Digital Signal Processing Solutions Manual

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Chapter 1

1.1
(a) One dimensional, multichannel, discrete time, and digital.
(b) Multi dimensional, single channel, continuous-time, analog.
(c) One dimensional, single channel, continuous-time, analog.
(d) One dimensional, single channel, continuous-time, analog.
(e) One dimensional, multichannel, discrete-time, digital.

1.2
1
(a) f = 0.01π
2π = 200 ⇒ periodic with Np = 200.
30π 1
(b) f = 105 ( 2π ) = 17 ⇒ periodic with Np = 7.
3π
(c) f = 2π
= 32 ⇒ periodic with Np = 2.
3
(d) f = 2π ⇒ non-periodic.
1
31
(e) f = 62π
10 ( 2π ) = 10 ⇒ periodic with Np = 10.

1.3
(a) Periodic with period Tp = 2π
5 .
5
⇒ non-periodic.
(b) f = 2π
1
(c) f = 12π
⇒ non-periodic.
n
(d) cos( 8 ) is non-periodic; cos( πn
8 ) is periodic; Their product is non-periodic.
(e) cos( πn
)
is
periodic
with
period
Np =4
2
sin( πn
)
is
periodic
with
period
N
p =16
8
π
cos( πn
+
)
is
periodic
with
period
Np =8
4
3
Therefore, x(n) is periodic with period Np =16. (16 is the least common multiple of 4,8,16).

1.4
(a) w =

2πk
N

implies that f =

k
N.

Let
α = GCD of (k, N ), i.e.,
k = k ′ α, N = N ′ α.

Then,
f=

k′
, which implies that
N′
N
N′ = .
α
3

© 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws
as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

(b)
N
k
GCD(k, N )
Np

= 7
= 01234567
= 71111117
=

17777771

(c)
N
k
GCD(k, N )
Np

=

16

= 0 1 2 3 4 5 6 7 8 9 10 11 12 . . . 16
= 16 1 2 1 4 1 2 1 8 1 2 1 4 . . . 16
= 1 6 8 16 4 16 8 16 2 16 8 16 4 . . . 1

1.5
(a) Refer to fig 1.5-1
(b)
3

2

−−−> xa(t)

1

0

−1

−2

−3
0

5

10

15
−−−> t (ms)

20

25

30

Figure 1.5-1:

x(n)

= xa (nT )
= xa (n/Fs )
=

f

=
=

3sin(πn/3) ⇒
1 π
( )
2π 3
1
, Np = 6
6
4

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

3

10

0

t (ms)
20

-3

Figure 1.5-2:
(c)Refer nto fig 1.5-2

o
x(n) = 0, √32 , √32 , 0, − √32 , − √32 , Np = 6.
(d) Yes.
100π
x(1) = 3 = 3sin(
) ⇒ Fs = 200 samples/sec.
Fs

1.6
(a)
x(n)

= Acos(2πF0 n/Fs + θ)
= Acos(2π(T /Tp )n + θ)

But T /Tp = f ⇒ x(n) is periodic if f is rational.
(b) If x(n) is periodic, then f=k/N where N is the period. Then,
Tp
k
Td = ( T ) = k( )T = kTp .
f
T
Thus, it takes k periods (kTp ) of the analog signal to make 1 period (Td ) of the discrete signal.
(c) Td = kTp ⇒ N T = kTp ⇒ f = k/N = T /Tp ⇒ f is rational ⇒ x(n) is periodic.

1.7
(a) Fmax = 10kHz ⇒ Fs ≥ 2Fmax = 20kHz.
(b) For Fs = 8kHz, Ffold = Fs /2 = 4kHz ⇒ 5kHz will alias to 3kHz.
(c) F=9kHz will alias to 1kHz.

1.8
(a) Fmax = 100kHz, Fs ≥ 2Fmax = 200Hz.
(b) Ffold = F2s = 125Hz.
5

© 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws
as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

1.9
(a) Fmax = 360Hz, FN = 2Fmax = 720Hz.
(b) Ffold = F2s = 300Hz.
(c)
x(n)

= xa (nT )
= xa (n/Fs )
= sin(480πn/600) + 3sin(720πn/600)

x(n)

= sin(4πn/5) − 3sin(4πn/5)
= −2sin(4πn/5).

Therefore, w = 4π/5.
(d) ya (t) = x(Fs t) = −2sin(480πt).

1.10
(a)
Number of bits/sample
Fs

Ffold

= log2 1024 = 10.
[10, 000 bits/sec]
=
[10 bits/sample]
= 1000 samples/sec.
=

500Hz.

(b)
Fmax

=

FN

=
=

1800π
2π
900Hz
2Fmax = 1800Hz.

(c)
f1

=
=

(d) △ =

xmax −x

f2

=

But f2

=
=

Hence, x(n)

=

min =

m−1

5−(−5)
1023

=

600π 1
( )
2π Fs
0.3;
1800π 1
( )
2π Fs
0.9;
0.9 > 0.5 ⇒ f2 = 0.1.

3cos[(2π)(0.3)n] + 2cos[(2π)(0.1)n]

10
1023 .

1.11
x(n)

= xa (nT )




250πn
100πn
+ 2sin
= 3cos
200
200
6

© 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws
as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

=

T′

ya (t)

3cos

 πn 

− 2sin



3πn
4



2
1
=
⇒ ya (t) = x(t/T ′ )
1000



3π1000t
π1000t
− 2sin
= 3cos
2
4
= 3cos(500πt) − 2sin(750πt)

1.12
(a) For Fs = 300Hz,
x(n)

=
=

3cos

 πn 

+ 10sin(πn) − cos
6 
 πn
 πn 
− 3cos
3cos
6
3

 πn 
3

(b) xr (t) = 3cos(10000πt/6) − cos(10000πt/3)

1.13
(a)
Range

xmax − xmin = 12.7.
range
m = 1+
△
= 127 + 1 = 128 ⇒ log2 (128)
= 7 bits.

(b) m = 1 +

127
0.02

=

= 636 ⇒ log2 (636) ⇒ 10 bit A/D.

1.14
R

Ffold
Resolution

samples
bits
) × (8
)
sec
sample
bits
= 160
sec
Fs
= 10Hz.
=
2
1volt
=
28 − 1
= 0.004.

=

(20

1.15
(a) Refer to fig 1.15-1. With a sampling frequency of 5kHz, the maximum frequency that can be
represented is 2.5kHz. Therefore, a frequency of 4.5kHz is aliased to 500Hz and the frequency of
3kHz is aliased to 2kHz.

7

© 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws
as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Fs = 5KHz, F0=500Hz

Fs = 5KHz, F0=2000Hz

1

1

0.5

0.5

0

0

−0.5

−0.5

−1
0

50

−1
0

100

Fs = 5KHz, F0=3000Hz
1

0.5

0.5

0

0

−0.5

−0.5

50

100

Fs = 5KHz, F0=4500Hz

1

−1
0

50

−1
0

100

50

100

Figure 1.15-1:
(b) Refer to fig 1.15-2. y(n) is a sinusoidal signal. By taking the even numbered samples, the
sampling frequency is reduced to half i.e., 25kHz which is still greater than the nyquist rate. The
frequency of the downsampled signal is 2kHz.

1.16
(a) for levels = 64, using truncation refer to fig 1.16-1.
for levels = 128, using truncation refer to fig 1.16-2.
for levels = 256, using truncation refer to fig 1.16-3.

8

© 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws
as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

F0 = 2KHz, Fs=50kHz
1
0.5
0
−0.5
−1
0

10

20

30

40

50

60

70

80

90

100

35

40

45

50

F0 = 2KHz, Fs=25kHz
1
0.5
0
−0.5
−1
0

5

10

15

20

25

30

Figure 1.15-2:

levels = 64, using truncation, SQNR = 31.3341dB
1

0.5
−−> xq(n)

−−> x(n)

0.5
0
−0.5
−1
0

1

0
−0.5

50

100
−−> n

150

200

50

100
−−> n

150

200

−1
0

50

100
−−> n

150

200

0

−−> e(n)

−0.01
−0.02
−0.03
−0.04
0

Figure 1.16-1:

9

© 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws
as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

levels = 128, using truncation, SQNR = 37.359dB
1

0.5
−−> xq(n)

−−> x(n)

0.5

1

0
−0.5

0
−0.5

−1
0

50

100
−−> n

150

200

50

100
−−> n

150

200

−1
0

50

100
−−> n

150

200

0

−−> e(n)

−0.005
−0.01
−0.015
−0.02
0

Figure 1.16-2:
levels = 256, using truncation, SQNR=43.7739dB
1

0.5
−−> xq(n)

−−> x(n)

0.5

1

0
−0.5

0
−0.5

−1
0

50

100
−−> n

150

200

50

100
−−> n

150

200

−1
0

50

100
−−> n

150

200

−3

0

x 10

−−> e(n)

−2
−4
−6
−8
0

Figure 1.16-3:
10

© 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws
as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

(b) for levels = 64, using rounding refer to fig 1.16-4.
for levels = 128, using rounding refer to fig 1.16-5.
for levels = 256, using rounding refer to fig 1.16-6.

levels = 64, using rounding, SQNR=32.754dB
1

1
0.5
−−> xq(n)

−−> x(n)

0.5
0
−0.5
−1
0

0
−0.5

50

100
−−> n

150

200

50

100
−−> n

150

200

−1
0

50

100
−−> n

150

200

0.04

−−> e(n)

0.02
0
−0.02
−0.04
0

Figure 1.16-4:

11

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

levels = 128, using rounding, SQNR=39.2008dB
1

1
0.5

−−> xq(n)

−−> x(n)

0.5
0
−0.5
−1
0

0
−0.5

50

100
−−> n

150

200

50

100
−−> n

150

200

−1
0

50

100
−−> n

150

200

50

100
−−> n

150

200

0.02

−−> e(n)

0.01
0
−0.01
−0.02
0

Figure 1.16-5:
levels = 256, using rounding, SQNR=44.0353dB
1

0.5
−−> xq(n)

−−> x(n)

0.5
0
−0.5
−1
0

1

0
−0.5

50

100
−−> n

150

200

50

100
−−> n

150

200

−1
0

0.01

−−> e(n)

0.005
0
−0.005
−0.01
0

Figure 1.16-6:
12

© 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws
as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

(c) The sqnr with rounding is greater than with truncation. But the sqnr improves as the number
of quantization levels are increased.
(d)
levels
64
128
256
theoretical sqnr
43.9000 49.9200 55.9400
sqnr with truncation 31.3341 37.359
43.7739
sqnr with rounding
32.754
39.2008 44.0353
The theoretical sqnr is given in the table above. It can be seen that theoretical sqnr is much
higher than those obtained by simulations. The decrease in the sqnr is because of the truncation
and rounding.

13

© 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws
as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

14

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Chapter 2

2.1
(a)

.



1 2
x(n) = . . . 0, , , 1, 1, 1, 1, 0, . . .
3 3 ↑

Refer to fig 2.1-1.
(b) After folding s(n) we have
1

1

1

1

0

1

2

3

2/3
1/3

-3

-2

-1

4

Figure 2.1-1:

x(−n) =




2 1
. . . 0, 1, 1, 1, 1, , , 0, . . . .
↑ 3 3

After delaying the folded signal by 4 samples, we have


2 1
x(−n + 4) = . . . 0, 0, 1, 1, 1, 1, , , 0, . . . .
3 3
↑
On the other hand, if we delay x(n) by 4 samples we have


1 2
x(n − 4) = . . . 0, 0, , , 1, 1, 1, 1, 0, . . . .
3 3
↑
Now, if we fold x(n − 4) we have
x(−n − 4) =



2 1
. . . 0, 1, 1, 1, 1, , , 0, 0, . . .
3 3
↑
15

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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

(c)
x(−n + 4) =



2 1
. . . 0, 1, 1, 1, 1, , , 0, . . .
3 3
↑



(d) To obtain x(−n + k), first we fold x(n). This yields x(−n). Then, we shift x(−n) by k
samples to the right if k > 0, or k samples to the left if k < 0.
(e) Yes.
2
1
x(n) = δ(n − 2) + δ(n + 1) + u(n) − u(n − 4)
3
3

2.2


1 1
x(n) = . . . 0, 1, 1, 1, 1, , , 0, . . .
2 2
↑
(a)

(b)



1 1
x(n − 2) = . . . 0, 0, 1, 1, 1, 1, , , 0, . . .
2 2
↑




1 1
x(4 − n) = . . . 0, , , 1, 1, 1, 1, 0, . . .


2 2
↑

(see 2.1(d))
(c)



1 1
x(n + 2) = . . . 0, 1, 1, 1, 1, , , 0, . . .
↑ 2 2

(d)
x(n)u(2 − n) =



. . . 0, 1, 1, 1, 1, 0, 0, . . .
↑

(e)
x(n − 1)δ(n − 3) =



. . . 0, 0, 1, 0, . . .
↑

(f)
x(n2 ) = {. . . 0, x(4), x(1), x(0), x(1), x(4), 0, . . .}


1
1
=
. . . 0, , 1, 1, 1, , 0, . . .
2
2
↑
(g)
xe (n)
x(−n)

x(n) + x(−n)
,
2


1 1
=
. . . 0, , , 1, 1, 1, 1, 0, . . .
2 2
↑


1 1 1
1 1 1
=
. . . 0, , , , 1, 1, 1, , , , 0, . . .
4 4 2
2 4 4
=

16

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

(h)
xo (n)

x(n) − x(−n)
2


1 1 1
1 1 1
=
. . . 0, − , − , − , 0, 0, 0, , , , 0, . . .
4 4 2
2 4 4
=

2.3
(a)

(b)


 0,
1,
u(n) − u(n − 1) = δ(n) =

0,
n
X

δ(k) = u(n) =

k=−∞
∞
X

k=0

δ(n − k) =




0,
1,

n<0
n=0
n>0

0, n < 0
1, n ≥ 0
n<0
n≥0

2.4
Let
xe (n) =

1
[x(n) + x(−n)],
2

xo (n) =

1
[x(n) − x(−n)].
2

Since
xe (−n) = xe (n)
and
xo (−n) = −xo (n),
it follows that
x(n) = xe (n) + xo (n).
The decomposition is unique. For


x(n) = 2, 3, 4, 5, 6 ,
↑

we have



xe (n) = 4, 4, 4, 4, 4
↑

and



xo (n) = −2, −1, 0, 1, 2 .
↑

17

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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2.5
First, we prove that
∞
X

xe (n)xo (n) = 0

n=−∞

∞
X

xe (n)xo (n)

∞
X

=

xe (−m)xo (−m)

m=−∞
∞
X

n=−∞

= −

xe (m)xo (m)

= −

xe (n)xo (n)

m=−∞
∞
X

n=−∞
∞
X

xe (n)xo (n)

=

n=−∞

=

0

Then,
∞
X

2

x (n)

=

n=−∞

=

∞
X

n=−∞
∞
X

[xe (n) + xo (n)]
x2e (n) +

∞
X

2

x2o (n) +

= Ee + Eo

2xe (n)xo (n)

n=−∞

n=−∞

n=−∞

∞
X

2.6
(a) No, the system is time variant. Proof: If
= x(n2 )

x(n) → y(n)

2

x(n − k) → y1 (n)

(b) (1)
x(n) =

↑

y(n) = x(n2 ) =

y(n − 2) =

= x(n2 + k 2 − 2nk)
6
=
y(n − k)



0, 1, 1, 1, 1, 0, . . .

(2)

(3)

= x[(n − k) ]



. . . , 0, 1, 1, 1, 0, . . .
↑



. . . , 0, 0, 1, 1, 1, 0, . . .
↑

18

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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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(4)



x(n − 2) = . . . , 0, 0, 1, 1, 1, 1, 0, . . .
↑

(5)
y2 (n) = T [x(n − 2)] =



. . . , 0, 1, 0, 0, 0, 1, 0, . . .
↑

(6)
y2 (n) 6= y(n − 2) ⇒ system is time variant.
(c) (1)



x(n) = 1, 1, 1, 1
↑

(2)



y(n) = 1, 0, 0, 0, 0, −1
↑

(3)
y(n − 2) =



0, 0, 1, 0, 0, 0, 0, −1
↑

(4)
x(n − 2) =
(5)
y2 (n) =




0, 0, 1, 1, 1, 1, 1
↑



0, 0, 1, 0, 0, 0, 0, −1
↑

(6)
y2 (n) = y(n − 2).
The system is time invariant, but this example alone does not constitute a proof.
(d) (1)
y(n) = nx(n),


x(n) = . . . , 0, 1, 1, 1, 1, 0, . . .
↑

(2)
y(n) =
(3)



. . . , 0, 1, 2, 3, . . .
↑





y(n − 2) = . . . , 0, 0, 0, 1, 2, 3, . . .
↑

(4)
x(n − 2) =



. . . , 0, 0, 0, 1, 1, 1, 1, . . .
↑

19

© 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws
as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

(5)
y2 (n) = T [x(n − 2)] = {. . . , 0, 0, 2, 3, 4, 5, . . .}
(6)
y2 (n) 6= y(n − 2) ⇒ the system is time variant.

2.7
(a) Static, nonlinear, time invariant, causal, stable.
(b) Dynamic, linear, time invariant, noncausal and unstable. The latter is easily proved.
For the bounded input x(k) = u(k), the output becomes
y(n) =

n+1
X

u(k) =

k=−∞



0,
n + 2,

n < −1
n ≥ −1

since y(n) → ∞ as n → ∞, the system is unstable.
(c) Static, linear, timevariant, causal, stable.
(d) Dynamic, linear, time invariant, noncausal, stable.
(e) Static, nonlinear, time invariant, causal, stable.
(f) Static, nonlinear, time invariant, causal, stable.
(g) Static, nonlinear, time invariant, causal, stable.
(h) Static, linear, time invariant, causal, stable.
(i) Dynamic, linear, time variant, noncausal, unstable. Note that the bounded input
x(n) = u(n) produces an unbounded output.
(j) Dynamic, linear, time variant, noncausal, stable.
(k) Static, nonlinear, time invariant, causal, stable.
(l) Dynamic, linear, time invariant, noncausal, stable.
(m) Static, nonlinear, time invariant, causal, stable.
(n) Static, linear, time invariant, causal, stable.

2.8
(a) True. If
v1 (n) = T1 [x1 (n)] and
v2 (n) = T1 [x2 (n)],

then

α1 x1 (n) + α2 x2 (n)
yields
α1 v1 (n) + α2 v2 (n)
by the linearity property of T1 . Similarly, if
y1 (n) = T2 [v1 (n)] and
y2 (n) = T2 [v2 (n)],

then

β1 v1 (n) + β2 v2 (n) → y(n) = β1 y1 (n) + β2 y2 (n)

by the linearity property of T2 . Since

v1 (n) = T1 [x1 (n)] and
20

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

v2 (n) = T2 [x2 (n)],
it follows that
A1 x1 (n) + A2 x2 (n)
yields the output
A1 T [x1 (n)] + A2 T [x2 (n)],
where T = T1 T2 . Hence T is linear.
(b) True. For T1 , if

x(n) → v(n) and
x(n − k) → v(n − k),

For T2 , if

v(n) → y(n)
andv(n − k) → y(n − k).

Hence, For T1 T2 , if

x(n) → y(n) and
x(n − k) → y(n − k)

Therefore, T = T1 T2 is time invariant.
(c) True. T1 is causal ⇒ v(n) depends only on x(k) for k ≤ n. T2 is causal ⇒ y(n) depends only on v(k) for k ≤
n. Therefore, y(n) depends only on x(k) for k ≤ n. Hence, T is causal.
(d) True. Combine (a) and (b).
(e) True. This follows from h1 (n) ∗ h2 (n) = h2 (n) ∗ h1 (n)
(f) False. For example, consider
T1 : y(n) = nx(n) and
T2 : y(n) = nx(n + 1).
Then,
T2 [T1 [δ(n)]]
T1 [T2 [δ(n)]]

= T2 (0) = 0.
= T1 [δ(n + 1)]

= −δ(n + 1)
6= 0.

(g) False. For example, consider
T1 : y(n) = x(n) + b and
T2 : y(n) = x(n) − b, where b 6= 0.
Then,
T [x(n)] = T2 [T1 [x(n)]] = T2 [x(n) + b] = x(n).
Hence T is linear.
(h) True.
T1 is stable ⇒ v(n) is bounded if x(n) is bounded.
T2 is stable ⇒ y(n) is bounded if v(n) is bounded .
21

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Hence, y(n) is bounded if x(n) is bounded ⇒ T = T1 T2 is stable.
(i) Inverse of (c). T1 and for T2 are noncausal ⇒ T is noncausal. Example:
T1 : y(n)
T2 : y(n)

= x(n + 1) and
= x(n − 2)

⇒ T : y(n)

= x(n − 1),

which is causal. Hence, the inverse of (c) is false.
Inverse of (h): T1 and/or T2 is unstable, implies T is unstable. Example:
T1 : y(n) = ex(n) , stable and T2 : y(n) = ln[x(n)], which is unstable.
But T : y(n) = x(n), which is stable. Hence, the inverse of (h) is false.

2.9
(a)
y(n)

=

n
X

k=−∞

y(n + N )

=

n+N
X

k=−∞

=

n
X

k=−∞

h(k)x(n − k), x(n) = 0, n < 0
h(k)x(n + N − k) =
h(k)x(n − k) +

= y(n) +

n+N
X

k=n+1

n+N
X

n+N
X

k=−∞

k=n+1

h(k)x(n − k)

h(k)x(n − k)

h(k)x(n − k)

For a BIBO system, limn→∞ |h(n)| = 0. Therefore,
n+N
X

limn→∞

k=n+1

h(k)x(n − k) = 0 and

limn→∞ y(n + N ) = y(N ).
(b) Let x(n) = xo (n) + au(n), where a is a constant and
xo (n) is a bounded signal with lim xo (n) = 0.
n→∞

Then,
y(n)

= a

∞
X

k=0
n
X

= a

h(k)u(n − k) +

∞
X

k=0

h(k)xo (n − k)

h(k) + yo (n)

k=0

clearly,

P

n

x2o (n) < ∞ ⇒

P

n

yo2 (n) < ∞ (from (c) below) Hence,
limn→∞ |yo (n)| = 0.
22

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

and, thus, limn→∞ y(n) = a
(c)

Pn

k=0

y(n)

h(k) = constant.

=

X
k

∞
X

2

y (n)

=

h(k)x(n − k)

"
∞
X
X
−∞

−∞

=

k

XX
k

But

X
n

Therefore,

h(k)h(l)

X

y 2 (n) ≤ Ex
X

X
n

l

k

Hence,

h(k)x(n − k)

x(n − k)x(n − l) ≤

n

For a BIBO stable system,

#2

X
k

X

x(n − k)x(n − l)

x2 (n) = Ex .

n

|h(k)|

X
l

|h(l)|.

|h(k)| < M.

Ey ≤ M 2 Ex , so that
Ey < 0 if Ex < 0.

2.10
The system is nonlinear. This is evident from observation of the pairs
x3 (n) ↔ y3 (n) and x2 (n) ↔ y2 (n).
If the system were linear, y2 (n) would be of the form
y2 (n) = {3, 6, 3}
because the system is time-invariant. However, this is not the case.

2.11
since
x1 (n) + x2 (n) = δ(n)
and the system is linear, the impulse response of the system is


y1 (n) + y2 (n) = 0, 3, −1, 2, 1 .
↑

If the system were time invariant, the response to x3 (n) would be


3, 2, 1, 3, 1 .
↑

But this is not the case.
23

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

2.12
(a) Any weighted linear combination of the signals xi (n), i = 1, 2, . . . , N .
(b) Any xi (n − k), where k is any integer and i = 1, 2, . . . , N .

2.13
A system is BIBO stable if and only if a bounded input produces a bounded output.
X
y(n) =
h(k)x(n − k)
k

|y(n)|

≤

X
k

≤ Mx

|h(k)||x(n − k)|
X
k

|h(k)|

where |x(n − k)| ≤ Mx . Therefore, |y(n)| < ∞ for all n, if and only if
X
|h(k)| < ∞.
k

2.14
(a) A system is causal ⇔ the output becomes nonzero after the input becomes non-zero. Hence,
x(n) = 0 for n < no ⇒ y(n) = 0 for n < no .
(b)

n
X

y(n) =

−∞

If h(k) = 0 for k < 0, then
y(n) =

n
X
0

h(k)x(n − k), where x(n) = 0 for n < 0.

h(k)x(n − k), and hence, y(n) = 0 for n < 0.

On the other hand, if y(n) = 0 for n < 0, then
n
X
−∞

h(k)x(n − k) ⇒ h(k) = 0, k < 0.

2.15
(a)
For a = 1,

N
X

an

= N −M +1

an

= aM + aM +1 + . . . + aN

an

= aM + aM +1 − aM +1 + . . . + aN − aN − aN +1

n=M

for a 6= 1,
(1 − a)

N
X

n=M
N
X

n=M

= aM − aN +1
24

© 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws
as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

(b) For M = 0, |a| < 1, and N → ∞,
∞
X

an =

n=0

1
, |a| < 1.
1−a

2.16
(a)
y(n) =

X
k

X

y(n) =

n

h(k)x(n − k)

XX

h(k)x(n − k) =

X

X

n

=

k

k

!

h(k)

n

X

h(k)

∞
X

n=−∞

k

!

x(n − k)

x(n)

(b) (1)
X
n

y(n) = h(n) ∗ x(n) = {1, 3, 7, 7, 7, 6, 4}
X
X
y(n) = 35,
h(k) = 5,
x(k) = 7
k

k

(2)
X
n

(3)

y(n) = {1, 4, 2, −4, 1}
X
X
y(n) = 4,
h(k) = 2,
x(k) = 2
k

k



5
1 1 3
y(n) = 0, , − , , −2, 0, − , −2
2 2 2
2
X
X
X
y(n) = −5,
h(n) = 2.5,
x(n) = −2
n

n

n

(4)
X
n

y(n) = {1, 2, 3, 4, 5}
X
X
y(n) = 15,
h(n) = 1,
x(n) = 15
n

n

(5)
X

y(n) = {0, 0, 1, −1, 2, 2, 1, 3}
X
X
y(n) = 8,
h(n) = 4,
x(n) = 2

X

y(n) = {0, 0, 1, −1, 2, 2, 1, 3}
X
X
y(n) = 8,
h(n) = 2,
x(n) = 4

n

n

n

(6)

n

n

n

(7)
y(n) = {0, 1, 4, −4, −5, −1, 3}
25

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

X
n

y(n) = −2,

X
n

h(n) = −1,

X

x(n) = 2

n

(8)
X
n

y(n) = u(n) + u(n − 1) + 2u(n − 2)
X
X
y(n) = ∞,
h(n) = ∞,
x(n) = 4
n

n

(9)
y(n) = {1, −1, −5, 2, 3, −5, 1, 4}
X
X
X
y(n) = 0,
h(n) = 0,
x(n) = 4
n

n

n

(10)
X
n

y(n) = {1, 4, 4, 4, 10, 4, 4, 4, 1}
X
X
y(n) = 36,
h(n) = 6,
x(n) = 6
n

n

(11)
1
1
y(n) = [2( )n − ( )n ]u(n)
2
4
X
X
X
8
4
y(n) = ,
h(n) = ,
x(n) = 2
3
3
n
n
n

2.17
(a)
x(n)
h(n)
y(n)



=
1, 1, 1, 1
↑


=
6, 5, 4, 3, 2, 1
=

↑
n
X

k=0

x(k)h(n − k)

y(0)
y(1)

= x(0)h(0) = 6,
= x(0)h(1) + x(1)h(0) = 11

y(2)
y(3)
y(4)

= x(0)h(2) + x(1)h(1) + x(2)h(0) = 15
= x(0)h(3) + x(1)h(2) + x(2)h(1) + x(3)h(0) = 18
= x(0)h(4) + x(1)h(3) + x(2)h(2) + x(3)h(1) + x(4)h(0) = 14

y(5)
y(6)

= x(0)h(5) + x(1)h(4) + x(2)h(3) + x(3)h(2) + x(4)h(1) + x(5)h(0) = 10
= x(1)h(5) + x(2)h(4) + x(3)h(2) = 6

y(7)
y(8)

= x(2)h(5) + x(3)h(4) = 3
= x(3)h(5) = 1

y(n)

=

y(n)

0, n ≥ 9


=
6, 11, 15, 18, 14, 10, 6, 3, 1
↑

26

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

(b) By following the same procedure as in (a), we obtain


y(n) = 6, 11, 15, 18, 14, 10, 6, 3, 1
↑

(c) By following the same procedure as in (a), we obtain


y(n) = 1, 2, 2, 2, 1
↑

(d) By following the same procedure as in (a), we obtain


y(n) = 1, 2, 2, 2, 1
↑

2.18
(a)
x(n)
h(n)




1 2
4 5
=
0, , , 1, , , 2
3 3
↑ 3 3


=
1, 1, 1, 1, 1
↑

y(n) = x(n) ∗ h(n)


20
11
10
1
, 1, 2, , 5, , 6, 5, , 2
=
3 ↑
3
3
3
(b)
x(n)
h(n)
y(n)

y(n)

1
n[u(n) − u(n − 7)],
3
= u(n + 2) − u(n − 3)

=

= x(n) ∗ h(n)
1
=
n[u(n) − u(n − 7)] ∗ [u(n + 2) − u(n − 3)]
3
1
n[u(n) ∗ u(n + 2) − u(n) ∗ u(n − 3) − u(n − 7) ∗ u(n + 2) + u(n − 7) ∗ u(n − 3)]
=
3
1
10
20
=
δ(n + 1) + δ(n) + 2δ(n − 1) + δ(n − 2) + 5δ(n − 3) + δ(n − 4) + 6δ(n − 5)
3
3
3
11
+5δ(n − 6) + 5δ(n − 6) + δ(n − 7) + δ(n − 8)
3

2.19

y(n)

=

4
X

k=0

x(n)
h(n)

h(k)x(n − k),



α−3 , α−2 , α−1 , 1, α, . . . , α5
↑


=
1, 1, 1, 1, 1

=

↑

27

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

4
X

y(n) =

k=0

=

x(n − k), −3 ≤ n ≤ 9

0, otherwise.

