATION 19-9 The reverse recursion equation. This is the same as Eq. 19-1, except the signal is filtered from left-to-right, instead of right-to-left. Figure (c) shows the result of this reverse filtering. This is analogous to passing an analog signal through an electronic RC circuit while running time backwards. !esrevinu eht pu-wercs nac lasrever emit -noituaC Filtering in the reverse direction does not produce any benefit in itself; the filtered signal still has left and right edges that do not look alike. The magic happens when forward and reverse filtering are combined. Figure (d) results from filtering the signal in the forward direction and then filtering again in the reverse direction. Voila! This produces a zero phase recursive filter. In fact, any recursive filter can be converted to zero phase with this bidirectional filtering technique. The only penalty for this improved performance is a factor of two in execution time and program complexity. Chapter 19- Recursive Filters 1.5 1.5 a. Original signal b. Filtered 1.0 Amplitude 1.0 0.5 0.0 0.5 0.0 -0.5 -0.5 0 25 50 75 100 Sample number 125 0 150 25 50 75 100 125 150 50 75 100 125 150 50 75 100 125 150 Sample number 1.5 c. Filtered FIGURE 19-8 Bidirectional recursive filtering. A rectangular pulse input signal is shown in (a). Figure (b) shows the signal after being filtered with a single pole recursive low-pass filter, passing from left-to-right. In (c), the signal has been processed in the same manner, except with the filter moving right-to-left. Figure (d) shows the signal after being filtered both left-to-right and then right-to-left. Any recursive filter can be made zero phase by using this technique. Amplitude 1.0 0.5 0.0 -0.5 0 25 Sample number 1.5 d. Filtered 1.0 Amplitude Amplitude 331 0.5 0.0 -0.5 0 25 Sample number How do you find the impulse and frequency responses of the overall filter? The magnitude of the frequency response is the same for each direction, while the phases are opposite in sign. When the two directions are combined, the magnitude becomes squared, while the phase cancels to zero. In the time domain, this corresponds to convolving the original impulse response with a left-for-right flipped version of itself. For instance, the impulse response of a 332 The Scientist and Engineer's Guide to Digital Signal Processing single pole low-pass filter is a one-sided exponential. The impulse response of the corresponding bidirectional filter is a one-sided exponential that decays to the right, convolved with a one-sided exponential that decays to the left. Going through the mathematics, this turns out to be a double-sided exponential that decays both to the left and right, with the same decay constant as the original filter. Some applications only have a portion of the signal in the computer at a particular time, such as systems that alternately input and output data on a continuing basis. Bidirectional filtering can be used in these cases by combining it with the overlap-add method described in the last chapter. When you come to the question of how long the impulse response is, don't say "infinite." If you do, you will need to pad each signal segment with an infinite number of zeros. Remember, the impulse response can be truncated when it has decayed below the round-off noise level, i.e., about 15 to 20 time constants. Each segment will need to be padded with zeros on both the left and right to allow for the expansion during the bidirectional filtering. Using Integers Single precision floating point is ideal to implement these simple recursive filters. The use of integers is possible, but it is much more difficult. There are two main problems. First, the round-off error from the limited number of bits can degrade the response of the filter, or even make it unstable. Second, the fractional values of the recursion coefficients must be handled with integer math. One way to attack this problem is to express each coefficient as a fraction. For example, 0.15 becomes 19/128. Instead of multiplying by 0.15, you first multiply by 19 and then divide by 128. Another way is to replace the multiplications with look-up tables. For example, a 12 bit ADC produces samples with a value between 0 and 4095. Instead of multiplying each sample by 0.15, you pass the samples through a look-up table that is 4096 entries long. The value obtained from the look-up table is equal to 0.15 times the value entering the look-up table. This method is very fast, but it does require extra memory; a separate look-up table is needed for each coefficient. Before you try either of these integer methods, make sure the recursive algorithm for the moving average filter will not suit your needs. It loves integers. CHAPTER 20 Chebyshev Filters Chebyshev filters are used to separate one band of frequencies from another. Although they cannot match the performance of the windowed-sinc filter, they are more than adequate for many applications. The primary attribute of Chebyshev filters is their speed, typically more than an order of magnitude faster than the windowed-sinc. This is because they are carried out by recursion rather than convolution. The design of these filters is based on a mathematical technique called the z-transform, discussed in Chapter 33. This chapter presents the information needed to use Chebyshev filters without wading through a mire of advanced mathematics. The Chebyshev and Butterworth Responses The Chebyshev response is a mathematical strategy for achieving a faster rolloff by allowing ripple in the frequency response. Analog and digital filters that use this approach are called Chebyshev filters. For instance, analog Chebyshev filters were used in Chapter 3 for analog-to-digital and digital-toanalog conversion. These filters are named from their use of the Chebyshev polynomials, developed by the Russian mathematician Pafnuti Chebyshev (1821-1894). This name has been translated from Russian and appears in the literature with different spellings, such as: Chebychev, Tschebyscheff, Tchebysheff and Tchebichef. Figure 20-1 shows the frequency response of low-pass Chebyshev filters with passband ripples of: 0%, 0.5% and 20%. As the ripple increases (bad), the roll-off becomes sharper (good). The Chebyshev response is an optimal tradeoff between these two parameters. When the ripple is set to 0%, the filter is called a maximally flat or Butterworth filter (after S. Butterworth, a British engineer who described this response in 1930). A ripple of 0.5% is a often good choice for digital filters. This matches the typical precision and accuracy of the analog electronics that the signal has passed through. The Chebyshev filters discussed in this chapter are called type 1 filters, meaning that the ripple is only allowed in the passband. In comparison, 333 334 The Scientist and Engineer's Guide to Digital Signal Processing 1.5 20% Ripple Amplitude FIGURE 20-1 The Chebyshev response. Chebyshev filters achieve a faster roll-off by allowing ripple in the passband. When the ripple is set to 0%, it is called a maximally flat or Butterworth filter. Consider using a ripple of 0.5% in your designs; this passband unflatness is so small that it cannot be seen in this graph, but the roll-off is much faster than the Butterworth. 1.0 0% 0.5% 0.5 0.0 0 0.1 0.2 0.3 Frequency 0.4 0.5 type 2 Chebyshev filters have ripple only in the stopband. Type 2 filters are seldom used, and we won't discuss them. There is, however, an important design called the elliptic filter, which has ripple in both the passband and the stopband. Elliptic filters provide the fastest roll-off for a given number of poles, but are much harder to design. We won't discuss the elliptic filter here, but be aware that it is frequently the first choice of professional filter designers, both in analog electronics and DSP. If you need this level of performance, buy a software package for designing digital filters. Designing the Filter You must select four parameters to design a Chebyshev filter: (1) a high-pass or low-pass response, (2) the cutoff frequency, (3) the percent ripple in the passband, and (4) the number of poles. Just what is a pole? Here are two answers. If you don't like one, maybe the other will help: Answer 1- The Laplace transform and z-transform are mathematical ways of breaking an impulse response into sinusoids and decaying exponentials. This is done by expressing the system's characteristics as one complex polynomial divided by another complex polynomial. The roots of the numerator are called zeros, while the roots of the denominator are called poles. Since poles and zeros can be complex numbers, it is common to say they have a "location" in the complex plane. Elaborate systems have more poles and zeros than simple ones. Recursive filters are designed by first selecting the location of the poles and zeros, and then finding the appropriate recursion coefficients (or analog components). For example, Butterworth filters have poles that lie on a circle in the complex plane, while in a Chebyshev filter they lie on an ellipse. This is the topic of Chapters 32 and 33. Answer 2- Poles are containers filled with magic powder. The more poles in a filter, the better the filter works. Chapter 20- Chebyshev Filters 1.25 1.25 a. Low-pass frequency response c. High-pass frequency response 1.00 Amplitude Amplitude 1.00 0.75 0.50 0.25 6 0.00 0 2 4 12 0.1 0.2 4 pole 8 6 0.3 Frequency 4 0.75 0.50 0.25 2 2 0.00 0.4 0.5 4 4 6 0 20 0.1 8 2 12 0.2 0.3 4 6 pole 0.4 Frequency 0.5 20 b. Low-pass frequency response (dB) d. High-pass frequency response (dB) 0 6 -40 2 4 pole 2 -20 8 4 6 4 12 -60 -80 Amplitude (dB) 0 Amplitude (dB) 335 4 2 -20 4 2 8 4 12 6 6 pole -40 -60 -80 -100 -100 0 0.1 0.2 0.3 Frequency 0.4 0.5 0 0.1 0.2 0.3 Frequency 0.4 0.5 FIGURE 20-2 Chebyshev frequency responses. Figures (a) and (b) show the frequency responses of low-pass Chebyshev filters with 0.5% ripple, while (c) and (d) show the corresponding high-pass filter responses. Kidding aside, the point is that you can use these filters very effectively without knowing the nasty mathematics behind them. Filter design is a specialty. In actual practice, more engineers, scientists and programmers think in terms of answer 2, than answer 1. Figure 20-2 shows the frequency response of several Chebyshev filters with 0.5% ripple. For the method used here, the number of poles must be even. The cutoff frequency of each filter is measured where the amplitude crosses 0.707 (-3dB). Filters with a cutoff frequency near 0 or 0.5 have a sharper roll-off than filters in the center of the frequency range. For example, a two pole filter at fC ' 0.05 has about the same roll-off as a four pole filter at fC ' 0.25 . This is fortunate; fewer poles can be used near 0 and 0.5 because of round-off noise. More about this later. There are two ways of finding the recursion coefficients without using the ztransform. First, the cowards way: use a table. Tables 20-1 and 20-2 provide the recursion coefficients for low-pass and high-pass filters with 0.5% passband ripple. If you only need a quick and dirty design, copy the appropriate coefficients into your program, and you're done. 336 fC The Scientist and Engineer's Guide to Digital Signal Processing 2 Pole 4 Pole a0= 8.663387E-04 a1= 1.732678E-03 b1= 1.919129E+00 a2= 8.663387E-04 b2= -9.225943E-01 a0= a1= a2= a3= a4= 4.149425E-07 1.659770E-06 2.489655E-06 1.659770E-06 4.149425E-07 (!! Unstable !!) b1= 3.893453E+00 b2= -5.688233E+00 b3= 3.695783E+00 b4= -9.010106E-01 0.025 a0= 5.112374E-03 a1= 1.022475E-02 b1= 1.797154E+00 a2= 5.112374E-03 b2= -8.176033E-01 a0= a1= a2= a3= a4= 1.504626E-05 6.018503E-05 9.027754E-05 6.018503E-05 1.504626E-05 0.05 a0= 1.868823E-02 a1= 3.737647E-02 b1= 1.593937E+00 a2= 1.868823E-02 b2= -6.686903E-01 a0= a1= a2= a3= a4= 0.075 a0= 3.869430E-02 a1= 7.738860E-02 b1= 1.392667E+00 a2= 3.869430E-02 b2= -5.474446E-01 0.1 6 Pole a0= a1= a2= a3= a4= a5= a6= 1.391351E-10 8.348109E-10 2.087027E-09 2.782703E-09 2.087027E-09 8.348109E-10 1.391351E-10 (!! Unstable !!) b1= 5.883343E+00 b2= -1.442798E+01 b3= 1.887786E+01 b4= -1.389914E+01 b5= 5.459909E+00 b6= -8.939932E-01 b1= 3.725385E+00 b2= -5.226004E+00 b3= 3.270902E+00 b4= -7.705239E-01 a0= a1= a2= a3= a4= a5= a6= 3.136210E-08 1.881726E-07 4.704314E-07 6.272419E-07 4.704314E-07 1.881726E-07 3.136210E-08 (!! Unstable !!) b1= 5.691653E+00 b2= -1.353172E+01 b3= 1.719986E+01 b4= -1.232689E+01 b5= 4.722721E+00 b6= -7.556340E-01 2.141509E-04 8.566037E-04 1.284906E-03 8.566037E-04 2.141509E-04 b1= 3.425455E+00 b2= -4.479272E+00 b3= 2.643718E+00 b4= -5.933269E-01 a0= a1= a2= a3= a4= a5= a6= 1.771089E-06 1.062654E-05 2.656634E-05 3.542179E-05 2.656634E-05 1.062654E-05 1.771089E-06 b1= 5.330512E+00 b2= -1.196611E+01 b3= 1.447067E+01 b4= -9.937710E+00 b5= 3.673283E+00 b6= -5.707561E-01 a0= a1= a2= a3= a4= 9.726342E-04 3.890537E-03 5.835806E-03 3.890537E-03 9.726342E-04 b1= 3.103944E+00 b2= -3.774453E+00 b3= 2.111238E+00 b4= -4.562908E-01 a0= a1= a2= a3= a4= a5= a6= 1.797538E-05 1.078523E-04 2.696307E-04 3.595076E-04 2.696307E-04 1.078523E-04 1.797538E-05 b1= 4.921746E+00 b2= -1.035734E+01 b3= 1.189764E+01 b4= -7.854533E+00 b5= 2.822109E+00 b6= -4.307710E-01 a0= 6.372802E-02 a1= 1.274560E-01 b1= 1.194365E+00 a2= 6.372802E-02 b2= -4.492774E-01 a0= a1= a2= a3= a4= 2.780755E-03 1.112302E-02 1.668453E-02 1.112302E-02 2.780755E-03 b1= 2.764031E+00 b2= -3.122854E+00 b3= 1.664554E+00 b4= -3.502232E-01 a0= a1= a2= a3= a4= a5= a6= 9.086148E-05 5.451688E-04 1.362922E-03 1.817229E-03 1.362922E-03 5.451688E-04 9.086148E-05 b1= 4.470118E+00 b2= -8.755594E+00 b3= 9.543712E+00 b4= -6.079376E+00 b5= 2.140062E+00 b6= -3.247363E-01 0.15 a0= 1.254285E-01 a1= 2.508570E-01 b1= 8.070778E-01 a2= 1.254285E-01 b2= -3.087918E-01 a0= a1= a2= a3= a4= 1.180009E-02 4.720034E-02 7.080051E-02 4.720034E-02 1.180009E-02 b1= 2.039039E+00 b2= -2.012961E+00 b3= 9.897915E-01 b4= -2.046700E-01 a0= a1= a2= a3= a4= a5= a6= 8.618665E-04 5.171199E-03 1.292800E-02 1.723733E-02 1.292800E-02 5.171199E-03 8.618665E-04 b1= 3.455239E+00 b2= -5.754735E+00 b3= 5.645387E+00 b4= -3.394902E+00 b5= 1.177469E+00 b6= -1.836195E-01 0.2 a0= 1.997396E-01 a1= 3.994792E-01 b1= 4.291048E-01 a2= 1.997396E-01 b2= -2.280633E-01 a0= a1= a2= a3= a4= 3.224554E-02 1.289821E-01 1.934732E-01 1.289821E-01 3.224554E-02 b1= 1.265912E+00 b2= -1.203878E+00 b3= 5.405908E-01 b4= -1.185538E-01 a0= a1= a2= a3= a4= a5= a6= 4.187408E-03 2.512445E-02 6.281112E-02 8.374816E-02 6.281112E-02 2.512445E-02 4.187408E-03 b1= 2.315806E+00 b2= -3.293726E+00 b3= 2.904826E+00 b4= -1.694128E+00 b5= 6.021426E-01 b6= -1.029147E-01 0.25 a0= 2.858110E-01 a1= 5.716221E-01 b1= 5.423258E-02 a2= 2.858110E-01 b2= -1.974768E-01 a0= a1= a2= a3= a4= 7.015301E-02 2.806120E-01 4.209180E-01 2.806120E-01 7.015301E-02 b1= 4.541481E-01 b2= -7.417536E-01 b3= 2.361222E-01 b4= -7.096476E-02 a0= a1= a2= a3= a4= a5= a6= 1.434449E-02 8.606697E-02 2.151674E-01 2.868899E-01 2.151674E-01 8.606697E-02 1.434449E-02 b1= 1.076052E+00 b2= -1.662847E+00 b3= 1.191063E+00 b4= -7.403087E-01 b5= 2.752158E-01 b6= -5.722251E-02 0.3 a0= 3.849163E-01 a1= 7.698326E-01 b1= -3.249116E-01 a2= 3.849163E-01 b2= -2.147536E-01 a0= a1= a2= a3= a4= 1.335566E-01 5.342263E-01 8.013394E-01 5.342263E-01 1.335566E-01 b1= -3.904486E-01 b2= -6.784138E-01 b3= -1.412021E-02 b4= -5.392238E-02 a0= a1= a2= a3= a4= a5= a6= 3.997487E-02 2.398492E-01 5.996231E-01 7.994975E-01 5.996231E-01 2.398492E-01 3.997487E-02 b1= -2.441152E-01 b2= -1.130306E+00 b3= 1.063167E-01 b4= -3.463299E-01 b5= 8.882992E-02 b6= -3.278741E-02 0.35 a0= 5.001024E-01 a1= 1.000205E+00 b1= -7.158993E-01 a2= 5.001024E-01 b2= -2.845103E-01 a0= a1= a2= a3= a4= 2.340973E-01 9.363892E-01 1.404584E+00 9.363892E-01 2.340973E-01 b1= -1.263672E+00 b2= -1.080487E+00 b3= -3.276296E-01 b4= -7.376791E-02 a0= a1= a2= a3= a4= a5= a6= 9.792321E-02 5.875393E-01 1.468848E+00 1.958464E+00 1.468848E+00 5.875393E-01 9.792321E-02 b1= -1.627573E+00 b2= -1.955020E+00 b3= -1.075051E+00 b4= -5.106501E-01 b5= -7.239843E-02 b6= -2.639193E-02 0.40 a0= 6.362308E-01 a1= 1.272462E+00 b1= -1.125379E+00 a2= 6.362308E-01 b2= -4.195441E-01 a0= a1= a2= a3= a4= 3.896966E-01 1.558787E+00 2.338180E+00 1.558787E+00 3.896966E-01 b1= -2.161179E+00 b2= -2.033992E+00 b3= -8.789098E-01 b4= -1.610655E-01 a0= a1= a2= a3= a4= a5= a6= 2.211834E-01 1.327100E+00 3.317751E+00 4.423668E+00 3.317751E+00 1.327100E+00 2.211834E-01 b1= -3.058672E+00 b2= -4.390465E+00 b3= -3.523254E+00 b4= -1.684185E+00 b5= -4.414881E-01 b6= -5.767513E-02 0.45 a0= 8.001101E-01 a1= 1.600220E+00 b1= -1.556269E+00 a2= 8.001101E-01 b2= -6.441713E-01 a0= a1= a2= a3= a4= 6.291693E-01 2.516677E+00 3.775016E+00 2.516677E+00 6.291693E-01 b1= -3.077062E+00 b2= -3.641323E+00 b3= -1.949229E+00 b4= -3.990945E-01 a0= a1= a2= a3= a4= a5= a6= 4.760635E-01 2.856381E+00 7.140952E+00 9.521270E+00 7.140952E+00 2.856381E+00 4.760635E-01 b1= -4.522403E+00 b2= -8.676844E+00 b3= -9.007512E+00 b4= -5.328429E+00 b5= -1.702543E+00 b6= -2.303303E-01 0.01 TABLE 20-1 Low-pass Chebyshev filters (0.5% ripple) Chapter 20- Chebyshev Filters fC 2 Pole 4 Pole a0= 9.567529E-01 a1= -1.913506E+00 b1= 1.911437E+00 a2= 9.567529E-01 b2= -9.155749E-01 a0= 9.121579E-01 a1= -3.648632E+00 a2= 5.472947E+00 a3= -3.648632E+00 a4= 9.121579E-01 (!! Unstable !!) b1= 3.815952E+00 b2= -5.465026E+00 b3= 3.481295E+00 b4= -8.322529E-01 0.025 a0= 8.950355E-01 a1= -1.790071E+00 b1= 1.777932E+00 a2= 8.950355E-01 b2= -8.022106E-01 a0= 7.941874E-01 a1= -3.176750E+00 a2= 4.765125E+00 a3= -3.176750E+00 a4= 7.941874E-01 0.05 a0= 8.001102E-01 a1= -1.600220E+00 b1= 1.556269E+00 a2= 8.001102E-01 b2= -6.441715E-01 0.075 337 6 Pole a0= 8.630195E-01 a1= -5.178118E+00 a2= 1.294529E+01 a3= -1.726039E+01 a4= 1.294529E+01 a5= -5.178118E+00 a6= 8.630195E-01 (!! Unstable !!) b1= 5.705102E+00 b2= -1.356935E+01 b3= 1.722231E+01 b4= -1.230214E+01 b5= 4.689218E+00 b6= -7.451429E-01 b1= 3.538919E+00 b2= -4.722213E+00 b3= 2.814036E+00 b4= -6.318300E-01 a0= 6.912863E-01 a1= -4.147718E+00 a2= 1.036929E+01 a3= -1.382573E+01 a4= 1.036929E+01 a5= -4.147718E+00 a6= 6.912863E-01 (!! Unstable !!) b1= 5.261399E+00 b2= -1.157800E+01 b3= 1.363599E+01 b4= -9.063840E+00 b5= 3.223738E+00 b6= -4.793541E-01 a0= 6.291694E-01 a1= -2.516678E+00 a2= 3.775016E+00 a3= -2.516678E+00 a4= 6.291694E-01 b1= 3.077062E+00 b2= -3.641324E+00 b3= 1.949230E+00 b4= -3.990947E-01 a0= 4.760636E-01 a1= -2.856382E+00 a2= 7.140954E+00 a3= -9.521272E+00 a4= 7.140954E+00 a5= -2.856382E+00 a6= 4.760636E-01 b1= 4.522403E+00 b2= -8.676846E+00 b3= 9.007515E+00 b4= -5.328431E+00 b5= 1.702544E+00 b6= -2.303304E-01 a0= 7.142028E-01 a1= -1.428406E+00 b1= 1.338264E+00 a2= 7.142028E-01 b2= -5.185469E-01 a0= 4.965350E-01 a1= -1.986140E+00 a2= 2.979210E+00 a3= -1.986140E+00 a4= 4.965350E-01 b1= 2.617304E+00 b2= -2.749252E+00 b3= 1.325548E+00 b4= -2.524546E-01 a0= 3.259100E-01 a1= -1.955460E+00 a2= 4.888651E+00 a3= -6.518201E+00 a4= 4.888651E+00 a5= -1.955460E+00 a6= 3.259100E-01 b1= 3.787397E+00 b2= -6.288362E+00 b3= 5.747801E+00 b4= -3.041570E+00 b5= 8.808669E-01 b6= -1.122464E-01 0.1 a0= 6.362307E-01 a1= -1.272461E+00 b1= 1.125379E+00 a2= 6.362307E-01 b2= -4.195440E-01 a0= 3.896966E-01 a1= -1.558786E+00 a2= 2.338179E+00 a3= -1.558786E+00 a4= 3.896966E-01 b1= 2.161179E+00 b2= -2.033991E+00 b3= 8.789094E-01 b4= -1.610655E-01 a0= 2.211833E-01 a1= -1.327100E+00 a2= 3.317750E+00 a3= -4.423667E+00 a4= 3.317750E+00 a5= -1.327100E+00 a6= 2.211833E-01 b1= 3.058671E+00 b2= -4.390464E+00 b3= 3.523252E+00 b4= -1.684184E+00 b5= 4.414878E-01 b6= -5.767508E-02 0.15 a0= 5.001024E-01 a1= -1.000205E+00 b1= 7.158993E-01 a2= 5.001024E-01 b2= -2.845103E-01 a0= 2.340973E-01 a1= -9.363892E-01 a2= 1.404584E+00 a3= -9.363892E-01 a4= 2.340973E-01 b1= 1.263672E+00 b2= -1.080487E+00 b3= 3.276296E-01 b4= -7.376791E-02 a0= 9.792321E-02 a1= -5.875393E-01 a2= 1.468848E+00 a3= -1.958464E+00 a4= 1.468848E+00 a5= -5.875393E-01 a6= 9.792321E-02 b1= 1.627573E+00 b2= -1.955020E+00 b3= 1.075051E+00 b4= -5.106501E-01 b5= 7.239843E-02 b6= -2.639193E-02 0.2 a0= 3.849163E-01 a1= -7.698326E-01 b1= 3.249116E-01 a2= 3.849163E-01 b2= -2.147536E-01 a0= 1.335566E-01 a1= -5.342262E-01 a2= 8.013393E-01 a3= -5.342262E-01 a4= 1.335566E-01 b1= 3.904484E-01 b2= -6.784138E-01 b3= 1.412016E-02 b4= -5.392238E-02 a0= 3.997486E-02 a1= -2.398492E-01 a2= 5.996230E-01 a3= -7.994973E-01 a4= 5.996230E-01 a5= -2.398492E-01 a6= 3.997486E-02 b1= 2.441149E-01 b2= -1.130306E+00 b3= -1.063169E-01 b4= -3.463299E-01 b5= -8.882996E-02 b6= -3.278741E-02 0.25 a0= 2.858111E-01 a1= -5.716222E-01 b1= -5.423243E-02 a2= 2.858111E-01 b2= -1.974768E-01 a0= 7.015302E-02 a1= -2.806121E-01 a2= 4.209182E-01 a3= -2.806121E-01 a4= 7.015302E-02 b1= -4.541478E-01 b2= -7.417535E-01 b3= -2.361221E-01 b4= -7.096475E-02 a0= 1.434450E-02 a1= -8.606701E-02 a2= 2.151675E-01 a3= -2.868900E-01 a4= 2.151675E-01 a5= -8.606701E-02 a6= 1.434450E-02 b1= -1.076051E+00 b2= -1.662847E+00 b3= -1.191062E+00 b4= -7.403085E-01 b5= -2.752156E-01 b6= -5.722250E-02 0.3 a0= 1.997396E-01 a1= -3.994792E-01 b1= -4.291049E-01 a2= 1.997396E-01 b2= -2.280633E-01 a0= 3.224553E-02 a1= -1.289821E-01 a2= 1.934732E-01 a3= -1.289821E-01 a4= 3.224553E-02 b1= -1.265912E+00 b2= -1.203878E+00 b3= -5.405908E-01 b4= -1.185538E-01 a0= 4.187407E-03 a1= -2.512444E-02 a2= 6.281111E-02 a3= -8.374815E-02 a4= 6.281111E-02 a5= -2.512444E-02 a6= 4.187407E-03 b1= -2.315806E+00 b2= -3.293726E+00 b3= -2.904827E+00 b4= -1.694129E+00 b5= -6.021426E-01 b6= -1.029147E-01 0.35 a0= 1.254285E-01 a1= -2.508570E-01 b1= -8.070777E-01 a2= 1.254285E-01 b2= -3.087918E-01 a0= 1.180009E-02 a1= -4.720035E-02 a2= 7.080051E-02 a3= -4.720035E-02 a4= 1.180009E-02 b1= -2.039039E+00 b2= -2.012961E+00 b3= -9.897915E-01 b4= -2.046700E-01 a0= 8.618665E-04 a1= -5.171200E-03 a2= 1.292800E-02 a3= -1.723733E-02 a4= 1.292800E-02 a5= -5.171200E-03 a6= 8.618665E-04 b1= -3.455239E+00 b2= -5.754734E+00 b3= -5.645387E+00 b4= -3.394902E+00 b5= -1.177469E+00 b6= -1.836195E-01 0.40 a0= 6.372801E-02 a1= -1.274560E-01 b1= -1.194365E+00 a2= 6.372801E-02 b2= -4.492774E-01 a0= 2.780754E-03 a1= -1.112302E-02 a2= 1.668453E-02 a3= -1.112302E-02 a4= 2.780754E-03 b1= -2.764031E+00 b2= -3.122854E+00 b3= -1.664554E+00 b4= -3.502233E-01 a0= 9.086141E-05 a1= -5.451685E-04 a2= 1.362921E-03 a3= -1.817228E-03 a4= 1.362921E-03 a5= -5.451685E-04 a6= 9.086141E-05 b1= -4.470118E+00 b2= -8.755595E+00 b3= -9.543712E+00 b4= -6.079377E+00 b5= -2.140062E+00 b6= -3.247363E-01 0.45 a0= 1.868823E-02 a1= -3.737647E-02 b1= -1.593937E+00 a2= 1.868823E-02 b2= -6.686903E-01 a0= 2.141509E-04 a1= -8.566037E-04 a2= 1.284906E-03 a3= -8.566037E-04 a4= 2.141509E-04 b1= -3.425455E+00 b2= -4.479272E+00 b3= -2.643718E+00 b4= -5.933269E-01 a0= 1.771089E-06 a1= -1.062654E-05 a2= 2.656634E-05 a3= -3.542179E-05 a4= 2.656634E-05 a5= -1.062654E-05 a6= 1.771089E-06 b1= -5.330512E+00 b2= -1.196611E+01 b3= -1.447067E+01 b4= -9.937710E+00 b5= -3.673283E+00 b6= -5.707561E-01 0.01 TABLE 20-2 High-pass Chebyshev filters (0.5% ripple) 338 The Scientist and Engineer's Guide to Digital Signal Processing There are two problems with using tables to design digital filters. First, tables have a limited choice of parameters. For instance, Table 20-1 only provides 12 different cutoff frequencies, a maximum of 6 poles per filter, and no choice of passband ripple. Without the ability to select parameters from a continuous range of values, the filter design cannot be optimized. Second, the coefficients must be manually transferred from the table into the program. This is very time consuming and will discourage you from trying alternative values. Instead of using tabulated values, consider including a subroutine in your program that calculates the coefficients. Such a program is shown in Table 204. The good news is that the program is relatively simple in structure. After the four filter parameters are entered, the program spits out the "a" and "b" coefficients in the arrays A[ ] and B[ ]. The bad news is that the program calls the subroutine in Table 20-5. At first glance this subroutine is really ugly. Don't despair; it isn't as bad as it seems! There is one simple branch in line 1120. Everything else in the subroutine is straightforward number crunching. Six variables enter the routine, five variables leave the routine, and fifteen temporary variables (plus indexes) are used within. Table 20-5 provides two sets of test data for debugging this subroutine. Chapter 31 discusses the operation of this program in detail. Step Response Overshoot Butterworth and Chebyshev filters have an overshoot of 5 to 30% in their step responses, becoming larger as the number of poles is increased. Figure 20-3a shows the step response for two example Chebyshev filters. Figure (b) shows something that is unique to digital filters and has no counterpart in analog electronics: the amount of overshoot in the step response depends to a small degree on the cutoff frequency of the filter. The excessive overshoot and ringing in the step response results from the Chebyshev filter being optimized for the frequency domain at the expense of the time domain. 30 1.5 b. Overshoot a. Step response 25 Percent overshoot 4 pole Amplitude 1.0 2 pole 0.5 20 15 6 pole 10 2 pole 5 0 0.0 -10 0 10 20 30 Sample number 40 50 60 0 0.1 0.2 0.3 Frequency 0.4 FIGURE 20-3 Chebyshev step response. The overshoot in the Chebyshev filter's step response is 5% to 30%, depending on the number of poles, as shown in (a), and the cutoff frequency, as shown in (b). Figure (a) is for a cutoff frequency of 0.05, and may be scaled to other cutoff frequencies. 0.5 Chapter 20- Chebyshev Filters 339 Stability The main limitation of digital filters carried out by convolution is execution time. It is possible to achieve nearly any filter response, provided you are willing to wait for the result. Recursive filters are just the opposite. They run like lightning; however, they are limited in performance. For example, consider a 6 pole, 0.5% ripple, low-pass filter with a 0.01 cutoff frequency. The recursion coefficients for this filter can be obtained from Table 20-1: a0= a1= a2= a3= a4= a5= a6= 1.391351E-10 8.348109E-10 2.087027E-09 2.782703E-09 2.087027E-09 8.348109E-10 1.391351E-10 b1= 5.883343E+00 b2= -1.442798E+01 b3= 1.887786E+01 b4= -1.389914E+01 b5= 5.459909E+00 b6= -8.939932E-01 Look carefully at these coefficients. The "b" coefficients have an absolute value of about ten. Using single precision, the round-off noise on each of these numbers is about one ten-millionth of the value, i.e., 10&6 . Now look at the "a" coefficients, with a value of about 10&9 . Something is obviously wrong here. The contribution from the input signal (via the "a" coefficients) will be 1000 times smaller than the noise from the previously calculated output signal (via the "b" coefficients). This filter won't work! In short, round-off noise limits the number of poles that can be used in a filter. The actual number will depend slightly on the ripple and if it is a high or low-pass filter. The approximate numbers for single precision are: TABLE 20-3 The maximum number of poles for single precision. Cutoff frequency 0.02 0.05 0.10 0.25 0.40 0.45 0.48 Maximum poles 4 6 10 20 10 6 4 The filter's performance will start to degrade as this limit is approached; the step response will show more overshoot, the stopband attenuation will be poor, and the frequency response will have excessive ripple. If the filter is pushed too far, or there is an error in the coefficients, the output will probably oscillate until an overflow occurs. There are two ways of extending the maximum number of poles that can be used. First, use double precision. This requires using double precision in the coefficient calculation as well (including the value for pi ). The second method is to implement the filter in stages. For example, a six pole filter starts out as a cascade of three stages of two poles each. The program in Table 20-4 combines these three stages into a single set of recursion coefficients for easier programming. However, the filter is more stable if carried out as the original three separate stages. This requires knowing the "a" and "b" coefficients for each of the stages. These can 340 The Scientist and Engineer's Guide to Digital Signal Processing 100 'CHEBYSHEV FILTER- RECURSION COEFFICIENT CALCULATION 110 ' 120 'INITIALIZE VARIABLES 130 DIM A[22] 'holds the "a" coefficients upon program completion 140 DIM B[22] 'holds the "b" coefficients upon program completion 150 DIM TA[22] 'internal use for combining stages 160 DIM TB[22] 'internal use for combining stages 170 ' 180 FOR I% = 0 TO 22 190 A[I%] = 0 200 B[I%] = 0 210 NEXT I% 220 ' 230 A[2] = 1 240 B[2] = 1 250 PI = 3.14159265 260 'ENTER THE FOUR FILTER PARAMETERS 270 INPUT "Enter cutoff frequency (0 to .5): ", FC 280 INPUT "Enter 0 for LP, 1 for HP filter: ", LH 290 INPUT "Enter percent ripple (0 to 29): ", PR 300 INPUT "Enter number of poles (2,4,...20): ", NP 310 ' 320 FOR P% = 1 TO NP/2 'LOOP FOR EACH POLE-PAIR 330 ' 340 GOSUB 1000 'The subroutine in TABLE 20-5 350 ' 360 FOR I% = 0 TO 22 'Add coefficients to the cascade 370 TA[I%] = A[I%] 380 TB[I%] = B[I%] 390 NEXT I% 400 ' 410 FOR I% = 2 TO 22 420 A[I%] = A0*TA[I%] + A1*TA[I%-1] + A2*TA[I%-2] 430 B[I%] = TB[I%] - B1*TB[I%-1] - B2*TB[I%-2] 440 NEXT I% 450 ' 460 NEXT P% 470 ' 480 B[2] = 0 'Finish combining coefficients 490 FOR I% = 0 TO 20 500 A[I%] = A[I%+2] 510 B[I%] = -B[I%+2] 520 NEXT I% 530 ' 540 SA = 0 'NORMALIZE THE GAIN 550 SB = 0 560 FOR I% = 0 TO 20 570 IF LH = 0 THEN SA = SA + A[I%] 580 IF LH = 0 THEN SB = SB + B[I%] 590 IF LH = 1 THEN SA = SA + A[I%] * (-1)^I% 600 IF LH = 1 THEN SB = SB + B[I%] * (-1)^I% 610 NEXT I% 620 ' 630 GAIN = SA / (1 - SB) 640 ' 650 FOR I% = 0 TO 20 660 A[I%] = A[I%] / GAIN 670 NEXT I% 680 ' 'The final recursion coefficients are in A[ ] and B[ ] 690 END TABLE 20-4 Chapter 20- Chebyshev Filters 341 1000 'THIS SUBROUTINE IS CALLED FROM TABLE 20-4, LINE 340 1010 ' 1020 ' Variables entering subroutine: PI, FC, LH, PR, HP, P% 1030 ' Variables exiting subroutine: A0, A1, A2, B1, B2 1040 ' Variables used internally: RP, IP, ES, VX, KX, T, W, M, D, K, 1050 ' X0, X1, X2, Y1, Y2 1060 ' 1070 ' 'Calculate the pole location on the unit circle 1080 RP = -COS(PI/(NP*2) + (P%-1) * PI/NP) 1090 IP = SIN(PI/(NP*2) + (P%-1) * PI/NP) 1100 ' 1110 ' 'Warp from a circle to an ellipse 1120 IF PR = 0 THEN GOTO 1210 1130 ES = SQR( (100 / (100-PR))^2 -1 ) 1140 VX = (1/NP) * LOG( (1/ES) + SQR( (1/ES^2) + 1) ) 1150 KX = (1/NP) * LOG( (1/ES) + SQR( (1/ES^2) - 1) ) 1160 KX = (EXP(KX) + EXP(-KX))/2 1170 RP = RP * ( (EXP(VX) - EXP(-VX) ) /2 ) / KX 1180 IP = IP * ( (EXP(VX) + EXP(-VX) ) /2 ) / KX 1190 ' 1200 ' 's-domain to z-domain conversion 1210 T = 2 * TAN(1/2) 1220 W = 2*PI*FC 1230 M = RP^2 + IP^2 1240 D = 4 - 4*RP*T + M*T^2 1250 X0 = T^2/D 1260 X1 = 2*T^2/D 1270 X2 = T^2/D 1280 Y1 = (8 - 2*M*T^2)/D 1290 Y2 = (-4 - 4*RP*T - M*T^2)/D 1300 ' 1310 ' 'LP TO LP, or LP TO HP transform 1320 IF LH = 1 THEN K = -COS(W/2 + 1/2) / COS(W/2 - 1/2) 1330 IF LH = 0 THEN K = SIN(1/2 - W/2) / SIN(1/2 + W/2) 1340 D = 1 + Y1*K - Y2*K^2 1350 A0 = (X0 - X1*K + X2*K^2)/D 1360 A1 = (-2*X0*K + X1 + X1*K^2 - 2*X2*K)/D 1370 A2 = (X0*K^2 - X1*K + X2)/D 1380 B1 = (2*K + Y1 + Y1*K^2 - 2*Y2*K)/D 1390 B2 = (-(K^2) - Y1*K + Y2)/D 1400 IF LH = 1 THEN A1 = -A1 1410 IF LH = 1 THEN B1 = -B1 1420 ' 1430 RETURN TABLE 20-5 TABLE 20-4 and 20-5 Program to calculate the "a" and "b" coefficients for Chebyshev recursive filters. In lines 270-300, four parameters are entered into the program. The cutoff frequency, FC, is expressed as a fraction of the sampling frequency, and therefore must be in the range: 0 to 0.5. The variable, LH, is set to a value of one for a high-pass filter, and zero for a low-pass filter. The value entered for PR must be in the range of 0 to 29, corresponding to 0 to 29% ripple in the filter's frequency response. The number of poles in the filter, entered in the variable NP, must be an even integer between 2 and 20. At the completion of the program, the "a" and "b" coefficients are stored in the arrays A[ ] and B[ ] (a0 = A[0], a1 = A[1], etc.). TABLE 20-5 is a subroutine called from line 340 of the main program. Six variables are passed to this subroutine, and five variables are returned. Table 20-6 (next page) contains two sets of data to help debug this subroutine. The functions: COS and SIN, use radians, not degrees. The function: LOG is the natural (base e) logarithm. Declaring all floating point variables (including the value of B) to be double precision will allow more poles to be used. Tables 20-1 and 20-2 were generated with this program and can be used to test for proper operation. Chapter 33 describes the mathematical operation of this program. 342 The Scientist and Engineer's Guide to Digital Signal Processing DATA SET 1 DATA SET 2 Enter the subroutine with these values: FC LH PR NP P% PI = = = = = = 0.1 0 0 4 1 3.141592 FC LH PR NP P% PI = = = = = = 0.1 1 10 4 2 3.141592 RP IP ES VX KX = = = = = -0.136178 0.933223 0.484322 0.368054 1.057802 T W M D X0 X1 X2 Y1 Y2 = = = = = = = = = 1.092605 0.628318 0.889450 5.656972 0.211029 0.422058 0.211029 1.038784 -0.789584 = = = = = 0.922919 -1.845840 0.922919 1.446913 -0.836653 These values should be present at line 1200: RP IP ES VX KX = = = = = -0.923879 0.382683 not used not used not used These values should be present at line 1310: T W M D X0 X1 X2 Y1 Y2 = = = = = = = = = 1.092605 0.628318 1.000000 9.231528 0.129316 0.258632 0.129316 0.607963 -0.125227 These values should be return to the main program: A0 A1 A2 B1 B2 = = = = = 0.061885 0.123770 0.061885 1.048600 -0.296140 A0 A1 A2 B1 B2 TABLE 20-6 Debugging data. This table contains two sets of data for debugging the subroutine listed in Table 20-5. be obtained from the program in Table 20-4. The subroutine in Table 20-5 is called once for each stage in the cascade. For example, it is called three times for a six pole filter. At the completion of the subroutine, five variables are return to the main program: A0, A1, A2, B1, & B2. These are the recursion coefficients for the two pole stage being worked on, and can be used to implement the filter in stages. CHAPTER 21 Filter Comparison Decisions, decisions, decisions! With all these filters to choose from, how do you know which to use? This chapter is a head-to-head competition between filters; we'll select champions from each side and let them fight it out. In the first match, digital filters are pitted against analog filters to see which technology is best. In the second round, the windowed-sinc is matched against the Chebyshev to find the king of the frequency domain filters. In the final battle, the moving average fights the single pole filter for the time domain championship. Enough talk; let the competition begin! Match #1: Analog vs. Digital Filters Most digital signals originate in analog electronics. If the signal needs to be filtered, is it better to use an analog filter before digitization, or a digital filter after? We will answer this question by letting two of the best contenders deliver their blows. The goal will be to provide a low-pass filter at 1 kHz. Fighting for the analog side is a six pole Chebyshev filter with 0.5 dB (6%) ripple. As described in Chapter 3, this can be constructed with 3 op amps, 12 resistors, and 6 capacitors. In the digital corner, the windowed-sinc is warming up and ready to fight. The analog signal is digitized at a 10 kHz sampling rate, making the cutoff frequency 0.1 on the digital frequency scale. The length of the windowed-sinc will be chosen to be 129 points, providing the same 90% to 10% roll-off as the analog filter. Fair is fair. Figure 21-1 shows the frequency and step responses for these two filters. Let's compare the two filters blow-by-blow. As shown in (a) and (b), the analog filter has a 6% ripple in the passband, while the digital filter is perfectly flat (within 0.02%). The analog designer might argue that the ripple can be selected in the design; however, this misses the point. The flatness achievable with analog filters is limited by the accuracy of their resistors and 343 344 The Scientist and Engineer's Guide to Digital Signal Processing capacitors. Even if a Butterworth response is designed (i.e., 0% ripple), filters of this complexity will have a residue ripple of, perhaps, 1%. On the other hand, the flatness of digital filters is primarily limited by round-off error, making them hundreds of times flatter than their analog counterparts. Score one point for the digital filter. Next, look at the frequency response on a log scale, as shown in (c) and (d). Again, the digital filter is clearly the victor in both roll-off and stopband attenuation. Even if the analog performance is improved by adding additional stages, it still can't compare to the digital filter. For instance, imagine that you need to improve these two parameters by a factor of 100. This can be done with simple modifications to the windowed-sinc, but is virtually impossible for the analog circuit. Score two more for the digital filter. The step response of the two filters is shown in (e) and (f). The digital filter's step response is symmetrical between the lower and upper portions of the step, i.e., it has a linear phase. The analog filter's step response is not symmetrical, i.e., it has a nonlinear phase. One more point for the digital filter. Lastly, the analog filter overshoots about 20% on one side of the step. The digital filter overshoots about 10%, but on both sides of the step. Since both are bad, no points are awarded. In spite of this beating, there are still many applications where analog filters should, or must, be used. This is not related to the actual performance of the filter (i.e., what goes in and what comes out), but to the general advantages that analog circuits have over digital techniques. The first advantage is speed: digital is slow; analog is fast. For example, a personal computer can only filter data at about 10,000 samples per second, using FFT convolution. Even simple op amps can operate at 100 kHz to 1 MHz, 10 to 100 times as fast as the digital system! The second inherent advantage of analog over digital is dynamic range. This comes in two flavors. Amplitude dynamic range is the ratio between the largest signal that can be passed through a system, and the inherent noise of the system. For instance, a 12 bit ADC has a saturation level of 4095, and an rms quantization noise of 0.29 digital numbers, for a dynamic range of about 14000. In comparison, a standard op amp has a saturation voltage of about 20 volts and an internal noise of about 2 microvolts, for a dynamic range of about ten million. Just as before, a simple op amp devastates the digital system. The other flavor is frequency dynamic range. For example, it is easy to design an op amp circuit to simultaneously handle frequencies between 0.01 Hz and 100 kHz (seven decades). When this is tried with a digital system, the computer becomes swamped with data. For instance, sampling at 200 kHz, it takes 20 million points to capture one complete cycle at 0.01 Hz. You may have noticed that the frequency response of digital filters is almost always plotted on a linear frequency scale, while analog filters are usually displayed with a logarithmic frequency. This is because digital filters need Chapter 21- Filter Comparison Analog Filter Digital Filter (6 pole 0.5dB Chebyshev) (129 point windowed-sinc) 1.50 1.50 a. Frequency response 1.25 b. Frequency response 1.25 1.00 Amplitude Amplitude 345 0.75 1.00 0.75 0.50 0.50 0.25 0.25 0.00 0.00 0 1000 2000 3000 4000 5000 0 0.1 0.2 40 c. Frequency response (dB) 0.5 0.4 0.5 d. Frequency response (dB) 20 0 Amplitude (dB) Amplitude (dB) 0.4 40 20 -20 -40 -60 -80 0 -20 -40 -60 -80 -100 -100 0 1000 2000 3000 Frequency (hertz) 4000 5000 0 2.0 0.1 0.2 0.3 Frequency 2.0 e. Step response f. Step response 1.5 Amplitude 1.5 Amplitude 0.3 Frequency Frequency (hertz) 1.0 0.5 0.0 1.0 0.5 0.0 -0.5 -0.5 -4 -2 0 2 4 Time (milliseconds) 6 8 -40 -20 0 20 40 Sample number 60 80 FIGURE 21-1 Comparison of analog and digital filters. Digital filters have better performance in many areas, such as: passband ripple, (a) vs. (b), roll-off and stopband attenuation, (c) vs. (d), and step response symmetry, (e) vs. (f). The digital filter in this example has a cutoff frequency of 0.1 of the 10 kHz sampling rate. This provides a fair comparison to the 1 kHz cutoff frequency of the analog filter. a linear scale to show their exceptional filter performance, while analog filters need the logarithmic scale to show their huge dynamic range. 346 The Scientist and Engineer's Guide to Digital Signal Processing Match #2: Windowed-Sinc vs. Chebyshev Both the windowed-sinc and the Chebyshev filters are designed to separate one band of frequencies from another. The windowed-sinc is an FIR filter implemented by convolution, while the Chebyshev is an IIR filter carried out by recursion. Which is the best digital filter in the frequency domain? We'll let them fight it out. The recursive filter contender will be a 0.5% ripple, 6 pole Chebyshev low-pass filter. A fair comparison is complicated by the fact that the Chebyshev's frequency response changes with the cutoff frequency. We will use a cutoff frequency of 0.2, and select the windowed-sinc's filter kernel to be 51 points. This makes both filters have the same 90% to 10% roll-off, as shown in Fig. 21-2(a). Now the pushing and shoving begins. The recursive filter has a 0.5% ripple in the passband, while the windowed-sinc is flat. However, we could easily set the recursive filter ripple to 0% if needed. No points. Figure 21-2b shows that the windowed-sinc has a much better stopband attenuation than the Chebyshev. One point for the windowed-sinc. Figure 21-3 shows the step response of the two filters. Both are bad, as you should expect for frequency domain filters. The recursive filter has a nonlinear phase, but this can be corrected with bidirectional filtering. Since both filters are so ugly in this parameter, we will call this a draw. So far, there isn't much difference between these two filters; either will work when moderate performance is needed. The heavy hitting comes over two critical issues: maximum performance and speed. The windowed-sinc is a powerhouse, while the Chebyshev is quick and agile. Suppose you have a really tough frequency separation problem, say, needing to isolate a 100 1.5 40 1.0 Chebyshev recursive 0.5 b. Frequency response (dB) 20 Amplitude (dB) Amplitude a. Frequency response windowed-sinc 0 -20 Chebyshev recursive -40 -60 windowed-sinc -80 0.0 -100 0 0.1 0.2 0.3 Frequency 0.4 0.5 0 0.1 0.2 0.3 Frequency 0.4 0.5 FIGURE 21-2 Windowed-sinc and Chebyshev frequency responses. Frequency responses are shown for a 51 point windowed-sinc filter and a 6 pole, 0.5% ripple Chebyshev recursive filter. The windowed-sinc has better stopband attenuation, but either will work in moderate performance applications. The cutoff frequency of both filters is 0.2, measured at an amplitude of 0.5 for the windowed-sinc, and 0.707 for the recursive. Chapter 21- Filter Comparison 2.0 2.0 a. Windowed-sinc step response b. Chebyshev step response 1.5 Amplitude Amplitude 1.5 1.0 0.5 0.0 -0.5 -30 347 1.0 0.5 0.0 -20 -10 0 10 Sample number 20 -0.5 -30 30 -20 -10 0 10 20 Sample number 30 FIGURE 21-3 Windowed--sinc and Chebyshev step responses. The step responses are shown for a 51 point windowed-sinc filter and a 6 pole, 0.5% ripple Chebyshev recursive filter. Each of these filters has a cutoff frequency of 0.2. The windowed-sinc has a slightly better step response because it has less overshoot and a zero phase. millivolt signal at 61 hertz that is riding on a 120 volt power line at 60 hertz. Figure 21-4 shows how these two filters compare when you need maximum performance. The recursive filter is a 6 pole Chebyshev with 0.5% ripple. This is the maximum number of poles that can be used at a 0.05 cutoff frequency with single precision. The windowed-sinc uses a 1001 point filter kernel, formed by convolving a 501 point windowed-sinc filter kernel with itself. As shown in Chapter 16, this provides greater stopband attenuation. How do these two filters compare when maximum performance is needed? The windowed-sinc crushes the Chebyshev! Even if the recursive filter were improved (more poles, multistage implementation, double precision, etc.), it is still no match for the FIR performance. This is especially impressive when you consider that the windowed-sinc has only begun to fight. There are strong limits on the maximum performance that recursive filters can provide. The windowed-sinc, in contrast, can be pushed to incredible levels. This is, of course, provided you are willing to wait for the result. Which brings up the second critical issue: speed. 20 0 Amplitude (dB) FIGURE 21-4 Maximum performance of FIR and IIR filters. The frequency response of the windowed-sinc can be virtually any shape needed, while the Chebyshev recursive filter is very limited. This graph compares the frequency response of a six pole Chebyshev recursive filter with a 1001 point windowed-sinc filter. -20 -40 Chebyshev (IIR) -60 Windowed-sinc (FIR) -80 -100 0 0.1 0.2 0.3 Frequency 0.4 0.5 348 The Scientist and Engineer's Guide to Digital Signal Processing 50 standard convolution 40 Relative execution time FIGURE 21-5 Comparing FIR and IIR execution speeds. These curves shows the relative execution times for a windowed-sinc filter compared with an equivalent six pole Chebyshev recursive filter. Curves are shown for implementing the FIR filter by both the standard and the FFT convolution algorithms. The windowed-sinc execution time rises at low and high frequencies because the filter kernel must be made longer to keep up with the greater performance of the recursive filter at these frequencies. In general, IIR filters are an order of magnitude faster than FIR filters of comparable performance. 30 FFT convolution 20 10 Recursive 0 0 0.1 0.2 0.3 Frequency 0.4 0.5 Comparing these filters for speed is like racing a Ferrari against a go-cart. Figure 21-5 shows how much longer the windowed-sinc takes to execute, compared to a six pole recursive filter. Since the recursive filter has a faster roll-off at low and high frequencies, the length of the windowed-sinc kernel must be made longer to match the performance (i.e., to keep the comparison fair). This accounts for the increased execution time for the windowed-sinc near frequencies 0 and 0.5. The important point is that FIR filters can be expected to be about an order of magnitude slower than comparable IIR filters (go-cart: 15 mph, Ferrari: 150 mph). Match #3: Moving Average vs. Single Pole Our third competition will be a battle of the time domain filters. The first fighter will be a nine point moving average filter. Its opponent for today's match will be a single pole recursive filter using the bidirectional technique. To achieve a comparable frequency response, the single pole filter will use a sample-to-sample decay of x ' 0.70 . The battle begins in Fig. 21-6 where the frequency response of each filter is shown. Neither one is very impressive, but of course, frequency separation isn't what these filters are used for. No points for either side. Figure 21-7 shows the step responses of the filters. In (a), the moving average step response is a straight line, the most rapid way of moving from one level to another. In (b), the recursive filter's step response is smoother, which may be better for some applications. One point for each side. These filters are quite equally matched in terms of performance and often the choice between the two is made on personal preference. However, there are Chapter 21- Filter Comparison 349 1.2 1.0 Amplitude FIGURE 21-6 Moving average and single pole frequency responses. Both of these filters have a poor frequency response, as you should expect for time domain filters. 0.8 0.6 Single pole recursive 0.4 Moving average 0.2 0.0 0 0.1 0.2 0.3 0.4 Frequency 0.5 two cases where one filter has a slight edge over the other. These are based on the trade-off between development time and execution time. In the first instance, you want to reduce development time and are willing to accept a slower filter. For example, you might have a one time need to filter a few thousand points. Since the entire program runs in only a few seconds, it is pointless to spend time optimizing the algorithm. Floating point will almost certainly be used. The choice is to use the moving average filter carried out by convolution, or a single pole recursive filter. The winner here is the recursive filter. It will be slightly easier to program and modify, and will execute much faster. The second case is just the opposite; your filter must operate as fast as possible and you are willing to spend the extra development time to get it. For instance, this filter might be a part of a commercial product, with the potential to be run millions of times. You will probably use integers for the highest possible speed. Your choice of filters will be the moving average 1.5 1.5 a. Moving average b. Bidirectional recursive 1.0 Amplitude Amplitude 1.0 0.5 0.0 0.5 0.0 -0.5 -0.5 -20 -10 0 Sample number 10 20 -20 -15 -10 -5 0 5 Sample number 10 15 FIGURE 21-7 Step responses of the moving average and the bidirectional single pole filter. The moving average step response occurs over a smaller number of samples, while the single pole filter's step response is smoother. 20 350 The Scientist and Engineer's Guide to Digital Signal Processing carried out by recursion, or the single pole recursive filter implemented with look-up tables or integer math. The winner is the moving average filter. It will execute faster and not be susceptible to the development and execution problems of integer arithmetic. CHAPTER Audio Processing 22 Audio processing covers many diverse fields, all involved in presenting sound to human listeners. Three areas are prominent: (1) high fidelity music reproduction, such as in audio compact discs, (2) voice telecommunications, another name for telephone networks, and (3) synthetic speech, where computers generate and recognize human voice patterns. While these applications have different goals and problems, they are linked by a common umpire: the human ear. Digital Signal Processing has produced revolutionary changes in these and other areas of audio processing. Human Hearing The human ear is an exceedingly complex organ. To make matters even more difficult, the information from two ears is combined in a perplexing neural network, the human brain. Keep in mind that the following is only a brief overview; there are many subtle effects and poorly understood phenomena related to human hearing. Figure 22-1 illustrates the major structures and processes that comprise the human ear. The outer ear is composed of two parts, the visible flap of skin and cartilage attached to the side of the head, and the ear canal, a tube about 0.5 cm in diameter extending about 3 cm into the head. These structures direct environmental sounds to the sensitive middle and inner ear organs located safely inside of the skull bones. Stretched across the end of the ear canal is a thin sheet of tissue called the tympanic membrane or ear drum. Sound waves striking the tympanic membrane cause it to vibrate. The middle ear is a set of small bones that transfer this vibration to the cochlea (inner ear) where it is converted to neural impulses. The cochlea is a liquid filled tube roughly 2 mm in diameter and 3 cm in length. Although shown straight in Fig. 22-1, the cochlea is curled up and looks like a small snail shell. In fact, cochlea is derived from the Greek word for snail. 351 352 The Scientist and Engineer's Guide to Digital Signal Processing When a sound wave tries to pass from air into liquid, only a small fraction of the sound is transmitted through the interface, while the remainder of the energy is reflected. This is because air has a low mechanical impedance (low acoustic pressure and high particle velocity resulting from low density and high compressibility), while liquid has a high mechanical impedance. In less technical terms, it requires more effort to wave your hand in water than it does to wave it in air. This difference in mechanical impedance results in most of the sound being reflected at an air/liquid interface. The middle ear is an impedance matching network that increases the fraction of sound energy entering the liquid of the inner ear. For example, fish do not have an ear drum or middle ear, because they have no need to hear in air. Most of the impedance conversion results from the difference in area between the ear drum (receiving sound from the air) and the oval window (transmitting sound into the liquid, see Fig. 22-1). The ear drum has an area of about 60 ( mm) 2, while the oval window has an area of roughly 4 ( mm) 2. Since pressure is equal to force divided by area, this difference in area increases the sound wave pressure by about 15 times. Contained within the cochlea is the basilar membrane, the supporting structure for about 12,000 sensory cells forming the cochlear nerve. The basilar membrane is stiffest near the oval window, and becomes more flexible toward the opposite end, allowing it to act as a frequency spectrum analyzer. When exposed to a high frequency signal, the basilar membrane resonates where it is stiff, resulting in the excitation of nerve cells close to the oval window. Likewise, low frequency sounds excite nerve cells at the far end of the basilar membrane. This makes specific fibers in the cochlear nerve respond to specific frequencies. This organization is called the place principle, and is preserved throughout the auditory pathway into the brain. Another information encoding scheme is also used in human hearing, called the volley principle. Nerve cells transmit information by generating brief electrical pulses called action potentials. A nerve cell on the basilar membrane can encode audio information by producing an action potential in response to each cycle of the vibration. For example, a 200 hertz sound wave can be represented by a neuron producing 200 action potentials per second. However, this only works at frequencies below about 500 hertz, the maximum rate that neurons can produce action potentials. The human ear overcomes this problem by allowing several nerve cells to take turns performing this single task. For example, a 3000 hertz tone might be represented by ten nerve cells alternately firing at 300 times per second. This extends the range of the volley principle to about 4 kHz, above which the place principle is exclusively used. Table 22-1 shows the relationship between sound intensity and perceived loudness. It is common to express sound intensity on a logarithmic scale, called decibel SPL (Sound Power Level). On this scale, 0 dB SPL is a sound wave power of 10-16 watts/cm 2 , about the weakest sound detectable by the human ear. Normal speech is at about 60 dB SPL, while painful damage to the ear occurs at about 140 dB SPL. Chapter 22- Audio Processing outer ear sound waves in air ear canal 353 tympanic membrane (ear drum) oval window sound waves in liquid middle ear bones high frequency detection cochlea medium frequency detection basilar membrane low frequency detection FIGURE 22-1 Functional diagram of the human ear. The outer ear collects sound waves from the environment and channels them to the tympanic membrane (ear drum), a thin sheet of tissue that vibrates in synchronization with the air waveform. The middle ear bones (hammer, anvil and stirrup) transmit these vibrations to the oval window, a flexible membrane in the fluid filled cochlea. Contained within the cochlea is the basilar membrane, the supporting structure for about 12,000 nerve cells that form the cochlear nerve. Due to the varying stiffness of the basilar membrane, each nerve cell only responses to a narrow range of audio frequencies, making the ear a frequency spectrum analyzer. The difference between the loudest and faintest sounds that humans can hear is about 120 dB, a range of one-million in amplitude. Listeners can detect a change in loudness when the signal is altered by about 1 dB (a 12% change in amplitude). In other words, there are only about 120 levels of loudness that can be perceived from the faintest whisper to the loudest thunder. The sensitivity of the ear is amazing; when listening to very weak sounds, the ear drum vibrates less than the diameter of a single molecule! The perception of loudness relates roughly to the sound power to an exponent of 1/3. For example, if you increase the sound power by a factor of ten, listeners will report that the loudness has increased by a factor of about two ( 101/3 . 2 ). This is a major problem for eliminating undesirable environmental sounds, for instance, the beefed-up stereo in the next door apartment. Suppose you diligently cover 99% of your wall with a perfect soundproof material, missing only 1% of the surface area due to doors, corners, vents, etc. Even though the sound power has been reduced to only 1% of its former value, the perceived loudness has only dropped to about 0.011/3 . 