Therefore,
y(−3)
y(−2)

= α−3 ,
= x(−3) + x(−2) = α−3 + α−2 ,

y(−1)
y(0)
y(1)

= α−3 + α−2 + α−1 ,
= α−3 + α−2 + α−1 + 1
= α−3 + α−2 + α−1 + 1 + α,

y(2)
y(3)

= α−3 + α−2 + α−1 + 1 + α + α2
= α−1 + 1 + α + α2 + α3 ,

y(4)
y(5)

= α4 + α3 + α2 + α + 1
= α + α2 + α3 + α4 + α5 ,

y(6)
y(7)

= α2 + α3 + α4 + α5
= α3 + α4 + α5 ,

y(8)
y(9)

= α4 + α5 ,
= α5

2.20
(a) 131 x 122 = 15982
(b) {1↑ , 3, 1} ∗ {1↑ , 2, 2} = {1, 5, 9, 8, 2}
(c) (1 + 3z + z 2 )(1 + 2z + 2z 2 ) = 1 + 5z + 9z 2 + 8z 3 + 2z 4
(d) 1.31 x 12.2 = 15.982.
(e) These are different ways to perform convolution.

2.21
(a)
y(n) =

n
X

k=0

ak u(k)bn−k u(n − k) = bn

y(n) =



bn+1 −an+1
u(n),
b−a
n

b (n + 1)u(n),

n
X

(ab−1 )k

k=0

a 6= b
a=b

(b)
x(n)
h(n)
y(n)



1, 2, 1, 1
↑


=
1, −1, 0, 0, 1, 1
↑


=
1, 1, − 1, 0, 0, 3, 3, 2, 1
=

↑

28

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

(c)


=
1, 1, 1, 1, 1, 0, −1 ,
↑


=
1, 2, 3, 2, 1
↑


=
1, 3, 6, 8, 9, 8, 5, 1, −2, −2, −1

x(n)
h(n)
y(n)

↑

(d)
x(n)
h′ (n)
h(n)
y(n)
y ′ (n)



1, 1, 1, 1, 1 ,
↑


=
0, 0, 1, 1, 1, 1, 1, 1

=

↑
′

= h (n) + h′ (n − 9),
= y ′ (n) + y ′ (n − 9), where


=
,
0,
1,
2,
3,
4,
5,
5,
4,
3,
2,
1
0
↑

2.22
(a)
yi (n)
y1 (n)
y2 (n)
y3 (n)
y4 (n)
y5 (n)

= x(n) ∗ hi (n)
= x(n) + x(n − 1)

= {1, 5, 6, 5, 8, 8, 6, 7, 9, 12, 12, 15, 9} , similarly
= {1, 6, 11, 11, 13, 16, 14, 13, 15, 21, 25, 28, 24, 9}

= {0.5, 2.5, 3, 2.5, 4, 4, 3, 3.5, 4.5, 6, 6, 7.5, 4.5}
= {0.25, 1.5, 2.75, 2.75, 3.25, 4, 3.5, 3.25, 3.75, 5.25, 6.25, 7, 6, 2.25}
= {0.25, 0.5, −1.25, 0.75, 0.25, −1, 0.5, 0.25, 0, 0.25, −0.75, 1, −3, −2.25}

(b)
y3 (n)

=

h3 (n)

=

y4 (n)

=

h4 (n)

=

1
y1 (n), because
2
1
h1 (n)
2
1
y2 (n), because
4
1
h2 (n)
4

(c) y2 (n) and y4 (n) are smoother than y1 (n), but y4 (n) will appear even smoother because of the
smaller scale factor.
(d) System 4 results in a smoother output. The negative value of h5 (0) is responsible for the
non-smooth characteristics of y5 (n)
(e)


1 3
1
1 1
1 3 9
y6 (n) =
, , −1, , 1, −1, 0, , , 1, − , , −
2 2
2
2 2
2 2 2
y2 (n) is smoother than y6 (n).
29

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

2.23
We can express the unit sample in terms of the unit step function as δ(n) = u(n) − u(n − 1).
Then,
h(n)

= h(n) ∗ δ(n)
= h(n) ∗ (u(n) − u(n − 1)
= h(n) ∗ u(n) − h(n) ∗ u(n − 1)

= s(n) − s(n − 1)
Using this definition of h(n)
y(n)

= h(n) ∗ x(n)
= (s(n) − s(n − 1)) ∗ x(n)

= s(n) ∗ x(n) − s(n − 1) ∗ x(n)

2.24
If
y1 (n)
y2 (n)
x(n)

= ny1 (n − 1) + x1 (n) and
= ny2 (n − 1) + x2 (n) then
= ax1 (n) + bx2 (n)

produces the output
y(n) = ny(n − 1) + x(n), where
y(n) = ay1 (n) + by2 (n).
Hence, the system is linear. If the input is x(n − 1), we have
y(n − 1)
y(n − 1)

= (n − 1)y(n − 2) + x(n − 1). But
= ny(n − 2) + x(n − 1).

Hence, the system is time variant. If x(n) = u(n), then |x(n)| ≤ 1. But for this bounded input,
the output is
y(0) = 1,
y(1) = 1 + 1 = 2,
y(2) = 2x2 + 1 = 5, . . .
which is unbounded. Hence, the system is unstable.

2.25
(a)
δ(n)
δ(n − k)
x(n)

= γ(n) − aγ(n − 1) and,
= γ(n − k) − aγ(n − k − 1). Then,
∞
X
x(k)δ(n − k)
=
=

k=−∞
∞
X

k=−∞

x(k)[γ(n − k) − aγ(n − k − 1)]
30

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

x(n)
x(n)

=
=
=

∞
X

k=−∞
∞
X

k=−∞
∞
X

x(k)γ(n − k) − a

∞
X

k=−∞
∞
X

x(k)γ(n − k) − a

k=−∞

x(k)γ(n − k − 1)
x(k − 1)γ(n − k)

[x(k) − ax(k − 1)]γ(n − k)

k=−∞

Thus, ck

= x(k) − ax(k − 1)

(b)
y(n)

= T [x(n)]
∞
X
ck γ(n − k)]
= T[
k=−∞

∞
X

=

k=−∞
∞
X

=

k=−∞

ck T [γ(n − k)]
ck g(n − k)

(c)
h(n)

= T [δ(n)]

= T [γ(n) − aγ(n − 1)]
= g(n) − ag(n − 1)

2.26
With x(n) = 0, we have
4
y(n − 1) + y(n − 1)
3

=

0

4
y(−1) = − y(−2)
3
4
y(0) = (− )2 y(−2)
3
4 3
y(1) = (− ) y(−2)
3
..
.
4
y(k) = (− )k+2 y(−2) ← zero-input response.
3

2.27
Consider the homogeneous equation:
5
1
y(n) − y(n − 1) + y(n − 2) = 0.
6
6
The characteristic equation is

1
1 1
5
λ2 − λ + = 0.λ = , .
6
6
2 3
31

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Hence,

1
1
yh (n) = c1 ( )n + c2 ( )n
2
3

The particular solution to
x(n) = 2n u(n) is
yp (n) = k(2n )u(n).
Substitute this solution into the difference equation. Then, we obtain
5
1
k(2n )u(n) − k( )(2n−1 )u(n − 1) + k( )(2n−2 )u(n − 2) = 2n u(n)
6
6
For n = 2,
4k −

5k k
8
+ =4⇒k= .
3
6
5

Therefore, the total solution is
y(n) = yp (n) + yh (n) =

1
1
8 n
(2 )u(n) + c1 ( )n u(n) + c2 ( )n u(n).
5
2
3

To determine c1 and c2 , assume that y(−2) = y(−1) = 0. Then,
y(0) = 1 and
y(1) =

5
17
y(0) + 2 =
6
6

Thus,
8
+ c1 + c2
5
1
16 1
+ c1 + c2
5
2
3

1 ⇒ c1 + c2 = −

=

17
11
⇒ 3c1 + 2c2 = −
6
5

and, therefore,
c1 = −1, c2 =
The total solution is
y(n) =



3
5

=

2
.
5


1
2 1
8 n
(2) − ( )n + ( )n u(n)
5
2
5 3

32

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

2.28
Fig. 2.28-1 shows the transient response, yzi (n), for y(−1) = 1 and the steady state response,
yzs (n).
1
0.8
0.6
0.4
0.2
0

0

5

10

15

20
25
30
Normalized Transient Response

35

40

45

50

0

5

10

15

20
25
30
Steady State Response

35

40

45

50

10
8
6
4
2
0

Figure 2.28-1:

2.29

h(n)

= h1 (n) ∗ h2 (n)
∞
X
ak [u(k) − u(k − N )][u(n − k) − u(n − k − M )]
=
=

k=−∞
∞
X

ak u(k)u(n − k) −

k=−∞
∞
X

−

=

k=−∞

n
X

k=0

=

0

k

∞
X

k=−∞

ak u(k)u(n − k − M )

ak u(k − N )u(n − k) +

a −

n−M
X
k=0

k

a

!

−

n
X

k=N

k

∞
X

k=−∞

a −

ak u(k − N )u(n − k − M )

n−M
X
k=N

k

a

!

33

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

2.30
y(n) − 3y(n − 1) − 4y(n − 2) = x(n) + 2x(n − 1)
The characteristic equation is
λ2 − 3λ − 4 = 0.
Hence, λ = 4, −1 and

yh (n) = c1 (n)4n + c2 (−1)n .

Since 4 is a characteristic root and the excitation is
x(n) = 4n u(n),
we assume a particular solution of the form
yp (n) = kn4n u(n).
Then
kn4n u(n) − 3k(n − 1)4n−1 u(n − 1) − 4k(n − 2)4n−2 u(n − 2)
= 4n u(n) + 2(4)n−1 u(n − 1)
. For n = 2,
k(32 − 12) = 42 + 8 = 24 → k =

6
.
5

The total solution is
y(n)

= yp (n) + yh (n)


6 n
n
n
=
n4 + c1 4 + c2 (−1) u(n)
5

To solve for c1 and c2 , we assume that y(−1) = y(−2) = 0. Then,
y(0) = 1 and
y(1) = 3y(0) + 4 + 2 = 9
Hence,
c1 + c2 = 1 and
24
+ 4c1 − c2 = 9
5
4c1 − c2 =

21
5

Therefore,
c1 =
The total solution is

26
1
and c2 = −
25
25


1
6 n 26 n
n
n4 + 4 − (−1) u(n)
y(n) =
5
25
25


34

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

2.31
From 2.30, the characteristic values are λ = 4, −1. Hence
yh (n) = c1 4n + c2 (−1)n
When x(n) = δ(N ), we find that
y(0) = 1 and
y(1) − 3y(0) = 2 or
y(1) = 5.
Hence,
c1 + c2 = 1 and 4c1 − c2 = 5
This yields, c1 =

6
5

and c2 = − 51 . Therefore,
h(n) =




6 n 1
4 − (−1)n u(n)
5
5

2.32
(a) L1 = N1 + M1 and L2 = N2 + M2
(b) Partial overlap from left:
low N1 + M1

high N1 + M2 − 1

Full overlap: low N1 + M2

high N2 + M1

Partial overlap from right:
low N2 + M1 + 1

high N2 + M2

(c)
x(n)
h(n)
N1
N2
M1
M2



1, 1, 1, 1, 1, 1, 1
↑


=
2, 2, 2, 2

=

= −2,

↑

= 4,
= −1,
=

2,

Partial overlap from left: n = −3
Full overlap: n = 0
Partial overlap from right:n = 4

n = −1

L1 = −3

n=3
n=6

L2 = 6

35

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

2.33
(a)
y(n) − 0.6y(n − 1) + 0.08y(n − 2) = x(n).
The characteristic equation is
λ2 − 0.6λ + 0.08 = 0.
λ = 0.2, 0.4 Hence,
yh (n) = c1

1n
2n
+ c2 .
5
5

With x(n) = δ(n), the initial conditions are
y(0)
y(1) − 0.6y(0)
Hence,c1 + c2
1
2
c1 +
5
5
Therefore h(n)

=

1,

= 0 ⇒ y(1) = 0.6.
= 1 and
=

0.6 ⇒ c1 = −1, c2 = 3.


2
1
=
−( )n + 2( )n u(n)
5
5

The step response is
s(n)

=

n
X

h(n − k), n ≥ 0

k=0
n 
X


2
1
2( )n−k − ( )n−k
5
5
k=0
 n+1
 n+1



2
1
1
1
(
(
=
−1 −
− 1 u(n)
0.12 5
0.16 5

=

(b)
y(n) − 0.7y(n − 1) + 0.1y(n − 2) = 2x(n) − x(n − 2).
The characteristic equation is
λ2 − 0.7λ + 0.1 = 0.

λ = 12 , 51 Hence,

yh (n) = c1

1n
1n
+ c2 .
2
5

With x(n) = δ(n), we have
y(0)
y(1) − 0.7y(0)

Hence,c1 + c2
1
1
c1 +
2
5
2
⇒ c1 + c2
5
These equations yield
4
10
, c2 = − .
c1 =
3
3
h(n)

=
=

2,
0 ⇒ y(1) = 1.4.

=

2 and

=

1.4 =

=

14
.
5

=



7
5


10 1 n 4 1 n
( ) − ( ) u(n)
3 2
3 5

36

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

The step response is
s(n)

=

n
X

h(n − k),

k=0

=

n

n

10 X 1 n−k 4 X 1 n−k
−
( )
( )
3
2
3
5
k=0

=
=

k=0

n

n

k=0

k=0

10 1 n X k 4 1 n X k
( )
2 − ( )
5
3 2
3 5

10 1 n n+1
1 1n
( (2
− 1)u(n) − ( (5n+1 − 1)u(n)
3 2
3 5

2.34

h(n)
y(n)
x(0)h(0)
1
x(0) + x(1)
2

=
=





1 1 1 1
1, , , ,
↑ 2 4 8 16




1, 2, 2.5, 3, 3, 3, 2, 1, 0
↑

= y(0) ⇒ x(0) = 1
3
= y(1) ⇒ x(1) =
2

By continuing this process, we obtain
x(n) =



3 3 7 3
1, , , , , . . .
2 2 4 2



2.35
(a) h(n) = h1 (n) ∗ [h2 (n) − h3 (n) ∗ h4 (n)]
(b)
h3 (n) ∗ h4 (n)
h2 (n) − h3 (n) ∗ h4 (n)
h1 (n)
Hence h(n)

=
=

(n − 1)u(n − 2)
2u(n) − δ(n)
1
1
1
=
δ(n) + δ(n − 1) + δ(n − 2)
2
4
2


1
1
1
δ(n) + δ(n − 1) + δ(n − 2) ∗ [2u(n) − δ(n)]
=
2
4
2
1
5
5
=
δ(n) + δ(n − 1) + 2δ(n − 2) + u(n − 3)
2
4
2

(c)
x(n)
y(n)



=
1, 0, 0, 3, 0, −4
↑


1 5
25 13
=
, , 2, , , 5, 2, 0, 0, . . .
2 4 ↑ 4 2
37

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

2.36
First, we determine
s(n)
s(n)

= u(n) ∗ h(n)
∞
X
u(k)h(n − k)
=
=
=

k=0
n
X

k=0
∞
X

h(n − k)
an−k

k=0
n+1

=

a

−1
,n ≥ 0
a−1

For x(n) = u(n + 5) − u(n − 10), we have the response
s(n + 5) − s(n − 10) =
From figure P2.33,
y(n)
Hence, y(n)

an−9 − 1
an+6 − 1
u(n + 5) −
u(n − 10)
a−1
a−1

= x(n) ∗ h(n) − x(n) ∗ h(n − 2)
an+6 − 1
an−9 − 1
=
u(n + 5) −
u(n − 10)
a−1
a−1
an−11 − 1
an+4 − 1
u(n + 3) +
u(n − 12)
−
a−1
a−1

2.37
h(n)

=

s(n)

=
=

[u(n) − u(n − M )] /M
∞
X
u(k)h(n − k)

k=−∞
n
X

k=0

h(n − k) =



n+1
M ,

1,

n y(n)

−−> x(n)

0.5
0

0.5
0
−0.5

−0.5
−1
−1
0

50

100
−−> n

150

−1.5
0

200

50

100
−−> n

150

200

15

−−> rxy(l)

10
5
0
−5

−20

0
−−> l

20

Figure 2.65-1: variance = 0.01
(c) variance = 0.1. Delay D = 20. Refer to fig 2.65-2.
(d) Variance = 1. delay D = 20. Refer to fig 2.65-3.
(e) x(n) = {−1, −1, −1, +1, +1, +1, +1, −1, +1, −1, +1, +1, −1, −1, +1}. Refer to fig 2.65-4.
(f) Refer to fig 2.65-5.

52

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

1

1.5
1
−−> y(n)

−−> x(n)

0.5
0

0.5
0
−0.5

−0.5
−1
−1
0

50

100
−−> n

150

−1.5
0

200

50

100
−−> n

150

200

20

−−> rxy(l)

15
10
5
0
−5

−20

0

20

Figure 2.65-2: variance = 0.1
1

3
2
−−> y(n)

−−> x(n)

0.5
0

1
0
−1

−0.5
−2
−1
0

50

100
−−> n

150

−3
0

200

50

100
−−> n

150

200

15

−−> rxy(l)

10
5
0
−5

−20

0
−−> l

20

Figure 2.65-3: variance = 1
53

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

1

1
0.5
−−> y(n)

−−> x(n)

0.5
0
−0.5

0
−0.5
−1

−1
0

50

100
−−> n

150

−1.5
0

200

50

100
−−> n

150

200

50

100
−−> n

150

200

20

−−> rxy(l)

15
10
5
0
−5
−10

−20

0
−−> n

20

Figure 2.65-4:
1

1.5
1
−−> y(n)

−−> x(n)

0.5
0

0.5
0
−0.5

−0.5
−1
−1
0

50

100
−−> n

150

−1.5
0

200

20

−−> rxy(l)

15
10
5
0
−5
−10

−20

0
−−> n

20

Figure 2.65-5:
54

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

2.66
(a) Refer to fig 2.66-1.
(b) Refer to fig 2.66-2.
impulse response h(n) of the system
1

−−> h(n)

0.5

0

−0.5
0

5

10

15

20

25
−−> n

30

35

40

45

50

Figure 2.66-1:
(c) Refer to fig 2.66-3.
(d) The step responses in fig 2.66-2 and fig 2.66-3 are similar except for the steady state value
after n=20.

55

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

zero−state step response s(n)
1.6
1.5
1.4

−−> s(n)

1.3
1.2
1.1
1
0.9
0.8
0.7
0

5

10

15

20

25
−−> n

30

35

40

45

50

35

40

45

50

Figure 2.66-2:

step response
1.6
1.5
1.4

−−> s(n)

1.3
1.2
1.1
1
0.9
0.8
0.7
0

5

10

15

20

25
−−> n

30

Figure 2.66-3:

56

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

2.67
Refer to fig 2.67-1.
7
6
5

−−> h(n)

4
3
2
1
0
−1
−2
0

10

20

30

40

50
−−> n

60

70

80

90

100

Figure 2.67-1:

57

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

58

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Chapter 3

3.1
(a)
X(z) =

X

x(n)z −n

n

=

3z 5 + 6 + z −1 − 4z −2 ROC: 0 < |z| < ∞

(b)
X(z) =

X

x(n)z −n

n

∞
X
1
( )n z −n
=
2
n=5

=
=

∞
X
1
( )n
2z
n=5

∞
X
1
( z −1 )m+5
2
m=0

=

(

=

(

1
z −1 5
)
2
1 − 12 z −1

1
1
z −5
ROC: |z| >
)
32 1 − 21 z −1
2

3.2
(a)
X(z) =
=
=

X

n
∞
X

n=0
∞
X

n=0

But

∞
X

n=0

z −n

=

x(n)z −n
(1 + n)z −n
z −n +

∞
X

nz −n

n=0

1
ROC: |z| > 1
1 − z −1

59

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

∞
X

nz −n

=

Therefore, X(z)

=

and

z −1
ROC: |z| > 1
(1 − z −1 )2

n=0

z −1
1 − z −1
+
(1 − z −1 )2
(1 − z −1 )2
1
(1 − z −1 )2

=
(b)
X(z)

∞
X

=

(an + a−n )z −n

n=0
∞
X

=

an z −n +

an z −n

=

a−n z −n

=

Hence, X(z)

=

But

1
ROC: |z| > |a|
1 − az −1

n=0

and

∞
X

a−n z −n

n=0

n=0

∞
X

∞
X

1
(1 −

n=0

1 −1 2
)
az

ROC: |z| >

1
1
+
1 − az −1
1 − a1 z −1

1
|a|

2 − (a + a1 )z −1
1
1 −1 ROC: |z| > max (|a|, |a| )
−1
(1 − az )(1 − a z )

=
(c)

X(z) =
=

∞
X

1
(− )n z −n
2
n=0
1
1
, |z| >
2
1 + 21 z −1

(d)
X(z) =
=

∞
X

n=0
∞
X

nan sinw0 nz −n
nan

n=0

=
=




ejw0 n − e−jw0 n −n
z
2j



1
aejw0 z −1
ae−jw0 z −1
−
2j (1 − aejw0 z −1 )2
(1 − ae−jw0 z −1 )2
 −1

az − (az −1 )3 sinw0
, |z| > a
(1 − 2acosw0 z −1 + a2 z −2 )2

(e)
X(z) =
=

∞
X

n=0
∞
X

n=0

nan cosw0 nz −n
nan




ejw0 n + e−jw0 n −n
z
2
60

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

=
=



1
aejw0 z −1
ae−jw0 z −1
+
2 (1 − aejw0 z −1 )2
(1 − ae−jw0 z −1 )2
 −1

−1 3
az + (az ) sinw0 − 2a2 z −2
,
|z| > a
(1 − 2acosw0 z −1 + a2 z −2 )2

(f)
X(z) = A

∞
X

n=0
∞
X

rn cos(w0 n + φ)z −n


ejw0 n ejφ + e−jw0 n e−jφ −n
z
r
= A
2
n=0


A
ejφ
e−jφ
=
+
2 1 − rejw0 z −1
1 − re−jw0 z −1


cosφ − rcos(w0 − φ)z −1
,
|z| > r
= A
1 − 2rcosw0 z −1 + r2 z −2
n



(g)

But

∞
X

∞
X
1 2
1
X(z) =
(n + n)( )n−1 z −n
2
3
n=1

1
n( )n−1 z −1
3
n=1

∞
X

1
n2 ( )n−1 z −n
3
n=1

( 31 )3z −1
z −1
=
(1 − 31 z −1 )2
(1 − 13 z −1 )2

=

z −1 + 31 z −2
(1 − 31 z −1 )3


z −1 + 31 z −2
1
z −1
+
2 (1 − 13 z −1 )2
(1 − 31 z −1 )3

=

Therefore, X(z) =

z −1
,
(1 − 31 z −1 )3

=

|z| >

1
3

(h)
X(z) =
=
=

∞
∞
X
X
1
1
( )n z −n
( )n z −n −
2
2
n=10
n=0

1

1 − 21 z −1

−

( 12 )10 z −10
1 − 21 z −1

1 − ( 21 z −1 )10
,
1 − 21 z −1

|z| >

1
2

The pole-zero patterns are as follows:
(a) Double pole at z = 1 and a zero at z = 0.
(b) Poles at z = a and z = a1 . Zeros at z = 0 and z = 12 (a + a1 ).
(c) Pole at z = − 21 and zero at z = 0.
(d) Double poles at z = aejw0 and z = ae−jw0 and zeros at z = 0, z = ±a.
(e) Double poles at z = aejw0 and z = ae−jw0 and zeros are obtained by solving the quadratic
acosw0 z 2 − 2a2 z + a3 cosw0 = 0.
(f) Poles at z = rejw0 and z = ae−jw0 and zeros at z = 0, and z = rcos(w0 − φ)/cosφ.
(g) Triple pole at z = 31 and zeros at z = 0 and z = 13 . Hence there is a pole-zero cancellation so
61

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

that in reality there is only a double pole at z = 13 and a zero at z = 0.
(h) X(z) has a pole of order 9 at z = 0. For nine zeros which we find from the roots of
1
1 − ( z −1 )10
2
1 10
or, equivalently, ( ) − z 10
2
Hence, zn

=

0

=

0

=

1 j2πn
e 10 , n = 1, 2, . . . , k.
2

Note the pole-zero cancellation at z = 21 .

3.3
(a)
X1 (z)

=
=
=
=

The ROC is
(b)

1
3

0
∞
X
X
1
1
( )n z −n − 1
( )n z −n +
3
2
n=−∞
n=0

1

+

1 − 31 z −1
1

(1 −

1
− 1,
1 − 12 z

+

1 −1
3z

1−

∞
X
1
( )n z n − 1
2
n=0

5
6
1 −1
)(1
3z

− 12 z)

< |z| < 2.
X2 (z)

=
=
=

∞
∞
X
X
1
( )n z −n −
2n z −n
3
n=0
n=0

1

1−

1 −1
3z

(1 −

−

1
,
1 − 2z −1

− 53 z −1
1 −1
)(1 −
3z

2z −1 )

The ROC is |z| > 2.
(c)
X3 (z)

=
=
=

The ROC is
(d)

1
3

∞
X

x1 (n + 4)z −n

n=−∞
z 4 X1 (z)

(1 −

5 4
6z
1 −1
)(1
3z

− 12 z)

< |z| < 2.
X4 (z)

=

∞
X

x1 (−n)z −n

n=−∞

62

© 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws
as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

∞
X

=

x1 (m)z m

m=−∞

= X1 (z −1 )
5
6

=
The ROC is

1
2

(1 − 31 z)(1 − 12 z −1 )

< |z| < 3.