0.2 , or 20%. The range of human hearing is generally considered to be 20 Hz to 20 kHz, but it is far more sensitive to sounds between 1 kHz and 4 kHz. For example, listeners can detect sounds as low as 0 dB SPL at 3 kHz, but require 40 dB SPL at 100 hertz (an amplitude increase of 100). Listeners can tell that two tones are different if their frequencies differ by more than about 0.3% at 3 kHz. This increases to 3% at 100 hertz. For comparison, adjacent keys on a piano differ by about 6% in frequency. TABLE 22-1 Units of sound intensity. Sound intensity is expressed as power per unit area (such as watts/cm 2), or more commonly on a logarithmic scale called decibels SPL. As this table shows, human hearing is the most sensitive between 1 kHz and 4 kHz. Louder The Scientist and Engineer's Guide to Digital Signal Processing Softer 354 Watts/cm2 Decibels SPL 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 10-10 10-11 10-12 10-13 10-14 10-15 10-16 10-17 10-18 140 dB 130 dB 120 dB 110 dB 100 dB 90 dB 80 dB 70 dB 60 dB 50 dB 40 dB 30 dB 20 dB 10 dB 0 dB -10 dB -20 dB Example sound Pain Discomfort Jack hammers and rock concerts OSHA limit for industrial noise Normal conversation Weakest audible at 100 hertz Weakest audible at 10kHz Weakest audible at 3 kHz The primary advantage of having two ears is the ability to identify the direction of the sound. Human listeners can detect the difference between two sound sources that are placed as little as three degrees apart, about the width of a person at 10 meters. This directional information is obtained in two separate ways. First, frequencies above about 1 kHz are strongly shadowed by the head. In other words, the ear nearest the sound receives a stronger signal than the ear on the opposite side of the head. The second clue to directionality is that the ear on the far side of the head hears the sound slightly later than the near ear, due to its greater distance from the source. Based on a typical head size (about 22 cm) and the speed of sound (about 340 meters per second), an angular discrimination of three degrees requires a timing precision of about 30 microseconds. Since this timing requires the volley principle, this clue to directionality is predominately used for sounds less than about 1 kHz. Both these sources of directional information are greatly aided by the ability to turn the head and observe the change in the signals. An interesting sensation occurs when a listener is presented with exactly the same sounds to both ears, such as listening to monaural sound through headphones. The brain concludes that the sound is coming from the center of the listener's head! While human hearing can determine the direction a sound is from, it does poorly in identifying the distance to the sound source. This is because there are few clues available in a sound wave that can provide this information. Human hearing weakly perceives that high frequency sounds are nearby, while low frequency sounds are distant. This is because sound waves dissipate their higher frequencies as they propagate long distances. Echo content is another weak clue to distance, providing a perception of the room size. For example, Chapter 22- Audio Processing 355 sounds in a large auditorium will contain echoes at about 100 millisecond intervals, while 10 milliseconds is typical for a small office. Some species have solved this ranging problem by using active sonar. For example, bats and dolphins produce clicks and squeaks that reflect from nearby objects. By measuring the interval between transmission and echo, these animals can locate objects with about 1 cm resolution. Experiments have shown that some humans, particularly the blind, can also use active echo localization to a small extent. Timbre The perception of a continuous sound, such as a note from a musical instrument, is often divided into three parts: loudness, pitch, and timbre (pronounced "timber"). Loudness is a measure of sound wave intensity, as previously described. Pitch is the frequency of the fundamental component in the sound, that is, the frequency with which the waveform repeats itself. While there are subtle effects in both these perceptions, they are a straightforward match with easily characterized physical quantities. Timbre is more complicated, being determined by the harmonic content of the signal. Figure 22-2 illustrates two waveforms, each formed by adding a 1 kHz sine wave with an amplitude of one, to a 3 kHz sine wave with an amplitude of one-half. The difference between the two waveforms is that the one shown in (b) has the higher frequency inverted before the addition. Put another way, the third harmonic (3 kHz) is phase shifted by 180 degrees compared to the first harmonic (1 kHz). In spite of the very different time domain waveforms, these two signals sound identical. This is because hearing is based on the amplitude of the frequencies, and is very insensitive to their phase. The shape of the time domain waveform is only indirectly related to hearing, and usually not considered in audio systems. 3 3 a. 1 kHz + 3 kHz sine waves b. 1 kHz - 3 kHz sine waves 2 Amplitude Amplitude 2 1 0 -1 1 0 -1 -2 -2 0 1 2 3 Time (milliseconds) 4 5 0 1 2 3 Time (milliseconds) 4 FIGURE 22-2 Phase detection of the human ear. The human ear is very insensitive to the relative phase of the component sinusoids. For example, these two waveforms would sound identical, because the amplitudes of their components are the same, even though their relative phases are different. 5 356 The Scientist and Engineer's Guide to Digital Signal Processing The ear's insensitivity to phase can be understood by examining how sound propagates through the environment. Suppose you are listening to a person speaking across a small room. Much of the sound reaching your ears is reflected from the walls, ceiling and floor. Since sound propagation depends on frequency (such as: attenuation, reflection, and resonance), different frequencies will reach your ear through different paths. This means that the relative phase of each frequency will change as you move about the room. Since the ear disregards these phase variations, you perceive the voice as unchanging as you move position. From a physics standpoint, the phase of an audio signal becomes randomized as it propagates through a complex environment. Put another way, the ear is insensitive to phase because it contains little useful information. However, it cannot be said that the ear is completely deaf to the phase. This is because a phase change can rearrange the time sequence of an audio signal. An example is the chirp system (Chapter 11) that changes an impulse into a much longer duration signal. Although they differ only in their phase, the ear can distinguish between the two sounds because of their difference in duration. For the most part, this is just a curiosity, not something that happens in the normal listening environment. Suppose that we ask a violinist to play a note, say, the A below middle C. When the waveform is displayed on an oscilloscope, it appear much as the sawtooth shown in Fig. 22-3a. This is a result of the sticky rosin applied to the fibers of the violinist's bow. As the bow is drawn across the string, the waveform is formed as the string sticks to the bow, is pulled back, and eventually breaks free. This cycle repeats itself over and over resulting in the sawtooth waveform. Figure 22-3b shows how this sound is perceived by the ear, a frequency of 220 hertz, plus harmonics at 440, 660, 880 hertz, etc. If this note were played on another instrument, the waveform would look different; however, the ear would still hear a frequency of 220 hertz plus the harmonics. Since the two instruments produce the same fundamental frequency for this note, they sound similar, and are said to have identical pitch. Since the relative amplitude of the harmonics is different, they will not sound identical, and will be said to have different timbre. It is often said that timbre is determined by the shape of the waveform. This is true, but slightly misleading. The perception of timbre results from the ear detecting harmonics. While harmonic content is determined by the shape of the waveform, the insensitivity of the ear to phase makes the relationship very onesided. That is, a particular waveform will have only one timbre, while a particular timbre has an infinite number of possible waveforms. The ear is very accustomed to hearing a fundamental plus harmonics. If a listener is presented with the combination of a 1 kHz and 3 kHz sine wave, they will report that it sounds natural and pleasant. If sine waves of 1 kHz and 3.1 kHz are used, it will sound objectionable. Chapter 22- Audio Processing 8 4 a. Time domain waveform b. Frequency spectrum 3 Amplitude 4 Amplitude 357 0 -4 fundamental harmonics 2 1 -8 0 0 5 10 15 Time (milliseconds) 20 25 0 200 400 600 800 1000 Frequency (hertz) 1200 1400 1600 FIGURE 22-3 Violin waveform. A bowed violin produces a sawtooth waveform, as illustrated in (a). The sound heard by the ear is shown in (b), the fundamental frequency plus harmonics. This is the basis of the standard musical scale, as illustrated by the piano keyboard in Fig. 22-4. Striking the farthest left key on the piano produces a fundamental frequency of 27.5 hertz, plus harmonics at 55, 110, 220, 440, 880 hertz, etc. (there are also harmonics between these frequencies, but they aren't important for this discussion). These harmonics correspond to the fundamental frequency produced by other keys on the keyboard. Specifically, every seventh white key is a harmonic of the far left key. That is, the eighth key from the left has a fundamental frequency of 55 hertz, the 15th key has a fundamental frequency of 110 hertz, etc. Being harmonics of each other, these keys sound similar when played, and are harmonious when played in unison. For this reason, they are all called the note, A. In this same manner, the white key immediate right of each A is called a B, and they are all harmonics of each other. This pattern repeats for the seven notes: A, B, C, D, E, F, and G. The term octave means a factor of two in frequency. On the piano, one octave comprises eight white keys, accounting for the name (octo is Latin for eight). In other words, the piano’s frequency doubles after every seven white keys, and the entire keyboard spans a little over seven octaves. The range of human hearing is generally quoted as 20 hertz to 20 kHz, BCDEF G A- 27.5 Hz BCDEF G A- 55 Hz BCDEF G A- 110 Hz BCDEF G A- 220 Hz BCDEF G A- 440 Hz BCDE F G A- 880 Hz BCDE F G A- 1760 Hz BC A- 3520 Hz C- 262 Hz (Middle C) FIGURE 22-4 The Piano keyboard. The keyboard of the piano is a logarithmic frequency scale, with the fundamental frequency doubling after every seven white keys. These white keys are the notes: A, B, C, D, E, F and G. 358 The Scientist and Engineer's Guide to Digital Signal Processing corresponding to about ½ octave to the left, and two octaves to the right of the piano keyboard. Since octaves are based on doubling the frequency every fixed number of keys, they are a logarithmic representation of frequency. This is important because audio information is generally distributed in this same way. For example, as much audio information is carried in the octave between 50 hertz and 100 hertz, as in the octave between 10 kHz and 20 kHz. Even though the piano only covers about 20% of the frequencies that humans can hear (4 kHz out of 20 kHz), it can produce more than 70% of the audio information that humans can perceive (7 out of 10 octaves). Likewise, the highest frequency a human can detect drops from about 20 kHz to 10 kHz over the course of an adult's lifetime. However, this is only a loss of about 10% of the hearing ability (one octave out of ten). As shown next, this logarithmic distribution of information directly affects the required sampling rate of audio signals. Sound Quality vs. Data Rate When designing a digital audio system there are two questions that need to be asked: (1) how good does it need to sound? and (2) what data rate can be tolerated? The answer to these questions usually results in one of three categories. First, high fidelity music, where sound quality is of the greatest importance, and almost any data rate will be acceptable. Second, telephone communication, requiring natural sounding speech and a low data rate to reduce the system cost. Third, compressed speech, where reducing the data rate is very important and some unnaturalness in the sound quality can be tolerated. This includes military communication, cellular telephones, and digitally stored speech for voice mail and multimedia. Table 22-2 shows the tradeoff between sound quality and data rate for these three categories. High fidelity music systems sample fast enough (44.1 kHz), and with enough precision (16 bits), that they can capture virtually all of the sounds that humans are capable of hearing. This magnificent sound quality comes at the price of a high data rate, 44.1 kHz × 16 bits = 706k bits/sec. This is pure brute force. Whereas music requires a bandwidth of 20 kHz, natural sounding speech only requires about 3.2 kHz. Even though the frequency range has been reduced to only 16% (3.2 kHz out of 20 kHz), the signal still contains 80% of the original sound information (8 out of 10 octaves). Telecommunication systems typically operate with a sampling rate of about 8 kHz, allowing natural sounding speech, but greatly reduced music quality. You are probably already familiar with this difference in sound quality: FM radio stations broadcast with a bandwidth of almost 20 kHz, while AM radio stations are limited to about 3.2 kHz. Voices sound normal on the AM stations, but the music is weak and unsatisfying. Voice-only systems also reduce the precision from 16 bits to 12 bits per sample, with little noticeable change in the sound quality. This can be reduced to only 8 bits per sample if the quantization step size is made unequal. This is a widespread procedure called companding, and will be Chapter 22- Audio Processing Sound Quality Required High fidelity music (compact disc) Telephone quality speech (with companding) Speech encoded by Linear Predictive Coding 359 Bandwidth Sampling rate Number of bits Data rate (bits/sec) 5 Hz to 20 kHz 44.1 kHz 16 bit 706k Satisfies even the most picky audiophile. Better than human hearing. 200 Hz to 3.2 kHz 8 kHz 12 bit 96k Good speech quality, but very poor for music. 200 Hz to 3.2 kHz 8 kHz 8 bit 64k Nonlinear ADC reduces the data rate by 50%. A very common technique. 200 Hz to 3.2 kHz 8 kHz 12 bit 4k DSP speech compression technique. Very low data rates, poor voice quality. Comments TABLE 22-2 Audio data rate vs. sound quality. The sound quality of a digitized audio signal depends on its data rate, the product of its sampling rate and number of bits per sample. This can be broken into three categories, high fidelity music (706 kbits/sec), telephone quality speech (64 kbits/sec), and compressed speech (4 kbits/sec). discussed later in this chapter. An 8 kHz sampling rate, with an ADC precision of 8 bits per sample, results in a data rate of 64k bits/sec. This is the brute force data rate for natural sounding speech. Notice that speech requires less than 10% of the data rate of high fidelity music. The data rate of 64k bits/sec represents the straightforward application of sampling and quantization theory to audio signals. Techniques for lowering the data rate further are based on compressing the data stream by removing the inherent redundancies in speech signals. Data compression is the topic of Chapter 27. One of the most efficient ways of compressing an audio signal is Linear Predictive Coding (LPC), of which there are several variations and subgroups. Depending on the speech quality required, LPC can reduce the data rate to as little as 2-6k bits/sec. We will revisit LPC later in this chapter with speech synthesis. High Fidelity Audio Audiophiles demand the utmost sound quality, and all other factors are treated as secondary. If you had to describe the mindset in one word, it would be: overkill. Rather than just matching the abilities of the human ear, these systems are designed to exceed the limits of hearing. It's the only way to be sure that the reproduced music is pristine. Digital audio was brought to the world by the compact laser disc, or CD. This was a revolution in music; the sound quality of the CD system far exceeds older systems, such as records and tapes. DSP has been at the forefront of this technology. 360 The Scientist and Engineer's Guide to Digital Signal Processing Figure 22-5 illustrates the surface of a compact laser disc, such as viewed through a high power microscope. The main surface is shiny (reflective of light), with the digital information stored as a series of dark pits burned on the surface with a laser. The information is arranged in a single track that spirals from the outside to the inside, the same as a phonograph record. The rotation of the CD is changed from about 210 to 480 rpm as the information is read from the outside to the inside of the spiral, making the scanning velocity a constant 1.2 meters per second. (In comparison, phonograph records spin at a fixed rate, such as 33, 45 or 78 rpm). During playback, an optical sensor detects if the surface is reflective or nonreflective, generating the corresponding binary information. As shown by the geometry in Fig. 22-5, the CD stores about 1 bit per ( µm ) 2, corresponding to 1 million bits per ( mm) 2, and 15 billion bits per disk. This is about the same feature size used in integrated circuit manufacturing, and for a good reason. One of the properties of light is that it cannot be focused to smaller than about one-half wavelength, or 0.3 µm. Since both integrated circuits and laser disks are created by optical means, the fuzziness of light below 0.3 µm limits how small of features can be used. Figure 22-6 shows a block diagram of a typical compact disc playback system. The raw data rate is 4.3 million bits per second, corresponding to 1 bit each 0.28 µm of track length. However, this is in conflict with the specified geometry of the CD; each pit must be no shorter than 0.8 µm, and no longer than 3.5 µm. In other words, each binary one must be part of a group of 3 to 13 ones. This has the advantage of reducing the error rate due to the optical pickup, but how do you force the binary data to comply with this strange bunching? The answer is an encoding scheme called eight-to-fourteen modulation (EFM). Instead of directly storing a byte of data on the disc, the 8 bits are passed through a look-up table that pops out 14 bits. These 14 bits have the desired bunching characteristics, and are stored on the laser disc. Upon playback, the binary values read from the disc are passed through the inverse of the EFM look-up table, resulting in each 14 bit group being turned back into the correct 8 bits. 0.5 µm pit width FIGURE 22-5 Compact disc surface. Micron size pits are burned into the surface of the CD to represent ones and zeros. This results in a data density of 1 bit per µm 2, or one million bits per mm2. The pit depth is 0.16 µm. 1.6 µm track spacing readout direction 0.8 µm minimum length 3.5 µm maximum length Chapter 22- Audio Processing 361 Compact disc Left channel Optical pickup EFM decoding serial data (4.3 Mbits/sec) ReedSolomon decoding Sample rate converter (×4) 14 bit DAC Bessel Filter Power Amplifier Speaker Sample rate converter (×4) 14 bit DAC Bessel Filter Power Amplifier Speaker 16 bit samples 14 bit samples at 44.1 kHz (706 Kbits/sec) at 176.4 kHz Right channel FIGURE 22-6 Compact disc playback block diagram. The digital information is retrieved from the disc with an optical sensor, corrected for EFM and Reed-Solomon encoding, and converted to stereo analog signals. In addition to EFM, the data are encoded in a format called two-level ReedSolomon coding. This involves combining the left and right stereo channels along with data for error detection and correction. Digital errors detected during playback are either: corrected by using the redundant data in the encoding scheme, concealed by interpolating between adjacent samples, or muted by setting the sample value to zero. These encoding schemes result in the data rate being tripled, i.e., 1.4 Mbits/sec for the stereo audio signals versus 4.3 Mbits/sec stored on the disc. After decoding and error correction, the audio signals are represented as 16 bit samples at a 44.1 kHz sampling rate. In the simplest system, these signals could be run through a 16 bit DAC, followed by a low-pass analog filter. However, this would require high performance analog electronics to pass frequencies below 20 kHz, while rejecting all frequencies above 22.05 kHz, ½ of the sampling rate. A more common method is to use a multirate technique, that is, convert the digital data to a higher sampling rate before the DAC. A factor of four is commonly used, converting from 44.1 kHz to 176.4 kHz. This is called interpolation, and can be explained as a two step process (although it may not actually be carried out this way). First, three samples with a value of zero are placed between the original samples, producing the higher sampling rate. In the frequency domain, this has the effect of duplicating the 0 to 22.05 kHz spectrum three times, at 22.05 to 44.1 kHz, 41 to 66.15 kHz, and 66.15 to 88.2 kHz. In the second step, an efficient digital filter is used to remove the newly added frequencies. The sample rate increase makes the sampling interval smaller, resulting in a smoother signal being generated by the DAC. The signal still contains frequencies between 20 Hz and 20 kHz; however, the Nyquist frequency has been increased by a factor of four. This means that the analog filter only needs to pass frequencies below 20 kHz, while blocking frequencies above 88.2 kHz. This is usually done with a three pole Bessel filter. Why use a Bessel filter if the ear is insensitive to phase? Overkill, remember? 362 The Scientist and Engineer's Guide to Digital Signal Processing Since there are four times as many samples, the number of bits per sample can be reduced from 16 bits to 14 bits, without degrading the sound quality. The sin(x)/x correction needed to compensate for the zeroth order hold of the DAC can be part of either the analog or digital filter. Audio systems with more than one channel are said to be in stereo (from the Greek word for solid, or three-dimensional). Multiple channels send sound to the listener from different directions, providing a more accurate reproduction of the original music. Music played through a monaural (one channel) system often sounds artificial and bland. In comparison, a good stereo reproduction makes the listener feel as if the musicians are only a few feet away. Since the 1960s, high fidelity music has used two channels (left and right), while motion pictures have used four channels (left, right, center, and surround). In early stereo recordings (say, the Beatles or the Mamas And The Papas), individual singers can often be heard in only one channel or the other. This rapidly progressed into a more sophisticated mix-down, where the sound from many microphones in the recording studio is combined into the two channels. Mix-down is an art, aimed at providing the listener with the perception of being there. The four channel sound used in motion pictures is called Dolby Stereo, with the home version called Dolby Surround Pro Logic. ("Dolby" and "Pro Logic" are trademarks of Dolby Laboratories Licensing Corp.). The four channels are encoded into the standard left and right channels, allowing regular two-channel stereo systems to reproduce the music. A Dolby decoder is used during playback to recreate the four channels of sound. The left and right channels, from speakers placed on each side of the movie or television screen, is similar to that of a regular two-channel stereo system. The speaker for the center channel is usually placed directly above or below the screen. Its purpose is to reproduce speech and other visually connected sounds, keeping them firmly centered on the screen, regardless of the seating position of the viewer/listener. The surround speakers are placed to the left and right of the listener, and may involve as many as twenty speakers in a large auditorium. The surround channel only contains midrange frequencies (say, 100 Hz to 7 kHz), and is delayed by 15 to 30 milliseconds. This delay makes the listener perceive that speech is coming from the screen, and not the sides. That is, the listener hears the speech coming from the front, followed by a delayed version of the speech coming from the sides. The listener's mind interprets the delayed signal as a reflection from the walls, and ignores it. Companding The data rate is important in telecommunication because it is directly proportional to the cost of transmitting the signal. Saving bits is the same as saving money. Companding is a common technique for reducing the data rate of audio signals by making the quantization levels unequal. As previously mentioned, the loudest sound that can be tolerated (120 dB SPL) is about onemillion times the amplitude of the weakest sound that can be detected (0 dB Chapter 22- Audio Processing 363 SPL). However, the ear cannot distinguish between sounds that are closer than about 1 dB (12% in amplitude) apart. In other words, there are only about 120 different loudness levels that can be detected, spaced logarithmically over an amplitude range of one-million. This is important for digitizing audio signals. If the quantization levels are equally spaced, 12 bits must be used to obtain telephone quality speech. However, only 8 bits are required if the quantization levels are made unequal, matching the characteristics of human hearing. This is quite intuitive: if the signal is small, the levels need to be very close together; if the signal is large, a larger spacing can be used. Companding can be carried out in three ways: (1) run the analog signal through a nonlinear circuit before reaching a linear 8 bit ADC, (2) use an 8 bit ADC that internally has unequally spaced steps, or (3) use a linear 12 bit ADC followed by a digital look-up table (12 bits in, 8 bits out). Each of these three options requires the same nonlinearity, just in a different place: an analog circuit, an ADC, or a digital circuit. Two nearly identical standards are used for companding curves: µ255 law (also called mu law), used in North America, and "A" law, used in Europe. Both use a logarithmic nonlinearity, since this is what converts the spacing detectable by the human ear into a linear spacing. In equation form, the curves used in µ255 law and "A" law are given by: EQUATION 22-1 Mu law companding. This equation provides the nonlinearity for µ255 law companding. The constant, µ, has a value of 255, accounting for the name of this standard. y ' ln(1% µx) ln(1% µ ) for 0 # x # 1 y ' 1% ln(Ax ) 1% ln(A ) for 1/A # x # 1 y ' Ax 1% ln(A ) for 0 # x # 1/A EQUATION 22-2 "A" law companding. The constant, A, has a value of 87.6. Figure 22-7 graphs these equations for the input variable, x, being between -1 and +1, resulting in the output variable also assuming values between -1 and +1. Equations 22-1 and 22-2 only handle positive input values; portions of the curves for negative input values are found from symmetry. As shown in (a), the curves for µ255 law and "A" law are nearly identical. The only significant difference is near the origin, shown in (b), where µ255 law is a smooth curve, and "A" law switches to a straight line. Producing a stable nonlinearity is a difficult task for analog electronics. One method is to use the logarithmic relationship between current and 364 The Scientist and Engineer's Guide to Digital Signal Processing 1.0 0.5 a. µ law and A law b. Zoom of zero crossing Output Output µ law 0.0 -1.0 -1.0 0.0 Input 1.0 A law 0.0 -0.5 -0.05 0.00 Input 0.05 FIGURE 22-7 Companding curves. The µ255 law and "A" law companding curves are nearly identical, differing only near the origin. Companding increases the amplitude when the signal is small, and decreases it when it is large. voltage across a pn diode junction, and then add circuitry to correct for the ghastly temperature drift. Most companding circuits take another strategy: approximate the nonlinearity with a group of straight lines. A typical scheme is to approximate the logarithmic curve with a group of 16 straight segments, called cords. The first bit of the 8 bit output indicates if the input is positive or negative. The next three bits identify which of the 8 positive or 8 negative cords is used. The last four bits break each cord into 16 equally spaced increments. As with most integrated circuits, companding chips have sophisticated and proprietary internal designs. Rather than worrying about what goes on inside of the chip, pay the most attention to the pinout and the specification sheet. Speech Synthesis and Recognition Computer generation and recognition of speech are formidable problems; many approaches have been tried, with only mild success. This is an active area of DSP research, and will undoubtedly remain so for many years to come. You will be very disappointed if you are expecting