3.4
(a)
X(z) =

∞
X

n(−1)n z −n

n=0
∞
d X
= −z
(−1)n z −n
dz n=0


1
d
= −z
dz 1 + z −1
z −1
, |z| > 1
= −
(1 + z −1 )2

(b)
X(z)

=

∞
X

n2 z −n

n=0
∞
d2 X −n
z
dz 2 n=0


2
1
2 d
= z
dz 2 1 − z −1
2z −1
z −1
+
= −
(1 − z −1 )2
(1 − z −1 )3
−1
−1
z (1 + z )
=
, |z| > 1
(1 − z −1 )3

= z2

(c)

X(z)

=

−1
X

n=−∞

−nan z −n

−1
d X
a(n)z −n
dz n=−∞


1
d
= −z
dz 1 − az −1
az −1
=
, |z| < |a|
(1 − az −1 )2

= −z

(d)

X(z) =

∞
X

π
(−1)n cos( n)z −n
3
n=0

63

© 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws
as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

From formula (9) in table 3.3 with a = −1,

1 + z −1 cos π3
1 + 2z −1 cos π3 + z −2

X(z) =

1 + 21 z −1
, ROC: |z| > 1
1 + z −1 + z −2

=
(e)
X(z)

∞
X

=

(−1)n z −n

n=0

1
,
1 + z −1

=

|z| > 1

(f)
x(n)



=
1, 0, −1, 0, 1, −1

X(z) =

↑

1 − z −2 + z −4 − z −5 ,

z 6= 0

3.5

Right-sided sequence :xr (n)

=

0, n < n0
−1
X

Xr (z) =

xr (n)z −n +

n=n0

∞
X

xr (n)z −n

n=0

P−1
The term n=n0 xr (n)z −n converges for all z except z = ∞.
P∞
The term n=0 xr (n)z −n converges for all |z| > r0 where some r0 . Hence Xr (z) converges for
r0 < |z| < ∞ when n0 < 0 and |z| > r0 for n0 > 0
Left-sided sequence :xl (n)

=

0, n > n0
0
X

Xl (z) =

xl (n)z −n +

n=−∞

n0
X

xl (n)z −n

n=1

The first term converges for some |z| < rl . The second term converges for all z, except z = 0.
Hence, Xl (z) converges for 0 < |z| < rl when n0 > 0, and for |z| < rl when n0 < 0.
Finite-Duration Two-sided sequence :x(n)

=

X(z) =

0, n > n0 and n < n1 , where n0 > n1
n0
X
x(n)z −n

n=n1

=

−1
X

n=n1

x(n)z −n +

n=n
X0

x(n)z −n

n=0

The first term converges everywhere except z = ∞.
The second term converges everywhere except z = 0. Therefore, X(z) converges for 0 < |z| < ∞.
64

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

3.6
y(n)

=

n
X

x(k)

k=−∞

⇒ y(n) − y(n − 1)

Hence,Y (z) − Y (z)z

−1

Y (z)

= x(n)

= X(z)
X(z)
=
1 − z −1

3.7
x1 (n) =

X1 (z)

=
=



1
1
+
−1
1 − 13 z −1
1 − 12 z
5
6
1 −1
z
)(1
3

(1 −
− 12 z)
∞
X
1
( )n z −n
=
2
n=0

=
Then,Y (z)
Hence,y(n)

n≥0
n<0

∞
−1
X
X
1
1
( )n z −n +
( )−n z −n
3
2
n=−∞
n=0

=
X2 (z)

( 31 )n ,
( 21 )−n ,

1

1−

1 −1 ,
2z

1
< |z| < 2
2

10
−4
−2
3
3
+
+
1 − 2z −1
1 − 13 z −1
1 − 12 z −1

1 n
−2( 13 )n + 10
3 (2) , n ≥ 0
=
4
n
n<0
3 (2) ,

=

3.8
(a)
y(n)

=
=

n
X

k=−∞
∞
X

k=−∞

Y (z)

(b)
u(n) ∗ u(n)

x(k)
x(k)u(n − k)

= x(n) ∗ u(n)
= X(z)U (z)
X(z)
=
1 − z −1
=

∞
X

k=−∞

u(k)u(n − k)

65

© 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws
as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

n
X

=

u(k) = (n + 1)u(n)

k=−∞

Hence,x(n)

= u(n) ∗ u(n)
1
, |z| > 1
=
(1 − z −1 )2

andX(z)

3.9
y(n) = x(n)ejw0 n . From the scaling theorem, we have Y (z) = X(e−jw0 z). Thus, the poles and
zeros are phase rotated by an angle w0 .

3.10
x(n)

=

X + (z) =
From the final value theorem
x(∞) =
=
=

1
[u(n) + (−1)n u(n)]
2
( 1−z1 −1 + 1+z1 −1 )
2
lim (z − 1)X + (z)

z→1

lim (z +

z→1

z(z − 1)
)
z+1

1
2

3.11
(a)
1 + 2z 4
1 − 2z −1 + z −2
= 1 + 4z −1 + 7z −2 + 10z −3 + . . .


=
,
4,
7,
10,
.
.
.
,
3n
+
1,
.
.
.
1

X(z) =

Therefore,x(n)

↑

(b)
2z + 5z 2 + 8z 3 + . . .


=
. . . , −(3n + 1), . . . , 11, 8, 5, 2, 0

X(z) =
Therefore,x(n)

↑

3.12
X(z) =

1

− z −1 )2
A
B
Cz −1
=
+
+
(1 − 2z −1 ) (1 − z −1 ) (1 − z −1 )2
A = 4, B = −3, C = −1
Hence,x(n) = [4(2)n − 3 − n] u(n)
(1 −

2z −1 )(1

66

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

3.13
(a)
x1 (n) =



X1 (z) =
=
=

x( n2 ),
0,
∞
X

n even
n odd
x1 (n)z −n

n=−∞
∞
X

n
x( )z −n
2
n=−∞
∞
X

x(k)z −2k

k=−∞
2

= X(z )
(b)
x2 (n)

= x(2n)
∞
X
x2 (n)z −n
X2 (z) =
=
=
=
=
=

n=−∞
∞
X

n=−∞
∞
X

x(2n)z −n
k

x(k)z − 2

k=−∞
∞ 
X

k=−∞
∞
X

1
2


x(k) + (−1)k x(k) − k
z 2 , k even
2
k

x(k)z − 2 +

k=−∞

∞
1
1 X
x(k)(−z 2 )−k
2
k=−∞

√ 
1 √
X( z + X(− z)
2

3.14
(a)
X(z) =
=
A =
Hence,x(n)

=

1 − 3z −1
1 + 3z −1 + 2z −2
A
B
+
(1 + z −1 ) (1 + 2z −1 )
2, B = −1
[2(−1)n − (−2)n ] u(n)

(b)
X(z) =
=

1
1 − z −1 + 21 z −2

A(1 − 12 z −1 ) + B( 21 z −1 )
1 − z −1 + 21 z −2
67

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

A =
Hence,X(z) =

1, B = 1
1−
1−

+

√1 (cos π )z −1
4
2
2 √12 (cos π4 )z −1 + ( √12 )2 z −2
√1 (sin π )z −1
4
2
1
π −1
√
2 2 (cos 4 )z + ( √12 )2 z −2

1−


π
π
1 n
1 n
=
( √ ) cos n + ( √ ) sin n u(n)
4
4
2
2

Hence,x(n)
(c)

X(z)
x(n)

z −7
z −6
+
1 − z −1
1 − z −1
= u(n − 6) + u(n − 7)

=

(d)
1
z −2
+
2
1 + z −2
1 + z −2
1
X(z) = 2 −
1 + z −2
π
π
x(n) = cos nu(n) + 2cos (n − 2)u(n − 2)
2
2
π
x(n) = 2δ(n) − cos nu(n)
2
X(z) =

(e)
X(z) =

1 + 6z −1 + z −2
1
4 (1 − 2z −1 + 2z −2 )(1 − 21 z −1 )

A(1 − z −1 )
Bz −1
C
+
+
−1
−2
−1
−2
1 − 2z + 2z
1 − 2z + 2z
1 − 21 z −1
23
17
3
,C =
A = − ,B =
10
20
5

π
π
3 1 n
23 1
17 1
Hence,x(n) =
− ( √ ) cos n + ( √ )n sin n + ( )n u(n)
5 2
4
10 2
4
20 2
=

(f)
X(z)

x(n)

2 − 1.5z −1
1 − 1.5z −1 + 0.5z −2
1
1
=
1 −1 + 1 − z −1
1− z
 2

1
=
( )n + 1 u(n)
2
=

(g)
X(z) =
=

1 + 2z −1 + z −2
1 + 4z −1 + 4z −2


2z −1 + 3z −2
1−
(1 + 2z −1 )(1 + 2z −1 )
68

© 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws
as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

=
x(n)

1−

2z −1
z −2
+
1 + 2z −1
(1 + 2z −1 )2

= δ(n) − 2(−2)n−1 u(n − 1) + (n − 1)(−2)n−1 u(n − 1)
= δ(n) + (n − 3)(−2)n−1 u(n − 1)

(h)
X(z) =

(z + 21 )(z + 14 )
1
1
4 (z − 2 )(z − √ 1j π )(z −
4
2e

=

√

1
π
2e−j 4

)

1 (1 + 43 z −1 + 18 z −2 )z −1
4 (1 − 21 z −1 )(1 − z −1 + 12 z −2 )

A( 12 z −1 )z −1
A(1 − 21 z −1 )z −1
Cz −1
1 −2 +
1 −2 +
−1
−1
1 − z + 2z
1 − z + 2z
1 − 12 z −1
1
7
3
A = − ,B = ,C =
2
8
4


1 1 n−1
7 1 n−1
3 1 n−1
π
π
2
2
Hence,x(n) =
− ( )
u(n − 1)
cos (n − 1) + ( )
sin (n − 1) + ( )
2 2
4
8 2
4
4 2
=

(i)
X(z)

=
=

x(n)

=

X(z)

=

1 − 41 z −1
1 + 21 z −1
1 z −1
4 1 + 21 z −1
1 + 21 z −1
1 1
1
(− )n u(n) + (− )n−1 u(n − 1)
2
4 2
1

−

(j)

=
=
x(n)

=
=

1 − az −1
z −1 − a


1 1 − az −1
−
a 1 − a1 z −1


az −1
1
1
−
−
a 1 − a1 z −1
1 − a1 z −1
1 1
1
− ( )n u(n) + ( )n−1 u(n − 1)
a a
a
1
1
(− )n+1 u(n) + ( )n−1 u(n − 1)
a
a

3.15
5z −1

X(z) =

If |z| > 2, x(n)

2z −1 )(3

(1 −
− z −1 )
1
1
=
+
−1
1 − 2z
1 − 31 z −1


1
=
2n − ( )n u(n)
3
69

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

If

1
< |z| < 2, x(n)
3
1
If |z| < , x(n)
3

1
= −( )n u(−n − 1) − 2n u(−n − 1)
3
1 n
= ( ) u(−n − 1) − 2n u(−n − 1)
3

3.16
(a)
x1 (n)

1 1 n−1
( )
u(n − 1)
4 4
( 14 )z −1
1
, |z| >
4
1 − 41 z −1


1
1 + ( )n u(n)
2
1
1
, |z| > 1
+
1 − z −1
1 − 21 z −1
X1 (z)X2 (z)
1
− 43
1
3
1 −1 + 1 − z −1 +
1 − 4z
1 − 12 z −1


1 n
4 1 n 1
− ( ) + + ( ) u(n)
3 4
3
2

=

⇒ X1 (z) =
x2 (n)

=

⇒ X2 (z) =
Y (z) =
=
y(n)

=

(b)
x1 (n)
⇒ X1 (z)
x2 (n)
⇒ X2 (z)
Y (z)

y(n)

= u(n)
1
=
,
1 − z −1
1
= δ(n) + ( )n u(n)
2
1
= 1+
1 − 21 z −1
= X1 (z)X2 (z)
1
3
−
=
−1
1−z
1 − 12 z −1


1
=
3 − ( )n u(n)
2

(c)
x1 (n)

=

⇒ X1 (z) =
x2 (n)

=

⇒ X2 (z) =
Y (z) =
=

1
( )n u(n)
2
1
,
1 − 12 z −1
cosπnu(n)
1 + z −1
1 + 2z −1 + z −2
X1 (z)X2 (z)
1 + z −1
(1 − 12 z −1 )(1 + 2z −1 + z −2 )
70

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

A(1 + z −1 )
B
+
1 + 2z −1 + z −2
1 − 12 z −1
1
2
,B =
A =
3
3


2
1 1 n
y(n) =
cosπn + ( ) u(n)
3
3 2
=

(d)
x1 (n)

= nu(n)
z −1
,
=
(1 − z −1 )2
= 2n u(n − 1)
2z −1
=
1 − 2z −1
= X1 (z)X2 (z)
2z −2
=
−1
(1 − z )2 (1 − 2z −1 )
−2z −1
2
−2
−
+
=
−1
1−z
(1 − z −1 )2
1 − 2z −1


= −2(n + 1) + 2n+1 u(n)

⇒ X1 (z)
x2 (n)
⇒ X2 (z)
Y (z)

y(n)

3.17


= z X + (z) − x(0)
= zX + (z) − zx(0)
∞
X
x(n + 1)z −n + zx(0)
Therefore, zX + (z) =
z + [x(n + 1)]

n=0
∞
X

(z − 1)X + (z) = −
limz→1 X + (z)(z − 1)

x(n)z −n +

∞
X

x(n + 1)z −n + zx(0)

n=0

n=0

= x(0) +

∞
X

n=0

x(n + 1) −

∞
X

x(n)

n=0

= limm→∞ [x(0) + x(1) + x(2) + . . . + x(m)
−x(0) − x(1)x(2) − . . . − x(m)]
= limm→∞ x(m + 1)
= x(∞)

3.18
(a)
∞
X

n=−∞

x∗ (n)z −n

∞
X


=

n=−∞

= X ∗ (z ∗ )

x(n)(z ∗ )−n

∗

71

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

(b)
1
[z {x(n)} + z {x∗ (n)}]
2

x(n) + x∗ (n)
= z
2
= z [Re {x(n)}]

1
[X(z) + X ∗ (z ∗ )]
2

=

(c)


x(n) − x∗ (n)
2j
= z [Im {x(n)}]

1
[X(z) − X ∗ (z ∗ )]
2j

= z



(d)
Xk (z)

∞
X

=

n=−∞,n/kinteger

=

∞
X

n
x( )z −n
k

x(m)z −mk

m=−∞

= X(z k )
(e)
∞
X

ejw0 n x(n)z −n

=

∞
X

x(n)(e−jw0 z)−n

n=−∞
−jw0

n=−∞

= X(ze

)

3.19
(a)
X(z) = log(1 − 2z), |z| <
Y (z)

1
2

dX(z)
dz
−1
1
, |z| <
2
1 − 21 z −1
1
( )n , n < 0
2
1
y(n)
n
1 1 n
( ) u(−n − 1)
n 2

= −z
=

⇒ y(n)

=

Then,x(n)

=
=

(b)
X(z)
Y (z)

1
1
= log(1 − z −1 ), |z| >
2
2
dX(z)
= −z
dz
72

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

− 21 z −1
1
1 −1 , |z| > 2
1 − 2z
1 1
= − ( )n−1 u(n − 1)
2 2
1
=
y(n)
n
1 1
= − ( )n u(n − 1)
n 2

=
Hence,y(n)
x(n)

3.20
(a)
= rn sinw0 nu(n),
0 1, x(n)
For 0.5 < |z| < 1, x(n)

= [2 − (0.5)n ] u(n)
= −(0.5)n u(n) − 2u(−n − 1)

(b)
X(z)

1
(1 − 0.5z −1 )2


0.5z −1
2z
=
(1 − 0.5z −1 )2
=

For |z| > 0.5, x(n)

= 2(n + 1)(0.5)n+1 u(n + 1)
= (n + 1)(0.5)n u(n)

For |z| < 0.5, x(n)

= −2(n + 1)(0.5)n+1 u(−n − 2)
= −(n + 1)(0.5)n u(−n − 1)

3.26
3

X(z) =

1

=
1
ROC: < |z| < 3, x(n)
3

−1 + z −2
− 10
3 z
27
− 83
8
+
1 − 3z −1
− 13 z −1

1
3 1 n
27
( ) u(n) − 3n u(−n − 1)
8 3
8

=

3.27

X(z)

=
=

∞
X

n=−∞
∞
X

x(n)z −n
x1 (n)x∗2 (n)z −n

n=−∞
∞
X

I
1
X1 (v)v n−1 dvx∗2 (n)z −n
2πj
c
n=−∞
#
" ∞
I
X
1
z −n −1
∗
=
v
X1 (v)dv
x2 (n)( )
2πj c
v
n=−∞
#∗
" ∞
I
X
z ∗ −n
1
v −1 dv
X1 (v)
x2 (n)( ∗ )
=
2πj c
v
n=−∞
I
1
z∗
=
X1 (v)X2∗ ( ∗ )v −1 dv
2πj c
v
=

3.28
Conjugation property:
75

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

∞
X

∗

x (n)z

−n

=

n=−∞

"

∞
X

∗ −n

x(n)(z )

n=−∞
∗ ∗

#∗

= X (z )

Parseval’s relation:
∞
X
x1 (n)x∗2 (n)

∞
X

I
1
X1 (v)v n−1 dvx∗2 (n)
2πj
c
n=−∞
" ∞
#
I
X
1
1 −n −1
∗
=
X1 (v)
v dv
x2 (n)( )
2πj c
v
n=−∞
I
1
1
X1 (v)X2∗ ( ∗ )dv
=
2πj c
v
=

n=−∞

3.29
x(n) =

1
2πj

I

c

z n dz
,
z−a

where the radius of the contour c is rc > |a|. For n < 0, let w = z1 . Then,
1
x(n) =
2πj
where the radius of c′ is
n < 0.

1
rc .

Since

1
rc

I

c′

1 −n−1
aw
dw,
w − a1

< |a|, there are no poles within c′ and, hence x(n) = 0 for

3.30
x(n) = x(N − 1 − n), since x(n) is even. Then
X(z) =

N
−1
X

x(n)z −n

n=0

= x(0) + x(1)z −1 + . . . + x(N − 2)z −N +2 + x(N − 1)z −N +1
N
2

= z

−(N −1)/2

−1
X

n=0

i
h
x(n) z (N −1−2n)/2 + z −(N −1−2n)/2

N even

If we substitute z −1 for z and multiply both sides by z −(N −1) we obtain
z −(N −1) X(z −1 ) = X(z)
Hence, X(z) and X(z −1 ) have identical roots. This means that if z1 is root (or a zero) of X(z)
then z11 is also a root. Since x(n) is real, then z1∗ must also be a root and so must z1∗
1

3.31
From the definition of the Fibonacci sequence, y(n) = y(n − 1) + y(n − 2), y(0) = 1. This is
equivalent to a system described by the difference equation y(n) = y(n − 1) + y(n − 2) + x(n),
76

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

where x(n) = δ(n) and y(n) = 0, n < 0. The z-transform of this difference equation is Y (z) =
z −1 Y (z) + z −2 Y (z) = X(z) Hence, for X(z) = 1, we have
Y (z) =
Y (z) =
where A =
Hence, y(n) =
=

1
1−

z −1
A

− z −2

B

+

√

√

−1
1 − 5+1
1 − 1−2 5 z −1
2 z
√
√
√
5+1
5−1
1− 5
√ ,B =
√
=− √
2 5
2 5
2 5
√
√
√
√
5+1 5+1 n
1− 5 1− 5 n
√ (
√
(
) u(n) −
) u(n)
2
2
2 5
2 5
"
#
√
√
1
1 + 5 n+1
1 − 5 n+1
√ (
)
−(
)
u(n)
2
2
5

3.32
(a)


Y (z) 1 − 0.2z −1
Y (z)
X(z)



= X(z) 1 − 0.3z −1 − 0.02z −2

=

=

(1 − 0.1z −1 )(1 − 0.2z −1 )
1 − 0.2z −1

1 − 0.1z −1

(b)
Y (z)
Y (z)
X(z)



= X(z) 1 − 0.1z −1

=

1 − 0.1z −1

Therefore, (a) and (b) are equivalent systems.

3.33
1
1 − az −1
= an u(n)

X(z) =
⇒ x1 (n)
or x2 (n)

= −an u(−n − 1)

Both x1 (n) and x2 (n) have the same autocorrelation sequence. Another sequence is obtained
1
from X(z −1 ) = 1−az
X(z −1 )

=

1
1 − az

= 1−
Hence x3 (n)

1

1 − a1 z −1
1
= δ(n) − ( )n u(n)
a

We observe that x3 (n) has the same autocorrelation as x1 (n) and x2 (n)
77

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

3.34

H(z)

−∞
X

=

3n z −n +

n=−1

∞
X
2
( )n z −n
5
n=0

1
−1
2
+
, ROC: < |z| < 3
2
−1
−1
1 − 3z
5
1 − 5z
1
=
1 − z −1
= H(z)X(z)
−1
− 13
5 z
=
, ROC: 1 < |z| < 2
−1
−1
(1 − z )(1 − 3z )(1 − 52 z −1 )
=

X(z)
Y (z)

13
6

=

1 − z −1

Therefore,
y(n)

−

3
2

1 − 3z −1

−

1−

2
3
2 −1
5z



3 n
13 2 2 n
3 u(−n − 1) +
− ( ) u(n)
2
6
3 5

=

3.35
(a)
h(n)

=

H(z)

=

x(n)

=

X(z)

=

Y (z)

=
=
=

Therefore,
y(n)

=

1
( )n u(n)
3
1
1 − 31 z −1
1
πn
( )n cos u(n)
2
3
1 − 14 z −1
1 − 21 z −1 + 14 z −2
H(z)X(z)
1 − 41 z −1
(1 − 31 z −1 )(1 − 12 z −1 + 14 z −2 )
1−
"

1
7
1 −1
3z

+

1

1 −1
6
7 (1 − 4 z
− 21 z −1 + 14 z −2

√
√
3 −1
3 3
4 z
+
7 1 − 21 z −1 + 14 z −2

#
√
1 1 n 6 1 n
πn 3 3 1 n
πn
u(n)
( ) + ( ) cos
+
( ) sin
7 3
7 2
3
7 2
3

(b)
h(n)

=

H(z)

=

x(n)

=

1
( )n u(n)
2
1
1 − 21 z −1
1
1
( )n u(n) + ( )−n u(−n − 1)
3
2
78

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

1

X(z) =

1 −1
3z

1−
Y (z) = H(z)X(z)

1
1 − 2z −1

− 35 z −1
(1 − 21 z −1 )(1 − 13 z −1 )(1 − 2z −1 )

=
=

1

Therefore,
y(n)

−



=

10
3
− 21 z −1

+

−4
−2
3
+
1 − 2z −1
1 − 31 z −1


4
1
10 1 n
( ) − 2( )n u(n) + 2n u(−n − 1)
3 2
3
3

(c)
y(n)

= −0.1y(n − 1) + 0.2y(n − 2) + x(n) + x(n − 1)
1 + z −1
H(z) =
1 + 0.1z −1 − 0.2z −2
1
x(n) = ( )n u(n)
3
1
X(z) =
1 − 31 z −1
Y (z) = H(z)X(z)
1 + z −1
=
1 −1
(1 − 3 z )(1 + 0.1z −1 − 0.2z −2 )
=
Therefore,
y(n)

−1
28
−8
3
3
+
+
1 − 0.4z −1
1 + 0.5z −1
1 − 31 z −1



1 n 28 2 n 1 1 n
=
−8( ) + ( ) − ( ) u(n)
3
3 5
3 2

(d)
y(n) =
⇒ Y (z) =
X(z) =
Hence, Y (z) =
y(n) =
=
=
=

1
1
x(n) − x(n − 1)
2
2
1
−1
(1 − z )X(z)
2
10
1 + z −2
(1 − z −1 )/2
10
1 + z −2
π(n − 1)
πn
u(n − 1)
5cos u(n) − 5cos
2
2
i
h
πn
πn
u(n − 1) + 5δ(n)
5cos
− 5sin
2
2
10
πn π
5δ(n) + √ sin(
+ )u(n − 1)
2
4
2
10
πn π
√ sin(
+ )u(n)
2
4
2

(e)
y(n) = −y(n − 2) + 10x(n)
79

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Y (z) =
X(z) =
Y (z) =
=
Therefore,

10
X(z)
1 + z −2
10
1 + z −2
100
(1 + z −2 )2
50
50
−25jz −1
25jz −1
+
+
+
1 + jz −1
1 − jz −1
(1 + jz −1 )2
(1 − jz −1 )2

y(n)

= {50 [j n + (−j)n ] − 25n [j n + (−j)n ]} u(n)
= (50 − 25n)(j n + (−j)n )u(n)
πn
= (50 − 25n)2cos u(n)
2

h(n)

=

H(z)

=

x(n)

=

X(z)

=

Y (z)

=

(f)

=
=
Therefore,
y(n)

=

2
( )n u(n)
5
1
1 − 25 z −1
u(n) − u(n − 7)
1 − z −n
1 − z −1
H(z)X(z)
1 − z −n
(1 − 52 z −1 )(1 − z −1 )
5
3

1 − z −1

+

1

−2
3
− 52 z −1

−



5
3

1 − z −1

+

1


−2
3
z −7
− 25 z −1





1
2
2
1
5 − 2( )n u(n) −
5 − 2( )n−7 u(n − 7)
3
5
3
5

(g)
1
( )n u(n)
2
1
H(z) =
1 − 21 z −1
x(n) = (−1)n ,
−∞ −1
< 1 and
> 1

Refer to fig 3.41-1.

=1/4
a2 a12

a2
real
real
complex

a 1-a 2 =1

a +a 2=-1
1

-2

-1

1

2

a1

real

-1

Figure 3.41-1:

3.42
H(z) =

z −1 + 12 z −2
2 −2
z
1 − 35 z −1 + 25
86

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

(a)

h(n)




9
− 72
2
+
1 − 51 z −1
1 − 25 z −1


9 2
7 1
=
− ( )n−1 + ( )n−1 u(n − 1)
2 5
2 5

H(z) = z −1

(b)
Y (z) = H(z)X(z)
1
X(z) =
1 − z −1

y(n)

25
8

7
8
1 −1
5z

−3
+
1 − z −1
1−
1 − 25 z −1


25 7 1 n
2 n
=
+ ( ) − 3( ) u(n)
8
8 5
5

Y (z) =

+

(c) Determine the response caused by the initial conditions and add it to the response in (b).
3
2
y(n) − y(n − 1) + y(n − 2)
5
25


2  +
3 +
Y (z)z −1 + 1 +
Y (z)z −2 + z −1 + 2
Y + (z) −
5
25

=

0

=

0

Y + (z) =
=
y + (n)

=




1 1 n 12 2 n
( ) − ( ) u(n)
25 5
25 5

=




25
33 1 n 87 2 n
+
( ) − ( ) u(n)
8
200 5
25 5

Therefore, the total step response is
y(n)

2 −1
− 11
25 z
25
(1 − 51 z −1 )(1 − 25 z −1 )
−12
1
25
25
+
1 − 51 z −1
1 − 25 z −1

3.43
[aY (z) + X(z)] z −2

= Y (z)
z −2
Y (z) =
X(z)
1 − az −2
Assume that a > 0. Then
1
1
√ −1 a
√
H(z) = − +
a (1 − az )(1 + az −1 )
1
1
1
1
1
√ −1 +
√ −1
= − +
a 2a 1 − az
2a 1 + az
√ 
1
1 √ n
h(n) = − δ(n) +
( a) + (− a)n u(n)
a
2a
1
Step Response: X(z) =
1 − z −1
z −2
√ −1
√
Y (z) =
−1
(1 − z )(1 − az )(1 + az −1 )
87

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
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Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

=
y(n)

=

1
(a−1)
1 − z −1



+

1

1√
2(a− a)
√ −1
− az

1
1
√
+
a − 1 2(a − a)

+

1√
2(a+ a)
√ −1
+ az

1
√ n
( a) +


√
1
√ (− a)n u(n)
2(a + a)

3.44
y(n)
Y (z)

= −a1 y(n − 1) + b0 x(n) + b1 x(n − 1)
b0 + b1 z −1
=
X(z)
1 + a1 z −1

(a)
H(z)

b0 + b1 z −1
⇒ h(n) = b0 (−a1 )n u(n) + b1 (−a1 )n−1 u(n − 1)
1 + a1 z −1
(b1 − b0 a1 )z −1
⇒ h(n) = b0 δ(n) + (b1 − b0 a1 )(−a1 )n−1 u(n − 1)
= b0 +
1 + a1 z −1
=

(b)
1
1 − z −1
b0 + b1 z −1
Y (z) =
(1 − z −1 )(1 + a1 z −1 )
a1 b0 − b1
1
1
b0 + b1
+
=
1 + a1 1 − z −1
1 + a1 1 + a1 z −1


b0 + b1
a1 b0 − b1
y(n) =
+
(−a1 )n u(n)
1 + a1
1 + a1

Step Response: X(z) =

(c) Let us compute the zero-input response and add it to the response in (b). Hence,


Y + (z) + a1 z −1 Y + (z) + A = 0
−a1 A
Y + (z) =
1 + a1 z −1
⇒ yzi (n) = −a1 A(−a1 )n u(n)
The total response to a unit step is


a1 b0 − b1 − a1 A(1 + a1 )
b0 + b 1
n
+
(−a1 ) u(n)
y(n) =
1 + a1
1 + a1
(d)
x(n)
X(z)
Y (z)

Then, y(n)

= cosw0 nu(n)
1 − z −1 cosw0
=
1 − 2z −1 cosw0 + z −2
(b0 + b1 z −1 )(1 − z −1 cosw0 )
=
(1 + a1 z −1 )(1 − 2z −1 cosw0 + z −2 )
A
B(1 − z −1 cosw0 )
C(z −1 cosw0 )
=
+
+
−1
−1
−2
1 + a1 z
1 − 2z cosw0 + z
1 − 2z −1 cosw0 + z −2
n
= [A(−a1 ) + Bcosw0 + Csinw0 ] u(n)
88

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

where A, B and C are determined from the equations
A+B
(2cosw0 )A + (a1 − cosw0 )B + (sinw0 )C
A − (a1 − cosw0 )B + (sinw0 )C

= b0
= b1 − b0 cosw0

= −b1 cosw0

3.45
y(n)

=

Y (z)

=

1
y(n − 1) + 4x(n) + 3x(n − 1)
2
4 + 3z −1
X(z)
1 − 12 z −1

= ejw0 n u(n)
1
X(z) =
1 − ejw0 z −1
4 + 3z −1
Y (z) =
1 −1
(1 − 2 z )(1 − ejw0 z −1 )
B
A
+
Y (z) =
1 − ejw0 z −1
1 − 12 z −1
5
where A = 1
−
ejw0
2
x(n)

B
Then y(n)
The steady state response is
limn→∞ y(n) ≡ yss (n)

4ejw0 + 3
ejw0 − 12


1 n
jw0 n
u(n)
=
A( ) + Be
2

=

= Bejw0 n

3.46
(a)
H(z)

H(z)|z=1 = 1 ⇒ C

(z − rejΘ )(z − re−jΘ )
z(z + 0.8)
1 − 2rcosΘz −1 + r2 z −2
= C
(1 + 0.8z 1 )
1.8
=
= 2.77
1 − 2rcosΘ + r2
= C

(b) The poles are inside the unit circle,√so the system is stable.
(c) y(n) = −0.8y(n − 1) + Cx(n) − 1.5 3Cx(n − 1) + 2.25Cx(n − 2). Refer to fig 3.46-1.

3.47
(a)
X1 (z)

= z 2 + z + 1 + z −1 + z −2
89

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

x(n)

+

c

+

y(n)

z -1

-0.8

-1.5 3

+

z -1
2.25

Figure 3.46-1:
X2 (z) = 1 + z −1 + z −2
Y (z) = X1 (z)X2 (z)
Hence, x1 (n) ∗ x2 (n)
By one-sided transform:

= z 2 + 2z + 3 + 3z −1 + 3z −2 + 2z −3 + z −4
= y(n)


=
1, 2, 3, 3, 3, 2, 1

X1+ (z) =
X2+ (z) =
Y + (z) =

↑

1 + z −1 + z −2
1 + z −1 + z −2
1 + 2z −1 + 3z −2 + 2z −3 + z −4

Hence, y(n) = {1, 2, 3, 2, 1}
(b) Since both x1 (n) and x2 (n) are causal, the one-sided and two-sided transform yield identical
results. Thus,
Y (z) = X1 (z)X2 (z)
1
=
1 −1
(1 − 2 z )(1 − 13 z −1 )
3
2
=
1 −1 −
1− z
1 − 13 z −1
 2

1
1
Therefore, y(n) =
3( )n − 2( )n u(n)
2
3
(c)
By convolution,
y(n)

= x1 (n) ∗ x2 (n)


=
4, 11, 20, 30, 20, 11, 4
↑

90

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

By one-sided z-transform,
X1+ (z) =

2 + 3z −1 + 4z −2

X2+ (z) = 2 + z −1
Y + (z) = X1+ (z)X2+ (z)
= 4 + 8z −1 + 11z −2 + 4z −3


Therefore, y(n) =
4, 8, 11, 4
↑

(d) Both x1 (n) and x2 (n) are causal. Hence, both types of transform yield the same result, i.e,
X1 (z)
X2 (z)
Then, Y (z)

1 + z −1 + z −2 + z −3 + z −4

=

= 1 + z −1 + z −2
= X1 (z)X2 (z)
1 + 2z −1 + 3z −2 + 3z −3 + 3z −4 + 2z −5 + z −6


=
1, 2, 3, 3, 3, 2, 1

=
Therefore, y(n)

↑

3.48
X + (z)

=
=

∞
X

n=0
∞
X

x(n)z −n
z −n

n=0

=

1
, |z| > 1
1 − z −1

3.49
(a)
Y + (z) +
(a)

 1  −2 +

1  −1 +
z Y (z) + y(−1) −
z Y (z) + z −1 y(−1) + y(−2) = 0
2
4
Hence, Y + (z) =
=
Therefore, y(n)

(b)

=

1 −1
−
4z
1 −1
z
−
2

1
4
1 −2
4z

1+
0.154
0.404
−
1 − 0.31z −1
1 + 0.81z −1
[0.154(0.31)n − 0.404(0.81)n ] u(n)





Y + (z) − 1.5 z −1 Y + (z) + 1 + 0.5 z −2 Y + (z) + z −1 + 0 = 0
1.5 − 0.5z −1
1 − 1.5z −1 + 0.5z −2
2
0.5
=
−
1 − z −1
1 − 0.5z −1
= [2 − 0.5(0.5)n ] u(n)


= 2 − (0.5)n+1 u(n)

Y + (z) =

Therefore, y(n)

91

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

(c)


Y + (z) − 0.5 z −1 Y + (z) + 1

1
1 − 13 z −1

=

1.5 − 61 z −1
(1 − 31 z −1 )(1 − 0.5z −1 )

Y + (z) =

7
2

=

2

−

1−
1 − 13 z −1


7
1
=
(0.5)n − 2( )n u(n)
2
3

Hence, y(n)

0.5z −1

(d)
Y + (z) −


1  −2 +
z Y (z) + 1 =
4

Y + (z) =

1
1 − z −1
5
1 −1
4 − 4z
(1 − z −1 )(1 − 14 z −2 )
4
3

=

z −1

+

−3
8
− 12 z −1

+

7
24
+ 21 z −1

1−
1
1


7
1 n
4 3 1 n
− ( ) + (− ) u(n)
Hence, y(n) =
3 8 2
24 2

3.50
If h(n) is real, even and has a finite duration 2N + 1, then (with M = 2N + 1)
H(z)
since h(n)
H(z)

= h(0) + h(1)z −1 + h(2)z −2 + . . . + h(M − 1)z −(M −1)/2
= h(M − n − 1), then
h
i
= z −(M −1)/2 (h(0) z (M −1)/2 + z −(M −1)/2
i
h
+h(1) z (M −3)/2 + z −(M −3)/2 + . . . + h(M − 1/2))

with M = 2N + 1, the expression becomes


H(z) = z −N (h(0) z N + z −N
i
h
+h(1) z N −1 + z −(N −1)
i
h
+h(2) z N −2 + z −(N −2) + . . . + h(N ))
)
(
N
−1
N
−1
X
X
−(N −n)
−N
N −n
h(n)z
= z
h(n)z
+
h(N ) +
n=0

n=0

= z

−N



h(N ) + P (z) + P (z

−1

)

Now, suppose z1 is a root of H(z), i.e.,
H(z1 )
Then, h(N ) + P (z1 ) + P (z1−1 )


= z1−N h(N ) + P (z1 ) + P (z1−1 ) = 0
=

0.

This implies that H( z11 ) = 0 since we again have
h(N ) + P (z1−1 ) + P (z1 ) =

0.

92

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

3.51
(a)
H(z) =

(z +

z−1
,
+ 3)(z − 2)

ROC:

1
2 )(z

1
< |z| < 2
2

(b) The system can be causal if the ROC is |z| > 3, but it cannot be stable.
(c)
C
B
A
H(z) =
1 −1 + 1 + 3z −1 + 1 − 2z −1
1 + 2z
(1) The system can be causal; (2) The system can be anti-causal; (3) There are two other
noncausal responses.The corresponding ROC for each of these possibilities are :
ROC4 : 2 < |z| < 3;
ROC1 : |z| > 3;
ROC2 : |z| < 3;
ROC3 : 21 < |z| < 2;

3.52
x(n) is causal.
(a)
X(z) =

∞
X

x(n)z −n

n=0

limz→∞ X(z) = x(0)
(b)(i) X(z) =

(z− 21 )4
⇒ limz→∞ X(z) = ∞ ⇒ x(n) is not causal.
(z− 13 )3
1 −2 2
(1− z )
= 1−21 z−1 ⇒ limz→∞ X(z) = 1 Hence X(z) can
3

(ii) X(z)
sequence.
(iii) X(z) =
sequence.

(z− 13 )2
(z− 12 )3

be associated wih a causal

⇒ limz→∞ X(z) = 0. Hence X(z) can be associated wih a causal

3.53
1
1−az −1 , |a| < 1. This system
a−3 z 3
= 1−az−1 the system is stable

The answer is no. For the given system h1 (n) = an u(n) ⇒ H1 (z) =
n

is causal and stable. However when h2 (n) = a u(n + 3) ⇒ H2 (z)
but is not causal.

3.54
Initial value theorem for anticausal signals: If x(n) is anticausal, then x(0) = limz→0 X(z)
P0
Proof: X(z) = n=−∞ x(n)z −n = x(0) + x(−1)z + x(−2)z 2 + . . . Then limz→0 X(z) = x(0)

3.55
1
s(n) = ( )n−2 u(n + 2)
3
(a)
h(n)

= s(n) − s(n − 1)
93

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

1
1
( )n−2 u(n + 2) − ( )n−3 u(n + 1)
3
3
1
4
= 3 δ(n + 2) − 54δ(n + 1) − 18( )n u(n)
3
−18
2
H(z) = 81z − 54z +
1 − 31 z −1
=

=

81z(z −1 )
1 − 13 z −1

H(z) has zeros at z = 0, 1 and a pole at z = 31 .
(b) h(n) = 81δ(n + 2) − 54δ(n + 1) − 18( 13 )n u(n)
(c) The system is not causal, but it is stable since the pole is inside the unit circle.

3.56
(a)
x(n)

=
=

for n ≥ 0, x(n)

=

for n < 0, x(−1)

=
=

x(−2)

=
=

I
z n−1
1
dz
2πj c 1 − 12 z −1
I
zn
1
dz
2πj c z − 21
1
( )n
2 I
1
1
dz
2πj c z(z − 21 )
1
1
|z=0 + |z= 21 = 0
z
z − 21
I
1
1
dz
2πj c z 2 (z − 12 )


1
d
1
|z=0 + 2 |z= 21 = 0
dz z − 21
z

By continuing this process, we find that x(n) = 0 for n < 0.
(b)
X(z) =
x(n)

=

1
1
, |z| <
2
1 − 21 z −1
I
n
1
z
1
dz, where c is contour of radius less than
2πj c z − 21
2

For n ≥ 0, there are no poles enclosed in c and, hence, x(n) = 0. For n < 0, we have
I
1
1
x(−1) =
dz
2πj c z(z − 21 )
1
|z=0 = −2
=
z − 21
I
1
1
dz
x(−2) =
2πj c z 2 (z − 21 )


d
1
|z=0 = −4
=
dz z − 21
94

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Alternatively, we may change variables by letting w = z −1 . Then,
I
1
1
w−n
(− 2 )dw,
x(n) = −
−1
2πj c′ w − 21
w
I
1
−1
= −
dw
2πj c′ wn+1 (1 − 21 w)
I
1
2w−n−1
= −
dw
2πj c′ w − 2)
= −(2)−n ,
n<0
(c)
X(z)
x(n)

For n ≥ 0, x(n)

For n = 0, x(n)

z−a
1
, |z| >
1 − az
|a|
I
1
z
−a
1
dz, c has a radius greater than
z n−1
=
2πj c
1 − az
|a|
I
1
−1 z n−1 (z − a)
=
dz
2πj c a
z − a1
−1 1 n−1 1
( )
( − a)
=
a a
a
1 n+1
1 n−1
−( )
= ( )
a I
a
1
−1 (z − a)
=
dz
2πj c a z(z − a1 )


1
−1 −a
a −a
+ 1
=
a −1
a
a
−1 2
=
(a + 1 − a2 )
a
−1
=
a
=

For n < 0, we let w = z −1 . Then
x(n)

I
1
1
−w−n−1 (w−1 − a)
=
(− 2 )dw,
−1
2πj c′
1 − aw
w
= 0, for n < 0

(d)
X(z) =
=
x(n)

=

1 − 14 z −1
1
, |z| >
2
1 − 61 z −1 − 16 z −2
7
3
10
10
+
1 − 21 z −1
1 + 13 z −1
I 3 n
1
1
10 z
dz +
2πj c z − 21
2πj

I

7 n
10 z
1
c z+ 3

dz

where the radius of the contour c is greater than |z| = 12 . Then, for n ≥ 0


7
1
3 1 n
( ) + (− )n u(n)
x(n) =
10 2
10 3
For n < 0, x(n) = 0
95

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

3.57

X(z)

=
=

x(n)
I
1
zn
dz
2πj c z − a
I
1
zn
dz
2πj c z − a1
I
1
zn
For n < 0,
dz
2πj c z − a
I
1
zn
dz
2πj c z − a1
For n ≥ 0,

=

1 − a2
1
,
a < |z| < , 0 < a < 1
(1 − az)(1 − az −1 )
a
−1
1
+
−1
1 − az
1 − a1 z −1
I
I
n
1
z
zn
1
dz
dz −
2πj c z − a
2πj c z − a1

= an and
=

0

=

0 and

= −a−n

3.58

X(z) =
x(n)

=

x(−18)

=
=
=
=
=

1
2 )(z

1
z 20
, < |z| < 2
− 2)5 (z + 52 )2 (z + 3) 2

(z −
I
z n−1 z 20
1
dz
2πj c (z − 12 )(z − 2)5 (z + 25 )2 (z + 3)
I
z
1
dz
1
5
2πj c (z − 2 )(z − 2) (z + 25 )2 (z + 3)
1
2
1
− 2)5 ( 2 + 52 )2 ( 12
− 21
( 23 )5 (3)2 ( 72 )
5

( 21

+ 3)

−2
(37 )(7)
−32
15309

96

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Chapter 4

4.1
(a) Since xa (t) is periodic, it can be represented by the fourier series
xa (t) =
where ck

=
=
=
=
=

Then, Xa (F ) =

∞
X

ck ej2πkt/τ

k=−∞
Z τ

1
Asin(πt/τ )ej2πkt/τ dt
τ 0
Z τh
i
A
ejπt/τ − e−jπt/τ e−j2πkt/τ dt
j2τ 0

τ
A ejπ(1−2k)t/τ
e−jπ(1+2k)t/τ
−
j2τ j π2 (1 − 2k)
−j π (1 + 2k) 0

 2
A
1
1
+
π 1 − 2k 1 + 2k
2A
π(1 − 4k 2 )
Z ∞
k
xa (t)e−j2π(F − τ )t dt
−∞

=
=

∞
X

k=−∞
∞
X

k=−∞

ck

Z

∞

k

e−j2π(F − τ )t dt

−∞

ck δ(F −

k
)
τ

Hence, the spectrum of xa (t) consists of spectral lines of frequencies τk , k = 0, ±1, ±2, . . . with
amplitude |ck | and phases 6 ck .
Rτ
Rτ
A2
(b) Px = τ1 0 x2a (t)dt = τ1 0 A2 sin2 ( πt
τ )dt = 2
(c) The power spectral density spectrum is |ck |2 , k = 0, ±1, ±2, . . .. Refer to fig 4.1-1.
(d) Parseval’s relation
Px
∞
X

k=−∞

|ck |2

Z
1 τ 2
x (t)dt
τ 0 a
= |ck |2
∞
1
4A2 X
=
2
2
π
(4k − 1)2

=

k=−∞

97

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
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Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

|c0|2
.
|c -1 | 2
.

|c1 | 2
.

|c-2 | 2
.
.

2

|c 2 |
.

. .

. . .
-2

-1

0

1

k

2

Figure 4.1-1:


4A2
2
2
1 + 2 + 2 + ...
π2
3
15

=


2
2
=
1 + 2 + 2 + ...
3
15
∞
X
|ck |2 =
Hence,

1.2337(Infinite series sum to

π2
)
8

4A2
(1.2337)
π2

k=−∞

A2
2

=

4.2
(a)
xa (t)
Xa (F )

= Ae−at u(t),
a>0
Z ∞
Ae−at e−j2πF t dt
=
0

=
=
|Xa (F )|

=

A
e−(a+j2πF )t
−a − j2πF
A
a + j2πF
A
p
2
a + (2πF )2

∞
0

98

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

6

Xa (F ) = −tan−1 (

2πF
)
a

Refer to fig 4.2-1

A = 2, a = 4
0.5

2
phase of Xa(F)

|Xa(F)|

0.4
0.3
0.2
0.1
0
−10

−5

0
−−> F

5

1
0
−1
−2
−10

10

−5

0
−−> F

5

10

Figure 4.2-1:
(b)
Z

Xa (F ) =

=

6

Aeat e−j2πF t dt +

0

Z

∞

Ae−at e−j2πF t dt

0

A
A
+
a − j2πF
a + j2πF
2aA
a2 + (2πF )2
2aA
2
a + (2πF )2
0

=

|Xa (F )|

∞

=

Xa (F ) =

Refer to fig 4.2-2

4.3
(a) Refer to fig 4.3-1.
x(t) =

Xa (F ) =

Z



0

(1 +
−τ

1−
0,

|t|
τ ,

|t| ≤ τ
otherwise

t −j2πF t
)e
dt +
τ

Z

τ
0

(1 −

t −j2πF t
)e
dt
τ

Alternatively, we may find the fourier transform of
 1
τ , −τ < t ≤ 0
y(t) = x′ (t) =
1
τ, 0 < t ≤ τ
99

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

A = 2, a = 6
0.7

0.6

0.5

|Xa(F)|

0.4

0.3

0.2

0.1

0
0

5

10

15

20

25

−−−> F

Figure 4.2-2:

x(t)

−τ

0

|X(F)|

τ

t

0

1/τ

2/τ

Figure 4.3-1:

100

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

F

Then,
Z

Y (F ) =

Z

=

and X(F ) =
=

6

y(t)e−j2πF t dt
−τ
0

1 −j2πF t
e
dt +
τ
−τ

2sin2 πF τ
−
jπF τ
1
Y (F )
j2πF
2

sinπF τ
τ
πF τ
2

sinπF τ
τ
πF τ
0

=

|X(F )|

τ

=

Xa (F ) =

Z

τ

(
0

−1 −j2πF t
)e
dt
τ

(b)
ck

=

1
Tp

Z

Tp /2

x(t)e−j2πkt/Tp dt

−Tp /2
Z 0

=

1
Tp

=


τ sinπkτ /Tp
Tp
πkτ /Tp

(1 +

−τ

(c) From (a) and (b), we have ck =

t −j2πkt/Tp
dt +
)e
τ
2

Z

τ
0

(1 −

t −j2πkt/Tp
dt
)e
τ



1
k
Tp Xa ( Tp )

4.4
(a)
x(n)

=

N

=

ck

=

Hence, c0



. . . , 1, 0, 1, 2, 3, 2, 1, 0, 1, . . .
↑

6



5
1X
x(n)e−j2πkn/6
6 n=0
i
h
−j2πk
−j4πk
−j10πk
−j2πk
= 3 + 2e 6 + e 3 + e 3 + 2e 6


1
πk
2πk
=
3 + 4cos
+ 2cos
6
3
3
4
1
4
9
, c1 = , c2 = 0, c3 = , c4 = 0, c5 =
=
6
6
6
6

(b)
Pt

=
=

5
1X
2
|x(n)|
6 n=0

1 2
(3 + 22 + 12 + 02 + 12 + 22 )
6
101

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

=
Pf

=

19
16
5
X

n=0

|c(n)|

2



4
1
4
9
=
( )2 + ( )2 + 02 + ( )2 + 02 + ( )2 )
16
6
6
6
19
=
16
= Pf
19
=
16

Thus, Pt

4.5
1
x(n) = 2 + 2cosπn/4 + cosπn/2 + cos3πn/4, ⇒ N = 8
2
(a)
ck
x(n)
Hence, c0

7
1X
x(n)e−jπkn/4
8 n=0


3√
3√
11
3√ 1
3√
,2 +
=
2, 1, 2 −
2, , 2 −
2, 1, 2 +
2
2
4
4
2
4
4
1
1
= 2, c1 = c7 = 1, c2 = c6 = , c3 = c5 = , c4 = 0
2
4

=

(b)
P

=

7
X
i=0

|c(i)|

2

=

4+1+1+

=

53
8

1
1
1 1
+ +
+
4 4 16 16

4.6
(a)
x(n)

=
=

ck

=
=
=

π(n − 2)
3
2π(n − 2)
4sin
6
5
1X
x(n)e−2jπkn/6
6 n=0
4sin

5
4X
2π(n − 2) −2jπkn/6
e
sin
6 n=0
6
i
1 h
√ −e−j2πk/3 − e−jπk/3 + e−jπk/3 + e−j2πk/3
3

102

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

=
Hence, c0

=

and |c1 | = |c5 | =
6

c1
c5

6
6

c0



1
2πk
πk −j2πk/3
√ (−j2) sin
e
+ sin
6
3
3
0, c1 = −j2e−j2π/3 , c2 = c3 = c4 = 0, c5 = c∗1

2, |c0 | = |c2 | = |c3 | = |c4 | = 0
π 2π
5π
= π+ −
=
2
3
6
−5π
=
6
= 6 c2 = 6 c3 = 6 c4 = 0

(b)
x(n)
ck

2πn
2πn
+ sin
⇒ N = 15
3
5
= c1k + c2k
= cos

2πn
where c1k is the DTFS coefficients of cos 2πn
3 and c2k is the DTFS coefficients of sin 5 . But

cos

−j2πn
1 j2πn
2πn
= (e 3 + e 3 )
3
2

Hence,
c1k =
Similarly,
sin



1
2,

0,

k = 5, 10
otherwise

−j2πn
2πn
1 j2πn
= (e 5 − e 5 ).
5
2j

Hence,
c2k =
Therefore,





1
2j ,
−1
2j ,

k=3
k = 12
otherwise

0,

ck = c1k + c2k













1
2j ,
1
2,
1
2,
−1
2j ,

0,

k=3
k=5
k = 10
k = 12
otherwise

2πn
1
16πn
1
4πn
(c) x(n) = cos 2πn
3 sin 5 = 2 sin 15 − 2 sin 15 . Hence, N = 15. Following the same method
as in (b) above, we find that
 −1
 4j , k = 2, 7
1
, k = 8, 13
ck =
 4j
0,
otherwise

(d)

N

=

ck

=
=
=

5
4
−j2πnk
1X
x(n)e 5
5 n=0
i
−j4πk
−j6πk
−j8πk
1 h −j2πk
e 5 + 2e 5 − 2e 5 − e 5
5 

2j
2πk
4πk
−sin(
) − 2sin(
)
5
5
5

103

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Therefore, c0
c1
c2
c3
c4

=

0,


2π
4π
2j
−sin( ) + 2sin( )
=
5
5
5


4π
2j
2π
sin( ) − 2sin( )
=
5
5
5
= −c2
= −c1

(e)
N

=

ck

=
=
=

Therefore, c0

=

c1

=

c2

=

c3

=

c4

=

c5

=

6
5
−j2πnk
1X
x(n)e 6
6 n=0
i
−jπk
−j2πk
−j4πk
−j5πk
1h
1 + 2e 3 − e 3 − e 3 + 2e 3
6

1
πk
2πk
1 + 4cos( ) − 2cos(
)
6
3
3
1
2
2
3
0
−5
6
0
2
3

(f)
N

=

ck

=
=
=

(g) N = 1
(h)

Therefore, c0

=

c1

=

c2

=

c3

=

c4

=

=

2

5
4
−j2πnk
1X
x(n)e 5
5 n=0
i
−j2πk
1h
1+e 5
5
πk −jπk
2
cos( )e 5
5
5
2
5
π −jπ
2
cos( )e 5
5
5
2
2π −j2π
cos( )e 5
5
5
3π −j3π
2
cos( )e 5
5
5
2
4π −j4π
cos( )e 5
5
5

ck = x(0) = 1 or c0 = 1
N

104

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

ck

=
=

⇒ c0

=

1
1X
x(n)e−jπnk
2 n=0

1
(1 − e−jπk )
2
0, c1 = 1

4.7
(a)
x(n)

=

7
X

ck e

j2πnk
8

k=0

Note that if ck
7
X
j2πpk
j2πnk
e 8 e 8

k=0

Since ck
We have x(n)

j2πpk

= e 8 , then
7
X
j2π(p+n)k
8
e
=
n=0

=
=

8,
p = −n
0,
p 6= −n
i
i
−j2πk
−j6πk
1 h j2πk
1 h j6πk
=
e 8 +e 8
e 8 −e 8
+
2
2j
= 4δ(n + 1) + 4δ(n − 1) − 4jδ(n + 3) + 4jδ(n − 3), −3 ≤ n ≤ 5

(b)
c0
x(n)

√
√
√
√
3
3
3
3
, c2 =
, c3 = 0, c4 = −
, c5 = −
, c6 = c7 = 0
= 0, c1 =
2
2
2
2
7
X
j2πnk
ck e 8
=
k=0

(c)

√ h
i
j2πn
j4πn
j5πn
3 jπn
e 4 +e 4 −e 4 −e 4
=
2
√ h
πn
πn i jπ(3n−2)
3 sin
e 4
=
+ sin
2
4

x(n)

=

4
X

ck e

j2πnk
8

k=−3

=
=

1 jπn
1 −jπn
1 j3πn
1 −j3πn
+ e 2 + e 2 + e 4 + e 4
2
2
4
4
πn
πn 1
3πn
2 + 2cos
+ cos
+ cos
4
2
2
4

2+e

jπn
4

+e

−jπn
4

4.8
(a)
If k
N
−1
X

ej2πkn/N

=
=

0, ±N, ±2N, . . .

N
−1
X

1=N

n=0

n=0

105

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

N
−1
X

If k

6=

ej2πkn/N

=

0, ±N, ±2N, . . .
1 − ej2πk
1 − ej2πk/N

n=0

=

0

(b) Refer to fig 4.8-1.
(c)
k=2

k=1

s 1(2)

k=3

s 2(4)
s 2(1)

s1(1)
s (0)
1

s1(3)
s1(4)

s 1(5)

s2(0)
s (3)
2

s2(2)

s

(0)
3
s 3 (2)

s (1)
3
s (3)
3
s (5)
3

s (4)
3

s (5)
2

k=4

k=5

s (5)
4
s (2)
4

s (1)
4
s (4)
4

s 5(5)

s (4)
5
s (0)
4
s (3)
4

k=6

s (0)
5

s(3)
5

s (1)
5

s (2)
5

s (0)
6
.
.
.
s6(5)

Figure 4.8-1:
N
−1
X

sk (n)s∗i (n)

=

N
−1
X

ej2πkn/N e−j2πin/N

n=0

n=0

=

N
−1
X

ej2π(k−i)n/N

n=0

= N, k = i
= 0, k =
6 i
Therefore, the {sk (n)} are orthogonal.

4.9
(a)
x(n)

= u(n) − u(n − 6)
106

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

X(w)

∞
X

=

x(n)e−jwn

n=−∞
5
X

=

e−jwn

n=0

1 − e−j6w
1 − e−jw

=
(b)
x(n)

=

X(w)

=
=

2n u(−n)
0
X

2n e−jwn

n=−∞
∞
jw
X

(

e

m=0

=
(c)
x(n)
X(w)

2

)n

2
2 − ejw

1
( )n u(n + 4)
4
∞
X
1
( )n e−jwn
=
4
n=−4
=

∞
X
1
( )m e−jwm )44 ej4w
4
m=0

=

(

=

44 ej4w
1 − 14 e−jw

(d)
x(n)
X(w)

= αn sinw0 nu(n), |α| < 1

 jw0 n
∞
X
e
− e−jw0 n −jwn
e
αn
=
2j
n=0
=
=
=

(e)

∞
∞
1 X h −j(w−w0 ) in
1 X h −j(w+w0 ) in
αe
−
αe
2j n=0
2j n=0


1
1
1
−
2j 1 − αe−j(w−w0 )
1 − αe−j(w+w0 )
αsinw0 e−jw
1 − 2αcosw0 e−jw + α2 e−j2w

x(n)
Note that

∞
X

n=−∞

|x(n)|

π
Suppose that w0 = , so that |sinw0 n| =
2
∞
X
n
|α|
=
n=−∞

n

= |α| sinw0 n, |α| < 1
∞
X
n
|α| |sinw0 n|
=
n=−∞

1.
∞
X

n=−∞

|x(n)| → ∞.