this section to describe how to build speech synthesis and recognition circuits. Only a brief introduction to the typical approaches can be presented here. Before starting, it should be pointed out that most commercial products that produce human sounding speech do not synthesize it, but merely play back a digitally recorded segment from a human speaker. This approach has great sound quality, but it is limited to the prerecorded words and phrases. Nearly all techniques for speech synthesis and recognition are based on the model of human speech production shown in Fig. 22-8. Most human speech sounds can be classified as either voiced or fricative. Voiced sounds occur when air is forced from the lungs, through the vocal cords, and out of the mouth and/or nose. The vocal cords are two thin flaps of tissue stretched across Chapter 22- Audio Processing 365 Noise Generator unvoiced Digital Filter voiced Pulse train Generator synthetic speech vocal tract response pitch FIGURE 22-8 Human speech model. Over a short segment of time, about 2 to 40 milliseconds, speech can be modeled by three parameters: (1) the selection of either a periodic or a noise excitation, (2) the pitch of the periodic excitation, and (3) the coefficients of a recursive linear filter mimicking the vocal tract response. the air flow, just behind the Adam's apple. In response to varying muscle tension, the vocal cords vibrate at frequencies between 50 and 1000 Hz, resulting in periodic puffs of air being injected into the throat. Vowels are an example of voiced sounds. In Fig. 22-8, voiced sounds are represented by the pulse train generator, with the pitch (i.e., the fundamental frequency of the waveform) being an adjustable parameter. In comparison, fricative sounds originate as random noise, not from vibration of the vocal cords. This occurs when the air flow is nearly blocked by the tongue, lips, and/or teeth, resulting in air turbulence near the constriction. Fricative sounds include: s, f, sh, z, v, and th. In the model of Fig. 22-8, fricatives are represented by a noise generator. Both these sound sources are modified by the acoustic cavities formed from the tongue, lips, mouth, throat, and nasal passages. Since sound propagation through these structures is a linear process, it can be represented as a linear filter with an appropriately chosen impulse response. In most cases, a recursive filter is used in the model, with the recursion coefficients specifying the filter's characteristics. Because the acoustic cavities have dimensions of several centimeters, the frequency response is primarily a series of resonances in the kilohertz range. In the jargon of audio processing, these resonance peaks are called the format frequencies. By changing the relative position of the tongue and lips, the format frequencies can be changed in both frequency and amplitude. Figure 22-9 shows a common way to display speech signals, the voice spectrogram, or voiceprint. The audio signal is broken into short segments, 366 The Scientist and Engineer's Guide to Digital Signal Processing say 2 to 40 milliseconds, and the FFT used to find the frequency spectrum of each segment. These spectra are placed side-by-side, and converted into a grayscale image (low amplitude becomes light, and high amplitude becomes dark). This provides a graphical way of observing how the frequency content of speech changes with time. The segment length is chosen as a tradeoff between frequency resolution (favored by longer segments) and time resolution (favored by shorter segments). As demonstrated by the a in rain, voiced sounds have a periodic time domain waveform, shown in (a), and a frequency spectrum that is a series of regularly spaced harmonics, shown in (b). In comparison, the s in storm, shows that fricatives have a noisy time domain signal, as in (c), and a noisy spectrum, displayed in (d). These spectra also show the shaping by the format frequencies for both sounds. Also notice that the time-frequency display of the word rain looks similar both times it is spoken. Over a short period, say 25 milliseconds, a speech signal can be approximated by specifying three parameters: (1) the selection of either a periodic or random noise excitation, (2) the frequency of the periodic wave (if used), and (3) the coefficients of the digital filter used to mimic the vocal tract response. Continuous speech can then be synthesized by continually updating these three parameters about 40 times a second. This approach was responsible for one the early commercial successes of DSP: the Speak & Spell, a widely marketed electronic learning aid for children. The sound quality of this type of speech synthesis is poor, sounding very mechanical and not quite human. However, it requires a very low data rate, typically only a few kbits/sec. This is also the basis for the linear predictive coding (LPC) method of speech compression. Digitally recorded human speech is broken into short segments, and each is characterized according to the three parameters of the model. This typically requires about a dozen bytes per segment, or 2 to 6 kbytes/sec. The segment information is transmitted or stored as needed, and then reconstructed with the speech synthesizer. Speech recognition algorithms take this a step further by trying to recognize patterns in the extracted parameters. This typically involves comparing the segment information with templates of previously stored sounds, in an attempt to identify the spoken words. The problem is, this method does not work very well. It is useful for some applications, but is far below the capabilities of human listeners. To understand why speech recognition is so difficult for computers, imagine someone unexpectedly speaking the following sentence: Larger run medical buy dogs fortunate almost when. Of course, you will not understand the meaning of this sentence, because it has none. More important, you will probably not even understand all of the individual words that were spoken. This is basic to the way that humans Chapter 22- Audio Processing The 5 rain bow was seen after 367 the rain storm Frequency (kHz) 4 3 2 1 0 0 0.5 1 1.5 2800 2.5 2100 a. Time domain: a in rain 2600 c. Time domain: s in storm 2080 2400 2200 Amplitude Amplitude 2 Time (seconds) 2000 1800 1600 2060 2040 2020 1400 1200 2000 0 2 4 6 8 10 12 14 Time (milliseconds) 16 18 20 22 0 2500 4 6 8 10 12 14 Time (milliseconds) 16 18 20 22 1000 b. Frequency domain: a in rain d. Frequency domain: s in storm 800 Amplitude 2000 Amplitude (×10) 2 1500 1000 500 600 400 200 0 0 0 1 2 3 Frequency (kHz) 4 5 0 1 2 3 Frequency (kHz) 4 FIGURE 22-9 Voice spectrogram. The spectrogram of the phrase: "The rainbow was seen after the rain storm." Figures (a) and (b) shows the time and frequency signals for the voiced a in rain. Figures (c) and (d) show the time and frequency signals for the fricative s in storm. 5 368 The Scientist and Engineer's Guide to Digital Signal Processing perceive and understand speech. Words are recognized by their sounds, but also by the context of the sentence, and the expectations of the listener. For example, imagine hearing the two sentences: The child wore a spider ring on Halloween. He was an American spy during the war. Even if exactly the same sounds were produced to convey the underlined words, listeners hear the correct words for the context. From your accumulated knowledge about the world, you know that children don't wear secret agents, and people don't become spooky jewelry during wartime. This usually isn't a conscious act, but an inherent part of human hearing. Most speech recognition algorithms rely only on the sound of the individual words, and not on their context. They attempt to recognize words, but not to understand speech. This places them at a tremendous disadvantage compared to human listeners. Three annoyances are common in speech recognition systems: (1) The recognized speech must have distinct pauses between the words. This eliminates the need for the algorithm to deal with phrases that sound alike, but are composed of different words (i.e., spider ring and spy during). This is slow and awkward for people accustomed to speaking in an overlapping flow. (2) The vocabulary is often limited to only a few hundred words. This means that the algorithm only has to search a limited set to find the best match. As the vocabulary is made larger, the recognition time and error rate both increase. (3) The algorithm must be trained on each speaker. This requires each person using the system to speak each word to be recognized, often needing to be repeated five to ten times. This personalized database greatly increases the accuracy of the word recognition, but it is inconvenient and time consuming. The prize for developing a successful speech recognition technology is enormous. Speech is the quickest and most efficient way for humans to communicate. Speech recognition has the potential of replacing writing, typing, keyboard entry, and the electronic control provided by switches and knobs. It just needs to work a little better to become accepted by the commercial marketplace. Progress in speech recognition will likely come from the areas of artificial intelligence and neural networks as much as through DSP itself. Don't think of this as a technical difficulty; think of it as a technical opportunity. Nonlinear Audio Processing Digital filtering can improve audio signals in many ways. For instance, Wiener filtering can be used to separate frequencies that are mainly signal, from frequencies that are mainly noise (see Chapter 17). Likewise, deconvolution can compensate for an undesired convolution, such as in the restoration of old Chapter 22- Audio Processing 369 recordings (also discussed in Chapter 17). These types of linear techniques are the backbone of DSP. Several nonlinear techniques are also useful for audio processing. Two will be briefly described here. The first nonlinear technique is used for reducing wideband noise in speech signals. This type of noise includes: magnetic tape hiss, electronic noise in analog circuits, wind blowing by microphones, cheering crowds, etc. Linear filtering is of little use, because the frequencies in the noise completely overlap the frequencies in the voice signal, both covering the range from 200 hertz to 3.2 kHz. How can two signals be separated when they overlap in both the time domain and the frequency domain? Here's how it is done. In a short segment of speech, the amplitude of the frequency components are greatly unequal. As an example, Fig. 22-10a illustrates the frequency spectrum of a 16 millisecond segment of speech (i.e., 128 samples at an 8 kHz sampling rate). Most of the signal is contained in a few large amplitude frequencies. In contrast, (b) illustrates the spectrum when only random noise is present; it is very irregular, but more uniformly distributed at a low amplitude. Now the key concept: if both signal and noise are present, the two can be partially separated by looking at the amplitude of each frequency. If the amplitude is large, it is probably mostly signal, and should therefore be retained. If the amplitude is small, it can be attributed to mostly noise, and should therefore be discarded, i.e., set to zero. Mid-size frequency components are adjusted in some smooth manner between the two extremes. Another way to view this technique is as a time varying Wiener filter. As you recall, the frequency response of the Wiener filter passes frequencies that are mostly signal, and rejects frequencies that are mostly noise. This 10 10 9 a. Signal spectrum 8 8 7 7 Amplitude Amplitude 9 6 5 4 6 5 4 3 3 2 2 1 1 0 b. Noise spectrum 0 0 0.1 0.2 0.3 Frequency 0.4 0.5 0 0.1 0.2 0.3 Frequency 0.4 0.5 FIGURE 22-10 Spectra of speech and noise. While the frequency spectra of speech and noise generally overlap, there is some separation if the signal segment is made short enough. Figure (a) illustrates the spectrum of a 16 millisecond speech segment, showing that many frequencies carry little speech information, in this particular segment. Figure (b) illustrates the spectrum of a random noise source; all the components have a small amplitude. (These graphs are not of real signals, but illustrations to show the noise reduction technique). 370 The Scientist and Engineer's Guide to Digital Signal Processing requires a knowledge of the signal and noise spectra beforehand, so that the filter's frequency response can be determined. This nonlinear technique uses the same idea, except that the Wiener filter's frequency response is recalculated for each segment, based on the spectrum of that segment. In other words, the filter's frequency response changes from segment-to-segment, as determined by the characteristics of the signal itself. One of the difficulties in implementing this (and other) nonlinear techniques is that the overlap-add method for filtering long signals is not valid. Since the frequency response changes, the time domain waveform of each segment will no longer align with the neighboring segments. This can be overcome by remembering that audio information is encoded in frequency patterns that change over time, and not in the shape of the time domain waveform. A typical approach is to divide the original time domain signal into overlapping segments. After processing, a smooth window is applied to each of the overlapping segments before they are recombined. This provides a smooth transition of the frequency spectrum from one segment to the next. The second nonlinear technique is called homomorphic signal processing. This term literally means: the same structure. Addition is not the only way that noise and interference can be combined with a signal of interest; multiplication and convolution are also common means of mixing signals together. If signals are combined in a nonlinear way (i.e., anything other than addition), they cannot be separated by linear filtering. Homomorphic techniques attempt to separate signals combined in a nonlinear way by making the problem become linear. That is, the problem is converted to the same structure as a linear system. For example, consider an audio signal transmitted via an AM radio wave. As atmospheric conditions change, the received amplitude of the signal increases and decreases, resulting in the loudness of the received audio signal slowly changing over time. This can be modeled as the audio signal, represented by a[ ] , being multiplied by a slowly varying signal, g[ ] , that represents the changing gain. This problem is usually handled in an electronic circuit called an automatic gain control (AGC), but it can also be corrected with nonlinear DSP. As shown in Fig. 22-11, the input signal, a [ ] ×g [ ] , is passed through the logarithm function. From the identity, log (x %y) ' log x % log y , this results in two signals that are combined by addition, i.e., log a [ ] % log g [ ] . In other words, the logarithm is the homomorphic transform that turns the nonlinear problem of multiplication into the linear problem of addition. Next, the added signals are separated by a conventional linear filter, that is, some frequencies are passed, while others are rejected. For the AGC, the gain signal, g[ ] , will be composed of very low frequencies, far below the 200 hertz to 3.2 kHz band of the voice signal. The logarithm of these signals will have more complicated spectra, but the idea is the same: a high-pass filter is used to eliminate the varying gain component from the signal. Chapter 22- Audio Processing a[ ]×g[ ] Logarithm Linear Filter log a [ ] % log g [ ] 371 AntiLogarithm a[ ] log a [ ] FIGURE 22-11 Homomorphic separation of multiplied signals. Taking the logarithm of the input signal transforms components that are multiplied into components that are added. These components can then be separated by linear filtering, and the effect of the logarithm undone. In effect, log a [ ] % log g [ ] is converted into log a [ ] . In the last step, the logarithm is undone by using the exponential function (the anti-logarithm, or e x ), producing the desired output signal, a [ ] Figure 22-12 shows a homomorphic system for separating signals that have been convolved. An application where this has proven useful is in removing echoes from audio signals. That is, the audio signal is convolved with an impulse response consisting of a delta function plus a shifted and scaled delta function. The homomorphic transform for convolution is composed of two stages, the Fourier transform, changing the convolution into a multi-plication, followed by the logarithm, turning the multiplication into an addition. As before, the signals are then separated by linear filtering, and the homomorphic transform undone. An interesting twist in Fig. 22-12 is that the linear filtering is dealing with frequency domain signals in the same way that time domain signals are usually processed. In other words, the time and frequency domains have been swapped from their normal use. For example, if FFT convolution were used to carry out the linear filtering stage, the "spectra" being multiplied would be in the time domain. This role reversal has given birth to a strange jargon. For instance, cepstrum (a rearrangment of spectrum) is the Fourier transform of the logarithm of the Fourier transform. Likewise, there are long-pass and shortpass filters, rather than low-pass and high-pass filters. Some authors even use Quefrency Alanysis and liftering. Homomorphic Transform x[ ]ty[ ] Fourier Transform Logarithm Inverse Homomorphic Transform Linear Filter X[ ]×Y[ ] Inverse Fourier Transform AntiLogarithm log X [ ] log X [ ] % log Y [ ] X[ ] FIGURE 22-12 Homomorphic separation of convolved signals. Components that have been convolved are converted into components that are added by taking the Fourier transform followed by the logarithm. After linear filtering to separate the added components, the original steps are undone. x[ ] 372 The Scientist and Engineer's Guide to Digital Signal Processing Keep in mind that these are simplified descriptions of sophisticated DSP algorithms; homomorphic processing is filled with subtle details. For example, the logarithm must be able to handle both negative and positive values in the input signal, since this is a characteristic of audio signals. This requires the use of the complex logarithm, a more advanced concept than the logarithm used in everyday science and engineering. When the linear filtering is restricted to be a zero phase filter, the complex log is found by taking the simple logarithm of the absolute value of the signal. After passing through the zero phase filter, the sign of the original signal is reapplied to the filtered signal. Another problem is aliasing that occurs when the logarithm is taken. For example, imagine digitizing a continuous sine wave. In accordance with the sampling theorem, two or more samples per cycle is sufficient. Now consider digitizing the logarithm of this continuous sine wave. The sharp corners require many more samples per cycle to capture the waveform, i.e., to prevent aliasing. The required sampling rate can easily be 100 times as great after the log, as before. Further, it doesn't matter if the logarithm is applied to the continuous signal, or to its digital representation; the result is the same. Aliasing will result unless the sampling rate is high enough to capture the sharp corners produced by the nonlinearity. The result is that audio signals may need to be sampled at 100 kHz or more, instead of only the standard 8 kHz. Even if these details are handled, there is no guarantee that the linearized signals can be separated by the linear filter. This is because the spectra of the linearized signals can overlap, even if the spectra of the original signals do not. For instance, imagine adding two sine waves, one at 1 kHz, and one at 2 kHz. Since these signals do not overlap in the frequency domain, they can be completely separated by linear filtering. Now imagine that these two sine waves are multiplied. Using homomorphic processing, the log is taken of the combined signal, resulting in the log of one sine wave plus the log of the other sine wave. The problem is, the logarithm of a sine wave contains many harmonics. Since the harmonics from the two signals overlap, their complete separation is not possible. In spite of these obstacles, homomorphic processing teaches an important lesson: signals should be processed in a manner consistent with how they are formed. Put another way, the first step in any DSP task is to understand how information is represented in the signals being processed. CHAPTER 23 Image Formation & Display Images are a description of how a parameter varies over a surface. For example, standard visual images result from light intensity variations across a two-dimensional plane. However, light is not the only parameter used in scientific imaging. For example, an image can be formed of the temperature of an integrated circuit, blood velocity in a patient's artery, x-ray emission from a distant galaxy, ground motion during an earthquake, etc. These exotic images are usually converted into conventional pictures (i.e., light images), so that they can be evaluated by the human eye. This first chapter on image processing describes how digital images are formed and presented to human observers. Digital Image Structure Figure 23-1 illustrates the structure of a digital image. This example image is of the planet Venus, acquired by microwave radar from an orbiting space probe. Microwave imaging is necessary because the dense atmosphere blocks visible light, making standard photography impossible. The image shown is represented by 40,000 samples arranged in a two-dimensional array of 200 columns by 200 rows. Just as with one-dimensional signals, these rows and columns can be numbered 0 through 199, or 1 through 200. In imaging jargon, each sample is called a pixel, a contraction of the phrase: picture element. Each pixel in this example is a single number between 0 and 255. When the image was acquired, this number related to the amount of microwave energy being reflected from the corresponding location on the planet's surface. To display this as a visual image, the value of each pixel is converted into a grayscale, where 0 is black, 255 is white, and the intermediate values are shades of gray. Images have their information encoded in the spatial domain, the image equivalent of the time domain. In other words, features in images are represented by edges, not sinusoids. This means that the spacing and number of pixels are determined by how small of features need to be seen, 373 374 The Scientist and Engineer's Guide to Digital Signal Processing rather than by the formal constraints of the sampling theorem. Aliasing can occur in images, but it is generally thought of as a nuisance rather than a major problem. For instance, pinstriped suits look terrible on television because the repetitive pattern is greater than the Nyquist frequency. The aliased frequencies appear as light and dark bands that move across the clothing as the person changes position. A "typical" digital image is composed of about 500 rows by 500 columns. This is the image quality encountered in television, personnel computer applications, and general scientific research. Images with fewer pixels, say 250 by 250, are regarded as having unusually poor resolution. This is frequently the case with new imaging modalities; as the technology matures, more pixels are added. These low resolution images look noticeably unnatural, and the individual pixels can often be seen. On the other end, images with more than 1000 by 1000 pixels are considered exceptionally good. This is the quality of the best computer graphics, high-definition television, and 35 mm motion pictures. There are also applications needing even higher resolution, requiring several thousand pixels per side: digitized x-ray images, space photographs, and glossy advertisements in magazines. The strongest motivation for using lower resolution images is that there are fewer pixels to handle. This is not trivial; one of the most difficult problems in image processing is managing massive amounts of data. For example, one second of digital audio requires about eight kilobytes. In comparison, one second of television requires about eight Megabytes. Transmitting a 500 by 500 pixel image over a 33.6 kbps modem requires nearly a minute! Jumping to an image size of 1000 by 1000 quadruples these problems. It is common for 256 gray levels (quantization levels) to be used in image processing, corresponding to a single byte per pixel. There are several reasons for this. First, a single byte is convenient for data management, since this is how computers usually store data. Second, the large number of pixels in an image compensate to a certain degree for a limited number of quantization steps. For example, imagine a group of adjacent pixels alternating in value between digital numbers (DN) 145 and 146. The human eye perceives the region as a brightness of 145.5. In other words, images are very dithered. Third, and most important, a brightness step size of 1/256 (0.39%) is smaller than the eye can perceive. An image presented to a human observer will not be improved by using more than 256 levels. However, some images need to be stored with more than 8 bits per pixel. Remember, most of the images encountered in DSP represent nonvisual parameters. The acquired image may be able to take advantage of more quantization levels to properly capture the subtle details of the signal. The point of this is, don't expect to human eye to see all the information contained in these finely spaced levels. We will consider ways around this problem during a later discussion of brightness and contrast. The value of each pixel in the digital image represents a small region in the continuous image being digitized. For example, imagine that the Venus Chapter 23- Image Formation and Display 375 Column 50 100 150 199 Column 0 0 155 160 165 Row 199 65 150 60 100 Row 55 50 50 150 Column 50 150 160 165 183 183 181 184 177 200 200 189 159 135 94 105 160 174 191 196 186 195 190 195 191 205 216 206 174 153 112 80 134 157 174 196 194 196 198 201 206 209 215 216 199 175 140 77 106 142 170 186 184 212 200 204 201 202 214 214 214 205 173 102 84 120 134 159 60 Row 55 202 215 203 179 165 165 199 207 202 208 197 129 73 112 131 146 65 FIGURE 23-1 Digital image structure. This example image is the planet Venus, as viewed in reflected microwaves. Digital images are represented by a two-dimensional array of numbers, each called a pixel. In this image, the array is 200 rows by 200 columns, with each pixel a number between 0 to 255. When this image was acquired, the value of each pixel corresponded to the level of reflected microwave energy. A grayscale image is formed by assigning each of the 0 to 255 values to varying shades of gray. 155 203 208 166 159 160 168 166 157 174 211 204 158 69 79 127 143 174 149 143 151 156 148 146 123 118 203 208 162 81 58 101 125 143 137 147 153 150 140 121 133 157 184 203 164 94 56 66 80 164 165 159 179 188 159 126 134 150 199 174 119 100 41 41 58 173 187 193 181 167 151 162 182 192 175 129 60 88 47 37 50 172 184 179 153 158 172 163 207 205 188 127 63 56 43 42 55 156 191 196 159 167 195 178 203 214 201 143 101 69 38 44 52 154 163 175 165 207 211 197 201 201 199 138 79 76 67 51 53 144 150 143 162 215 212 211 209 197 198 133 71 69 77 63 53 140 151 150 185 215 214 210 210 211 209 135 80 45 69 66 60 135 143 151 179 213 216 214 191 201 205 138 61 59 61 77 63 probe takes samples every 10 meters along the planet's surface as it orbits overhead. This defines a square sample spacing and sampling grid, with each pixel representing a 10 meter by 10 meter area. Now, imagine what happens in a single microwave reflection measurement. The space probe emits 376 The Scientist and Engineer's Guide to Digital Signal Processing a highly focused burst of microwave energy, striking the surface in, for example, a circular area 15 meters in diameter. Each pixel therefore contains information about this circular area, regardless of the size of the sampling grid. This region of the continuous image that contributes to the pixel value is called the sampling aperture. The size of the sampling aperture is often related to the inherent capabilities of the particular imaging system being used. For example, microscopes are limited by the quality of the optics and the wavelength of light, electronic cameras are limited by random electron diffusion in the image sensor, and so on. In most cases, the sampling grid is made approximately the same as the sampling aperture of the system. Resolution in the final digital image will be limited primary by the larger of the two, the sampling