107

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Therefore, the fourier transform does not exist.
(f)
x(n) =

X(w)

=

4
X



2 − ( 12 )n , |n| ≤ 4
0,
otherwise

x(n)e−jwn

n=−4

=

4 
X


1 n −jwn
2−( ) e
2

n=−4
j4w

2e
1 − e−jw

1
− −4ej4w + 4e−j4w − 3ej3w + e−j3w − 2ej2w + 2e−j2w − ejw + e−jw
2
2ej4w
=
+ j [4sin4w + 3sin3w + 2sin2w + sinw]
1 − e−jw

=

(g)
X(w)

∞
X

=

x(n)e−jwn

n=−∞
j2w

= −2e
− ejw + ejw + 2e−j2w
= −2j [2sin2w + sinw]
(h)
x(n) =

X(w)

=

M
X



A(2M + 1 − |n|),
0,

|n| ≤ M
|n| > M

x(n)e−jwn

n=−M

= A

M
X

n=−M

=

(2M + 1 − |n|)e−jwn

(2M + 1)A + A

M
X

k=1

=

(2M + 1)A + 2A

(2M + 1 − k)(e−jwk + ejwk )

M
X

k=1

(2M + 1 − k)coswk

4.10
(a)
x(n)

=
=

x(0)

=

1
2π

Z

π

X(w)ejwn dw

−π
Z −w0

Z π
1
1
ejwn dw
ejwn dw +
2π −π
2π w0
1
1
(π − w0 ) +
(π − w0 )
2π
2π
108

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

π − w0
π
1 jwn −w0
e
|−π
jn
1 −jw0 n
(e
− e−jπn )
jn
1 jwn π
e
|w0
jn
1 jπn
(e
− ejw0 n )
jn
sinnw0
, n 6= 0
−
nπ

=
For n 6= 0,

Z

−w0

ejwn dw

=

−π

=
Z

π

ejwn dw

=

w0

=
Hence, x(n)

=

(b)
X(w)

x(n)

= cos2 (w)
1
1
= ( ejw + e−jw )2
2
2
1 j2w
=
(e
+ 2 + e−j2w )
4 Z
π
1
X(w)ejwn dw
=
2π −π
1
=
[2πδ(n + 2) + 4πδ(n) + 2πδ(n − 2)]
8π
1
[δ(n + 2) + 2δ(n) + δ(n − 2)]
=
4

(c)
x(n)

=

1
2π

=

1
2π

=

Z

X(w)ejwn dw

−π

Z

2
δw
π

π

w0 + δw
2

ejwn dw

w0 − δw
2



sin(nδw/2)
nδw/2



ejnw0

(d)
x(n)

=
=
=

)
(Z
Z π
Z 7π/8
Z 3π/8
π/8
1
jwn
jwn
jwn
jwn
e
dw
e
dw +
e
dw +
Re
2e
dw +
2π
7π/8
6π/8
π/8
0
#
"Z
Z π
Z 7π/8
Z 3π/8
π/8
1
2coswndw
coswndw +
coswndw +
2coswndw +
π 0
7π/8
6π/8
π/8


7πn
6πn
3πn
πn
1
sin
+ sin
− sin
− sin
nπ
8
8
8
8

4.11

xe (n)

=

x(n) + x(−n)
2
109

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.




1
1
, 0, 1, 2, 1, 0,
2
2
↑
x(n) − x(−n)
=
2


1
1
=
, 0, −2, 0, 2, 0,
2
2
↑
=

xo (n)

Then, XR (w)

3
X

=

xe (n)e−jwn

n=−3

jXI (w)

3
X

=

xo (n)e−jwn

n=−3

Now, Y (w) = XI (w) + XR (w)ej2w . Therefore,

y(n) = F −1 {XI (w)} + F −1 XR (w)ej2w

= −jxo (n) + xe (n + 2)


1
j
j
j
1
=
, 0, 1 − , 2, 1 + , 0, − j2, 0,
2
2
2 ↑ 2
2

4.12
(a)
x(n)

=

1
2π

"Z

9π/10
jwn

e

dw +

Z

−8π/10

jwn

e

dw + 2

−9π/10

8π/10

Z

π
jwn

e

dw + 2

9π/10


1
1 j9πn/10
(e
− e−j9πn/10 − ej8πn/10 + e−j8πn/10 )
2π jn

2
+ (−ej9πn/10 + e−j9πn/10 + ejπn − e−jπn )
jn
1
=
[sinπn − sin8πn/10 − sin9πn/10]
nπ
1
[sin4πn/5 + sin9πn/10]
= −
nπ

Z

−9π/10
−π

jwn

e

dw

#

=

(b)
x(n)

=
=
=
=

1
2π

Z

0

−π
Z 0

X(w)ejwn dw +

1
2π

Z

π

X(w)ejwn dw

0

Z π
w
w jwn
1
+ 1)ejwn dw +
e
dw
π
2π
−π
0 π


w jwn π
ejwn 0
1
e
|−π +
|
2π jnπ
jn −π
1
πn
sin e−jnπ/2
πn
2
1
2π

(

(c)
x(n)

=
=

Z wc + w2
Z −wc + w2
1
1
jwn
2e
dw +
2ejwn dw
2π wc − w2
2π −wc − w2


1 jwn wc + w2
1
ejwn −wc + w2
e
|wc − w +
|−wc − w
2
2
π jnπ
jn
110

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.



w
w
w
w
2 ej(wc + 2 )n − ej(wc − 2 )n + e−j(wc − 2 )n − e−j(wc + 2 )n
πn
2j
h
2
w
w i
sin(wc + )n − sin(wc − )n
πn
2
2

=
=

4.13
x1 (n) =



X1 (w)

=

1,
0,

0≤n≤M
otherwise

M
X

e−jwn

n=0

=

x2 (n) =

X2 (w)

−1
X

=



1 − e−jw(M +1)
1 − e−jw

1,
0,

−M ≤ n ≤ −1
otherwise

e−jwn

n=−M

=

M
X

ejwn

n=1

1 − ejwM jw
e
1 − ejw
= X1 (w) + X2 (w)

=
X(w)

=
=
=
=

1 + ejw − ejw − 1 − e−jw(M +1) − ejw(M +1) + ejwM + e−jwM
2 − e−jw − ejw
2coswM − 2cosw(M + 1)
2 − 2cosw
2sin(wM + w2 )cos w2
2sin2 w2
sin(M + 12 )w
sin( w2 )

4.14
P
(a) X(0) = n x(n) = −1
(b) 6 X(w) = Rπ for all w
Rπ
π
1
(c) x(0) = 2π
X(w)dw Hence, −π X(w)dw = 2πx(0) = −6π
−π
(d)
∞
X
X
x(n)e−jnπ =
X(π) =
(−1)n x(n) = −3 − 4 − 2 = −9
n=−∞

(e)

Rπ

−π

|X(w)|2 dw = 2π

P

n

n

|x(n)|2 = (2π)(19) = 38π
111

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

4.15
(a)
X(w)

=

X

x(n)e−jwn

n

X(0)

=

X

x(n)

n

dX(w)
|w=0
dw

= −j
= −j

n

X

nx(n)e−jwn |w=0
nx(n)

n
j dX(w)
dw |w=0

Therefore, c =
(b) See fig 4.15-1

X

X(0)

X(0) = 1 Therefore, c =

0
1

= 0.

dX(w)
dw

w

Figure 4.15-1:

4.16
x1 (n)

≡
F

↔

an u(n)
1
1 − ae−jw
112

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Now, suppose that
xk (n)

(n + k − 1)! n
a u(n)
n!(k − 1)!
1
(1 − ae−jw )k

=
F

↔
holds. Then
xk+1 (n)

(n + k)! n
a u(n)
n!k!
n+k
xk (n)
k
X
X
1
nxk (n)e−jwn +
xk (n)e−jwn
k n
n

=
=

Xk+1 (w)

=

1 dXk (w)
j
+ Xk (w)
k
dw
ae−jw
1
+
−jw
k+1
(1 − ae
)
(1 − ae−jw )k

=
=

4.17
(a)

X
n

X
x∗ (n)e−jwn = (
x(n)e−j(−w)n )∗ = X ∗ (−w)
n

(b)
X

x∗ (−n)e−jwn =

∞
X

x∗ (n)ejwn = X ∗ (w)

n=−∞

n

(c)
X

y(n)e−jwn

X

=

n

n

Y (w)

x(n)e−jwn −

X

n
−jw

= X(w) + X(w)e
= (1 − e−jw )X(w)

x(n − 1)e−jwn

(d)
y(n)

=

n
X

x(k)

k=−∞

= y(n) − y(n − 1)
= x(n)
Hence, X(w)
⇒ Y (w)
(e)
Y (w)

=

(1 − e−jw )Y (w)
X(w)
=
1 − e−jw

=

X

x(2n)e−jwn

n

=

X

w

x(n)e−j 2 n

n

w
= X( )
2
113

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

(f)
Y (w)

=

X
n

=

X

n
x( )e−jwn
2
x(n)e−j2wn

n

= X(2w)

4.18
(a)
X1 (w)

X

=

x(n)e−jwn

n

= ej2w + ejw + 1 + e−jw + e−j2w
=

1 + 2cosw + 2cos2w

(b)
X2 (w)

=

X

x2 (n)e−jwn

X

x3 (n)e−jwn

n

= ej4w + ej2w + 1 + e−j2w + e−j4w
= 1 + 2cos2w + 2cos4w
(c)
X3 (w)

=

n
j6w

= e
=

+ ej3w + 1 + e−j3w + e−j6w

1 + 2cos3w + 2cos6w

(d) X2 (w) = X1 (2w) and X3 (w) = X1 (3w). Refer to fig 4.18-1
(e) If

x( nk ), nk an integer
xk (n) =
0,
otherwise
Then,
Xk (w)

=
=

X

xk (n)e−jwn

n, n
k

an integer

X

x(n)e−jkwn

n

= X(kw)

4.19
(a)
x1 (n)

=

X1 (w)

=

1 jπn/4
(e
+ e−jπn/4 )x(n)
2
π
π i
1h
X(w − ) + X(w + )
2
4
4
114

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

X(w)
1

−π

0

X(w)
2

π

−0.5π

0

0.5π

π

w

X(w)
3

−π/3

0

π/3

w

π

Figure 4.18-1:
(b)
x2 (n)

=

X2 (w)

=

x3 (n)

=

X3 (w)

=

(c)

(d)
x4 (n)
X4 (w)

1 jπn/2
(e
+ e−jπn/2 )x(n)
2j
1 h
π i
π
X(w − ) + X(w + )
2j
2
2
1 jπn/2
(e
+ e−jπn/2 )x(n)
2
1h
π
π i
X(w − ) + X(w + )
2
2
2

1 jπn
(e
+ e−jπn )x(n)
2
1
[X(w − π) + X(w + π)]
=
2
= X(w − π)
=

4.20
cyk

=

N −1
1 X
y(n)e−j2πkn/N
N n=0

115

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

N −1
1 X
N n=0

=

1
N

=

But

∞ N −1−lN
X
X

x(m)e−jw(m+lN )

"

∞
X

l=−∞

#

x(n − lN ) e−j2πkn/N

∞ N −1−lN
X
X

x(m)e−j2πk(m+lN )/N

l=−∞ m=−lN

= X(w)

l=−∞ m=−lN

Therefore, cyk

1
2πk
X(
)
N
N

=

4.21
Let xN (n)

sinwc n
,
−N ≤n≤N
πn
x(n)w(n)
sinwc n
,
−∞≤n≤∞
πn
1,
−N ≤n≤N
0,
otherwise

=
=

where x(n)

=

w(n)

=
=

Then

sinwc n
πn

XN (w)

↔

F

X(w)

=
=

1,
0,

=

X(w) ∗ W (w)
Z π
X(Θ)W (w − Θ)dΘ
−π
Z wc
sin(2N + 1)(w − Θ)/2
dΘ
sin(w − Θ)/2
−wc

=
=

|w| ≤ wc
otherwise

4.22
(a)
X1 (w)

X

=

x(2n + 1)e−jwn

n

X

=

x(k)e−jwk/2 ejw/2

k

w
= X( )ejw/2
2
ejw/2
=
1 − aejw/2
(b)
X2 (w)

=

X

x(n + 2)eπn/2 e−jwn

n

= −

X

x(k)e−jk(w+jπ/2) ej2w

k

116

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

= −X(w +

jπ j2w
)e
2

(c)
X3 (w)

=

X

x(−2n)e−jwn

n

= −

X

x(k)e−jkw/2)

k

w
= X(− )
2
(d)
X4 (w)

=
=
=

X1

2
n
X
1

2

n

(ej0.3πn + e−j0.3πn )x(n)e−jwn
i
h
x(n) e−j(w−0.3π)n + e−j(w+0.3π)n

1
[X(w − 0.3π) + X(w + 0.3π)]
2



(e) X5 (w) = X(w) X(w)e−jw = X 2 (w)e−jw
(f)
X6 (w)

= X(w)X(−w)
1
=
−jw
(1 − ae
)(1 − aejw )
1
=
(1 − 2acosw + a2 )

4.23
P
P
(a) Y1 (w) = n y1 (n)e−jwn = n,n even x(n)e−jwn The fourier transform Y1 (w) can easily be
obtained by combining the results of (b) and (c).
(b)
y2 (n)

= x(2n)
X
=
y2 (n)e−jwn

Y2 (w)

n

=

X

x(2n)e−jwn

n

=

X

x(m)e−jwm/2

m

w
= X( )
2
Refer to fig 4.23-1.
(c)
y3 (n) =



Y3 (w)

=

x(n/2),
0,
X

n even
otherwise

y3 (n)e−jwn

n

117

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Y(w)
2

−π

−π/2

π/2

0

π

3π/2

2π

Figure 4.23-1:
=

X

x(n/2)e−jwn

even
X
=
x(m)e−j2wm
n

m

= X(2w)

We now return to part(a). Note that y1 (n) may be expressed as

y2 (n/2), n even
y1 (n) =
0,
n odd
Hence, Y1 (w) = Y2 (2w). Refer to fig 4.23-2.

Y(w)
3

−π −7π/8

−π/8

0

π/8

π/2 7π/8

π

π/2 3π/4

π

Y(w)
1

−π −3π/4 −π/2 −π/4

0

π/4

Figure 4.23-2:

118

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Chapter 5

5.1
(a) Because the range of n is (−∞, ∞), the fourier transforms of x(n) and y(n) do not exist.
However, the relationship implied by the forms of x(n) and y(n) is y(n) = x3 (n). In this case,
the system H1 is non-linear.
(b) In this case,
X(w)

=

Y (w)

=

Hence, H(w)

=
=
⇒

1
,
1 − 12 e−jw
1
,
1 − 18 e−jw
Y (w)
X(w)
1 − 12 e−jw
1 − 18 e−jw
System is LTI

Note however that the system may also be nonlinear, e.g., y(n) = x3 (n).
(c) and (d). Clearly, there is an LTI system that produces y(n) when excited by x(n), e.g.
H(w) = 3, for all w, or H( π5 ) = 3.
(e) If this system is LTI, the period of the output signal would be the same as the period of the
input signal, i.e., N1 = N2 . Since this is not the case, the system is nonlinear.

5.2
(a)
WR (w)

=

M
X

wR (n)e−jwn

n=0

=

M
X

e−jwn

n=0

1 − e−j(M +1)w
1 − e−jw
sin( M2+1 )w
= e−jM w/2
sin w2
=

119

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

(b) Let wT (n) = hR (n) ∗ hR (n − 1),
hR (n) =



1,
0,

0≤n≤ M
2 −1
otherwise

Hence,
WT (w)

2
= HR
(w)e−jw
!2
sin M
4 w
=
e−jwM/2
sin w2

(c)
Let c(n)
Then, C(w)
Wc (w)

2πn
1
(1 + cos
)
2
M

1
2π
1
2π
= π δ(w) + δ(w −
) + δ(w +
)
−π ≤w ≤π
2
M
2
M
Z π
1
c(Θ)WR (w − Θ)dΘ
=
2π −π
1
1
2π
1
2π
=
WR (w) + WR (w −
) + WR (w +
)
2
2
M
2
M
=

Refer to fig 5.2-1

|W(w)|
T

|W(w)|
R

−2π/Μ+1

0

w

2π/Μ+1

−4π/Μ

|W(w)|
c

−2π/Μ

−2π/Μ+1

2π/Μ+1

0

0

2π/Μ

4π/Μ

w

w

Figure 5.2-1:

5.3
(a)
h(n)

=

1
( )n u(n)
2
120

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

H(w)

∞
X
1
( )n e−jwn
2
n=0

=

∞
X
1
( e−jw )n
2
n=0

=

1

=
|H(w)|

1−

1

1
1
(1 − 2 cosw)2 + ( 21 sinw)2 2
1
5
 12
4 − cosw

=
=

6

1 −jw
2e

= −tan−1

H(w)

1

≡ Θ(w)

1
2 sinw
− 21 cosw

(b) (1)

For the input x(n)

= cos
=

X(w)

=

Y (w)

=
=

y(n)

=

3π
n
10

−j3πn
1 j3πn
(e 10 + e 10 )
2

3π
3π
π δ(w −
) + δ(w +
) , |w| ≤ π
10
10
H(w)X(w)


3π
3π
3π
) + δ(w +
)
H( )π δ(w −
10
10
10


3π
3π
3πn
+ Θ( )
|H( )|cos
10
10
10

(2)

x(n)
First, determine xe (n)
and xo (n)
Then, XR (w)

=



. . . , 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, . . .
↑

x(n) + x(−n)
2
x(n) − x(−n)
=
X 2
=
xe (n)e−jwn



=

n

XI (w)

=

X

xo (n)e−jwn

n

|H(w)|
Θ(w)
and Y (w)


2
XR
(w) + XI2 (w) ,
XI (w)
= tan−1
XR (w)
= H(w)X(w)

=



121

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

5.4
(a)
y(n)
Y (w)
H(w)

x(n) + x(n − 1)
2
1
=
(1 + e−jw )X(w)
2
1
(1 + e−jw )
=
2
w
= (cos )e−jw/2
2
=

Refer to fig 5.4-1.
(b)
1

−−> |H(w)|

0.8
0.6
0.4
0.2
0
−4

−3

−2

−1

0
−−> w

1

2

3

4

−3

−2

−1

0
−−> w

1

2

3

4

−−> theta(w)

2
1
0
−1
−2
−4

Figure 5.4-1:
x(n) − x(n − 1)
2
1
=
(1 − e−jw )X(w)
2
1
(1 − e−jw )
=
2
w
= (sin )e−jw/2 ejπ/2
2

y(n) =
Y (w)
H(w)

Refer to fig 5.4-2.
(c)
y(n) =

x(n + 1) − x(n − 1)
2
122

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1

−−> |H(w)|

0.8
0.6
0.4
0.2
0
0

0.5

1

1.5

2

2.5

3

3.5

2

2.5

3

3.5

−−> w

−−> theta(w)

2
1.5
1
0.5
0
0

0.5

1

1.5
−−> w

Figure 5.4-2:
Y (w)
H(w)

1 jw
(e − e−jw )X(w)
2
1 jw
=
(e − e−jw )
2
= (sinw)ejπ/2
=

Refer to fig 5.4-3.
(d)
y(n)
Y (w)
H(w)

x(n + 1) + x(n − 1)
2
1 jw
(e + e−jw )X(w)
=
2
1 jw
=
(e + e−jw )
2
= cosw
=

Refer to fig 5.4-4
(e)
y(n)
Y (w)
H(w)

x(n) + x(n − 2)
2
1
=
(1 + e−j2w )X(w)
2
1
(1 + e−j2w )
=
2
= (cosw)e−jw
=

Refer to fig 5.4-5.
(f)
123

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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

1

−−> |H(w)|

0.8
0.6
0.4
0.2
0
0

0.5

1

1.5

2

2.5

3

3.5

2

2.5

3

3.5

2

2.5

3

3.5

2

2.5

3

3.5

−−> w
3

−−> theta(w)

2.5
2
1.5
1
0.5
0

0.5

1

1.5
−−> w

Figure 5.4-3:

1

−−> |H(w)|

0.8
0.6
0.4
0.2
0
0

0.5

1

1.5
−−> w

−−> theta(w)

4
3
2
1
0
0

0.5

1

1.5
−−> w

Figure 5.4-4:

124

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

1

−−> |H(w)|

0.8
0.6
0.4
0.2
0
0

0.5

1

1.5

2

2.5

3

3.5

2

2.5

3

3.5

−−> w

−−> theta(w)

2
1
0
−1
−2
0

0.5

1

1.5
−−> w

Figure 5.4-5:

y(n)
Y (w)
H(w)

x(n) − x(n − 2)
2
1
=
(1 − e−j2w )X(w)
2
1
(1 − e−j2w )
=
2
= (sinw)e−jw+jπ/2
=

Refer to fig 5.4-6
(g)
x(n) + x(n − 1) + x(n − 2)
3
1
−jw
(1 + e
+ e−j2w )X(w)
Y (w) =
3
1
H(w) =
(1 + e−jw + e−j2w )
3
1
(1 + ejw + e−jw )e−jw
=
3
1
(1 + 2cosw)e−jw
=
3
1
|H(w)| = | (1 + 2cosw)|
3

−w,
1 + 2cosw > 0
6 H(w) =
π − w, 1 + 2cosw < 0
y(n)

=

Refer to fig 5.4-7.
(h)
125

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

1

−−> |H(w)|

0.8
0.6
0.4
0.2
0
0

0.5

1

1.5

2

2.5

3

3.5

2

2.5

3

3.5

2

2.5

3

3.5

2

2.5

3

3.5

−−> w

−−> theta(w)

2
1
0
−1
−2
0

0.5

1

1.5
−−> w

Figure 5.4-6:

1

−−> |H(w)|

0.8
0.6
0.4
0.2
0
0

0.5

1

1.5
−−> w

2

−−> theta(w)

1
0
−1
−2
−3
0

0.5

1

1.5
−−> w

Figure 5.4-7:

126

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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y(n)
Y (w)
H(w)

= x(n) − x(n − 8)
= (1 − e−j8w )X(w)

= (1 − e−j8w )
= 2(sin4w)ej(π/2−4w)

Refer to fig 5.4-8.
(i)
2

−−> |H(w)|

1.5
1
0.5
0
0

0.5

1

1.5

2

2.5

3

3.5

2

2.5

3

3.5

−−> w

−−> theta(w)

2
1
0
−1
−2
0

0.5

1

1.5
−−> w

Figure 5.4-8:
y(n) = 2x(n − 1) − x(n − 2)
Y (w) = (2e−jw − e−j2w )X(w)
H(w) = (2e−jw − e−j2w )
=

|H(w)|
Θ(w)

2cosw − cos2w − j(2sinw − sin2w)

1
= (2cosw − cos2w)2 + (2sinw − sin2w)2 2


2sinw − sin2w
= −tan−1
2cosw − cos2w

Refer to fig 5.4-9.
(j)
y(n) =
Y (w)

=

H(w)

=
=

x(n) + x(n − 1) + x(n − 2) + x(n − 3)
4
1
−jw
−j2w
(1 + e
+e
+ e−j3w )X(w)
4

1  −jw jw
e
(e + e−jw ) + e−j2w (ejw + e−jw )
3
1 −jw
(e
+ e−j2w )cosw
2
127

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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

3

−−> |H(w)|

2.5
2
1.5
1
0

0.5

1

1.5

2

2.5

3

3.5

2

2.5

3

3.5

−−> w

−−> theta(w)

2
1
0
−1
−2
0

0.5

1

1.5
−−> w

Figure 5.4-9:
=

w
(cosw)(cos )e−j3w/2
2

Refer to fig 5.4-10.
(k)
y(n)
Y (w)
H(w)

x(n) + 3x(n − 1) + 3x(n − 2) + x(n − 3)
8
1
−jw
−j2w
=
(1 + 3e
+ 3e
+ e−j3w )X(w)
8
1
(1 + e−jw )3
=
8
= (cosw/2)3 e−j3w/2
=

Refer to fig 5.4-11.
(l)
y(n)
Y (w)
H(w)
|H(w)|
Θ(w)

= x(n − 4)
= e−j4w X(w)
= e−j4w
= 1
= −4w

Refer to fig 5.4-12.
(m)
y(n)
Y (w)

= x(n + 4)
= ej4w X(w)

H(w)

= ej4w
128

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

1

−−> |H(w)|

0.8
0.6
0.4
0.2
0
0

0.5

1

1.5

2

2.5

3

3.5

2

2.5

3

3.5

2

2.5

3

3.5

2

2.5

3

3.5

−−> w

−−> theta(w)

2
1
0
−1
−2
0

0.5

1

1.5
−−> w

Figure 5.4-10:

1

−−> |H(w)|

0.8
0.6
0.4
0.2
0
0

0.5

1

1.5
−−> w

−−> theta(w)

2
1
0
−1
−2
0

0.5

1

1.5
−−> w

Figure 5.4-11:

129

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

2

−−> |H(w)|

1.5
1
0.5
0
0

0.5

1

1.5

2

2.5

3

3.5

2

2.5

3

3.5

−−> w

−−> theta(w)

2
1
0
−1
−2
0

0.5

1

1.5
−−> w

Figure 5.4-12:
|H(w)|
Θ(w)

= 1
= 4w

Refer to fig 5.4-13.
(n)
y(n)
Y (w)
H(w)

x(n) − 2x(n − 1) + x(n − 2)
4
1
−jw
(1 − 2e
+ e−j2w )X(w)
=
4
1
=
(1 − e−jw )2
4
= (sin2 w/2)e−j(w−π)
=

Refer to fig 5.4-14.

5.5
(a)
y(n)
Y (w)
H(w)

= x(n) + x(n − 10)
= (1 + e−j10w )X(w)
= (2cos5w)e−j5w

Refer to fig 5.5-1.
(b)
130

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

2

−−> |H(w)|

1.5
1
0.5
0
0

0.5

1

1.5

2

2.5

3

3.5

2

2.5

3

3.5

2

2.5

3

3.5

2

2.5

3

3.5

−−> w

−−> theta(w)

2
1
0
−1
−2
0

0.5

1

1.5
−−> w

Figure 5.4-13:

1

−−> |H(w)|

0.8
0.6
0.4
0.2
0
0

0.5

1

1.5
−−> w

−−> theta(w)

4
3
2
1
0
0

0.5

1

1.5
−−> w

Figure 5.4-14:

131

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

2

−−> |H(w)|

1.5
1
0.5
0
0

0.5

1

1.5

2

2.5

3

3.5

2

2.5

3

3.5

−−> w

−−> theta(w)

2
1
0
−1
−2
0

0.5

1

1.5
−−> w

Figure 5.5-1:
π
)
10
π
H( )
3

H(

y(n)

=

0

5π −j 5π
)e 3
3
π
π
5π
5π
−
)
= (6cos )sin( +
3
3
10
3
5π
π 47π
= (6cos )sin( −
)
3
3
30
=

(2cos

(c)
H(0) = 2
4π
H( ) = 2
10
y(n) =

20 + 10cos

2πn π
+
5
2

5.6
h(n)
H(w)

π
Steady State Response: H( )
2
Therefore, yss (n)

= δ(n) + 2δ(n − 2) + δ(n − 4)

= 1 + 2e−j2w + e−j4w
= (1 + e−j2w )2
= 4(cosw)2 e−j2w
=

0

=

0, (n ≥ 4)
132

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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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Transient Response:
ytr (n)

πn
2

π(n−2)
2

π(n−4)
2

=

10e

=

10δ(n) + j10δ(n − 1) + 10δ(n − 2) + j10δ(n − 3)

u(n) + 20e

u(n − 2) + 10e

u(n − 4)

5.7
(a)
y(n)
Y (w)

= x(n) + x(n − 4)
= (1 + e−j4w )X(w)

H(w)

=

(2cos2w)e−j2w

Refer to fig 5.7-1.
(b)
2

−−> |H(w)|

1.5
1
0.5
0
0

0.5

1

1.5

2

2.5

3

3.5

2

2.5

3

3.5

−−> w

−−> theta(w)

2
1
0
−1
−2
0

0.5

1

1.5
−−> w

Figure 5.7-1:
π
π
π
π
= cos n + cos n + cos (n − 4) + cos (n − 4)
2
4
2
4
π
π
π
But cos (n − 4) = cos ncos2π + sin nsin2π
2
2
2
π
= cos n
2
π
π
π
and cos (n − 4) = cos ncosπ − sin nsinπ
4
4
4
π
= −cos n
4
π
Therefore, y(n) = 2cos n
2
(c) Note that H( π2 ) = 2 and H( π4 ) = 0. Therefore, the filter does not pass the signal cos( π4 n).
y(n)

133

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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

5.8
y(n)

=

Y (w)

=

H(w)

=
=

H(0)

=

Hence, yss (n)

=

ytr (n)

=

1
[x(n) − x(n − 2)]
2
1
(1 − e−j2w )X(w)
2
1
(1 − e−j2w )
2
π
(sinw)ej( 2 −w)
π
0, H( ) = 1
2
π
3cos( n + 60o )
2
0

5.9
x(n) = Acos π4 n
(a) y(n) = x(2n) = Acos π2 n ⇒ w = π2
(b) y(n) = x2 (n) = A2 cos2 π4 n = 21 A2 + 21 A2 cos π2 n. Hence, w = 0 and w =
(c)

π
2

y(n) = x(n)cosπn
π
= Acos ncosπn
4
A
5π
A
3π
=
cos n + cos n
2
4
2
4
5π
3π
and w =
Hence, w =
4
4

5.10
(a)
y(n)
Y (w)
H(w)

1
[x(n) + x(n − 1)]
2
1
=
(1 + e−jw )X(w)
2
1
(1 + e−jw )
=
2
w
w
= cos( )e−j 2
2
=

Refer to fig 5.10-1.
(b)
y(n)
Y (w)
|H(w)|
Θ(w)

1
= − [x(n) − x(n − 1)]
2
1
= − (1 − e−jw )X(w)
2
w
= sin
2
w
π
= ej( 2 − 2 )
134

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

1

−−> |H(w)|

0.8
0.6
0.4
0.2
0
0

0.5

1

1.5

2

2.5

3

3.5

2

2.5

3

3.5

−−> w

−−> theta(w)

0
−0.5
−1
−1.5
−2
0

0.5

1

1.5
−−> w

Figure 5.10-1:
Refer to fig 5.10-2.
(c)
1
[x(n) + 3x(n − 1) + 3x(n − 2) + x(n − 3)]
8
1
(1 + e−jw )3 X(w)
=
8
1
=
(1 + e−jw )3
8
3w
w
= cos3 ( )e−j 2
2

y(n) =
Y (w)
H(w)

Refer to fig 5.10-3.