grid or the sampling aperture. We will return to this topic in Chapter 25 when discussing the spatial resolution of digital images. Color is added to digital images by using three numbers for each pixel, representing the intensity of the three primary colors: red, green and blue. Mixing these three colors generates all possible colors that the human eye can perceive. A single byte is frequently used to store each of the color intensities, allowing the image to capture a total of 256×256×256 = 16.8 million different colors. Color is very important when the goal is to present the viewer with a true picture of the world, such as in television and still photography. However, this is usually not how images are used in science and engineering. The purpose here is to analyze a two-dimensional signal by using the human visual system as a tool. Black and white images are sufficient for this. Cameras and Eyes The structure and operation of the eye is very similar to an electronic camera, and it is natural to discuss them together. Both are based on two major components: a lens assembly, and an imaging sensor. The lens assembly captures a portion of the light emanating from an object, and focus it onto the imaging sensor. The imaging sensor then transforms the pattern of light into a video signal, either electronic or neural. Figure 23-2 shows the operation of the lens. In this example, the image of an ice skater is focused onto a screen. The term focus means there is a oneto-one match of every point on the ice skater with a corresponding point on the screen. For example, consider a 1 mm × 1 mm region on the tip of the toe. In bright light, there are roughly 100 trillion photons of light striking this one square millimeter area each second. Depending on the characteristics of the surface, between 1 and 99 percent of these incident light photons will be reflected in random directions. Only a small portion of these reflected photons will pass through the lens. For example, only about one-millionth of the reflected light will pass through a one centimeter diameter lens located 3 meters from the object. Chapter 23- Image Formation and Display 377 lens projected image FIGURE 23-2 Focusing by a lens. A lens gathers light expanding from a point source, and force it to return to a point at another location. This allows a lens to project an image onto a surface. Refraction in the lens changes the direction of the individual photons, depending on the location and angle they strike the glass/air interface. These direction changes cause light expanding from a single point to return to a single point on the projection screen. All of the photons that reflect from the toe and pass through the lens are brought back together at the "toe" in the projected image. In a similar way, a portion of the light coming from any point on the object will pass through the lens, and be focused to a corresponding point in the projected image. Figures 23-3 and 23-4 illustrate the major structures in an electronic camera and the human eye, respectively. Both are light tight enclosures with a lens mounted at one end and an image sensor at the other. The camera is filled with air, while the eye is filled with a transparent liquid. Each lens system has two adjustable parameters: focus and iris diameter. If the lens is not properly focused, each point on the object will project to a circular region on the imaging sensor, causing the image to be blurry. In the camera, focusing is achieved by physically moving the lens toward or away from the imaging sensor. In comparison, the eye contains two lenses, a bulge on the front of the eyeball called the cornea, and an adjustable lens inside the eye. The cornea does most of the light refraction, but is fixed in shape and location. Adjustment to the focusing is accomplished by the inner lens, a flexible structure that can be deformed by the action of the ciliary muscles. As these muscles contract, the lens flattens to bring the object into a sharp focus. In both systems, the iris is used to control how much of the lens is exposed to light, and therefore the brightness of the image projected onto the imaging sensor. The iris of the eye is formed from opaque muscle tissue that can be contracted to make the pupil (the light opening) larger. The iris in a camera is a mechanical assembly that performs the same function. 378 The Scientist and Engineer's Guide to Digital Signal Processing The parameters in optical systems interact in many unexpected ways. For example, consider how the amount of available light and the sensitivity of the light sensor affects the sharpness of the acquired image. This is because the iris diameter and the exposure time are adjusted to transfer the proper amount of light from the scene being viewed to the image sensor. If more than enough light is available, the diameter of the iris can be reduced, resulting in a greater depth-of-field (the range of distance from the camera where an object remains in focus). A greater depth-of-field provides a sharper image when objects are at various distances. In addition, an abundance of light allows the exposure time to be reduced, resulting in less blur from camera shaking and object motion. Optical systems are full of these kinds of trade-offs. An adjustable iris is necessary in both the camera and eye because the range of light intensities in the environment is much larger than can be directly handled by the light sensors. For example, the difference in light intensities between sunlight and moonlight is about one-million. Adding to this that reflectance can vary between 1% and 99%, results in a light intensity range of almost one-hundred million. The dynamic range of an electronic camera is typically 300 to 1000, defined as the largest signal that can be measured, divided by the inherent noise of the device. Put another way, the maximum signal produced is 1 volt, and the rms noise in the dark is about 1 millivolt. Typical camera lenses have an iris that change the area of the light opening by a factor of about 300. This results in a typical electronic camera having a dynamic range of a few hundred thousand. Clearly, the same camera and lens assembly used in bright sunlight will be useless on a dark night. In comparison, the eye operates over a dynamic range that nearly covers the large environmental variations. Surprisingly, the iris is not the main way that this tremendous dynamic range is achieved. From dark to light, the area of the pupil only changes by a factor of about 20. The light detecting nerve cells gradually adjust their sensitivity to handle the remaining dynamic range. For instance, it takes several minutes for your eyes to adjust to the low light after walking into a dark movie theater. One way that DSP can improve images is by reducing the dynamic range an observer is required to view. That is, we do not want very light and very dark areas in the same image. A reflection image is formed from two image signals: the two-dimensional pattern of how the scene is illuminated, multiplied by the two-dimensional pattern of reflectance in the scene. The pattern of reflectance has a dynamic range of less than 100, because all ordinary materials reflect between 1% and 99% of the incident light. This is where most of the image information is contained, such as where objects are located in the scene and what their surface characteristics are. In comparison, the illumination signal depends on the light sources around the objects, but not on the objects themselves. The illumination signal can have a dynamic range of millions, although 10 to 100 is more typical within a single image. The illumination signal carries little interesting information, Chapter 23- Image Formation and Display iris 379 focus FIGURE 23-3 Diagram of an electronic camera. Focusing is achieved by moving the lens toward or away from the imaging sensor. The amount of light reaching the sensor is controlled by the iris, a mechanical device that changes the effective diameter of the lens. The most common imaging sensor in present day cameras is the CCD, a two-dimensional array of light sensitive elements. lens CCD serial output sclera FIGURE 23-4 Diagram of the human eye. The eye is a liquid filled sphere about 3 cm in diameter, enclosed by a tough outer case called the sclera. Focusing is mainly provided by the cornea, a fixed lens on the front of the eye. The focus is adjusted by contracting muscles attached to a flexible lens within the eye. The amount of light entering the eye is controlled by the iris, formed from opaque muscle tissue covering a portion of the lens. The rear hemisphere of the eye contains the retina, a layer of light sensitive nerve cells that converts the image to a neural signal in the optic nerve. TO EAR ciliary muscle iris fovea lens cornea TO NOSE (top view) clear liquid retina optic nerve but can degrade the final image by increasing its dynamic range. DSP can improve this situation by suppressing the illumination signal, allowing the reflectance signal to dominate the image. The next chapter presents an approach for implementing this algorithm. The light sensitive surface that covers the rear of the eye is called the retina. As shown in Fig. 23-5, the retina can be divided into three main layers of specialized nerve cells: one for converting light into neural signals, one for image processing, and one for transferring information to the optic nerve leading to the brain. In nearly all animals, these layers are seemingly backward. That is, the light sensitive cells are in last layer, requiring light to pass through the other layers before being detected. There are two types of cells that detect light: rods and cones, named for their physical appearance under the microscope. The rods are specialized in operating with very little light, such as under the nighttime sky. Vision appears very noisy in near darkness, that is, the image appears to be filled with a continually changing grainy pattern. This results from the image signal being very weak, and is not a limitation of the eye. There is so little light entering 380 The Scientist and Engineer's Guide to Digital Signal Processing the eye, the random detection of individual photons can be seen. This is called statistical noise, and is encountered in all low-light imaging, such as military night vision systems. Chapter 25 will revisit this topic. Since rods cannot detect color, low-light vision is in black and white. The cone receptors are specialized in distinguishing color, but can only operate when a reasonable amount of light is present. There are three types of cones in the eye: red sensitive, green sensitive, and blue sensitive. This results from their containing different photopigments, chemicals that absorbs different wavelengths (colors) of light. Figure 23-6 shows the wavelengths of light that trigger each of these three receptors. This is called RGB encoding, and is how color information leaves the eye through the optic nerve. The human perception of color is made more complicated by neural processing in the lower levels of the brain. The RGB encoding is converted into another encoding scheme, where colors are classified as: red or green, blue or yellow, and light or dark. RGB encoding is an important limitation of human vision; the wavelengths that exist in the environment are lumped into only three broad categories. In comparison, specialized cameras can separate the optical spectrum into hundreds or thousands of individual colors. For example, these might be used to classify cells as cancerous or healthy, understand the physics of a distant star, or see camouflaged soldiers hiding in a forest. Why is the eye so limited in detecting color? Apparently, all humans need for survival is to find a red apple, among the green leaves, silhouetted against the blue sky. Rods and cones are roughly 3 µm wide, and are closely packed over the entire 3 cm by 3 cm surface of the retina. This results in the retina being composed of an array of roughly 10,000 × 10,000 = 100 million receptors. In comparison, the optic nerve only has about one-million nerve fibers that connect to these cells. On the average, each optic nerve fiber is connected to roughly 100 light receptors through the connecting layer. In addition to consolidating information, the connecting layer enhances the image by sharpening edges and suppressing the illumination component of the scene. This biological image processing will be discussed in the next chapter. Directly in the center of the retina is a small region called the fovea (Latin for pit), which is used for high resolution vision (see Fig. 23-4). The fovea is different from the remainder of the retina in several respects. First, the optic nerve and interconnecting layers are pushed to the side of the fovea, allowing the receptors to be more directly exposed to the incoming light. This results in the fovea appearing as a small depression in the retina. Second, only cones are located in the fovea, and they are more tightly packed that in the remainder of the retina. This absence of rods in the fovea explains why night vision is often better when looking to the side of an object, rather than directly at it. Third, each optic nerve fiber is influenced by only a few cones, proving good localization ability. The fovea is surprisingly small. At normal reading distance, the fovea only sees about a 1 mm diameter area, less than the size of a single letter! The resolution is equivalent to about a 20×20 grid of pixels within this region. Chapter 23- Image Formation and Display 381 light optic nerve connecting layer rods and cones optic nerve to brain sclera FIGURE 23-5 The human retina. The retina contains three principle layers: (1) the rod and cone light receptors, (2) an intermediate layer for data reduction and image processing, and (3) the optic nerve fibers that lead to the brain. The structure of these layers is seemingly backward, requiring light to pass through the other layers before reaching the light receptors. Human vision overcomes the small size of the fovea by jerky eye movements called saccades. These abrupt motions allow the high resolution fovea to rapidly scan the field of vision for pertinent information. In addition, saccades present the rods and cones with a continually changing pattern of light. This is important because of the natural ability of the retina to adapt to changing levels of light intensity. In fact, if the eye is forced to remain fixed on the same scene, detail and color begin to fade in a few seconds. red yellow orange green 1 green cones rods Relative sensitivity FIGURE 23-6 Spectral response of the eye. The three types of cones in the human eye respond to different sections of the optical spectrum, roughly corresponding to red, green, and blue. Combinations of these three form all colors that humans can perceive. The cones do not have enough sensitivity to be used in low-light environments, where the rods are used to detect the image. This is why colors are difficult to perceive at night. blue perception of wavelength violet The most common image sensor used in electronic cameras is the charge coupled device (CCD). The CCD is an integrated circuit that replaced most vacuum tube cameras in the 1980s, just as transistors replaced vacuum tube amplifiers twenty years before. The heart of the CCD is a thin wafer of blue cones red cones 0 300 400 500 Wavelength (nm) 600 700 382 The Scientist and Engineer's Guide to Digital Signal Processing silicon, typically about 1 cm square. As shown by the cross-sectional view in Fig. 23-7, the backside is coated with a thin layer of metal connected to ground potential. The topside is covered with a thin electrical insulator, and a repetitive pattern of electrodes. The most common type of CCD is the three phase readout, where every third electrode is connected together. The silicon used is called p-type, meaning it has an excess of positive charge carriers called holes. For this discussion, a hole can be thought of as a positively charged particle that is free to move around in the silicon. Holes are represented in this figure by the "+" symbol. In (a), +10 volts is applied to one of the three phases, while the other two are held at 0 volts. This causes the holes to move away from every third electrode, since positive charges are repelled by a positive voltage. This forms a region under these electrodes called a well, a shortened version of the physics term: potential well. Each well in the CCD is a very efficient light sensor. As shown in (b), a single photon of light striking the silicon converts its energy into the formation of two charged particles, one electron, and one hole. The hole moves away, leaving the electron stuck in the well, held by the positive voltage on the electrode. Electrons in this illustration are represented by the "-" symbol. During the integration period, the pattern of light striking the CCD is transferred into a pattern of charge within the CCD wells. Dimmer light sources require longer integration periods. For example, the integration period for standard television is 1/60th of a second, while astrophotography can accumulate light for many hours. Readout of the electronic image is quite clever; the accumulated electrons in each well are pushed to the output amplifier. As shown in (c), a positive voltage is placed on two of the phase lines. This results in each well expanding to the right. As shown in (d), the next step is to remove the voltage from the first phase, causing the original wells to collapse. This leaves the accumulated electrons in one well to the right of where they started. By repeating this pulsing sequence among the three phase lines, the accumulated electrons are pushed to the right until they reach a charge sensitive amplifier. This is a fancy name for a capacitor followed by a unity gain buffer. As the electrons are pushed from the last well, they flow onto the capacitor where they produce a voltage. To achieve high sensitivity, the capacitors are made extremely small, usually less than 1 DF. This capacitor and amplifier are an integral part of the CCD, and are made on the same piece of silicon. The signal leaving the CCD is a sequence of voltage levels proportional to the amount of light that has fallen on sequential wells. Figure 23-8 shows how the two-dimensional image is read from the CCD. After the integration period, the charge accumulated in each well is moved up the column, one row at a time. For example, all the wells in row 15 are first moved into row 14, then row 13, then row 12, etc. Each time the rows are moved up, all the wells in row number 1 are transferred into the horizontal register. This is a group of specialized CCD wells that rapidly move the charge in a horizontal direction to the charge sensitive amplifier. Chapter 23- Image Formation and Display insulator electrode well output amplifier N1 (10v) N2 (0v) N3 (0v) a. 383 + ++ + + + + + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + + + + + ++ + + + + + + + + + ++ + + + + + + + + + ++ + + + + + + ++ + + ++ + + + + + ++ + + + + + + + + +++ + + + + ++ + + + ++ + + + + + grounded back surface p type silicon light photon N1 (10v) N2 (0v) N3 (0v) b. + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + + + + + + ++ + + ++ + ++ + + + + + + + + + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + ++ + + + + N1 (10v) N2 (0v) N3 (10v) c. + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + + + + + + + + ++ + + ++ + + + + + ++ + + ++ + + ++ ++ + + + + + + + + + + + + + + + + + + + ++ + + + + + + + ++ + + ++ + + + + + N1 (0v) N2 (0v) N3 (10v) d. + ++ + + + + + + + + + + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ ++ + + + + ++ + + + + + ++ + + + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + ++ + + + + + FIGURE 23-7 Operation of the charge coupled device (CCD). As shown in this cross-sectional view, a thin sheet of p-type silicon is covered with an insulating layer and an array of electrodes. The electrodes are connected in groups of three, allowing three separate voltages to be applied: N1, N2, and N3. When a positive voltage is applied to an electrode, the holes (i.e., the positive charge carriers indicated by the "+") are pushed away. This results in an area depleted of holes, called a well. Incoming light generates holes and electrons, resulting in an accumulation of electrons confined to each well (indicated by the "-"). By manipulating the three electrode voltages, the electrons in each well can be moved to the edge of the silicon where a charge sensitive amplifier converts the charge into a voltage. 384 The Scientist and Engineer's Guide to Digital Signal Processing horizontal register serial video output row 1 row 2 last row column 2 column 1 last column FIGURE 23-8 Architecture of the CCD. The imaging wells of the CCD are arranged in columns. During readout, the charge from each well is moved up the column into a horizontal register. The horizontal register is then readout into the charge sensitive preamplifier. Notice that this architecture converts a two-dimensional array into a serial data stream in a particular sequence. The first pixel to be read is at the top-left corner of the image. The readout then proceeds from left-to-right on the first line, and then continues from left-to-right on subsequent lines. This is called row major order, and is almost always followed when a two-dimensional array (image) is converted to sequential data. Television Video Signals Although over 50 years old, the standard television signal is still one of the most common way to transmit an image. Figure 23-9 shows how the television signal appears on an oscilloscope. This is called composite video, meaning that there are vertical and horizontal synchronization (sync) pulses mixed with the actual picture information. These pulses are used in the television receiver to synchronize the vertical and horizontal deflection circuits to match the video being displayed. Each second of standard video contains 30 complete images, commonly called frames. A video engineer would say that each frame contains 525 lines, the television jargon for what programmers call rows. This number is a little deceptive because only 480 to 486 of these lines contain video information; the remaining 39 to 45 lines are reserved for sync pulses to keep the television's circuits synchronized with the video signal. Standard television uses an interlaced format to reduce flicker in the displayed image. This means that all the odd lines of each frame are transmitted first, followed by the even lines. The group of odd lines is called the odd field, and the group of even lines is called the even field. Since Signal level (volts) Chapter 23- Image Formation and Display EVEN FIELD ODD FIELD 0.75 385 0.50 0.25 0.00 -0.25 vertical sync pulses line 1 line 3 line 5 line 485 vertical sync pulses line 2 line 4 horizontal sync pulses FIGURE 23-9 Composite video. The NTSC video signal consists of 30 complete frames (images) per second, with each frame containing 480 to 486 lines of video. Each frame is broken into two fields, one containing the odd lines and the other containing the even lines. Each field starts with a group of vertical sync pulses, followed by successive lines of video information separated by horizontal sync pulses. (The horizontal axis of this figure is not drawn to scale). each frame consists of two fields, the video signal transmits 60 fields per second. Each field starts with a complex series of vertical sync pulses lasting 1.3 milliseconds. This is followed by either the even or odd lines of video. Each line lasts for 63.5 microseconds, including a 10.2 microsecond horizontal sync pulse, separating one line from the next. Within each line, the analog voltage corresponds to the grayscale of the image, with brighter values being in the direction away from the sync pulses. This places the sync pulses beyond the black range. In video jargon, the sync pulses are said to be blacker than black. The hardware used for analog-to-digital conversion of video signals is called a frame grabber. This is usually in the form of an electronics card that plugs into a computer, and connects to a camera through a coaxial cable. Upon command from software, the frame grabber waits for the beginning of the next frame, as indicated by the vertical sync pulses. During the following two fields, each line of video is sampled many times, typically 512, 640 or 720 samples per line, at 8 bits per sample. These samples are stored in memory as one row of the digital image. This way of acquiring a digital image results in an important difference between the vertical and horizontal directions. Each row in the digital image corresponds to one line in the video signal, and therefore to one row of wells in the CCD. Unfortunately, the columns are not so straightforward. In the CCD, each row contains between about 400 and 800 wells (columns), depending on the particular device used. When a row of wells is read from the CCD, the resulting line of video is filtered into a smooth analog signal, such as in Fig. 23-9. In other words, the video signal does not depend on how many columns are present in the CCD. The resolution in the horizontal direction is limited by how rapidly the analog signal is allowed to change. This is usually set at 3.2 MHz for color television, resulting in a risetime of about 100 nanoseconds, i.e., about 1/500th of the 53.2 microsecond video line. 386 The Scientist and Engineer's Guide to Digital Signal Processing When the video signal is digitized in the frame grabber, it is converted back into columns. However, these columns in the digitized image have no relation to the columns in the CCD. The number of columns in the digital image depends solely on how many times the frame grabber samples each line of video. For example, a CCD might have 800 wells per row, while the digitized image might only have 512 pixels (i.e., columns) per row. The number of columns in the digitized image is also important for another reason. The standard television image has an aspect ratio of 4 to 3, i.e., it is slightly wider than it is high. Motion pictures have the wider aspect ratio of 25 to 9. CCDs used for scientific applications often have an aspect ratio of 1 to 1, i.e., a perfect square. In any event, the aspect ratio of a CCD is fixed by the placement of the electrodes, and cannot be altered. However, the aspect ratio of the digitized image depends on the number of samples per line. This becomes a problem when the image is displayed, either on a video monitor or in a hardcopy. If the aspect ratio isn't properly reproduced, the image looks squashed horizontally or vertically. The 525 line video signal described here is called NTSC (National Television Systems Committee), a standard defined way back in 1954. This is the system used in the United States and Japan. In Europe there are two similar standards called P A L (Phase Alternation by Line) and S E C A M (Sequential Chrominance And Memory). The basic concepts are the same, just the numbers are different. Both PAL and SECAM operate with 25 interlaced frames per second, with 625 lines per frame. Just as with NTSC, some of these lines occur during the vertical sync, resulting in about 576 lines that carry picture information. Other more subtle differences relate to how color and sound are added to the signal. The most straightforward way of transmitting color television would be to have three separate analog signals, one for each of the three colors the human eye can detect: red, green and blue. Unfortunately, the historical development of television did not allow such a simple scheme. The color television signal was developed to allow existing black and white television sets to remain in use without modification. This was done by retaining the same signal for brightness information, but adding a separate signal for color information. In video jargon, the brightness is called the luminance signal, while the color is the chrominance signal. The chrominance signal is contained on a 3.58 MHz carrier wave added to the black and white video signal. Sound is added in this same way, on a 4.5 MHz carrier wave. The television receiver separates these three signals, processes them individually, and recombines them in the final display. Other Image Acquisition and Display Not all images are acquired an entire frame at a time. Another very common way is by line scanning. This involves using a detector containing a onedimensional array of pixels, say, 2048 pixels long by 1 pixel wide. As an object is moved past the detector, the image is acquired line-by-line. Line Chapter 23- Image Formation and Display 387 scanning is used by fax machines and airport x-ray baggage scanners. As a variation, the object can be kept stationary while the detector is moved. This is very convenient when the detector is already mounted on a moving object, such as an aircraft taking images of the ground beneath it. The advantage of line scanning is that speed is traded for detector simplicity. For example, a fax machine may take several seconds to scan an entire page of text, but the resulting image contains thousands of rows and columns. An even more simplified approach is to acquire the image point-by-point. For example, the microwave image of Venus was acquired one pixel at a time. Another example is the scanning probe microscope, capable of imaging individual atoms. A small probe, often consisting of only a single atom at its tip, is brought exceedingly close to the sample being imaged. Quantum mechanical effects can be detected between the probe and the sample, allowing the probe to be stopped an exact distance from the sample's surface. The probe is then moved over the surface of the sample, keeping a constant distance, tracing out the peaks and valleys. In the final image, each pixel's value represents the elevation of the corresponding location on the sample's surface. Printed images are divided into two categories: grayscale and halftone. Each pixel in a grayscale image is a shade of gray between black and white, such as in a photograph. In comparison, each pixel in a halftone image is formed from many individual dots, with each dot being completely black or completely white. Shades of gray are produced by alternating various numbers of these black and white dots. For example, imagine a laser printer with a resolution of 600 dots-per-inch. To reproduce 256 levels of brightness between black and white, each pixel would correspond to an array of 16 by 16 printable dots. Black pixels are formed by making all of these 256 dots black. Likewise, white pixels are formed making all of these 256 dots white. Midgray has one-half of the dots white and one-half black. Since the individual dots are too small to be seen when viewed at a normal distance, the eye is fooled into thinking a grayscale has been formed. Halftone images are easier for printers to handle, including photocopy machines. The disadvantage is that the image quality is often worse than grayscale pictures. Brightness and Contrast Adjustments An image must have the proper brightness and contrast for easy viewing. Brightness refers to the overall lightness or darkness of the image. Contrast is the difference in brightness between objects or regions. For example, a white rabbit running across a snowy field has poor contrast, while a black dog against the same white background has good contrast. Figure 23-10 shows four possible ways that brightness and contrast can be misadjusted. When the brightness is too high, as in (a), the whitest pixels are saturated, destroying the detail in these areas. The reverse is shown in (b), where the brightness is set too low, saturating the blackest pixels. Figure (c) shows 388 The Scientist and Engineer's Guide to Digital Signal Processing a. Brightness too high b. Brightness too low c. Contrast too high d. Contrast too low FIGURE 23-10 Brightness and contrast adjustments. Increasing the brightness makes every pixel in the image becomes lighter. In comparison, increasing the contrast makes the light areas become lighter, and the dark areas become darker. These images show the effect of misadjusting the brightness and contrast. the contrast set to high, resulting in the blacks being too black, and the whites being too white. Lastly, (d) has the contrast set too low; all of the pixels are a mid-shade of gray making the objects fade into each other. Figures 23-11 and 23-12 illustrate brightness and contrast in more detail. A test image is displayed in Fig. 23-12, using six different brightness and contrast levels. Figure 23-11 shows the construction of the test image, an array of 80×32 pixels, with each pixel having a value between 0 and 255. The backgound of the test image is filled with random noise, uniformly distributed between 0 and 255. The three square boxes have pixel values of 75, 150 and 225, from left-to-right. Each square contains two triangles with pixel values only slightly different from their surroundings. In other Chapter 23- Image Formation and Display random 0 to 255 32 pixels FIGURE 23-11 Brightness and contrast test image. This is the structure of the digital image used in Fig. 23-12. The three squares form dark, medium, and bright objects, each containing two low contrast triangles. This figure indicates the digital numbers (DN) of the pixels in each region. 389 75 71 150 146 79 225 221 154 229 80 pixels words, there is a dark region in the image with faint detail, this is a medium region in the image with faint detail, and there is a bright region in the image with faint detail. Figure 23-12 shows how adjustment of the contrast and brightness allows different features in the image to be visualized. In (a), the brightness and contrast are set at the normal level, as indicated by the B and C slide bars at the left side of the image. Now turn your attention to the graph shown with each image, called an output transform, an output look-up table, or a gamma curve. This controls the hardware that displays the image. The value of each pixel in the stored image, a number between 0 and 255, is passed through this look-up table to produces another number between 0 and 255. This new digital number drives the video intensity circuit, with 0 through 255 being transformed into black through white, respectively. That is, the look-up table maps the stored numbers into the displayed brightness. Figure (a) shows how the image appears when the output transform is set to do nothing, i.e., the digital output is identical to the digital input. Each pixel in the noisy background is a random shade of gray, equally distributed between black and white. The three boxes are displayed as dark, medium and light, clearly distinct from each other. The problem is, the triangles inside each square cannot be easily seen; the contrast is too low for the eye to distinguished these regions from their surroundings. Figures (b) & (c) shows the effect of changing the brightness. Increasing the brightness shifts the output transform to the left, while decreasing the brightness shifts it to the right. Increasing the brightness makes every pixel in the image appear lighter. Conversely, decreasing the brightness makes every pixel in the image appear darker. These changes can improve the viewability of excessively dark or light areas in the image, but will saturate the image if taken too far. For example, all of the pixels in the far right square in (b) are displayed with full intensity, i.e., 255. The opposite effect is shown in (c), where all of the pixels in the far left square are displayed as blackest black, or digital number 0. Since all the pixels in these regions have the same value, the triangles are completely wiped out. Also notice that none of the triangles in (b) and (c) are easier to see than in (a). Changing the brightness provides little (if any) help in distinguishing low contrast objects from their surroundings. 390 The Scientist and Engineer's Guide to Digital Signal Processing Figure (d) shows the display optimized to view pixel values around digital number 75. This is done by turning up the contrast, resulting in the output transform increasing in slope. For example, the stored pixel values of 71 and 75 become 100 and 116 in the display, making the contrast a factor of four greater. Pixel values between 46 and 109 are displayed as the blackest black, to the whitest white. The price for this increased contrast is that pixel values 0 to 45 are saturated at black, and pixel values 110 to 255 are saturated at white. As shown in (d), the increased contrast allows the triangles in the left square to be seen, at the cost of saturating the middle and right squares. Figure (e) shows the effect of increasing the contrast even further, resulting in only 16 of the possible 256 stored levels being displayed as nonsaturated. The brightness has also been decreased so that the 16 usable levels are centered on digital number 175. The details in the center square are now very visible; however, almost everything else in the image is saturated. For example, look at the noise around the border of the image. There are very few pixels with an intermediate gray shade; almost every pixel is either pure black or pure white. This technique of using high contrast to view only a few levels is sometimes called a grayscale stretch. The contrast adjustment is a way of zooming in on a smaller range of pixel values. The brightness control centers the zoomed section on the pixel values of interest. Most digital imaging systems allow the brightness and contrast to be adjusted in just this manner, and often provide a graphical display of the output transform (as in Fig. 23-12). In comparison, the brightness and contrast controls on television and video monitors are analog circuits, and may operate differently. For example, the contrast control of a monitor may adjust the gain of the analog signal, while the brightness might add or subtract a DC offset. The moral is, don't be surprised if these analog controls don't respond in the way you think they should. Grayscale Transforms The last image, Fig. 23-12f, is different from the rest. Rather than having a slope in the curve over one range of input values, it has a slope in the curve over two ranges. This allows the display to simultaneously show the triangles in both the left and the right squares. Of course, this results in saturation of the pixel values that are not near these digital numbers. Notice that the slide bars for contrast and brightness are not shown in (f); this display is beyond what brightness and contrast adjustments can provide. Taking this approach further results in a powerful technique for improving the appearance of images: the grayscale transform. The idea is to increase the contrast at pixel values of interest, at the expense of the pixel values we don't care about. This is done by defining the relative importance of each of the 0 to 255 possible pixel values. The more important the value, the greater its contrast is made in the displayed image. An example will show a systematic way of implementing this procedure. Chapter 23- Image Formation and Display B C 250 Output value a. Normal 200 150 100 50 0 Output value C 50 0 50 0 50 0 50 0 50 0 50 100 150 200 250 100 150 200 250 100 150 200 250 100 150 200 250 100 150 200 250 100 150 200 250 Input value 250 b. Increased brightness B 0 200 150 100 50 0 250 C Output value c. Decreased brightness B Input value 200 150 100 50 0 B C 250 Output value d. Slightly increased contrast at DN 75 Input value 200 150 100 50 0 B C 250 Output value e. Greatly increased contrast at DN 150 Input value 200 150 100 50 0 B C 250 Output value f. Increased contrast at both DN 75 and 225 Input value 200 150 100 50 0 Input value FIGURE 23-12 391 392 The Scientist and Engineer's Guide to Digital Signal Processing a. Original IR image b. With grayscale transform FIGURE 23-13 Grayscale processing. Image (a) was acquired with an infrared camera in total darkness. Brightness in the image is related to the temperature, accounting for the appearance of the warm human body and the hot truck grill. Image (b) was processed with the manual grayscale transform shown in Fig. 23-14c. The image in Fig. 23-13a was acquired in total darkness by using a CCD camera that is sensitive in the far infrared. The parameter being imaged is temperature: the hotter the object, the more infrared energy it emits and the brighter it appears in the image. This accounts for the background being very black (cold), the body being gray (warm), and the truck grill being white (hot). These systems are great for the military and police; you can see the other guy when he can't even see himself! The image in (a) is difficult to view because of the uneven distribution of pixel values. Most of the image is so dark that details cannot be seen in the scene. On the other end, the grill is near white saturation. The histogram of this image is displayed in Fig. 23-14a, showing that the background, human, and grill have reasonably separate values. In this example, we will increase the contrast in the background and the grill, at the expense of everything else, including the human body. Figure (b) represents this strategy. We declare that the lowest pixel values, the background, will have a relative contrast of twelve. Likewise, the highest pixel values, the grill, will have a relative contrast of six. The body will have a relative contrast of one, with a staircase transition between the regions. All these values are determined by trial and error. The grayscale transform resulting from this strategy is shown in (c), labeled manual. It is found by taking the running sum (i.e., the discrete integral) of the curve in (b), and then normalizing so that it has a value of 255 at the Chapter 23- Image Formation and Display 16 1600 a. Histogram 1400 1200 12 Desired contrast background 1000 800 600 400 b. Desired contrast 14 10 8 6 4 human body grill 200 2 0 0 0 50 100 150 Value of pixel 200 250 FIGURE 23-14 Developing a grayscale transform. Figure (a) is the histogram of the raw image in Fig. 23-13a. In (b), a curve is manually generated indicating the desired contrast at each pixel value. The LUT for the output transform is then found by integration and normalization of (b), resulting in the curve labeled manual in (c). In histogram equalization, the histogram of the raw image, shown in (a), is integrated and normalized to find the LUT, shown in (c). 0 50 100 150 Value of pixel 200 250 250 from histogram 200 Display value Number of pixels 393 manual 150 100 c. Output transform 50 0 0 50 100 150 Input value 200 250 right side. Why take the integral to find the required curve? Think of it this way: The contrast at a particular pixel value is equal to the slope of the output transform. That is, we want (b) to be the derivative (slope) of (c). This means that (c) must be the integral of (b). Passing the image in Fig. 23-13a through this manually determined grayscale transform produces the image in (b). The background has been made lighter, the grill has been made darker, and both have better contrast. These improvements are at the expense of the body's contrast, producing a less detailed image of the intruder (although it can't get much worse than in the original image). Grayscale transforms can significantly improve the viewability of an image. The problem is, they can require a great deal of trial and error. Histogram equalization is a way to automate the procedure. Notice that the histogram in (a) and the contrast weighting curve in (b) have the same general shape. Histogram equalization blindly uses the histogram as the contrast weighing curve, eliminating the need for human judgement. That is, the output transform is found by integration and normalization of the histogram, rather than a manually generated curve. This results in the greatest contrast being given to those values that have the greatest number of pixels. 394 The Scientist and Engineer's Guide to Digital Signal Processing Histogram equalization is an interesting mathematical procedure because it maximizes the entropy of the image, a measure of how much information is transmitted by a fixed number of bits. The fault with histogram equalization is that it mistakes the shear number of pixels at a certain value with the importance of the pixels at that value. For example, the truck grill and human intruder are the most prominent features in Fig. 23-13. In spite of this, histogram equalization would almost completely ignore these objects because they contain relatively few pixels. Histogram equalization is quick and easy. Just remember, if it doesn't work well, a manually generated curve will probably do much better. Warping One of the problems in photographing a planet's surface is the distortion from the curvature of the spherical shape. For example, suppose you use a telescope to photograph a square region near the center of a planet, as illustrated in Fig. 23-15a. After a few hours, the planet will have rotated on its axis, appearing as in (b). The previously photographed region appears highly distorted because it is curved near the horizon of the planet. Each of the two images contain complete information about the region, just from a different perspective. It is quite common to acquire a photograph such as (a), but really want the image to look like (b), or vice versa. For example, a satellite mapping the surface of a planet may take thousands of images from straight above, as in (a). To make a natural looking picture of the entire planet, such as the image of Venus in Fig. 23-1, each image must be distorted and placed in the proper position. On the other hand, consider a weather satellite looking at a hurricane that is not directly below it. There is no choice but to acquire the image obliquely, as in (b). The image is then converted into how it would appear from above, as in (a). These spatial transformations are called warping. Space photography is the most common use for warping, but there are others. For example, many vacuum tube imaging detectors have various amounts of spatial distortion. This includes night vision cameras used by the military and x-ray detectors used in the medical field. Digital warping (or dewarping if you prefer) can be used to correct the inherent distortion in these devices. Special effects artists for motion pictures love to warp images. For example, a technique called morphing gradually warps one object into another over a series of frames. This can produces illusions such as a child turning into an adult, or a man turning into a werewolf. Warping takes the original image (a two-dimensional array) and generates a warped image (another two-dimensional array). This is done by looping through each pixel in the warped image and asking: What is the proper pixel value that should be placed here? Given the particular row and column being calculated in the warped image, there is a corresponding row and column in the original image. The pixel value from the original image is transferred to the warped image to carry out the algorithm. In the jargon of image processing, the row and column that the pixel comes from in the Chapter 23- Image Formation and Display 395 a. Normal View N S b. Oblique View N S FIGURE 23-15 Image warping. As shown in (a), a normal view of a small section of a planet appears relatively distortion free. In comparison, an oblique view presents significant spatial distortion. Warping is the technique of changing one of these images into the other. original image is called the comes-from address. Transferring each pixel from the original to the warped image is the easy part. The hard part is calculating the comes-from address associated with each pixel in the warped image. This is usually a pure math problem, and can become quite involved. Simply stretching the image in the horizontal or vertical direction is easier, involving only a multiplication of the row and/or column number to find the comes-from address. One of the techniques used in warping is subpixel interpolation. For example, suppose you have developed a set of equations that turns a row and column address in the warped image into the comes-from address in the original 396 The Scientist and Engineer's Guide to Digital Signal Processing image. Consider what might happen when you try to find the value of the pixel at row 10 and column 20 in the warped image. You pass the information: row = 10, column = 20, into your equations, and out pops: comes-from row = 20.2, comes-from column = 14.5. The point being, your calculations will likely use floating point, and therefore the comes-from addresses will not be integers. The easiest method to use is the nearest neighbor algorithm, that is, simply round the addresses to the nearest integer. This is simple, but can produce a very grainy appearance at the edges of objects where pixels may appear to be slightly misplaced. Bilinear interpolation requires a little more effort, but provides significantly better images. Figure 23-16 shows how it works. You know the value of the four pixels around the fractional address, i.e., the value of the pixels at row 20 & 21, and column 14 and 15. In this example we will assume the pixels values are 91, 210, 162 and 95. The problem is to interpolate between these four values. This is done in two steps. First, interpolate in the horizontal direction between column 14 and 15. This produces two intermediate values, 150.5 on line 20, and 128.5 on line 21. Second, interpolate between these intermediate values in the vertical direction. This produces the bilinear interpolated pixel value of 139.5, which is then transferred to the warped image. Why interpolate in the horizontal direction and then the vertical direction instead of the reverse? It doesn't matter; the final answer is the same regardless of which order is used. a. 91 20.2 14.5 150.5 b. 210 15 210 139.5 91 95 value row 20 14 column col 162 21 row 162 128.5 95 FIGURE 23-16 Subpixel interpolation. Subpixel interpolation for image warping is usually accomplished with bilinear interpolation. As shown in (a), two intermediate values are calculated by linear interpolation in the horizontal direction. The final value is then found by using linear interpolation in the vertical direction between the intermediate values. As shown by the three-dimensional illustration in (b), this procedure uniquely defines all values between the four known pixels at each of the corners. CHAPTER 24 Linear Image Processing Linear image processing is based on the same two techniques as conventional DSP: convolution and Fourier analysis. Convolution is the more important of these two, since images have their information encoded in the spatial domain rather than the frequency domain. Linear filtering can improve images in many ways: sharpening the edges of objects, reducing random noise, correcting for unequal illumination, deconvolution to correct for blur and motion, etc. These procedures are carried out by convolving the original image with an appropriate filter kernel, producing the filtered image. A serious problem with image convolution is the enormous number of calculations that need to be performed, often resulting in unacceptably long execution times. This chapter presents strategies for designing filter kernels for various image processing tasks. Two important techniques for reducing the execution time are also described: convolution by separability and FFT convolution. Convolution Image convolution works in the same way as one-dimensional convolution. For instance, images can be viewed as a summation of impulses, i.e., scaled and shifted delta functions. Likewise, linear systems are characterized by how they respond to impulses; that is, by their impulse responses. As you should expect, the output image from a system is equal to the input image convolved with the system's impulse response. The two-dimensional delta function is an image composed of all zeros, except for a single pixel at: row = 0, column = 0, which has a value of one. For now, assume that the row and column indexes can have both positive and negative values, such that the one is centered in a vast sea of zeros. When the delta function is passed through a linear system, the single nonzero point will be changed into some other two-dimensional pattern. Since the only thing that can happen to a point is that it spreads out, the impulse response is often called the point spread function (PSF) in image processing jargon. 397 398 The Scientist and Engineer's Guide to Digital Signal Processing The human eye provides an excellent example of these concepts. As described in the last chapter, the first layer of the retina transforms an image represented as a pattern of light into an image represented as a pattern of nerve impulses. The second layer of the retina processes this neural image and passes it to the third layer, the fibers forming the optic nerve. Imagine that the image being projected onto the retina is a very small spot of light in the center of a dark background. That is, an impulse is fed into the eye. Assuming that the system is linear, the image processing taking place in the retina can be determined by inspecting the image appearing at the optic nerve. In other words, we want to find the point spread function of the processing. We will revisit the assumption about linearity of the eye later in this chapter. Figure 24-1 outlines this experiment. Figure (a) illustrates the impulse striking the retina while (b) shows the image appearing at the optic nerve. The middle layer of the eye passes the bright spike, but produces a circular region of increased darkness. The eye accomplishes this by a process known as lateral inhibition. If a nerve cell in the middle layer is activated, it decreases the ability of its nearby neighbors to become active. When a complete image is viewed by the eye, each point in the image contributes a scaled and shifted version of this impulse response to the image appearing at the optic nerve. In other words, the visual image is convolved with this PSF to produce the neural image transmitted to the brain. The obvious question is: how does convolving a viewed image with this PSF improve the ability of the eye to understand the world? a. Image at first layer b. Image at third layer FIGURE 24-1 The PSF of the eye. The middle layer of the retina changes an impulse, shown in (a), into an impulse surrounded by a dark area, shown in (b). This point spread function enhances the edges of objects. Chapter 24- Linear Image Processing 399 a. True brightness FIGURE 24-2 Mach bands. Image processing in the retina results in a slowly changing edge, as in (a), being sharpened, as in (b). This makes it easier to separate objects in the image, but produces an optical illusion called Mach bands. Near the edge, the overshoot makes the dark region look darker, and the light region look lighter. This produces dark and light bands that run parallel to the edge. b. Perceived brightness Humans and other animals use vision to identify nearby objects, such as enemies, food, and mates. This is done by distinguishing one region in the image from another, based on differences in brightness and color. In other words, the first step in recognizing an object is to identify its edges, the discontinuity that separates an object from its background. The middle layer of the retina helps this task by sharpening the edges in the viewed image. As an illustration of how this works, Fig. 24-2 shows an image that slowly changes from dark to light, producing a blurry and poorly defined edge. Figure (a) shows the intensity profile of this image, the pattern of brightness entering the eye. Figure (b) shows the brightness profile appearing on the optic nerve, the image transmitted to the brain. The processing in the retina makes the edge between the light and dark areas appear more abrupt, reinforcing that the two regions are different. The overshoot in the edge response creates an interesting optical illusion. Next to the edge, the dark region appears to be unusually dark, and the light region appears to be unusually light. The resulting light and dark strips are called Mach bands, after Ernst Mach (1838-1916), an Austrian physicist who first described them. As with one-dimensional signals, image convolution can be viewed in two ways: from the input, and from the output. From the input side, each pixel in 400 The Scientist and Engineer's Guide to Digital Signal Processing the input image contributes a scaled and shifted version of the point spread function to the output image. As viewed from the output side, each pixel in the output image is influenced by a group of pixels from the input signal. For one-dimensional signals, this region of influence is the impulse response flipped left-for-right. For image signals, it is the PSF flipped left-for-right and topfor-bottom. Since most of the PSFs used in DSP are symmetrical around the vertical and horizonal axes, these flips do nothing and can be ignored. Later in this chapter we will look at nonsymmetrical PSFs that must have the flips taken into account. Figure 24-3 shows several common PSFs. In (a), the pillbox has a circular top and straight sides. For example, if the lens of a camera is not properly focused, each point in the image will be projected to a circular spot on the image sensor (look back at Fig. 23-2 and consider the effect of moving the projection screen toward or away from the lens). In other words, the pillbox is the point spread function of an out-of-focus lens. The Gaussian, shown in (b), is the PSF of imaging systems limited by random imperfections. For instance, the image from a telescope is blurred by atmospheric turbulence, causing each point of light to become a Gaussian in the final image. Image sensors, such as the CCD and retina, are often limited by the scattering of light and/or electrons. The Central Limit Theorem dictates that a Gaussian blur results from these types of random processes. The pillbox and Gaussian are used in image processing the same as the moving average filter is used with one-dimensional signals. An image convolved with these PSFs will appear blurry and have less defined edges, but will be lower in random noise. These are called smoothing filters, for their action in the time domain, or low-pass filters, for how they treat the frequency domain. The square PSF, shown in (c), can also be used as a smoothing filter, but it is not circularly symmetric. This results in the blurring being different in the diagonal directions compared to the vertical and horizontal. This may or may not be important, depending on the use. The opposite of a smoothing filter is an edge enhancement or high-pass filter. The spectral inversion technique, discussed in Chapter 14, is used to change between the two. As illustrated in (d), an edge enhancement filter kernel is formed by taking the negative of a smoothing filter, and adding a delta function in the center. The image processing which occurs in the retina is an example of this type of filter. Figure (e) shows the two-dimensional sinc function. One-dimensional signal processing uses the windowed-sinc to separate frequency bands. Since images do not have their information encoded in the frequency domain, the sinc function is seldom used as an imaging filter kernel, although it does find use in some theoretical problems. The sinc function can be hard to use because its tails decrease very slowly in amplitude ( 1/x ), meaning it must be treated as infinitely wide. In comparison, the Gaussian's tails decrease very rapidly 2 ( e &x ) and can eventually be truncated with no ill effect. Chapter 24- Linear Image Processing b. Gaussian 4 4 3 3 value value a. Pillbox 2 1 2 1 0 0 -8 -6 -4 -2 0 row 2 4 6 8 -8 -6 -4 -2 0 2 4 6 -8 8 -6 -4 col -2 0 row c. Square 2 4 6 8 -8 -6 -4 -2 0 2 4 6 8 col d. Edge enhancement 4 3 3 value 4 2 1 2 1 0 0 -8 -6 -4 -2 0 row 2 4 6 8 -8 -6 -4 -2 0 2 4 6 -8 8 -6 -4 col -2 0 row 2 4 6 8 -8 -6 -4 -2 0 2 4 6 8 col e. Sinc FIGURE 24-3 Common point spread functions. The pillbox, Gaussian, and square, shown in (a), (b), & (c), are common smoothing (low-pass) filters. Edge enhancement (high-pass) filters are formed by subtracting a low-pass kernel from an impulse, as shown in (d). The sinc function, (e), is used very little in image processing because images have their information encoded in the spatial domain, not the frequency domain. 