135

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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1

−−> |H(w)|

0.8
0.6
0.4
0.2
0
0

0.5

1

1.5

2

2.5

3

3.5

2

2.5

3

3.5

2

2.5

3

3.5

2

2.5

3

3.5

−−> w

−−> theta(w)

2
1.5
1
0.5
0
0

0.5

1

1.5
−−> w

Figure 5.10-2:

1

−−> |H(w)|

0.8
0.6
0.4
0.2
0
0

0.5

1

1.5
−−> w

−−> theta(w)

2
1
0
−1
−2
0

0.5

1

1.5
−−> w

Figure 5.10-3:

136

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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5.11
y(n)
Y (w)
H(w)
H(w)
or w
|H(w)|

= x(n) + x(n − M )
= (1 + e−jwM )X(w)
= (1 + e−jwM )
1
wM
= (k + )π,
k = 0, 1, . . .
= 0, at
2
2
= (2k + 1)π/M,
k = 0, 1, . . .
wM
= |2cos
|
2

5.12
y(n) = 0.9y(n − 1) + bx(n)

(a)

0.9e−jw Y (w) + bX(w)
Y (w)
H(w) =
X(w)
b
=
1 − 0.9e−jw
|H(0)| = 1, ⇒ b = ±0.1
 wM
cos wM
− 2 ,
2 >0
Θ(w) =
wM
π − 2 , cos wM
2 <0
Y (w)

2

=

b
= 21 ⇒ w0 = 0.105
(b) |H(w0 )|2 = 21 ⇒ 1.81−1.8cosw
0
(c) The filter is lowpass.
(d) For |H(w0 )|2 = 12 ⇒ w0 = 3.036. This filter is a highpass filter.

5.13
(a)
Px

=
=

N −1
1 X
|x(n)|2
N n=0

N
−1
X
k=0

=

c20

|ck |2

+2

N
2

−1
X

k=1

Spurious power
THD

|ck |2

= Px − 2|ck0 |2
Px − 2|ck0 |2
=
Px
2|ck0 |2
= 1−
Px
137

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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1
(b) for f 0 = 96
, refer to fig 5.13-1
1
for f 0 = 32 , refer to fig 5.13-2
1
for f 0 = 256
, refer to fig 5.13-3
1
(c) for f 0 = 96 , refer to fig 5.13-4
1
, refer to fig 5.13-5
for f 0 = 32
1
for f 0 = 256
, refer to fig 5.13-6
The total harmonic distortion(THD) reduces as the number of terms in the Taylor approximation is increased.

terms= 2

terms= 3

−10

50
terms= 5

100

0

−50
0

50
terms= 6

100

0
−20
−40
0

10
0

50
terms= 8

100

50

100

50
terms= 7

100

50

100

4
−−> x(n)

5
−−> x(n)

−−> x(n)

20

−10
0

20
−−> x(n)

0

−20
0

terms= 4

50
−−> x(n)

−−> x(n)

10

0

−5
0

50

100

2
0
−2
0

−−> x(n)

1
0
−1
−2
0

Figure 5.13-1:

138

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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terms= 2

terms= 3

−10

20
terms= 5

20
0
−20
0

40

20
terms= 6

0
−20
−40
0

40

10
0

20
terms= 8

40

20

40

20
terms= 7

40

20

40

2
−−> x(n)

2
−−> x(n)

−−> x(n)

20

−10
0

20
−−> x(n)

0

−20
0

terms= 4

40
−−> x(n)

−−> x(n)

10

0
−2
−4
0

20

1
0
−1
0

40

−−> x(n)

1
0
−1
−2
0

Figure 5.13-2:

terms= 2

terms= 3

−10

100
200
terms= 5

300

−50
0

100
200
terms= 6

300

0
−20
−40
0

0

100
200
terms= 8

300

100

300

100
200
terms= 7

300

100

300

4
−−> x(n)

20

−20
0

0

5
−−> x(n)

−−> x(n)

40

20
−−> x(n)

0

−20
0

terms= 4

50
−−> x(n)

−−> x(n)

10

0
−5
−10
0

100

200

300

2
0
−2
0

200

−−> x(n)

1
0
−1
−2
0

200

Figure 5.13-3:

139

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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0

0

psd

0

−200
0

−200

0.5
1
terms=4 thd=0.5283

−400
0

−50

0.5
1
terms=5 thd=0.6054

−100
0

50

50

0

0

0

−50
−100
0

−50

0.5
1
terms=7 thd=0.06924

100

psd

0
−100
−200
0

psd

50

psd

psd

terms=3 thd=0.4379
50

−100

psd

terms=2 thd=0.06186
200

psd

psd

orig cos thd=2.22e−16
100

0.5
1
terms=6 thd=0.6295

−50

−100
−100
0
0.5
1
0
terms=8 thd=0.002657
100

0.5

1

0
−100

0.5

1

−200
0

0.5

1

Figure 5.13-4:

terms=2 thd=0.07905

0

0

0

−50

0.5
1
terms=4 thd=0.5312

−100
0

−50

0.5
1
terms=5 thd=0.5953

−100
0

50

50

0

0

0

−50
−100
0

−50

0.5
1
terms=7 thd=0.05309

50

psd

0
−50
−100
0

psd

50

psd

psd

−100
0

psd

50

−50

psd

terms=3 thd=0.4439

50

psd

psd

orig cos thd=0
50

0.5
1
terms=6 thd=0.6509

−50

−100
−100
0
0.5
1
0
terms=8 thd=0.001794
50

0.5

1

0
−50

0.5

1

−100
0

0.5

1

Figure 5.13-5:

140

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

−100

−100

0.5
1
terms=4 thd=0.5271

−100

0.5
1
terms=5 thd=0.6077

−200
0
100

0

0

0

−200
0

psd

100

−100

−100

0.5
1
terms=7 thd=0.07458

100
0

psd

psd

−200
0

0

100

psd

psd

−200
0

0

terms=3 thd=0.4357
100

psd

0

psd

psd

orig cos thd=−6.661e−16
terms=2 thd=0.05647
100
100

−100

0.5
1
terms=6 thd=0.6238

−100

−200
−200
0
0.5
1
0
terms=8 thd=0.002976
100

0.5

1

0
−100

−200
0

0.5

1

−200
0

0.5

1

Figure 5.13-6:

5.14
(a) Refer to fig 5.14-1
1
(b) f0 = 50
1

1.5
1
−−> xq(n)

−−> x(n)

0.5
0

0.5
0
−0.5

−0.5
−1
−1
0

100

200

−1.5
0

300

−−> n

100

200

300

−−> n

Figure 5.14-1:

(c) f0 =

bits
THD

4
9.4616e − 04

6
5.3431e − 05

8
3.5650e − 06

16
4.2848e − 11

bits
THD

4
9.1993e − 04

6
5.5965e − 05

8
3.0308e − 06

16
4.5383e − 11

1
100

141

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

(d) As the number of bits are increased, THD is reduced considerably.

5.15
(a) Refer to fig 5.15-1
(b) Refer to fig 5.15-2
f=0.25

f=0.2

1

1

0.5

0.5

0

0

−0.5

−0.5

−1
0

50

−1
0

100

f=0.1

−14

1

4

0.5

2

0

0

−0.5

−2

−1
0

50

50

x 10

−4
0

100

100

f=0.5

50

100

Figure 5.15-1:
The response of the system to xi (n) can be seen from fig 5.15-3

5.16
(a)
H(w)

=

∞
X

h(n)e−jwn

n=−∞

=
=
=

−1
X

∞
X
1
1
( )−n e−jwn +
( )n e−jwn
3
3
n=−∞
n=0

1

1 jw
2e
− 31 ejw

+

4
5 − 3cosw

1
1 − 31 ejw

142

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

1

magnitude

0.8
0.6
0.4
0.2
0
0

0.05

0.1

0.15

0.2

0.25
freq(Hz)

0.3

0.35

0.4

0.45

0.5

0.05

0.1

0.15

0.2

0.25
freq(Hz)

0.3

0.35

0.4

0.45

0.5

phase

0

−0.5

−1
0

Figure 5.15-2:
f=0.25

f=0.2

0.1

0.15
0.1

0.05

0.05
0
0
−0.05
−0.1
0

−0.05
50

−0.1
0

100

f=0.1

50
−15

0.3

2

0.2

x 10

100

f=0.5

0

0.1
−2
0
−4

−0.1
−0.2
0

50

−6
0

100

50

100

Figure 5.15-3:
143

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

6

|H(w)|

=

H(w)

=

4
5 − 3cosw
0

(b) (1)
3πn
= cos
 8

3πn
3πn
X(w) = π δ(w −
) + δ(w +
) , −π ≤ w ≤ π
8
8
Y (w) = H(w)X(w)


3πn
4π
3πn
δ(w −
=
) + δ(w +
)
8
8
5 − 3cos 3π
8
3πn
Hence, the output is simply y(n) = Acos
8
3π
= H( )
where A = H(w)|w= 3π
8
8
(2)


x(n) =
. . . , −1, 1, −1, 1, −1, 1, −1, 1, −1, . . .
x(n)

↑

H(w)|w=π
y(n)
Y (w)

= cosπn, −∞ < n < ∞
4
1
4
= =
=
5 − 3cosπ
8
2
1
=
cosπn
2
π
=
[δ(w − π) + δ(w + π)]
2

5.17
(a)
y(n)
h(n)

= x(n) − 2cosw0 x(n − 1) + x(n − 2)
= δ(n) − 2cosw0 δ(n − 1) + δ(n − 2)

(b)
H(w)

|H(w)|

⇒ |H(w)|

= 1 − 2cosw0 e−jw + e−j2w
= (1 − e−jw0 e−jw )(1 − ejw0 ejw )
w − w0
w + w0
sin
= −4e−jw sin
2
2
= −2e−jw (cosw − cosw0 )
= 2|cosw − cosw0 |
= 0 at w = ±w0

Refer to fig 5.17-1.
(c)
when w0 = π/2, H(w)
at w = π/3, H(π/3)
y(n)

=

1 − ej2w

1 − ej2π/3 = 1ejπ/3
π
= |H(π/3)|3cos( n + 30o − 60o )
3
π
o
= 3cos( n − 30 )
3
=

144

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

w0 = pi/3
3

−−> |H(w)|

2.5
2
1.5
1
0.5
0
0

0.5

1

1.5

2

2.5

3

3.5

2

2.5

3

3.5

−−> w

−−> theta(w)

2
1
0
−1
−2
0

0.5

1

1.5
−−> w

Figure 5.17-1:

5.18
(a)
y(n)
H(w)

= x(n) − x(n − 4)

= 1 − e−j4w
= 2e−j2w ejπ/2 sin2w

Refer to fig 5.18-1.
(b)
x(n)
y(n)

π
π
π
= cos n + cos n,
H( n) = 0
2
4
2
π
π
π
6 H( ) = 0
= 2cos n,
H( ) = 2,
4
4
4

(c) The filter blocks the frequency at w =

π
2.

5.19
y(n)
H(w)

1
[x(n) − x(n − 2)]
2
1
=
(1 − e−j2w )
2
= e−jw ejπ/2 sinw
=

145

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

2

−−> |H(w)|

1.5
1
0.5
0
0

0.5

1

1.5

2

2.5

3

3.5

2

2.5

3

3.5

−−> w

−−> theta(w)

2
1
0
−1
−2
0

0.5

1

1.5
−−> w

Figure 5.18-1:
x(n)
H(0)
y(n)

π
5 + 3sin( n + 60o ) + 4sin(πn + 45o )
2
π
H(π) = 0
= 0,
H( ) = 1,
2
π
= 3sin( n + 60o )
2
=

5.20
(a)
y(n)
Y (w)

= x(2n) ⇒ This is a linear, time-varying system
∞
X
y(n)e−jwn
=
=

n=−∞
∞
X

x(2n)e−jwn

n=−∞

w
= X( )
2
=

1,

=

0,

|w| ≤

π
2

π
≤ |w| ≤ π
2

(b)
y(n)

= x2 (n) ⇒ This is a non-linear, time-invariant system
146

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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Y (w)

=

1
X(w) ∗ X(w)
2π

Refer to fig 5.20-1.
(c)

Y(w)
1/4

0

−π/2

π/2

w

Figure 5.20-1:
y(n)
Y (w)

=

(cosπn)x(n) ⇒ This is a time-varying system
1
[πδ(w − π) + πδ(w + π)] ∗ X(w)
=
2π
1
[X(w − π) + X(w + π)]
=
2
3π
= 0,
|w| ≤
4
3π
1
,
≤ |w| ≤ π
=
2
4

5.21


1 n
π
h(n) = ( ) cos nu(n)
4
4

(a)

1 − 14 cos π4 z −1
1 − 2( 14 )cos π4 z −1 + ( 41 )2 z −2

H(z) =
=
(b) Yes. Refer to fig 5.21-1
π
(c) Poles at z = 41 e±j 4 , zeros at z =
H(w) =
(d)

√
1− 82 e−jw
√
1 −j2w
e
1− 42 e−jw + 16

1−

√

2 −1
8 z
1 −2
2 −1
+ 16
z
4 z

1−

√

√

2
8 .

. Refer to fig 5.21-2.

x(n)

=

X(z)

=

1
( )n u(n)
4
1
1 − 14 z −1
147

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

x(n)

+

y(n)

+

z -1

z -1
- 2 /8

2 /4

+

z

-1

-1/16

Figure 5.21-1:
Y (z) = X(z)H(z)
=

y(n)

=

√
1
2 −1
1
(1
−
)
2
2√
8 z
+
1 −1
1
2
−1
1 − 4z
1 − 4 z + 16 z −2
√ √
1+ 2 2 −1
8 z
√2
+
1 −2
2 −1
z
1 − 4 z + 16

√
1 1 nh
π i
π
( ) 1 + cos n + (1 + 2)sin n u(n)
2 4
4
4

5.22
y(n) = x(n) − x(n − 10)
(a)
H(w)

=

1 − e−j10w
π

|H(w)|
Θ(w)

= 2e−j5w ej 2 sin5w
= 2|sin5w|,
π
=
− 5w, for sin5w > 0
2
π
=
− 5w + π, for sin5w < 0
2

Refer to fig 5.22-1.
(b)
148

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
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1.4

1.3

−−> |H(w)|

1.2

1.1

1

0.9

0.8
0

0.5

1

1.5

2

2.5

3

3.5

−−> w

Figure 5.21-2:

2

−−> |H(w)|

1.5
1
0.5
0
0

0.5

1

1.5

2

2.5

3

3.5

2

2.5

3

3.5

−−> w

−−> theta(w)

2
1
0
−1
−2
0

0.5

1

1.5
−−> w

Figure 5.22-1:

149

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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π
)|
10
π
|H( )|
3

π
)=0
10
π
π
π
3,
Θ( ) = 6 H( ) = −
=
3
3
6
√
π
π
π
= 2cos n + 3 3sin( n − )
10
3
15
2π
= 0,
H( ) = 0
5
= 0

|H(

(1)

=

Hence, y(n)
H(0)

(2)

Hence, y(n)

2,
√

6

H(

5.23
(a)
h(n)

=
=
=
=

Z

1
2π

π

X(w)ejwn dw

−π

"Z

1
2π

3π
8

jwn

e

− 3π
8

dw −

Z

π
8

−jwn

e

dw

−π
8

#



1
3π
π
sin n − sin n
πn
8
8
π
π
2
sin ncos n
πn
8
4

(b) Let
h1 (n) =
Then,
H1 (w) =
and



2,
0,

2sin π8 n
nπ
|w| ≤ π8
π
8 < |w| < π

π
h(n) = h1 (n)cos n
4

5.24
y(n)

=

Y (z)

=

H(z)

=
=

1
1
y(n − 1) + x(n) + x(n − 1)
2
2
1
1 −1
z Y (z) + X(z) + z −1 X(z)
2
2
Y (z)
X(z)
1 + 21 z −1
1 − 21 z −1

(a)
H(z) =
h(n)

=

2

−1
1 − 21 z −1
1
2( )n u(n) − δ(n)
2
150

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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(b)
H(w)

=

∞
X

h(n)e−jwn

n=0

=

1−

2
1 −jw
2e
1 −jw
2e
1 −jw
2e

−1

1+
1−
= H(z)|z=ejw

=

(c)
π
H( ) =
2
=

π

1 + 12 e−j 2
π
1 − 12 e−j 2
1 − j 12
1 + j 12

−1 1

1e−j2tan 2
π
π
1
= cos( n + − 2tan−1 )
2
4
2

=
Hence, y(n)

5.25
Refer to fig 5.25-1.

5.26

H(z) =
H(w)

=
=
=

y(n) =
for x(n)

=

y(0)

=

y(1)

=

y(2)

=

y(3)

=

y(4)

=

π

π

(1 − ej 4 z −1 )(1 − e−j 4 z −1 )
√
1 − 2z −1 + z −2
√
1 − 2e−jw + e−2jw
√
2
−jw
2e
(cosw −
)
2
√
x(n) − 2x(n − 1) + x(n − 2)
π
sin u(n)
4
x(0) = 0
√
√
2
x(1) − 2x(0) + x(−1) =
2 √
√ 2
√
x(2) − 2x(1) + x(0) = 1 − 2
+0=0
2 √
√
√
2 √
2
x(3) − 2x(2) + x(1) =
− 2+
=0
2
2
0
151

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|X(w)| for (b)
8

3

6
magnitude

magnitude

|X(w)| for (a)
4

2
1

4
2

0
0

1

2

3

0
0

4

1

|X(w)| for (c)

2

3

4

3

4

|X(w)| for (d)

1.5

12

magnitude

magnitude

10

1

8
6
4
2

0.5
0

1

2

3

0
0

4

1

2

Figure 5.25-1:

5.27
−1

1−z
(a) H(z) = k 1+0.9z
−1 . Refer to fig 5.27-1.
(b)

H(w)
|H(w)|
Θ(w)

1 − e−jw
1 + 0.9e−jw
2|sin w2 |
= k√
1.81 + 1.8cosw
0.9sinw
sinw
+ tan−1
= tan−1
1 − cosw
1 + 0.9cosw
= k

−jπ

1−e
2
(c) H(π) = k 1+0.9e
−jπ = k 0.1 = 20k = 1 ⇒ k =
1
[x(n) − x(n − 1)]
(d) y(n) = −0.9y(n − 1) + 20
(e)

π
H( ) =
6
y(n)

=

1
20

π

0.014ejΘ( 6 )
π
0.028cos( n + 134.2o )
6

5.28
−1

1+bz
(a) H(z) = b0 1+az
−1 . Refer to fig 5.28-1.

(b) For a = 0.5, b = −0.6,

−1

H(z) = b0 1−0.6z
1+0.5z −1 . Since the pole is inside the unit circle and the
152

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Figure 5.27-1:
filter is causal, it is also stable. Refer to fig 5.28-2.
(c)
H(z)
⇒ |H(w)|

2

1 + 0.5z −1
1 − 0.5z −1
5
+ cosw
= b20 45
4 − cosw
= b0

The maximum occurs at w = 0. Hence,
9

H(w)|w=0

= b20 41
4

9b20 = 1
1
= ±
3

=
⇒ b0

(d) Refer to fig 5.28-3.
(e) Refer to fig 5.28-4.
obviously, this is a highpass filter. By selecting b = −1, the frequency response of the
highpass filter is improved.

5.29
|H(w)|

2

d
1
dw |H(w)|2

=
=

A
[1 + r2 − 2rcos(w − Θ)] [1 + r2 − 2rcos(w + Θ)]
1
[2rsin(w − Θ)(1 + r2 − 2rcos(w + Θ))
A
+2rsin(w + Θ)(1 + r2 − 2rcos(w − Θ))]
153

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Direct form I:
x(n)

b0

y(n)
+

+

-1
z

-1
z

b b0

-a

Direct form II :
x(n)

+

+

b0

y(n)

-1
z
-a

b

Figure 5.28-1:

z-plane

Figure 5.28-2:

154

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

|H(w)|
1

|H(w)|

0.8
0.6
0.4
0.2
0
0

0.5

1

1.5

2

2.5

3

3.5

2

2.5

3

3.5

w
phase
0

phase

−0.2
−0.4
−0.6
−0.8
−1
0

0.5

1

1.5
w

Figure 5.28-3:

-0.8

-b
z-plane

Figure 5.28-4:

155

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
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= 0
(1 + r )(sin(w − Θ) + sin(w + Θ)) = 2r [sin(w − Θ)cos(w + Θ) + sin(w + Θ)cos(w − Θ)]
2

(1 + r2 )2sinwcosΘ
Therefore, cosw
wr

= 2rsin2w
= 4rsinwcosw
1 + r2
=
cosΘ
2r 

2
−1 1 + r
= cos
cosΘ
2r

5.30

y(n) =
H(w)

=
=
=

|H(w)|

=

Θ(w)

=

1
1
1
x(n) + x(n − 1) + x(n − 2)
4
2
4
1 1 −jw 1 −j2w
+ e
+ e
4 2
4
1 + e−jw 2
)
(
2
w
e−jw cos2
2
2w
cos
2
6 H(w) = −w

Refer to fig 5.30-1

5.31
(a)
x(n)

=

X(z) =
Hence, H(z) =
=

1
( )n u(n) + u(−n − 1)
4
−1
1
1
+
, ROC: < |z| < 1
1 − z −1
4
1 − 14 z −1
Y (z)
X(z)
1 − z −1
, ROC: |z| < 1
1 + z −1

(b)
Y (z)

=

− 34 z −1
(1 − 41 z −1 )(1 + z −1 )

3
− 53
5
1 −1 + 1 + z −1
1 − 4z
3 1
3
= − ( )n u(n) − (−1)n u(−n − 1)
5 4
5

=
y(n)

156

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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1

−−> |H(w)|

0.8
0.6
0.4
0.2
0
−4

−3

−2

−1

0
−−> w

1

2

3

4

−3

−2

−1

0
−−> w

1

2

3

4

−−> theta(w)

4
2
0
−2
−4
−4

Figure 5.30-1:

5.32
y(n)
H(w)

= b0 x(n) + b1 x(n − 1) + b2 x(n − 2)
= b0 + b1 e−jw + b2 e−j2w

(a)
2π
)
3
H(0)
For linear phase, b0
H(

select b0
These conditions yield

= b0 + b1 e−j

2π
3

+ b2 e−j

4π
3

=0

= b0 + b1 + b2 = 1
= ±b2 .

= b2 (otherwise b1 = 0).

b 0 = b1 = b2

=

Hence, H(w)

=

1
3
1 −jw
e
(1 + 2cosw)
3

(b)
H(w) =
Θ(w) =



1
(1 + 2cosw)
3

−w,
−w + π,

for 1 + 2cosw > 0
for 1 + 2cosw < 0

157

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Refer to fig 5.32-1.

1

−−> |H(w)|

0.8
0.6
0.4
0.2
0
0

0.5

1

1.5

2

2.5

3

3.5

2

2.5

3

3.5

−−> w
2

−−> theta(w)

1
0
−1
−2
−3
0

0.5

1

1.5
−−> w

Figure 5.32-1:

5.33
(a)
y(n)

=

M
X
1
x(n − k)
2M + 1
k=−M

H(w)

=

=

M
X
1
e−jwk
2M + 1
k=−M
#
"
M
X
1
coswk
1+2
2M + 1
k=1

(b)
y(n)

=

H(w)

=

M
−1
X
1
1
1
x(n + M ) +
x(n − M )
x(n − k) +
4M
2M
4M
k=−M +1
#
"
M
−1
X
1
1
coswk
1+2
cosM w +
2M
2M
k=1

158

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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The filter in (b) provides somewhat better smoothing because of its sharper attenuation at
the high frequencies.

5.34
H(z) =
=
H(w)

=
=
=

|H(w)|
Θ(w)

H(w)

=
=

1 + z + z2 + . . . + z8
1 − z9
1 − z −1
1 − e−j9w
1 − e−jw
e−j9w/2 sin9w/2
e−jw/2 sinw/2
sin9w/2
e−j4w
sinw/2
sin9w/2
|
|
sinw/2
−4w, when sin9w/2 > 0

= −4w + π, when sin9w/2 < 0
2πk
= 0, at w =
, k = 1, 2, . . . , 8
9

The corresponding analog frequencies are

kFs
9 ,

k = 1, 2, 3, 4, or 19 kHz, 92 kHz, 93 kHz, 94 kHz.

5.35
Refer to fig 5.35-1.

l
1/2

Figure 5.35-1:
159

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

H(w)

(1 − ej3π/4 z −1 )(1 − e−j3π/4 z −1 )
(1 − 12 z −1 )2
= H(z)|z=ejw

H(0)

= G

H(z) = G

(1 − ej3π/4 )(1 − e−j3π/4 )
(1 − 12 )2

|H(w)|

=

1⇒G

l2

=

2+

√

l2
1
4

=1

2
1
√ = 0.073
4(2 + 2)

G =

5.36
Hz (w)

=
=

1 − rejθ e−jw

1 − rcos(w − θ) + jrsin(w − θ)

(a)
|Hz (w)|
20log10 |Hz (w)|

1

= {[1 − rcos(w − θ)]2 + [rsin(w − θ)]2 } 2
=
=

1

[1 + r2 − 2rcos(w − θ)] 2
10log10 [1 − 2rcos(w − θ) + r2 ]

Hence proved.
(b)
Θz (w)

imag. part
real part
rsin(w − θ)
= tan−1
1 − rcos(w − θ)
= tan−1

Hence proved.
(c)
τgz (w)

= −
= −
=

dΘz (w)
dw
1
1+

r 2 sin2 (w−θ)
[1−rcos(w−θ)]2

[1 − rcos(w − θ)]rcos(w − θ) − rsin(w − θ)(rsin(w − θ))
[1 − rcos(w − θ)]2

r2 − rcos(w − θ)
1 + r2 − 2rcos(w − θ)

Hence proved.
(d) Refer to fig 5.36-1.

5.37
Hp (w) =

1
,
1 − rejθ e−jw

r<1

160

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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magnitude theta=0

phase theta=0

10

group delay theta=0

1

20

0

10
0

−10

0

−20
−5
0
5
magnitude theta=1.571
10

−1
−5

0
5
phase theta=1.571

1

0

−10
−5
0
5
group delay theta=1.571
20
10

0
−10

0

−20
−5
0
5
magnitude theta=3.142
10

−1
−5

0
5
phase theta=3.142

1

0

−10
−5
0
5
group delay theta=3.142
20
10

0
−10
−20
−5

0

0

5

−1
−5

0

5

−10
−5

0

5

Figure 5.36-1:

161

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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(a)
|Hp (w)|

1

=

θ)]2

{[1 − rcos(w −
1
|Hz (w)|
1
20log10 (
)
|Hz (w)|
−20log10 |Hz (w)|
−|Hz (w)|dB

=
|Hp (w)|dB

=
=
=

1

+ [rsin(w − θ)]2 } 2

Hence proved.
(b)
1 − rcos(w − θ) − jrsin(w − θ)
[1 − rcos(w − θ)]2 + [rsin(w − θ)]2
rsin(w − θ)
= −tan−1
1 − rcos(w − θ)
= −Θz (w)

Hp (w)

=

Θp (w)

Hence proved.
(c)
τgp (w)

dΘp (w)
dw
d(−Θz (w))
= −
dw
dΘz (w)
=
dw
= −τgz (w)
= −

Hence proved.

5.38
Hz (w)

= (1 − rejθ e−jw )(1 − re−jθ e−jw )
= (1 − re−j(w−θ) )(1 − re−j(w+θ) )
= A(w)B(w)

(a)
|Hz (w)|
|Hz (w)|dB

= |A(w)b(w)|

= |A(w)||B(w)|
= 20log10 |Hz (w)|
=

10log10 [1 − 2rcos(w − θ) + r2 ] + 10log10 [1 − 2rcos(w + θ) + r2 ]

(b)
6

Hz (w)

=

A(w) + 6 B(w)
rsin(w + θ)
rsin(w − θ)
+ tan−1
= tan−1
1 − rcos(w − θ)
1 − rcos(w + θ)
6

162

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
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(c)
τgz (w)

dΘz (w)
dw
z
= τA (w) + τgB (w)
= −

r2 − rcos(w + θ)
r2 − rcos(w − θ)
+
2
1 + r − 2rcos(w − θ) 1 + r2 − 2rcos(w + θ)

=
(d)

Hp (w)
Therefore, |Hp (w)|
|Hp (w)|dB
on the same lines of prob4.62
Θp (w)
τgp (w)

1
Hz (w)
1
=
|Hz (w)|
= −|Hz (w)|dB
=

= −Θz (w) and
= −τgz (w)

(e) Refer to fig 5.38-1.
magnitude theta=0

phase theta=0

10

1.5

5

1

0

0.5

−5

0

−10

−0.5

−15

−1

group delay theta=0
200
150
100
50

−20
−5

0

5

−1.5
−5

magnitude theta=1.571

0

5

phase theta=1.571

10

1.5

5

1

0

0.5

−5

0

−10

−0.5

−15

−1

0
−5

0

5

group delay theta=1.571
200
150
100
50

−20
−5

0

5

−1.5
−5

0

5

0
−5

0

5

Figure 5.38-1:

163

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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5.39
(a)
|H1 (w)|

2

=
=

2

|H1 (w)| =

1
⇒ cosw1
2

=

(1 − a)2
(1 − acosw)2 + a2 sin2 w
(1 − a)2
1 + a2 − 2acosw
4a − 1 − a2
2a

(b)
|H2 (w)|

2

=
=

2

|H2 (w)| =

1
⇒ cosw2
2

=

1 − a 2 (1 + cosw)2 + sin2 w
)
2
(1 − acosw)2 + a2 sin2 w
(1 − a)2 2(1 + cosw)
2
1 + a2 − 2acosw
2a
1 + a2

(

By comparing the results of (a) and (b), we find that cosw2 > cosw1 and, hence w2 < w1
Therefore, the second filter has a smaller 3dB bandwidth.

5.40
h(n)

= cos(w0 n + Θ)
= cosw0 ncosΘ − sinw0 nsinΘ

use the coupled-form oscillator shown in figure 5.38 and multiply the two outputs by cosΘ
and sinΘ, respectively, and add the products, i.e.,
yc (n)cosΘ + ys (n)sinΘ = cos(w0 n + Θ)

5.41
(a)
y(n)
yR (n − 1) + jyI (n − 1)

= ejw0 y(n − 1) + x(n)

= (cosw0 + jsinw0 ) [yR (n − 1) + jyI (n − 1)] + x(n)
= yR (n − 1)cosw0 − yI (n − 1)sinw0 + x(n)
+j [yR (n − 1)sinw0 + yI (n − 1)cosw0 ]

(b)Refer to fig 5.41-1.
(c)
Y (z) = ejw0 z −1 Y (z) + 1
1
=
jw
1 − e 0 z −1
y(n) = ejnw0 u(n)
Hence, yR (n)
yI (n)

= [cosw0 n + jsinw0 n] u(n)
= cosw0 nu(n)
= sinw0 nu(n)
164

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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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x(n)

+

y (n)
R

+

cos w 0

-sin w0

sin w0

z -1

z -1
yI (n)

+

cos w0

Figure 5.41-1:
(d)
n
yR (n)
yI (n)

0 1√
1 23
0 12

2
1
2
√

3
2

3
0
1

4
1
−
√2
3
2

5√
− 23
1
2

6
7√
−1 − 23
0
− 21

8
1
−√
2
− 23

9
0
1

5.42
(a) poles: p1,2 = re±jw0
zeros: z1,2 = e±jw0
(b) For w = w0 , H(w0 ) = 0 For w 6= w0 , the poles and zeros factors in H(w) cancel, so that
H(w) = 1. Refer to fig 5.42-1.
(c)
2

|H(w)|

2

where w0
d|H(w)|
dw

= G2

|1 − ejw0 e−jw | |1 − e−jw0 e−jw |
2

2
2

|1 − rejw0 e−jw | |1 − re−jw0 e−jw |



2(1 − rcos(w + w0 ))
2(1 − cos(w − w0 ))
= G2
1 + r2 − 2rcos(w − w0 ) 1 + r2 − 2rcos(w + w0 )
π
. Then
=
3

2

|H(π)|

2

=

0⇒w=π

=

4G2 (

3
2

1 + r + r2

)2 = 1
165

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Figure 5.42-1:

G =

1
(1 + r + r2 )
3

(d) Refer to fig 5.42-2.
(e)

2

|H(w)|

2

= G2

In the vicinity of w = w0 , we have

2

|1 − ejw0 e−jw | |1 − e−jw0 e−jw |
2

2

|1 − rejw0 e−jw | |1 − re−jw0 e−jw |
2

|H(w)|

2

≈ G2
=

cos(w − w0 )

=

w1,2

=

B3dB = w1 − w2

=
=
=
=

|1 − ejw0 e−jw |

2

|1 − rejw0 e−jw |


2(1 − cos(w − w0 ))
1
G2
=
1 + r2 − 2rcos(w − w0 )
2
2
2
1 + r − 4G
2r − 4G2
1 + r2 − 4G2
)
w0 ± cos−1 (
2r − 4G2
1 + r2 − 4G2
2cos−1 (
)
2r − 4G2
r−1
2cos−1 (1 − ( √ )2 )
2
s
1−r
2 2( √ )2
2
√
2 1−r

166

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x(n)

+

+

1+r+r2
3

y(n)

z -1
+

-2r cos w0

-2cos w0

+

z -1
r2

Figure 5.42-2:

5.43
For the sampling frequency Fs = 500samples/sec., the rejected frequency should be w1 =
60
6
4
2π( 100
) = 25
π. The filter should have unity gain at w2 = 2π( 200
500 ) = 5 π. Hence,
6
π) = 0
25
4
and H( π) = 1
5
6π
6π
H(w) = G(1 − ej 25 e−jw )(1 − e−j 25 e−jw )
6π
= Ge−jw [2cosw − 2cos ]
25
4
4
6
H( π) = 2G|[cos( π) − cos( π)]| = 1
5
5
25
H(

Hence, G =

1
2
6
cos 25
π

− cos 45 π

5.44
From (5.4.22) we have,
H(w)

= b0

2

= b20

1 − e−j2w
(1 − rej(w0 −w) )(1 − re−j(w0 −w) )
2

|H(w0 )|

|1 − e−j2w |
=1
2
(1 − r) [(1 − rcos2w0 )2 + (rsin2w0 )2 ]
167

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Hence, b0

=

p

(1 − r)2 (1 − 2rcos2w0 + r2 )
2|sinw0 |

5.45

From α
β
and cosα + cosβ
cos(n + 1)w0 + cos(n − 1)w0

with y(n)
y(n + 1) + y(n − 1)
y(n)

= (n + 1)w0
= (n − 1)w0
α+β
α−β
= 2cos
cos
, we obtain
2
2
= 2cosnw0 cosw0
= cosw0 n, it follows that
= 2cosw0 y(n) or equivalently,
= 2cosw0 y(n − 1) − y(n − 2)

5.46

sinα + sinβ
when α
sinnw0 + sin(n − 2)w0

If y(n)
y(n)
Initial conditions: y(−1)

α−β
α+β
cos
, we obtain
2
2
= nw0 and β = (n − 2)w0 , we obtain
= 2sin(n − 1)w0 cosw0

=

2sin

= Asinw0 n, then
= 2cosw0 y(n − 1) − y(n − 2)
= −Asinw0 , y(−2) = −Asin2w0

5.47

For h(n)
H(z)
Hence, y(n)
For h(n)
H(z)
Hence, y(n)

= Acosw0 nu(n)
1 − z −1 cosw0
= A
1 − 2cosw0 z −1 + z −2
= 2cosw0 y(n − 1) − y(n − 2) + Ax(n) − Acosw0 x(n − 1)
= Asinnw0 u(n)
z −1 sinw0
= A
1 − 2cosw0 z −1 + z −2
= 2cosw0 y(n − 1) + y(n − 2) + Ax(n) − Asinw0 x(n − 1)

5.48
Refer to fig 5.48-1. y1 (n) = Acosnw0 u(n), y2 (n) = Asinnw0 u(n)
168

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x(n)

+

+
z -1

+

y (n)
1
-A cos w 0

2r cos w0
z -1

A sin w0

-1

y (n)
2

Figure 5.48-1:

5.49
(a) Replace z by z 8 . We need 8 zeros at the frequencies w = 0, ± π4 , ± π2 , ± 3π
4 , π Hence,
H(z)

Hence, y(n)
π

π

1 − z −8
1 − az −8
Y (z)
=
X(z)
= ay(n − 8) + x(n) − x(n − 8)
=

3π

(b) Zeros at 1, e±j 4 , e±j 2 , e±j 4 , −1
1
1
π
1
π
1
3π
Poles at a 8 , a 8 e±j 4 , a 8 e±j 2 , a 8 e±j 4 , −1. Refer to fig 5.49-1.
(c)
2|cos4w|
|H(w)| = √
1 − 2acos8w + a2

asin8w
−tan−1 1−acos8w
,
cos4w ≥ 0
6 H(w) =
−1 asin8w
π − tan 1−acos8w , cos4w < 0
Refer to fig 5.49-2.

5.50
1
= 0.071, which results in a
We use Fs /L = 1cycle/day. We also choose nulls of multiples of 14
narrow passband of k±0.067. Thus, M + 1 = 14 or, equivalently M = 13

169

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X
-1

X

X

X

Unit circle

X
X

1

X
X

Figure 5.49-1:
magnitude of notch filter
10

−−> |H(f)|

8
6
4
2
0
0

0.05

0.1

0.15

0.2

0.25
−−> f

0.3

0.35

0.4

0.45

0.5

0.4

0.45

0.5

magnitude of a high pass filter
10

−−> |H(f)|

8
6
4
2
0
0

0.05

0.1

0.15

0.2

0.25
−−> f

0.3

0.35

Figure 5.49-2:
170

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

5.51
(a)
H(w)

=

2

=

|H(w)|

=
=
Hence, |H(w)|

=

1 − a1 e−jw
1 − ae−jw
(1 − a1 cosw)2 + ( a1 sinw)2
(1 − acosw)2 + (asinw)2
1 + a12 − a2 cosw
1 + a2 − 2acosw
1
for all w
a2
1
a

For the two-pole, two-zero system,
H(w)

=
=

Hence, |H(w)|
1− 2 cosw0 z −1 +

1

=

(1 − 1r ejw0 e−jw )(1 − 1r e−jw0 e−jw )
(1 − re−jw0 e−jw )(1 − rejw0 e−jw )
1 − 2r cosw0 e−jw + r12 e−j2w
1 − 2rcosw0 e−jw + r2 e−j2w
1
r2

z −2

r
r2
(b) H(z) = 1−2rcosw
−1 +r 2 z −2
0z
Hence, we need two delays and four multiplies per output point.

5.52
(a)
w0

=

H(z)

=
=

H(w)

=

|H(0)|

=

b0

=

6π
60
.2π =
200
50
6π
6π
(1 − ej 50 z −1 )(1 − e−j 50 z −1 )b0
6π
b0 (1 − 2cos z −1 + z −2 )
50
6π
−jw
2b0 e
(cosw − cos )
50
6π
2b0 (1 − cos ) = 1
25
1
2(1 − cos 6π
25 )

(b)
6π

H(z)

= b0

|H(0)|

=

b0

=

6π

(1 − ej 25 z −1 )(1 − e−j 25 z −1 )
6π

6π

(1 − rej 25 z −1 )(1 − re−j 25 z −1 )
2b0 (1 − cos 6π
25 )
=1
6π
1 − 2rcos 25 + r2
2
1 − 2rcos 6π
25 + r
6π
2(1 − cos 25 )

171

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

5.53
h(n)
Hence, Hr (w)
π
Hr ( )
4
3π
Hr ( )
4
1.85h(0) + 0.765h(1)
−0.765h(0) + 1.85h(1)

h(1)

= {h(0), h(1), h(2), h(3)} where h(0) = −h(3), h(1) = −h(2)
3w
w
= 2(h(0)sin
+ h(1)sin )
2
2
3π
π
1
= 2h(0)sin
+ 2h(1)sin ) =
8
8
2
9π
3π
= 2h(0)sin
+ 2h(1)sin ) = 1
8
8
1
=
2
= 1
=

0.56, h(0) = 0.04

5.54
(a)
H(z) = b0
H(w)

= b0

|H(w)|

= b0

−1
(1 − z −1 )(1 + z −1 )(1 − 2cos 3π
+ z −2 )
4 z
4π −1
2π −1
−2
(1 − 1.6cos 9 z + 0.64z )(1 − 1.6cos 9 z + 0.64z −2 )

(2je−jw sinw)(2e−jw )(cosw − cos 3π
4 )
2π −jw
4π −jw
−j2w
(1 − 1.6cos 9 e
+ 0.64e
)(1 − 1.6cos 9 e
+ 0.64e−j2w )
|1 −

−jw
1.6cos 2π
9 e

4|sinw||cosw − cos 3π
4 |
−jw + 0.64e−j2w |
+ 0.64e−j2w ||1 − 1.6cos 4π
9 e

5π
)| = 1 ⇒ b0 = 0.089
12
(b) H(z) as given above.
(c) Refer to fig 5.54-1.
The filter designed is not a good approximation of the desired response.
|H(

5.55
Y (w) = e−jw X(w) +

dX(w)
dw

(a)
For x(n)
dX(w)
Hence,
dw
h(n)

= δ(n), X(w) = 1.
0, and Y (w) = e−jw
Z π
1
Y (w)ejwn dw
=
2π −π
Z π
1
=
ejw(n−1) dw
2π −π
1
=
ejw(n−1) |π−π
2πj(n − 1)
sinπ(n − 1)
=
π(n − 1)
=

(b) y(n) = x(n − 1) − jnx(n). the system is unstable and time-variant.
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−−> |H(f)|

1.5

1

0.5

0
0

0.05

0.1

0.15

0.2

0.25
−−> f

0.3

0.35

0.4

0.45

0.5

0.05

0.1

0.15

0.2

0.25
−−> f

0.3

0.35

0.4

0.45

0.5

4

−−> phase

2
0
−2
−4
0

Figure 5.54-1:

5.56

H(w)

=

∞
X

h(n)e−jwn

n=−∞

=
=
G(w)

1,
|w| ≤ wc
0,
wc < |w|π
∞
X
g(n)e−jwn
=

=
=

n=−∞
∞
X

n
h( )e−jwn
2
n=−∞
∞
X

h(m)e−j2wm

m=−∞

= H(2w)
Hence,
G(w) =



1,
0,

|w| ≤ w2c and |w| ≥ π −
wc
wc
2 < |w| < π − 2

wc
2

173

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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5.57
y(n) = x(n) − x(n) ∗ h(n) = [δ(n) − h(n)] ∗ x(n) The overall system function is 1 − H(z) and the
frequency response is 1 − H(w). Refer to fig 5.57-1.

H(w)

1-H(w)

1

1

0

w

w
c

0

w

1-H(w)

H(w)

1

1

0

π

w
c

w
c

π

w

w

0

wc

Figure 5.57-1:

5.58
(a) Since X(w) and Y (w) are periodic, it is observed that Y (w) = X(w − π). Therefore,
y(n) = ejπn x(n) = (−1)n x(n)
(b) x(n) = (−1)n y(n).

5.59
y(n) = 0.9y(n − 1) + 0.1x(n)
(a)
H(z)

=

π
)
2

=

Hbp (w) = H(w −

=

0.1
1 − 0.9z −1
0.1
π
1 − 0.9e−j(w− 2 )
0.1
1 − j0.9e−jw

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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

π

(b) h(n) = 0.1(0.9ej 2 )n u(n)
(c) Since the impulse response is complex, a real input signal produces a complex-valued output
signal. For the output to be real, the bandpass filter should have a complex conjugate pole.

5.60
(a)
Let g(n)
Then, G(w)
D

dH(w)
dw
Therefore,

But

D

= nh(n)
dH(w)
= j
dw
∞
X
2
|g(n)|
=
=

n=−∞
Z π

=

1
2π

1
2
|G(w)| dw
2π −π
Z π
1
=
G(w)G∗ (w)dw
2π −π
∗ 

Z π 
dH(w)
dH(w)
1
dw
j
(−j)
=
2π −π
dw
dw


dΘ(w) jΘ(w)
dH(w)
e
=
+ j|H(w)|
dw
dw
Z

π

−π

(

dH(w)
dw

2

+ |H(w)|

2



dΘ(w)
dw

2 )

dw

(b) D consists of two terms, both of which are positive. For |H(w)| =
6 0, D is minimized by
selecting Θ(w) = 0, in which case the second term becomes zero.

5.61
y(n) = ay(n − 1) + bx(n), 0 < a < 1
H(z) =
(a)
H(w)
|H(0)|
b

b
1 − az −1

b
1 − ae−jw
|b|
=1
=
1−a
= ±(1 − a)
=

(b)
|H(w)|

2

=

⇒ 2b2

=

cosw

=

1
b2
=
1 + − 2acosw
2
1 + a2 − 2acosw

1 
1 + a2 − 2(1 − a)2
2a
a2

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1
(4a − 1 − a2 )
2a
4a − 1 − a2
= cos−1 (
)
2a
=

w3
(c)

w3
Let f (a)
Then f ′ (a)

(a − 1)2
)
2a
(a − 1)2
= 1−
2a
a2 − 1
= −
2a2
1 − a2
=
>0
2a2

= cos−1 (1 −

Therefore f (a) is maximum at a = 1 and decreases monotonically as a → 0. Consequently,
w3 increases as a → 0.
(d)
b
w3

= ±(1 − a)
4a − 1 − a2
= cos−1 (
)
2a

The 3-dB bandwidth increases as a → 0.

5.62

y(n) = x(n) + αx(n − M ), α > 0

H(w)
|H(w)|
Θ(w)

1 + αe−jwM
p
1 + 2αcoswM + α2
=
−αsinwM
= tan−1
1 + αcoswM
=

Refer to fig 5.62-1.

5.63
(a)
Y (z)

=

H(z)

=
=
=


1
X(z) + z −1 X(z)
2
Y (z)
X(z)
1
(1 + z −1 )
2
z+1
2z
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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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M=10, alpha = 0.1
2

−−> |H(f)|

1.5
1
0.5
0
0

0.05

0.1

0.15

0.2

0.25
−−> f

0.3

0.35

0.4

0.45

0.5

0.05

0.1

0.15

0.2

0.25
−−> f

0.3

0.35

0.4

0.45

0.5

1

−−> phase

0.5
0
−0.5
−1
0

Figure 5.62-1:

(b)

Zero at z = −1 and a pole at z = 0. The system is stable.

1
−X(z) + z −1 X(z)
2
Y (z)
H(z) =
X(z)
1
(−1 + z −1 )
=
2
z−1
= −
2z
Zero at z = 1 and a pole at z = 0. The system is stable.
Y (z)

=

(c)
Y (z) =
=

1
(1 + z −1 )3
8
1 (1 + z)3
8 z3

Three zeros at z = −1 and three poles at z = 0. The system is stable.

5.64
Y (z) = X(z) + bz −2 X(z) + z −4 X(z)
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H(z)

=

Y (z)
X(z)

For b = 1, H(w)

= 1 + bz −2 + z −4
= 1 + ej2w + e−j4w

|H(w)|

= (1 + 2cosw)e−jw
= |1 + 2cosw|

6

H(w) =



−w,
1 + 2cosw ≥ 0
π − w, 1 + 2cosw < 0

Refer to fig 5.64-1.

3

−−> |H(w)|

2.5
2
1.5
1
0.5
0
0

0.5

1

1.5

2

2.5

3

3.5

2

2.5

3

3.5

−−> w
2

−−> phase

1
0
−1
−2
−3
0

0.5

1

1.5
−−> w

Figure 5.64-1:

b = −1, H(w)
|H(w)|
6

H(w) =



= 1 − e−jw + e−j2w
= (2cosw − 1)e−jw
= |2cosw − 1|

−w,
π − w,

−1 + 2cosw ≥ 0
−1 + 2cosw < 0

Refer to fig 5.64-2.

178

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3

−−> |H(w)|

2.5
2
1.5
1
0.5
0
0

0.5

1

1.5

2

2.5

3

3.5

2

2.5

3

3.5

−−> w
3

−−> phase

2
1
0
−1
−2
0

0.5

1

1.5
−−> w

Figure 5.64-2:

5.65
y(n) = x(n) − 0.95x(n − 6)
(a)
Y (z)
H(z)

z6
z

= X(z)(1 − 0.95z −6 )
= (1 − 0.95z −6 )
z 6 − 0.95
=
z6
= 0.95
=

1

(0.95) 6 ej2πk/6 , k = 0, 1, . . . , 5

6th order pole at z = 0. Refer to fig 5.65-1.
(b)Refer to fig 5.65-2.
1
z6
. r = (0.95) 6 . Refer to fig 5.65-3.
(c) Hin (z) = z6 −0.95
(d)Refer to fig 5.65-4.

179

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r=(0.95)1/6

r

X

Figure 5.65-1:

2

−−> |H(f)|

1.5
1
0.5
0
0

0.05

0.1

0.15

0.2

0.25
−−> f

0.3

0.35

0.4

0.45

0.5

Figure 5.65-2:

180

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

X

X

r=(0.95)1/6

r

X

X
X

X

Figure 5.65-3:

20

−−> |H(f)|

15
10
5
0
0

0.05

0.1

0.15

0.2

0.25
−−> f

0.3

0.35

0.4

0.45

0.5

Figure 5.65-4:

181

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

5.66
(a)
H(z)

z −1
1 − z −1 − z −2
z −1

=
=
=

If |z|
h(n)
√
√
5−1
5+1
If ROC is
< |z| <
, then
2
2
h(n)
If |z|
h(n)
From H(z), the difference equation is
y(n)

√
√
1+ 5 −1
)(1 − 1−2 5 z −1 )
2 z
√1
− √15
5
√
√
+
1+ 5 −1
1 − 1−2 5 z −1
2 z

(1 −
1−

√
1+ 5
> 1−
is ROC, then
2
#
"
√
√
1 1− 5 n
1 1+ 5 n
√ (
) −√ (
) u(n)
=
2
2
5
5
√
√
1 1− 5 n
1 1+ 5 n
= −√ (
) u(n) − √ (
) u(−n − 1)
2
2
5
5
√
5−1
< 1−
is ROC, then
2
#
"
√
√
1 1− 5 n
1 1+ 5 n
) +√ (
) u(−n − 1)
=
−√ (
2
2
5
5
= y(n − 1) + y(n − 2) + x(n − 1)

(b)
H(z)
The difference equation is
y(n)
H(z)

=

1
1 − e−4a z −4

= e−4a y(n − 1) + x(n)
=

(1 −

1
4
e−a z −1

=
If ROC is |z|
h(n)
If ROC is |z|
h(n)

e−a z −1 )(1
+

−

1

π
ej 2 e−a z −1 )(1

1
4
je−a z −1

+

+ e−a z −1 )(1 + je−a z −1 )
1
4
e−a z −1

+

1−
1−
1+
1+
> 1, then
1
[1 + (j)n + (−1)n + (−j)n ] e−an u(n)
=
4
< 1, then
1
= − [1 + (j)n + (−1)n + (−j)n ] e−an u(−n − 1)
4

1
4
je−a z −1

5.67
Y (z) =
=

1 − z −1 + 3z −2 − z −3 + 6z −4

(1 + z −1 + 2z −2 )(1 − 2z −1 + 3z −2 )
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X(z)

=

Therefore, H(z)

=

1 + z −1 + 2z −2
Y (z)
X(z)

1 − 2z −1 + 3z −2


=
1, −2, 3

=
h(n)

↑

5.68
y(n)

=

x(n)

=

H(z)

=
=

X(z)

=

Y (z)

=

Rxx (z)

= X(z)X(z −1 )
1
=
(1 − 14 z −1 )(1 − 14 z)
=
=

Hence, rxx (n)

=

Rhh (z)

=
=
=
=

Hence, rhh (n)

=

Rxy (z)

=
=
=

Hence, rxy (n)

1
y(n − 1) + x(n)
2
1
( )n u(n)
4
Y (z)
X(z)
1
1 − 12 z −1
1
1 − 14 z −1
1
1 −1
(1 − 4 z )(1 − 12 z −1 )

=

−4z −1
(1 − 14 z −1 )(1 − 4z −1 )
16
16
1
1
−
1
−1
15 1 − 4 z
15 1 − 4z −1
16 1 n
16
( ) u(n) + (4)n u(−n − 1)
15 4
15
H(z)H(z −1 )
1
(1 − 12 z −1 )(1 − 12 z)

−2z −1
(1 − 12 z −1 )(1 − 2z −1 )
4
4
1
1
−
1
−1
3 1 − 2z
3 1 − 2z −1
4 1 n
4
( ) u(n) + (2)n u(−n − 1)
3 2
3
X(z)Y (z −1 )
1
1 −1
(1 − 4 z )(1 − 14 z)(1 − 21 z)
1
1
1
16
128
16
+
+
−
−1
−1
17 1 − 2z
15 1 − 4z
105 1 − 41 z −1
16
128 1 n
16 n
(2) u(−n − 1) − (4)n u(−n − 1) +
( ) u(n)
17
15
105 4
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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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Ryy (z) = Y (z)Y (z −1 )
1
(1 − 14 z −1 )(1 − 12 z −1 )(1 − 14 z)(1 − 21 z)
64
128
1
1
1
1
128
64
= −
−
+
+
21 1 − 2z −1
105 1 − 4z −1
21 1 − 21 z −1
105 1 − 14 z −1
64 n
128 n
64 1
128 1 n
=
(2) u(−n − 1) −
(4) u(−n − 1) + ( )n u(n) −
( ) u(n)
21
105
21 2
105 4
=

Hence, ryy (n)

5.69
(a)



h(n) = 10, 9, −7, −8, 0, 5, 3
↑

The roots(zeros) are 0.8084 ± j0.3370, −0.3750 ± j0.6074, −1.0, −0.7667
All the roots of H(z) are inside the unit circle. Hence, the system is minimum phase.
(b) h(n) = {5, 4, −3, −4, 0, 2, 1} H(z) = 5 + 4z −1 − 3z −2 − 4z −3 + 2z −5 + z −6
The roots(zeros) are 0.7753 ± j0.2963, −0.4219 ± j0.5503, −0.7534 ± j0.1900
All the roots of H(z) are inside the unit circle. Hence, the system is minimum phase.

5.70
The impulse response satisfies the difference equation
N
X

=

δ(n), a0 = 1

ak h(−k)

=

a0 h(0) = 1

a0

=

n = 1, ⇒ a0 h(1) + a0 h(0)

=

a1

=

k=0

n = 0, ⇒

ak h(n − k)

N
X

k=0

1
h(0)
0
−a0 h(1)
−h(1)
= 2
h(0)
h (0)

..
.
n = N, ⇒ a0 h(N ) + a1 h(N − 1) + . . . + aN h(0)

⇒

yields aN

It is apparent that the coefficients {an } can be determined if we know the order N and the values
h(0), h(1), . . . , h(N ). If we do not know the filter order N, we cannot determine the {an }.