4 3 value value 401 2 1 0 -8 -6 -4 -2 0 row 2 4 6 8 -8 -6 -4 -2 0 2 4 6 8 col All these filter kernels use negative indexes in the rows and columns, allowing the PSF to be centered at row = 0 and column = 0. Negative indexes are often eliminated in one-dimensional DSP by shifting the filter kernel to the right until all the nonzero samples have a positive index. This shift moves the output signal by an equal amount, which is usually of no concern. In comparison, a shift between the input and output images is generally not acceptable. Correspondingly, negative indexes are the norm for filter kernels in image processing. 402 The Scientist and Engineer's Guide to Digital Signal Processing A problem with image convolution is that a large number of calculations are involved. For instance, when a 512 by 512 pixel image is convolved with a 64 by 64 pixel PSF, more than a billion multiplications and additions are needed (i.e., 64 ×64 ×512 ×512 ). The long execution times can make the techniques impractical. Three approaches are used to speed things up. The first strategy is to use a very small PSF, often only 3×3 pixels. This is carried out by looping through each sample in the output image, using optimized code to multiply and accumulate the corresponding nine pixels from the input image. A surprising amount of processing can be achieved with a mere 3×3 PSF, because it is large enough to affect the edges in an image. The second strategy is used when a large PSF is needed, but its shape isn't critical. This calls for a filter kernel that is separable, a property that allows the image convolution to be carried out as a series of one-dimensional operations. This can improve the execution speed by hundreds of times. The third strategy is FFT convolution, used when the filter kernel is large and has a specific shape. Even with the speed improvements provided by the highly efficient FFT, the execution time will be hideous. Let's take a closer look at the details of these three strategies, and examples of how they are used in image processing. 3×3 Edge Modification Figure 24-4 shows several 3×3 operations. Figure (a) is an image acquired by an airport x-ray baggage scanner. When this image is convolved with a 3×3 delta function (a one surrounded by 8 zeros), the image remains unchanged. While this is not interesting by itself, it forms the baseline for the other filter kernels. Figure (b) shows the image convolved with a 3×3 kernel consisting of a one, a negative one, and 7 zeros. This is called the shift and subtract operation, because a shifted version of the image (corresponding to the -1) is subtracted from the original image (corresponding to the 1). This processing produces the optical illusion that some objects are closer or farther away than the background, making a 3D or embossed effect. The brain interprets images as if the lighting is from above, the normal way the world presents itself. If the edges of an object are bright on the top and dark on the bottom, the object is perceived to be poking out from the background. To see another interesting effect, turn the picture upside down, and the objects will be pushed into the background. Figure (c) shows an edge detection PSF, and the resulting image. Every edge in the original image is transformed into narrow dark and light bands that run parallel to the original edge. Thresholding this image can isolate either the dark or light band, providing a simple algorithm for detecting the edges in an image. Chapter 24- Linear Image Processing a. Delta function 0 0 0 0 0 0 0 1 0 0 1 0 0 0 -1 0 0 0 b. Shift and subtract -1/8 -1/8 -1/8 -1/8 c. Edge detection 403 1 -k/8 -k/8 -k/8 -1/8 -1/8 -1/8 -1/8 -k/8 k+1 -k/8 d. Edge enhancement -k/8 -k/8 -k/8 FIGURE 24-4 3×3 edge modification. The original image, (a), was acquired on an airport x-ray baggage scanner. The shift and subtract operation, shown in (b), results in a pseudo three-dimensional effect. The edge detection operator in (c) removes all contrast, leaving only the edge information. The edge enhancement filter, (d), adds various ratios of images (a) and (c), determined by the parameter, k. A value of k = 2 was used to create this image. A common image processing technique is shown in (d): edge enhancement. This is sometimes called a sharpening operation. In (a), the objects have good contrast (an appropriate level of darkness and lightness) but very blurry edges. In (c), the objects have absolutely no contrast, but very sharp edges. The 404 The Scientist and Engineer's Guide to Digital Signal Processing strategy is to multiply the image with good edges by a constant, k, and add it to the image with good contrast. This is equivalent to convolving the original image with the 3×3 PSF shown in (d). If k is set to 0, the PSF becomes a delta function, and the image is left unchanged. As k is made larger, the image shows better edge definition. For the image in (d), a value of k = 2 was used: two parts of image (c) to one part of image (a). This operation mimics the eye's ability to sharpen edges, allowing objects to be more easily separated from the background. Convolution with any of these PSFs can result in negative pixel values appearing in the final image. Even if the program can handle negative values for pixels, the image display cannot. The most common way around this is to add an offset to each of the calculated pixels, as is done in these images. An alternative is to truncate out-of-range values. Convolution by Separability This is a technique for fast convolution, as long as the PSF is separable. A PSF is said to be separable if it can be broken into two one-dimensional signals: a vertical and a horizontal projection. Figure 24-5 shows an example of a separable image, the square PSF. Specifically, the value of each pixel in the image is equal to the corresponding point in the horizontal projection multiplied by the corresponding point in the vertical projection. In mathematical form: EQUATION 24-1 Image separation. An image is referred to as separable if it can be decomposed into horizontal and vertical projections. x[r,c ] ' vert [r] × horz [c ] where x[r, c] is the two-dimensional image, and vert[r] & horz[c] are the onedimensional projections. Obviously, most images do not satisfy this requirement. For example, the pillbox is not separable. There are, however, an infinite number of separable images. This can be understood by generating arbitrary horizontal and vertical projections, and finding the image that corresponds to them. For example, Fig. 24-6 illustrates this with profiles that are double-sided exponentials. The image that corresponds to these profiles is then found from Eq. 24-1. When displayed, the image appears as a diamond shape that exponentially decays to zero as the distance from the origin increases. In most image processing tasks, the ideal PSF is circularly symmetric, such as the pillbox. Even though digitized images are usually stored and processed in the rectangular format of rows and columns, it is desired to modify the image the same in all directions. This raises the question: is there a PSF that is circularly symmetric and separable? The answer is, yes, Chapter 24- Linear Image Processing 405 1.5 horz[c] FIGURE 24-5 Separation of the rectangular PSF. A PSF is said to be separable if it can be decomposed into horizontal and vertical profiles. Separable PSFs are important because they can be rapidly convolved. 1.0 0.5 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 1 1 1 0 0 0 1 0 0 0 1 1 1 1 1 1 1 0 0 0 1 0 0 0 1 1 1 1 1 1 1 0 0 0 1 0 0 0 1 1 1 1 1 1 1 0 0 0 1 0 0 0 1 1 1 1 1 1 1 0 0 0 1 0 0 0 1 1 1 1 1 1 1 0 0 0 1 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0 0.5 1.0 1.5 vert[r] 0.0 horz[c] 4 value 3 2 1 0 vert[r] -32 32 -24 -16 16 -8 0 row 24 8 8 16 24 32 -32 -24 -16 -8 0 col FIGURE 24-6 Creation of a separable PSF. An infinite number of separable PSFs can be generated by defining arbitrary projections, and then calculating the two-dimensional function that corresponds to them. In this example, the profiles are chosen to be double-sided exponentials, resulting in a diamond shaped PSF. 406 The Scientist and Engineer's Guide to Digital Signal Processing 20 FIGURE 24-7 Separation of the Gaussian. The Gaussian is the only PSF that is circularly symmetric and separable. This makes it a common filter kernel in image processing. horz[c] 15 10 5 0 0.04 0 0 0 0 0 1 1 1 0 0 0 0 0 0.25 0 0 0 1 2 3 4 3 2 1 0 0 0 1.11 0 0 1 4 9 15 18 15 9 4 1 0 0 3.56 0 1 4 13 29 48 57 48 29 13 4 1 0 8.20 0 2 9 29 67 111 131 111 67 29 9 2 0 13.5 1 3 15 48 111 183 216 183 111 48 15 3 1 16.0 1 4 18 57 131 216 255 216 131 57 18 4 1 13.5 1 3 15 48 111 183 216 183 111 48 15 3 1 8.20 0 2 9 29 67 111 131 111 67 29 9 2 0 3.56 0 1 4 13 29 48 57 48 29 13 4 1 0 1.11 0 0 1 4 9 15 18 15 9 4 1 0 0 0.25 0 0 0 1 2 3 4 3 2 1 0 0 0 0.04 0 0 0 0 0 1 1 1 0 0 0 0 0 0 5 10 15 20 vert[r] 0.04 0.25 1.11 3.56 8.20 13.5 16.0 13.5 8.20 3.56 1.11 0.25 0.04 but there is only one, the Gaussian. As is shown in Fig. 24-7, a two-dimensional Gaussian image has projections that are also Gaussians. The image and projection Gaussians have the same standard deviation. To convolve an image with a separable filter kernel, convolve each row in the image with the horizontal projection, resulting in an intermediate image. Next, convolve each column of this intermediate image with the vertical projection of the PSF. The resulting image is identical to the direct convolution of the original image and the filter kernel. If you like, convolve the columns first and then the rows; the result is the same. The convolution of an N×N image with an M×M filter kernel requires a time proportional to N 2M 2 . In other words, each pixel in the output image depends on all the pixels in the filter kernel. In comparison, convolution by separability only requires a time proportional to N 2M . For filter kernels that are hundreds of pixels wide, this technique will reduce the execution time by a factor of hundreds. Things can get even better. If you are willing to use a rectangular PSF (Fig. 24-5) or a double-sided exponential PSF (Fig. 24-6), the calculations are even more efficient. This is because the one-dimensional convolutions are the moving average filter (Chapter 15) and the bidirectional single pole filter Chapter 24- Linear Image Processing 407 (Chapter 19), respectively. Both of these one-dimensional filters can be rapidly carried out by recursion. This results in an image convolution time proportional to only N 2 , completely independent of the size of the PSF. In other words, an image can be convolved with as large a PSF as needed, with only a few integer operations per pixel. For example, the convolution of a 512×512 image requires only a few hundred milliseconds on a personal computer. That's fast! Don't like the shape of these two filter kernels? Convolve the image with one of them several times to approximate a Gaussian PSF (guaranteed by the Central Limit Theorem, Chapter 7). These are great algorithms, capable of snatching success from the jaws of failure. They are well worth remembering. Example of a Large PSF: Illumination Flattening A common application requiring a large PSF is the enhancement of images with unequal illumination. Convolution by separability is an ideal algorithm to carry out this processing. With only a few exceptions, the images seen by the eye are formed from reflected light. This means that a viewed image is equal to the reflectance of the objects multiplied by the ambient illumination. Figure 24-8 shows how this works. Figure (a) represents the reflectance of a scene being viewed, in this case, a series of light and dark bands. Figure (b) illustrates an example illumination signal, the pattern of light falling on (a). As in the real world, the illumination slowly varies over the imaging area. Figure (c) is the image seen by the eye, equal to the reflectance image, (a), multiplied by the illumination image, (b). The regions of poor illumination are difficult to view in (c) for two reasons: they are too dark and their contrast is too low (the difference between the peaks and the valleys). To understand how this relates to the problem of every day vision, imagine you are looking at two identically dressed men. One of them is standing in the bright sunlight, while the other is standing in the shade of a nearby tree. The percent of the incident light reflected from both men is the same. For instance, their faces might reflect 80% of the incident light, their gray shirts 40% and their dark pants 5%. The problem is, the illumination of the two might be, say, ten times different. This makes the image of the man in the shade ten times darker than the person in the sunlight, and the contrast (between the face, shirt, and pants) ten times less. The goal of the image processing is to flatten the illumination component in the acquired image. In other words, we want the final image to be representative of the objects' reflectance, not the lighting conditions. In terms of Fig. 24-8, given (c), find (a). This is a nonlinear filtering problem, since the component images were combined by multiplication, not addition. While this separation cannot be performed perfectly, the improvement can be dramatic. To start, we will convolve image (c) with a large PSF, one-fifth the size of the entire image. The goal is to eliminate the sharp features in (c), resulting 408 The Scientist and Engineer's Guide to Digital Signal Processing a. Reflectance c. Viewed image b. Illumination 1 1 0 Brightness Reflectance Illumination 1 0 0 40 80 120 Column number 160 200 0 0 40 80 120 Column number 160 200 0 40 80 120 160 200 Column number FIGURE 24-8 Model of image formation. A viewed image, (c), results from the multiplication of an illumination pattern, (b), by a reflectance pattern, (a). The goal of the image processing is to modify (c) to make it look more like (a). This is performed in Figs. (d), (e) and (f) on the opposite page. in an approximation to the original illumination signal, (b). This is where convolution by separability is used. The exact shape of the PSF is not important, only that it is much wider than the features in the reflectance image. Figure (d) is the result, using a Gaussian filter kernel. Since a smoothing filter provides an estimate of the illumination image, we will use an edge enhancement filter to find the reflectance image. That is, image (c) will be convolved with a filter kernel consisting of a delta function minus a Gaussian. To reduce execution time, this is done by subtracting the smoothed image in (d) from the original image in (c). Figure (e) shows the result. It doesn't work! While the dark areas have been properly lightened, the contrast in these areas is still terrible. Linear filtering performs poorly in this application because the reflectance and illumination signals were original combined by multiplication, not addition. Linear filtering cannot correctly separate signals combined by a nonlinear operation. To separate these signals, they must be unmultiplied. In other words, the original image should be divided by the smoothed image, as is shown in (f). This corrects the brightness and restores the contrast to the proper level. This procedure of dividing the images is closely related to homomorphic processing, previously described in Chapter 22. Homomorphic processing is a way of handling signals combined through a nonlinear operation. The strategy is to change the nonlinear problem into a linear one, through an appropriate mathematical operation. When two signals are combined by Chapter 24- Linear Image Processing d. Smoothed e. (c) - (d) f. (c)÷(d) 0.5 1 Brightness 1 Brightness Brightness 1 0 -0.5 0 0 40 80 120 Column number 160 409 200 0 0 40 80 120 Column number 160 200 0 40 80 120 160 200 Column number FIGURE 24-8 (continued) Figure (d) is a smoothed version of (c), used as an approximation to the illumination signal. Figure (e) shows an approximation to the reflectance image, created by subtracting the smoothed image from the viewed image. A better approximation is shown in (f), obtained by the nonlinear process of dividing the two images. multiplication, homomorphic processing starts by taking the logarithm of the acquired signal. With the identity: log(a×b) ' log(a) % log(b) , the problem of separating multiplied signals is converted into the problem of separating added signals. For example, after taking the logarithm of the image in (c), a linear high-pass filter can be used to isolate the logarithm of the reflectance image. As before, the quickest way to carry out the high-pass filter is to subtract a smoothed version of the image. The antilogarithm (exponent) is then used to undo the logarithm, resulting in the desired approximation to the reflectance image. Which is better, dividing or going along the homomorphic path? They are nearly the same, since taking the logarithm and subtracting is equal to dividing. The only difference is the approximation used for the illumination image. One method uses a smoothed version of the acquired image, while the other uses a smoothed version of the logarithm of the acquired image. This technique of flattening the illumination signal is so useful it has been incorporated into the neural structure of the eye. The processing in the middle layer of the retina was previously described as an edge enhancement or high-pass filter. While this is true, it doesn't tell the whole story. The first layer of the eye is nonlinear, approximately taking the logarithm of the incoming image. This makes the eye a homomorphic processor. Just as described above, the logarithm followed by a linear edge enhancement filter flattens the illumination component, allowing the eye to see under poor lighting conditions. Another interesting use of homomorphic processing 410 The Scientist and Engineer's Guide to Digital Signal Processing occurs in photography. The density (darkness) of a negative is equal to the logarithm of the brightness in the final photograph. This means that any manipulation of the negative during the development stage is a type of homomorphic processing. Before leaving this example, there is a nuisance that needs to be mentioned. As discussed in Chapter 6, when an N point signal is convolved with an M point filter kernel, the resulting signal is N%M&1 points long. Likewise, when an M×M image is convolved with an N×N filter kernel, the result is an ( M%N&1) × ( M%N&1) image. The problem is, it is often difficult to manage a changing image size. For instance, the allocated memory must change, the video display must be adjusted, the array indexing may need altering, etc. The common way around this is to ignore it; if we start with a 512×512 image, we want to end up with a 512×512 image. The pixels that do not fit within the original boundaries are discarded. While this keeps the image size the same, it doesn't solve the whole problem; these is still the boundary condition for convolution. For example, imagine trying to calculate the pixel at the upper-right corner of (d). This is done by centering the Gaussian PSF on the upper-right corner of (c). Each pixel in (c) is then multiplied by the corresponding pixel in the overlaying PSF, and the products are added. The problem is, three-quarters of the PSF lies outside the defined image. The easiest approach is to assign the undefined pixels a value of zero. This is how (d) was created, accounting for the dark band around the perimeter of the image. That is, the brightness smoothly decreases to the pixel value of zero, exterior to the defined image. Fortunately, this dark region around the boarder can be corrected (although it hasn't been in this example). This is done by dividing each pixel in (d) by a correction factor. The correction factor is the fraction of the PSF that was immersed in the input image when the pixel was calculated. That is, to correct an individual pixel in (d), imagine that the PSF is centered on the corresponding pixel in (c). For example, the upper-right pixel in (c) results from only 25% of the PSF overlapping the input image. Therefore, correct this pixel in (d) by dividing it by a factor of 0.25. This means that the pixels in the center of (d) will not be changed, but the dark pixels around the perimeter will be brightened. To find the correction factors, imagine convolving the filter kernel with an image having all the pixel values equal to one. The pixels in the resulting image are the correction factors needed to eliminate the edge effect. Fourier Image Analysis Fourier analysis is used in image processing in much the same way as with one-dimensional signals. However, images do not have their information encoded in the frequency domain, making the techniques much less useful. For example, when the Fourier transform is taken of an audio signal, the confusing time domain waveform is converted into an easy to understand frequency Chapter 24- Linear Image Processing 411 spectrum. In comparison, taking the Fourier transform of an image converts the straightforward information in the spatial domain into a scrambled form in the frequency domain. In short, don't expect the Fourier transform to help you understand the information encoded in images. Likewise, don't look to the frequency domain for filter design. The basic feature in images is the edge, the line separating one object or region from another object or region. Since an edge is composed of a wide range of frequency components, trying to modify an image by manipulating the frequency spectrum is generally not productive. Image filters are normally designed in the spatial domain, where the information is encoded in its simplest form. Think in terms of smoothing and edge enhancement operations (the spatial domain) rather than high-pass and low-pass filters (the frequency domain). In spite of this, Fourier image analysis does have several useful properties. For instance, convolution in the spatial domain corresponds to multiplication in the frequency domain. This is important because multiplication is a simpler mathematical operation than convolution. As with one-dimensional signals, this property enables FFT convolution and various deconvolution techniques. Another useful property of the frequency domain is the Fourier Slice Theorem, the relationship between an image and its projections (the image viewed from its sides). This is the basis of computed tomography, an x-ray imaging technique widely used medicine and industry. The frequency spectrum of an image can be calculated in several ways, but the FFT method presented here is the only one that is practical. The original image must be composed of N rows by N columns, where N is a power of two, i.e., 256, 512, 1024, etc. If the size of the original image is not a power of two, pixels with a value of zero are added to make it the correct size. We will call the two-dimensional array that holds the image the real array. In addition, another array of the same size is needed, which we will call the imaginary array. The recipe for calculating the Fourier transform of an image is quite simple: take the one-dimensional FFT of each of the rows, followed by the onedimensional FFT of each of the columns. Specifically, start by taking the FFT of the N pixel values in row 0 of the real array. The real part of the FFT's output is placed back into row 0 of the real array, while the imaginary part of the FFT's output is placed into row 0 of the imaginary array. After repeating this procedure on rows 1 through N&1 , both the real and imaginary arrays contain an intermediate image. Next, the procedure is repeated on each of the columns of the intermediate data. Take the N pixel values from column 0 of the real array, and the N pixel values from column 0 of the imaginary array, and calculate the FFT. The real part of the FFT's output is placed back into column 0 of the real array, while the imaginary part of the FFT's output is placed back into column 0 of the imaginary array. After this is repeated on columns 1 through N&1 , both arrays have been overwritten with the image's frequency spectrum. 412 The Scientist and Engineer's Guide to Digital Signal Processing Since the vertical and horizontal directions are equivalent in an image, this algorithm can also be carried out by transforming the columns first and then transforming the rows. Regardless of the order used, the result is the same. From the way that the FFT keeps track of the data, the amplitudes of the low frequency components end up being at the corners of the two-dimensional spectrum, while the high frequencies are at the center. The inverse Fourier transform of an image is calculated by taking the inverse FFT of each row, followed by the inverse FFT of each column (or vice versa). Figure 24-9 shows an example Fourier transform of an image. Figure (a) is the original image, a microscopic view of the input stage of a 741 op amp integrated circuit. Figure (b) shows the real and imaginary parts of the frequency spectrum of this image. Since the frequency domain can contain negative pixel values, the grayscale values of these images are offset such that negative values are dark, zero is gray, and positive values are light. The lowfrequency components in an image are normally much larger in amplitude than the high-frequency components. This accounts for the very bright and dark pixels at the four corners of (b). Other than this, the spectra of typical images have no discernable order, appearing random. Of course, images can be contrived to have any spectrum you desire. As shown in (c), the polar form of an image spectrum is only slightly easier to understand. The low-frequencies in the magnitude have large positive values (the white corners), while the high-frequencies have small positive values (the black center). The phase looks the same at low and high-frequencies, appearing to run randomly between -B and B radians. Figure (d) shows an alternative way of displaying an image spectrum. Since the spatial domain contains a discrete signal, the frequency domain is periodic. In other words, the frequency domain arrays are duplicated an infinite number of times to the left, right, top and bottom. For instance, imagine a tile wall, with each tile being the N×N magnitude shown in (c). Figure (d) is also an N×N section of this tile wall, but it straddles four tiles; the center of the image being where the four tiles touch. In other words, (c) is the same image as (d), except it has been shifted N/2 pixels horizontally (either left or right) and N/2 pixels vertically (either up or down) in the periodic frequency spectrum. This brings the bright pixels at the four corners of (c) together in the center of (d). Figure 24-10 illustrates how the two-dimensional frequency domain is organized (with the low-frequencies placed at the corners). Row N/2 and column N/2 break the frequency spectrum into four quadrants. For the real part and the magnitude, the upper-right quadrant is a mirror image of the lower-left, while the upper-left is a mirror image of the lower-right. This symmetry also holds for the imaginary part and the phase, except that the values of the mirrored pixels are opposite in sign. In other words, every point in the frequency spectrum has a matching point placed symmetrically on the other side of the center of the image (row N/2 and column N/2 ). One of the points is the positive frequency, while the other is the matching Chapter 24- Linear Image Processing 413 a. Image FIGURE 24-9 Frequency spectrum of an image. The example image, shown in (a), is a microscopic photograph of the silicon surface of an integrated circuit. The frequency spectrum can be displayed as the real and imaginary parts, shown in (b), or as the magnitude and phase, shown in (c). Figures (b) & (c) are displayed with the low-frequencies at the corners and the high-frequencies at the center. Since the frequency domain is periodic, the display can be rearranged to reverse these positions. This is shown in (d), where the magnitude and phase are displayed with the low-frequencies located at the center and the high-frequencies at the corners. Real Imaginary Magnitude Phase Magnitude Phase b. Frequency spectrum displayed in rectangular form (as the real and imaginary parts). c. Frequency spectrum displayed in polar form (as the magnitude and phase). d. Frequency spectrum displayed in polar form, with the spectrum shifted to place zero frequency at the center. 414 The Scientist and Engineer's Guide to Digital Signal Processing negative frequency, as discussed in Chapter 10 for one-dimensional signals. In equation form, this symmetry is expressed as: EQUATION 24-2 Symmetry of the two-dimensional frequency domain. These equations can be used in both formats, when the low-frequencies are displayed at the corners, or when shifting places them at the center. In polar form, the magnitude has the same symmetry as the real part, while the phase has the same symmetry as the imaginary part. Re X [r, c ] ' Re X [N & r, N & c] Im X [r, c ] ' & Im X [N & r, N & c] These equations take into account that the frequency spectrum is periodic, repeating itself every N samples with indexes running from 0 to N&1 . In other words, X [r,N ] should be interpreted as X [r,0 ] , X [N,c ] as X [0,c ] , and X [N,N ] as X [0,0 ] . This symmetry makes four points in the spectrum match with themselves. These points are located at: [0, 0] , [0, N/2] , [N/2, 0] and [N/2, N/2] . Each pair of points in the frequency domain corresponds to a sinusoid in the spatial domain. As shown in (a), the value at [0, 0] corresponds to the zero frequency sinusoid in the spatial domain, i.e., the DC component of the image. There is only one point shown in this figure, because this is one of the points that is its own match. As shown in (b), (c), and (d), other pairs of points correspond to two-dimensional sinusoids that look like waves on the ocean. One-dimensional sinusoids have a frequency, phase, and amplitude. Two dimensional sinusoids also have a direction. The frequency and direction of each sinusoid is determined by the location of the pair of points in the frequency domain. As shown, draw a line from each point to the zero frequency location at the outside corner of the quadrant that the point is in, i.e., [0, 0], [0,N/2], [N/2, 0], or [N/2,N/2] (as indicated by the circles in this figure). The direction of this line determines the direction of the spatial sinusoid, while its length is proportional to the frequency of the wave. This results in the low frequencies being positioned near the corners, and the high frequencies near the center. When the spectrum is displayed with zero frequency at the center ( Fig. 24-9d), the line from each pair of points is drawn to the DC value at the center of the image, i.e., [ N/2 , N/2 ]. This organization is simpler to understand and work with, since all the lines are drawn to the same point. Another advantage of placing zero at the center is that it matches the frequency spectra of continuous images. When the spatial domain is continuous, the frequency domain is aperiodic. This places zero frequency at the center, with the frequency becoming higher in all directions out to infinity. In general, the frequency spectra of discrete images are displayed with zero frequency at the center whenever people will view the data, in textbooks, journal articles, and algorithm documentation. However, most calculations are carried out with the computer arrays storing data in the other format (low-frequencies at the corners). This is because the FFT has this format. Chapter 24- Linear Image Processing 415 Spatial Domain Frequency Domain column 0 0 column N-1 0 0 N/2 N-1 row row a. N/2 N-1 N-1 column 0 0 column N-1 0 0 N/2 N-1 FIGURE 24-10 Two-dimensional sinusoids. Image sine and cosine waves have both a frequency and a direction. Four examples are shown here. These spectra are displayed with the lowfrequencies at the corners. The circles in these spectra show the location of zero frequency. row row b. N/2 N-1 N-1 column column 0 0 N-1 0 0 N/2 N-1 row row c. N/2 N-1 N-1 column column 0 0 N-1 0 N-1 row row d. N/2 N-1 0 N/2 N-1 416 The Scientist and Engineer's Guide to Digital Signal Processing Even with the FFT, the time required to calculate the Fourier transform is a tremendous bottleneck in image processing. For example, the Fourier transform of a 512×512 image requires several minutes on a personal computer. This is roughly 10,000 times slower than needed for real time image processing, 30 frames per second. This long execution time results from the massive amount of information contained in images. For comparison, there are about the same number of pixels in a typical image, as there are words in this book. Image processing via the frequency domain will become more popular as computers become faster. This is a twentyfirst century technology; watch it emerge! FFT Convolution Even though the Fourier transform is slow, it is still the fastest way to convolve an image with a large filter kernel. For example, convolving a 512×512 image with a 50×50 PSF is about 20 times faster using the FFT compared with conventional convolution. Chapter 18 discusses how FFT convolution works for one-dimensional signals. The two-dimensional version is a simple extension. We will demonstrate FFT convolution with an example, an algorithm to locate a predetermined pattern in an image. Suppose we build a system for inspecting one-dollar bills, such as might be used for printing quality control, counterfeiting detection, or payment verification in a vending machine. As shown in Fig. 24-11, a 100×100 pixel image is acquired of the bill, centered on the portrait of George Washington. The goal is to search this image for a known pattern, in this example, the 29×29 pixel image of the face. The problem is this: given an acquired image and a known pattern, what is the most effective way to locate where (or if) the pattern appears in the image? If you paid attention in Chapter 6, you know that the solution to this problem is correlation (a matched filter) and that it can be implemented by using convolution. Before performing the actual convolution, there are two modifications that need to be made to turn the target image into a PSF. These are illustrated in Fig. 24-12. Figure (a) shows the target signal, the pattern we are trying to detect. In (b), the image has been rotated by 180E, the same as being flipped left-forright and then flipped top-for-bottom. As discussed in Chapter 7, when performing correlation by using convolution, the target signal needs to be reversed to counteract the reversal that occurs during convolution. We will return to this issue shortly. The second modification is a trick for improving the effectiveness of the algorithm. Rather than trying to detect the face in the original image, it is more effective to detect the edges of the face in the edges of the original image. This is because the edges are sharper than the original features, making the correlation have a sharper peak. This step isn't required, but it makes the results significantly better. In the simplest form, a 3×3 edge detection filter is applied to both the original image and the target signal Chapter 24- Linear Image Processing 417 100 pixels 29 pixels 100 pixels 29 pixels b. Target a. Image to be searched FIGURE 24-11 Target detection example. The problem is to search the 100×100 pixel image of George Washington, (a), for the target pattern, (b), the 29×29 pixel face. The optimal solution is correlation, which can be carried out by convolution. before the correlation is performed. From the associative property of convolution, this is the same as applying the edge detection filter to the target signal twice, and leaving the original image alone. In actual practice, applying the edge detection 3×3 kernel only once is generally sufficient. This is how (b) is changed into (c) in Fig. 24-12. This makes (c) the PSF to be used in the convolution Figure 24-13 illustrates the details of FFT convolution. In this example, we will convolve image (a) with image (b) to produce image (c). The fact that these images have been chosen and preprocessed to implement correlation is irrelevant; this is a flow diagram of convolution. The first step is to pad both signals being convolved with enough zeros to make them a power of two in size, and big enough to hold the final image. For instance, when images of 100×100 and 29×29 pixels are convolved, the resulting image will be 128×128 pixels. Therefore, enough zeros must be added to (a) and (b) to make them each 128×128 pixels in size. If this isn't done, circular a. Original b. Rotated c. Edge detection FIGURE 24-12 Development of a correlation filter kernel. The target signal is shown in (a). In (b) it is rotated by 180E to undo the rotation inherent in convolution, allowing correlation to be performed. Applying an edge detection filter results in (c), the filter kernel used for this example. 418 The Scientist and Engineer's Guide to Digital Signal Processing convolution takes place and the final image will be distorted. If you are having trouble understanding these concepts, go back and review Chapter 18, where the one-dimensional case is discussed in more detail. The FFT algorithm is used to transform (a) and (b) into the frequency domain. This results in four 128×128 arrays, the real and imaginary parts of the two images being convolved. Multiplying the real and imaginary parts of (a) with the real and imaginary parts of (b), generates the real and imaginary parts of (c). (If you need to be reminded how this is done, see Eq. 9-1). Taking the Inverse FFT completes the algorithm by producing the final convolved image. The value of each pixel in a correlation image is a measure of how well the target image matches the searched image at that point. In this particular example, the correlation image in (c) is composed of noise plus a single bright peak, indicating a good match to the target signal. Simply locating the brightest pixel in this image would specify the detected coordinates of the face. If we had not used the edge detection modification on the target signal, the peak would still be present, but much less distinct. While correlation is a powerful tool in image processing, it suffers from a significant limitation: the target image must be exactly the same size and rotational orientation as the corresponding area in the searched image. Noise and other variations in the amplitude of each pixel are relatively unimportant, but an exact spatial match is critical. For example, this makes the method almost useless for finding enemy tanks in military reconnaissance photos, tumors in medical images, and handguns in airport baggage scans. One approach is to correlate the image multiple times with a variety of shapes and rotations of the target image. This works in principle, but the execution time will make you loose interest in a hurry. A Closer Look at Image Convolution Let's use this last example to explore two-dimensional convolution in more detail. Just as with one dimensional signals, image convolution can be viewed from either the input side or the output side. As you recall from Chapter 6, the input viewpoint is the best description of how convolution works, while the output viewpoint is how most of the mathematics and algorithms are written. You need to become comfortable with both these ways of looking at the operation. Figure 24-14 shows the input side description of image convolution. Every pixel in the input image results in a scaled and shifted PSF being added to the output image. The output image is then calculated as the sum of all the contributing PSFs. This illustration show the contribution to the output image from the point at location [r,c] in the input image. The PSF is shifted such that pixel [0,0] in the PSF aligns with pixel [r,c] in the output image. If the PSF is defined with only positive indexes, such as in this example, the shifted PSF will be entirely to the lower-right of [r,c]. Don't Chapter 24- Linear Image Processing Spatial Domain 419 Frequency Domain a. Kernel, h[r,c] H[r,c] DFT Im Re × b. Image, x[r,c] X[r,c] DFT Re Im = c. Correlation, y[r,c] Y[r,c] IDFT Re Im FIGURE 24-13 Flow diagram of FFT image convolution. The images in (a) and (b) are transformed into the frequency domain by using the FFT. These two frequency spectra are multiplied, and the Inverse FFT is used to move back into the spatial domain. In this example, the original images have been chosen and preprocessed to implement correlation through the action of convolution. 420 The Scientist and Engineer's Guide to Digital Signal Processing Output image Input image column c N-1 0 0 r r N-1 column 0 row row 0 c N-1 N-1 FIGURE 24-14 Image convolution viewed from the input side. Each pixel in the input image contributes a scaled and shifted PSF to the output image. The output image is the sum of these contributions. The face is inverted in this illustration because this is the PSF we are using. be confused by the face appearing upside down in this figure; this upside down face is the PSF we are using in this example (Fig. 24-13a). In the input side view, there is no rotation of the PSF, it is simply shifted. Image convolution viewed from the output is illustrated in Fig. 24-15. Each pixel in the output image, such as shown by the sample at [r,c], receives a contribution from many pixels in the input image. The PSF is rotated by 180E around pixel [0,0], and then shifted such that pixel [0,0] in the PSF is aligned with pixel [r,c] in the input image. If the PSF only uses positive indexes, it will be to the upper-left of pixel [r,c] in the input image. The value of the pixel at [r,c] in the output image is found by multiplying the pixels in the rotated PSF with the corresponding pixels in the input image, and summing the products. This procedure is given by Eq. 24-3, and in the program of Table 24-1. EQUATION 24-3 Image convolution. The images x[ , ] and h[ , ] are convolved to produce image, y[ , ] . The size of h[ , ] is M×M pixels, with the indexes running from 0 to M& 1 . In this equation, an individual pixel in the output image, y[r,c] , is calculated according to the output side view. The indexes j and k are used to loop through the rows and columns of h[ , ] to calculate the sum-of-products. y[r,c ] ' j j h[k, j ] x [r&k, c &j ] M &1 M&1 k'0 j'0 Notice that the PSF rotation resulting from the convolution has undone the rotation made in the design of the PSF. This makes the face appear upright in Fig. 24-15, allowing it to be in the same orientation as the pattern being detected in the input image. That is, we have successfully used convolution to implement correlation. Compare Fig. 24-13c with Fig. 24-15 to see how the bright spot in the correlation image signifies that the target has been detected. Chapter 24- Linear Image Processing Output image Input image column c N-1 0 0 0 r r N-1 row row 0 421 column c N-1 N-1 FIGURE 24-15 Image convolution viewed from the output side. Each pixel in the output signal is equal to the sum of the pixels in the rotated PSF multiplied by the corresponding pixels in the input image. FFT convolution provides the same output image as the conventional convolution program of Table 24-1. Is the reduced execution time provided by FFT convolution really worth the additional program complexity? Let's take a closer look. Figure 24-16 shows an execution time comparison between conventional convolution using floating point (labeled FP), conventional convolution using integers (labeled INT), and FFT convolution using floating point (labeled FFT). Data for two different image sizes are presented, 512×512 and 128×128. First, notice that the execution time required for FFT convolution does not depend on the size of the kernel, resulting in flat lines in this graph. On a 100 MHz Pentium personal computer, a 128×128 image can be convolved 100 CONVENTIONAL IMAGE CONVOLUTION 110 ' 120 DIM X[99,99] 'holds the input image, 100×100 pixels 130 DIM H[28,28] 'holds the filter kernel, 29×29 pixels 140 DIM Y[127,127] 'holds the output image, 128×128 pixels 150 ' 160 FOR R% = 0 TO 127 'loop through each row and column in the output 170 FOR C% = 0 TO 127 'image calculating the pixel value via Eq. 24-3 180 ' 190 Y[R%,C%] = 0 'zero the pixel so it can be used as an accumulator 200 ' 210 FOR J% = 0 TO 28 'multiply each pixel in the kernel by the corresponding 220 FOR K% = 0 TO 28 'pixel in the input image, and add to the accumulator 230 Y[R%,C%] = Y[R%,C%] + H[J%,K%] * X[R%-J%,C%-J%] 240 NEXT K% 250 NEXT J% 260 ' 270 NEXT C% 280 NEXT R% 290 ' 300 END TABLE 24-1 The Scientist and Engineer's Guide to Digital Signal Processing 5 FP INT FFT 4 FIGURE 24-16 Execution time for image convolution. This graph shows the execution time on a 100 MHz Pentium processor for three image convolution methods: conventional convolution carried out with floating point math (FP), conventional convolution using integers (INT), and FFT convolution using floating point (FFT). The two sets of curves are for input image sizes of 512×512 and 128×128 pixels. Using FFT convolution, the time depends only on the image size, and not the size of the kernel. In contrast, conventional convolution depends on both the image and the kernel size. Execution time (minutes) 422 512 × 512 3 2 1 128 × 128 FP INT FFT 0 0 10 20 30 40 Kernel width (pixels) 50 in about 15 seconds using FFT convolution, while a 512×512 image requires more than 4 minutes. Adding up the number of calculations shows that the execution time for FFT convolution is proportional to N 2 Log2(N) , for an N×N image. That is, a 512×512 image requires about 20 times as long as a 128×128 image. Conventional convolution has an execution time proportional to N 2M 2 for an N×N image convolved with an M×M kernel. This can be understood by examining the program in Table 24-1. In other words, the execution time for conventional convolution depends very strongly on the size of the kernel used. As shown in the graph, FFT convolution is faster than conventional convolution using floating point if the kernel is larger than about 10×10 pixels. In most cases, integers can be used for conventional convolution, increasing the breakeven point to about 30×30 pixels. These break-even points depend slightly on the size of the image being convolved, as shown in the graph. The concept to remember is that FFT convolution is only useful for large filter kernels. CHAPTER 25 Special Imaging Techniques This chapter presents four specific aspects of image processing. First, ways to characterize the spatial resolution are discussed. This describes the minimum size an object must be to be seen in an image. Second, the signal-to-noise ratio is examined, explaining how faint an object can be and still be detected. Third, morphological techniques are introduced. These are nonlinear operations used to manipulate binary images (where each pixel is either black or white). Fourth, the remarkable technique of computed tomography is described. This has revolutionized medical diagnosis by providing detailed images of the interior of the human body. Spatial Resolution Suppose we want to compare two imaging systems, with the goal of determining which has the best spatial resolution. In other words, we want to know which system can detect the smallest object. To simplify things, we would like the answer to be a single number for each system. This allows a direct comparison upon which to base design decisions. Unfortunately, a single parameter is not always sufficient to characterize all the subtle aspects of imaging. This is complicated by the fact that spatial resolution is limited by two distinct but interrelated effects: sample spacing and sampling aperture size. This section contains two main topics: (1) how a single parameter can best be used to characterize spatial resolution, and (2) the relationship between sample spacing and sampling aperture size. Figure 25-1a shows profiles from three circularly symmetric PSFs: the pillbox, the Gaussian, and the exponential. These are representative of the PSFs commonly found in imaging systems. As described in the last chapter, the pillbox can result from an improperly focused lens system. Likewise, the Gaussian is formed when random errors are combined, such as viewing stars through a turbulent atmosphere. An exponential PSF is generated when electrons or x-rays strike a phosphor layer and are converted into 423 424 The Scientist and Engineer's Guide to Digital Signal Processing 1.25 1.25 a. PSF b. MTF 1.00 1.00 Amplitude Amplitude P 0.75 0.50 0.75 0.50 0.25 0.25 G E P E G 0.00 0.00 -2 -1.5 -1 -0.5 0 Distance 0.5 1 1.5 2 0 0.5 1 1.5 2 Spatial frequency (lp per unit distance) FIGURE 25-1 FWHM versus MTF. Figure (a) shows profiles of three PSFs commonly found in imaging systems: (P) pillbox, (G) Gaussian, and (E) exponential. Each of these has a FWHM of one unit. The corresponding MTFs are shown in (b). Unfortunately, similar values of FWHM do not correspond to similar MTF curves. light. This is used in radiation detectors, night vision light amplifiers, and CRT displays. The exact shape of these three PSFs is not important for this discussion, only that they broadly represent the PSFs seen in real world applications. The PSF contains complete information about the spatial resolution. To express the spatial resolution by a single number, we can ignore the shape of the PSF and simply measure its width. The most common way to specify this is by the Full-Width-at-Half-Maximum (FWHM) value. For example, all the PSFs in (a) have an FWHM of 1 unit. Unfortunately, this method has two significant drawbacks. First, it does not match other measures of spatial resolution, including the subjective judgement of observers viewing the images. Second, it is usually very difficult to directly measure the PSF. Imagine feeding an impulse into an imaging system; that is, taking an image of a very small white dot on a black background. By definition, the acquired image will be the PSF of the system. The problem is, the measured PSF will only contain a few pixels, and its contrast will be low. Unless you are very careful, random noise will swamp the measurement. For instance, imagine that the impulse image is a 512×512 array of all zeros except for a single pixel having a value of 255. Now compare this to a normal image where all of the 512×512 pixels have an average value of about 128. In loose terms, the signal in the impulse image is about 100,000 times weaker than a normal image. No wonder the signal-to-noise ratio will be bad; there's hardly any signal! A basic theme throughout this book is that signals should be understood in the domain where the information is encoded. For instance, audio signals should be dealt with in the frequency domain, while image signals should be handled in the spatial domain. In spite of this, one way to measure image resolution is by looking at the frequency response. This goes against the fundamental Chapter 25- Special Imaging Techniques 425 philosophy of this book; however, it is a common method and you need to become familiar with it. Taking the two-dimensional Fourier transform of the PSF provides the twodimensional frequency response. If the PSF is circularly symmetric, its frequency response will also be circularly symmetric. In this case, complete information about the frequency response is contained in its profile. That is, after calculating the frequency domain via the FFT method, columns 0 to N/2 in row 0 are all that is needed. In imaging jargon, this display of the frequency response is called the Modulation Transfer Function (MTF). Figure 25-1b shows the MTFs for the three PSFs in (a). In cases where the PSF is not circularly symmetric, the entire two-dimensional frequency response contains information. However, it is usually sufficient to know the MTF curves in the vertical and horizontal directions (i.e., columns 0 to N/2 in row 0, and rows 0 to N/2 in column 0). Take note: this procedure of extracting a row or column from the two-dimensional frequency spectrum is not equivalent to taking the one-dimensional FFT of the profiles shown in (a). We will come back to this issue shortly. As shown in Fig. 25-1, similar values of FWHM do not correspond to similar MTF curves. Figure 25-2 shows a line pair gauge, a device used to measure image resolution via the MTF. Line pair gauges come in different forms depending on the particular application. For example, the black and white pattern shown in this figure could be directly used to test video cameras. For an x-ray imaging system, the ribs might be made from lead, with an x-ray transparent material between. The key feature is that the black and white lines have a closer spacing toward one end. When an image is taken of a line pair gauge, the lines at the closely spaced end will be blurred together, while at the other end they will be distinct. Somewhere in the middle the lines will be just barely separable. An observer looks at the image, identifies this location, and reads the corresponding resolution on the calibrated scale. a. Example profile at 12 lp/mm 10 200 150 100 50 7 0 0 60 120 180 240 Pixel number 5 b. Example profile at 3 lp/mm 4 3 250 Pixel value line pairs / mm FIGURE 25-2 Line pair gauge. The line pair gauge is a tool used to measure the resolution of imaging systems. A series of black and white ribs move together, creating a continuum of spatial frequencies. The resolution of a system is taken as the frequency where the eye can no longer distinguish the individual ribs. This example line pair gauge is shown several times larger than the calibrated scale indicates. 250 Pixel value 40 30 20 15 200 150 100 50 0 0 2 60 120 180 Pixel number 240 426 The Scientist and Engineer's Guide to Digital Signal Processing The way that the ribs blur together is important in understanding the limitations of this measurement. Imagine acquiring an image of the line pair gauge in Fig. 25-2. Figures (a) and (b) show examples of the profiles at low and high spatial frequencies. At the low frequency, shown in (b), the curve is flat on the top and bottom, but the edges are blurred, At the higher spatial frequency, (a), the amplitude of the modulation has been reduced. This is exactly what the MTF curve in Fig. 25-1b describes: higher spatial frequencies are reduced in amplitude. The individual ribs will be distinguishable in the image as long as the amplitude is greater than about 3% to 10% of the original