5.71
h(n) = b0 δ(n) + b1 δ(n − D) + b2 δ(n − 2D) (a) If the input to the system is x(n), the output is
y(n) = b0 x(n) + b1 x(n − D) + b2 x(n − 2D). Hence, the output consists of x(n), which is the input
signal, and the delayed signals x(n − D) and x(n − 2D). The latter may be thought of as echoes
of x(n).
(b)
H(w)

= b0 + b1 e−jwD + b2 e−j2wD
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|H(w)|
Θ(w)

= b0 + b1 coswD + b2 cos2wD − j(b1 sinwD + b2 sin2wD)
q
b0 2 + b1 2 + b2 2 + 2b1 (b0 + b2 )coswD + 2b0 b2 cos2wD
=
= −tan−1

b1 sinwD + b2 sin2wD
b0 + b1 coswD + b2 cos2wD

(c) If |b0 + b2 | << |b1 |, then the dominant term is b1 e−jwD and
q
|H(w)| = b0 2 + b1 2 + b2 2 + 2b1 (b0 + b2 )coswD

k
and |H(w)| has maxima and minima at w = ± D
π, k = 0, 1, 2, . . .
(d) The phase Θ(w) is approximately linear with a slope of −D. Refer to fig 5.71-1.

1.2

−−> |H(f)|

1.1
1
0.9
0.8
0

0.05

0.1

0.15

0.2

0.25
−−> f

0.3

0.35

0.4

0.45

0.5

0.05

0.1

0.15

0.2

0.25
−−> f

0.3

0.35

0.4

0.45

0.5

4

−−> phase

2
0
−2
−4
0

Figure 5.71-1:

5.72
H(z) =

∞
X
B(z)
1 + bz 1
h(n)z −n
=
=
A(z)
1 + az 1
n=0

(a)
H(z)
Hence, h(0)

= 1 + (b − a)z −1 + (a2 − ab)z −2 + (a2 b − a3 )z −3 + (a4 − a3 b)z −5 + . . .
= 1,

185

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
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h(1)
h(2)
h(3)
h(4)

= b − a,
= a2 − ab,

= a2 b − a3 ,
= a4 − a3 b

(b)
y(n) + ay(n − 1)

For x(n)
h(n) + ah(n − 1)

Multiply both sides by h(n) and sum. Then
rhh (0) + arhh (1)

rhh (1) + arhh (0)
rhh (2) + arhh (1)
rhh (3) + arhh (2)

= x(n) + bx(n − 1)
= δ(n),
= δ(n) + bδ(n − 1)
= h(0) + bh(1)
= bh(0)
= 0
= 0

By solving these equations recursively, we obtain
rhh (0)
rhh (1)
rhh (2)
rhh (3)

b2 − 2ab + 1
1 − a2
(ab − 1)(a − b)
=
1 − a2
(ab − 1)(a − b)
= −a
1 − a2
(ab
−
1)(a − b)
= a2
1 − a2
=

5.73
x(n) is a real-valued, minimum-phase sequence. The sequence y(n) must satisfy the conditions,
y(0) = x(0), |y(n)| = |x(n)|, and must be minimum phase. The solution that satisfies the
condition is y(n) = (−1)n x(n). The proof that y(n) is minimum phase proceeds as follows:
Y (z)

=

X

y(n)z −n

n

=

X

(−1)n x(n)z −n

n

=

X

x(n)(−z −1 )n

n

= X(−z)
This preserves the minimum phase property since a factor (1 − αz −1 ) → (1 + αz −1 )

5.74

Consider the system with real and even impulse response h(n) = 14 , 1, 41 and frequency response
√
H(w) = 1 + 12 cosw. Then H(z) = z −1 ( 41 z 2 + z + 41 ). The system has zeros at z = −2 ± 3.
We observe that the system is stable, and its frequency response is real and even. However, the
inverse system is unstable. Therefore, the stability of the inverse system is not guaranteed.
186

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5.75
(a)
g(n)
f (n)
Y (w)
Then, Y (w)

= h(n) ∗ x(n) ⇒ G(w) = H(w)X(w)
= h(n) ∗ g(−n) ⇒ F (w) = H(w)G(−w)
= F (−w)
= H(−w)G(w)

= H(−w)H(w)X(w)
= H ∗ (w)H(w)X(w)
2

= |H(w)| X(w)
2

(b)

But Ha (w) ≡ |H(w)| is a zero-phase system.
G(w)

= H(w)X(w)

F (w) = H(w)X(−w)
Y (w) = G(w) + F (−w)
= H(w)X(w) + H(−w)X(w)
= X(w)(H(w) + H ∗ (−w))
= 2X(w)Re(H(w))
But Hb (w) = 2Re {H(w)} is a zero-phase system.

5.76
(a) Correct. The zeros of the resulting system are the combination of the zeros of the two systems.
Hence, the resulting system is minimum phase if the inividual system are minimum phase.
(b) Incorrect. For example, consider the two minimum-phase systems.
H1 (z) =
and H2 (z) =
Their sum is H1 (z) + H2 (z) =

1 − 21 z −1
1 − 31 z −1

−2(1 + 31 z −1 )
1 − 31 z −1

−1 − 76 z −1
, which is not minimum phase.
1 − 13 z −1

5.77
(a)
|H(w)|

2

=

5
4
10
9

− cosw
− 23 cosw

= H(z)H(z −1 )|z=e−jw
5
1
−1
)
4 − 2 (z + z
Hence, H(z)H(z −1 ) = 10
1
−1
)
9 − 3 (z + z
=

1 − 12 z −1
1 − 13 z −1

187

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(b)
2

=

H(z)H(z −1 )

=

H(z)H(z −1 )

=

Hence, H(z)

=

or H(z)

=

|H(w)|

2(1 − a2 )
1 + a2 − 2acosw
2(1 − a2 )
1 + a2 − a(z + z −1 )
2(1 + a)(1 − a)
(1 − az −1 )(1 − az)
p
2(1 − a2 )
1 − az −1
p
2(1 − a2 )
1 − az

5.78
H(z) =
=

(1 − 0.8ejπ/2 z −1 )(1 − 0.8e−jπ/2 z −1 )(1 − 1.5ejπ/4 z −1 )(1 − 1.5e−jπ/4 z −1 )
3
(1 + 0.64z −2 )(1 − √ z −1 + 2.25z −2 )
2

(a) There are four different FIR systems with real coefficients:
H1 (z)

=

H2 (z)

=

H3 (z)

=

H4 (z)

=

3
(1 + 0.64z −2 )(1 − √ z −1 + 2.25z −2 )
2
3 −1
−2
(1 + 0.64z )(1 − √ z + 2.25z −2 )
2
3
(1 + 0.64z −2 )(1 − √ z −1 + 2.25z −2 )
2
3 −1
−2
(1 + 0.64z )(1 − √ z + 2.25z −2 )
2

H(z) is the minimum-phase system.
(b)
H1 (z) =
h1 (n)

=

H2 (z) =
h2 (n)

=

H3 (z) =
h3 (n)

=

H4 (z) =
h4 (n)

=

1.92
3
1 − √ z −1 + 2.89z −2 − √ z −3 + 1.44z −4
2
2


3
−1.92
1, − √ , 2.89, √ , 1.44
↑
2
2
3
1.92
0.64z 2 − √ z + 2.44 − √ z −1 + 2.25z −2
2
2


−1.92
3
0.64, √ , 2.44, − √ , 2.25
↑
2
2
3
1.92
2.25z 2 − √ z + 2.44 − √ z −1 + 0.64z −2
2
2


−3
1.92
√
√
2.25,
, 2.44, −
, 0.64
2 ↑
2
3
1.92
1.44z 4 − √ z 3 + 2.89z 2 − √ z + 1
2
2


−1.92
3
1.44, √ , 2.89, − √ , 1,
2
2 ↑
188

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

(c)
E1 (n)
E2 (n)
E3 (n)
E4 (n)

= {1, 5.5, 13.85, 15.70, 17.77}
= {0.64, 2.48, 8.44, 12.94, 18.0}
= {2.25, 6.75, 12.70, 14.55, 14.96}

= {1.44, 3.28, 11.64, 16.14, 17.14}

Clearly, h3 (n) is minimum phase and h2 (n) is maximum phase.

5.79
H(z) =

1+

1
PN

k=1

ak z −k

(a) The new system function is H ′ (z) = H(λ−1 z)
H ′ (z) =

1
1+

PN

k=1

ak λk z −k

If pk is a pole of H(z), then λpk is a pole of H ′ (z).
1
Hence, λ < |pmax
| is selected then |pk λ| < 1 for all k and, hence the system is stable.
PN
(b) y(n) = − k=1 ak λk y(n − k) = x(n)

5.80
(a) The impulse response is given in pr10fig 5.80-1.
(b) Reverberator 1: refer to fig 5.80-2.

1.2

1

−−> magnitude

0.8

0.6

0.4

0.2

0
0

500

1000

1500

2000

2500

−−> n

Figure 5.80-1:
189

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

impulse response for unit1

−−> magnitude

1

0.5

0
0

100

200

300
−−> n

400

500

600

500

600

impulse response for unit2

−−> magnitude

1.5

1

0.5

0
0

100

200

300
−−> n

400

Figure 5.80-2:
Reverberator 2: refer to fig 5.80-2.
(c) Unit 2 is a better reverberator.
(d) For prime number of D1 , D2 , D3 , the reverberations of the signal in the different sections do
not overlap which results in the impulse response of the unit being more dense.
(e) Refer to fig 5.80-3.
(f) Refer to fig 5.80-4 for the delays being prime numbers.

5.81
(a) Refer to fig 5.81-1.
(b) Refer to fig 5.81-2.

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phase response for unit1
3

−−> phase

2
1
0
−1
−2
−3
0

1

2

3

4

5

6

7

5

6

7

−−> w
phase response for unit2
3

−−> phase

2
1
0
−1
−2
−3
0

1

2

3

4
−−> w

Figure 5.80-3:

1.2

1

−−> magnitude

0.8

0.6

0.4

0.2

0
0

500

1000

1500

2000

2500

−−> n

Figure 5.80-4:
191

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

30
−−> magnitude

20
10
0
−10
−20
0

0.5

1

1.5
2
−−> w(rad)

2.5

3

3.5

0.5

1

1.5
2
−−> w(rad)

2.5

3

3.5

0

−−> phase

−0.5
−1
−1.5
−2
−2.5
0

Figure 5.81-1:

−−> magnitude

100
50
0
−50
−100
0

0.5

1

1.5
−−> w(rad)

2

2.5

3

4

−−> phase

2
0
−2
−4
0

0.5

1

1.5
2
−−> w(rad)

2.5

3

3.5

Figure 5.81-2:
192

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5.82
(a)
B
Fs
z1
z2
z3
z4
H(z)

=
=

10kHz
20kHz
10k
=
= 0.5
20k
7.778k
= 0.3889
=
20k
8.889k
=
= 0.4445
20k
6.667k
= 0.3334
=
20k
= (z − 0.5)(z − 0.3889)(z − 0.4445)(z − 0.3334)

(b) Refer to fig 5.82-1.
(c) It satisfies the objectives but this filter is not recommended in a practical application because

−−> magnitude

0

−50

−100

−150
0

0.5

1

1.5
2
−−> w(rad)

2.5

3

3.5

0.5

1

1.5
2
−−> w(rad)

2.5

3

3.5

4

−−> phase

2
0
−2
−4
0

Figure 5.82-1:
in a speech application linear phase for the filter is desired and this filter does not provide linear
phase for all frequencies.

5.83
Refer to fig 5.83-1.
Practical:
193

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r = 0.99
r = 0.9
r = 0.6

wr = π6
wr = π6
wr = 0

π
Bandwidth = 128
= 0.0245
5π
Bandwidth = 32 = 0.49
Bandwidth = 1.1536

Theoretical:
r = 0.99,
r = 0.9,

wr =
wr =

π
6
π
6

Bandwidth = 2(1 − r) = 0.02
Bandwidth = 2(1 − r) = 0.2

For r very close to 1, the theoretical and practical values match.

−−> magnitude

40
30

.... r = 0.99

20

−−−− r = 0.9
__ r = 0.6

10
0
−10
−20
−4

−3

−2

−1

0
−−> w(rad)

1

2

3

4

3

4

4

−−> phase

2
0
.... r = 0.99
−−−− r = 0.9
__ r = 0.6

−2
−4
−4

−3

−2

−1

0
−−> w(rad)

1

2

Figure 5.83-1:

5.84
H(z)

=

H(z)

=
=

Let B1 (z)
B2 (z)
A(z)

=
=

(1 − 0.9ej0.4π z −1 )(1 − 0.9e−j0.4π z −1 )(1 − 1.5ej0.6π z −1 )(1 − 1.5e−j0.6π z −1 )
B(z)
A(z)
(z − 0.9ej0.4π )(z − 0.9e−j0.4π )(z − 1.5ej0.6π )(z − 1.5e−j0.6π )
z4
j0.4π
−j0.4π
(z − 0.9e
)(z − 0.9e
)
j0.6π
−j0.6π
(z − 1.5e
)(z − 1.5e
)

= z4

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B1 (z) B2 (z)
A(z)
(z − 0.9ej0.4π )(z − 0.9e−j0.4π )(z −1 − 1.5ej0.6π )(z −1 − 1.5e−j0.6π )
z4
B2 (z)
B2 (z −1 )
(z − 1.5ej0.6π )(z − 1.5e−j0.6π )
(z −1 − 1.5ej0.6π )(z −1 − 1.5e−j0.6π )

Hmin (z) =
=
Hap (z) =
=

Hap (z) has a flat magnitude response. To get a flat magnitude response for the system, connect
a system which is the inverse of Hmin (z), i.e.,
Hc (z) =
=

1
Hmin (z)
z4
(z − 0.9ej0.4π )(z − 0.9e−j0.4π )(z −1 − 1.5ej0.6π )(z −1 − 1.5e−j0.6π )

(b) Refer to fig 5.84-1 and fig 5.84-2.
pole−zero plots for Hc(z)
901.5
120
60
1
150
30
0.5
180

0
330

210
240

270

300

pole−zero plots for compensated system
901.5
60
120
1
30
150
0.5
180

0

210

330
240

270

300

Figure 5.84-1:

195

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
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Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

mag for Hc(z)

mag of compensated system

10

2
−−> magnitude

−−> magnitude

5
0
−5
−10
−15
−4

−2

0
−−> w(rad)

2

1
0
−1
−2

4

phase for Hc(z)

−2

0
−−> w(rad)

2

phase of compensated system

4

1.5
1
−−> phase

−−> phase

2
0

0.5
0
−0.5

−2
−1
−4
−4

−2

0
−−> w(rad)

2

−1.5
−4

4

−2

0
−−> w(rad)

2

4

Figure 5.84-2:

196

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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Chapter 6

6.1
(a) Fourier transform of dxa (t)/dt is X̂a (F ) = j2πF Xa (F ), then Fs ≥ 2B
(b) Fourier transform of x2a (t) is X̂a (F ) = Xa (F ) ∗ Xa (F ), then Fs ≥ 4B
(c) Fourier transform of xa (2t) is X̂a (F ) = 2Xa (F/2), then Fs ≥ 4B
(d) Fourier transform of xa (t) cos(6πBt) is X̂a (F ) = 21 Xa (F + 3B) + 21 Xa (F − 3B) resulting in
FL = 2B and FH = 4B. Hence, Fs = 2B
(d) Fourier transform of xa (t) cos(7πBt) is X̂a (F ) = 21 Xa (F + 3.5B) + 21 Xa (F − 3.5B) resulting
in FL = 5B/2 and FH = 9B/2. Hence, kmax = ⌊ FBH ⌋ = 2 and Fs = 2FH /kmax = 9B/2

6.2
(a) Fs = 1/T ≥ 2B ⇒ A = T, Fc = B.
(b) Xa (F ) = 0 for |F | ≥ 3B. Fs = 1/T ≥ 6B ⇒ A = T, Fc = 3B.
(c) Xa (F ) = 0 for |F | ≥ 5B. Fs = 1/T ≥ 10B ⇒ A = T, Fc = 5B.

6.3
xa (t) =

∞  |k|
X
1
ej2πkt/Tp
2

(6.1)

k=−∞

Since filter cut-off frequency, Fc = 102.5, then terms with |n|/Tp > Fc will be filtered resulting
10  |k|
X
1
ya (t) =
ej2πkt/Tp
2
k=−10
10  |k|
X
1
δ(F − k/Tp )
Ya (F ) =
2
k=−10

Sampling this signal with F s = 1/T = 1/0.005 = 200 = 20/Tp results in aliasing
Y (F ) =

=

∞
1 X
Xa (F − nF s)
3 n=−∞
!
 9
∞
9  |k|
X
1 X
1
1
δ(F − k/Tp − nFs ) +
δ(F − 10/Tp − nFs )
3 n=−∞
2
2
k=−9

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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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6.4
(a)
= nT e−nT ua (nT )
= nT an ua (nT )

x(n) = xa (nT )

where a = e−T .
Define x1 (n) = an ua (n). The Fourier transform of x1 (n) is
∞
X

X1 (F ) =

an e−j2πF n

n=0

1
1 − ae−j2πF

=

Using the differentiation in frequency domain property of the Fourier transform
X(F )

= Tj
=

X1 (F )
dF
T ae−j2πF
2

(1 − ae−j2πF )
T
e(T +j2πF ) + e−(T +j2πF ) − 2

=
(b) The Fourier transform of xa (t) is

1
(1 + j2πF )2

Xa (F ) =

Fig. 6.4-1(a) shows the original signal xa (t) and its spectrum Xa (F ). Sampled signal x(n) and
its spectrum X(F ) are shown for Fs = 3 Hz and Fs = 1 Hz in Fig. 6.4-1(b) and Fig. 6.4-1(c),
respectively.
(c) Fig. 6.4-2 illustrates the reconstructed sugnal x̂a (t) and its spectrum for Fs = 3 Hz and
Fs = 1 Hz.
x̂a (t) =

∞
X

xa (nT )

n=−∞

sin (π(t − nT )/T )
π(t − nT )/T

6.5
The Fourier transfrom of y(t) =

Rt

−∞

x(τ )dτ is

Y (w) =

Then,
H(w) =



X(w)
+ πX(j0)δ(w)
jw

1
jw

+ πδ(w),

0,

0≤n≤I
otherwise

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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

0.4

1

xa(t)

Xa(F)

0.3
0.2

0.5

0.1
0
−5

0

5

0
−4

10

−2

t(sec)
0.4

0
F(Hz)

2

4

1
|Xa(F)|

x(n)=xa(nT)

0.3

|X(F)|

0.2

0.5

0.1
0
−5

0

5

0
−4

10

−2

t(sec)
0.4

0
F(Hz)

2

4

1
|Xa(F)|

x(n)=xa(nT)

0.3

|X(F)|

0.2

0.5

0.1
0
−5

0

5

0
−4

10

t(sec)

−2

0
F(Hz)

2

4

Figure 6.4-1:

199

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0.4

1

xa(t)

Xa(F)

0.3
0.2

0.5

0.1
0
−5

0

5

0
−4

10

−2

t(sec)
0.6

0
F(Hz)

2

4

1
|Xa(F)|

0.4

|X(F)|

0.2

0.5

0
−0.2
−5

0

5

0
−4

10

−2

t(sec)

0
F(Hz)

2

4

1
|Xa(F)|

0.3

|X(F)|

0.2
0.5
0.1
0
−0.1
−5

0

5

0
−4

10

t(sec)

−2

0
F(Hz)

2

4

Figure 6.4-2:

200

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

6.6
(a) B = F2 −F1 is the bandwidth of the signal. Based on arbitrary band positioning for first-order
sampling,
2FH
Fs,min =
kmax
where
kmax = ⌊

F2
⌋.
B

(b)
x̂a (t) =

∞
X

n=−∞

where
ga (t) =

xa (nT )ga (t − nT )

sin πBt
cos 2πFc t
πBt

and Fc = (F1 + F2 )/2.

6.7

ga (t)

=
=

Z

∞

Ga (F )ej2πF t dF

−∞
FL −mB

Z

−(FL −B)

Z

1
ej2πF t dF +
1 − γ m+1

−FL +mB

Z

FL −mB

1
ej2πF t dF +
+
1
−
γ −m
FL
= A+B+C +D

A =
=
B

=

C

=

D

=

−FL

Z

1
ej2πF t dF
1 − γm

FL +B

−FL +mB

1
ej2πF t dF
1 − γ −(m+1)



1
ej2π(FL −mB)t − e−j2π(FL +B)t
m+1
j2πBt(1 − γ
)


jπB∆(m+1)
e
j2π(FL −mB)t
−j2π(FL +B)t
e
−
e
j2πBt(ejπB∆(m+1) − e−jπB∆(m+1) )


ejπB∆m
−j2πFL t
j2π(FL −mB)t
e
−
e
j2πBt(ejπB∆m − e−jπB∆m )


e−jπB∆m
j2πFL t
−j2π(FL −mB)t
e
−
e
j2πBt(ejπB∆m − e−jπB∆m )


e−jπB∆(m+1)
−j2π(FL −mB)t
j2π(FL +B)t
e
−
e
j2πBt(ejπB∆(m+1) − e−jπB∆(m+1) )

Combining A and D, and B and C, we obtain,

A+D

=

1
ej[2π(FL +B)t−πB∆(m+1)] + e−j[2π(FL +B)t−πB∆(m+1)]
πBt sin(πB∆(m + 1))


−ej[2π(FL −mB)t+πB∆(m+1)] − e−j[2π(FL −mB)t+πB∆(m+1)]
201

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cos [2π(FL + B)t − π(m + 1)B∆] − cos [2π(mB − FL )t − π(m + 1)B∆]
2πBt sin [π(m + 1)B∆]
1
ej[2π(FL −mB)t+πB∆m] + e−j[2π(FL −mB)t+πB∆m]
=
πBt sin(πB∆m))


−ej[2πFL t−πB∆m] − e−j[2πFL t−πB∆m]
cos [2π(mB − FL )t − πmB∆] − cos [2πFL t − πmB∆]
=
2πBt sin(πmB∆)
=

B+C

We observe that a(t) = B + C and b(t) = A + D. Q.E.D.

6.8
1.
gSH (n) =
2.
GSH (w)



1,
0,
∞
X

=

0≤n≤I
otherwise

gSH (n)e−jwn

n=−∞
I
X

=

e−jwn

n=0

= e−jw(I−1)/2

sin [wI/2]
sin(w/2)

3. The linear interpolator is defined as
glin [n] =



1 − |n|/I,
0,

|n| ≤ I
otherwise

Taking the Fourier transform, we obtain
Glin (w) =


2
1 sin(wI/2)
I sin(w/2)

Fig. 6.8-1 shows magnitude and phase responses of the ideal interpolator (dashed-dotted line),
the linear interpolator (dashed line), and the sample-and-hold interpolator (solid line).

6.9
(a)
xa (t)
Xa (F )

= e−j2πF0 t
Z ∞
xa (t)e−j2πF t dt
=
0
Z ∞
e−j2πF0 t e−j2πt dt
=
0
Z ∞
e−j2π(F +F0 )t dt
=
0

=
Xa (F )

=

e−j2π(F +F0 )t ∞
|
−j2π(F + F0 ) 0
1
j2π(F + F0 )
202

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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

4

|G|

3

2

1

0
−4

−3

−2

−1

0
F

1

2

3

4

−3

−2

−1

0
F

1

2

3

4

3

angle(G)

2
1
0
−1
−2
−3
−4

Figure 6.8-1:

(b)
x(n)
X(f )

j2πF0 n

= e− Fs
∞
X
x(n)e−j2πf n
=
=
=

n=−∞
∞
X
j2πF n
− Fs0
−j2πf n

e

n=0
∞
X

e

F0

e−j2π(F + Fs )n

n=0

=

1
F

−j2π(F + F0s )

1−e
(c) Refer to fig 6.9-1
(d) Refer to fig 6.9-2
(e) Aliasing occurs at Fs = 10Hz.

6.10
Since

Fc + B
2
B

=

50+10
20

= 3 is an integer, then Fs = 2B = 40Hz
203

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

1200

1000

−−> |Xa(F)|

800

600

400

200

0
0

200

400

600

800

1000

1200

Figure 6.9-1:

Fs= 10

Fs= 20

12

15

8

−−> |X(F)|

−−> |X(F)|

10

6
4

10

5

2
0
0

5

10

0
0

15

10

Fs= 40

100

150

100
80
−−> |X(F)|

30
−−> |X(F)|

30

Fs= 100

40

20
10
0
0

20

60
40
20

20

40

0
0

60

50

Figure 6.9-2:

204

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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6.11
Fc
B
r

B′

=
=

100
12
Fc + B2
= ⌈
⌉
B
106
⌉
= ⌈
12
= ⌈8.83⌉ = 8
=
=
=

Fs

=
=

Fc + B2
r
106
8
53
4
2B ′
53
Hz
2

6.12
(a)
x(n)
x2 (n)

↔
↔

X(w)
X(w) ∗ X(w)

The output y1 (t) is basically the square of the input signal ya (t). For the second system,

X(w) * X(w)

X(w)

−3π −2π

−π

0

π

2π

3π

w

−2π

−π

0

0

2π

w

2
spectrum of sampled xa (t),
(i.e.), s(n) = x2 (nT)
a

spectrum of x 2 (t)
a

-2B

π

2B

-2B

0

2B

Figure 6.12-3:
x2a (t) ↔ X(w) ∗ X(w), the bandwidth is basically 2B. The spectrum of the sampled signal is
205

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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

given in fig 6.12-3.
(b)

xa (t)
x(n)

y(n)

y1 (t)
sa (t)

s(n)

Hence, y2 (t)
For Fs
x(n)

y(n)

y1 (t)
sa (t)

s(n)

= cos40πt
40πn
= cos
50
4πn
= cos
5
= x2 (n)
4πn
= cos2
5
1 1
8πn
=
+ cos
2 2
5
2πn
1 1
+ cos
=
2 2
5
1 1
=
+ cos20πt
2 2
= x2a (t)
= cos2 40πt
1 1
=
+ cos80πt
2 2
80πn
1 1
+ cos
=
2 2
50
1 1
8πn
=
+ cos
2 2
5
2πn
1 1
+ cos
=
2 2
5
1 1
=
+ cos20πt
2 2
= 30,
4πn
= cos
3
2πn
= cos
3
= x2 (n)
2πn
= cos2
3
1 1
4πn
=
+ cos
2 2
3
2πn
1 1
+ cos
=
2 2
3
1 1
+ cos20πt
=
2 2
= x2a (t)
= cos2 40πt
1 1
=
+ cos80πt
2 2
80πn
1 1
+ cos
=
2 2
30
2πn
1 1
+ cos
=
2 2
3
206

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Hence, y2 (t)

1 1
+ cos20πt
2 2

=

6.13
sa (t)

= xa (t) + αxa (t − τ ),
τ
= xa (n) + αxa (n − )
T

sa (n)
Sa (w)
Xa (w)
If

|α| < 1

τw
T

=

1 + αe−j

=

τ
1
where
=L
1 − αz −2
T

τ
is an integer, then we may select
T
H(z)

6.14
∞
X

x2 (n)

=

n=−∞

X(w)

=
=

∞
X

x2 (n)

=

n=−∞

=
=
Also, Ea

=
=

1
2π

Z

π

−π

|X(w)|2 dw



∞
1 X
w − 2πk
Xa
T
T
1
T

k=−∞
∞
X
k=−∞

1
2π

Z

π

−π

Xa

w
T

,

|w| ≤ π

1
w
|Xa ( )|2 dw
T2
T

Z Tπ
1
|Xa (λ)|2 T dλ
2πT 2 − Tπ
Z Tπ
1
|Xa (λ)|2 dλ
2πT − Tπ
Z ∞
x2a (t)dt
−∞
Z ∞
|Xa (f )|2 df
−∞

Therefore,

∞
X

x2 (n)

Fs
2

=

Z

=

Ea
T

n=−∞

− F2s

|Xa (f )|2 df

6.15
(a)
H(F )

=

Z

∞

h(t)e−j2πf t dt

−∞

207

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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Z

=

|0

Substituting a = −j2πf

T

Z 2T
Z 2T
t −j2πf t
t −j2πf t
−j2πf t
2e
dt −
e
dt +
e
dt
T
T
{z
} |T
{z
} |T
{z
}
A

A(F ) =
=

B

1
T



C


eaT
1
(aT
−
1)
−
(−1)
a2
a2

1
eaT
eaT
+
−
2
a
T
a
T
a2
|{z} |{z} |{z}
A2

A1

A3


2  a2T
e
− eaT
B(F ) =
a
2ea3T /2
=
sin(πf T )
πf


eaT
1 ea2T
(a2T
−
1)
−
(aT
−
1)
C(F ) = −
T
a2
a2
= −

ea2T
ea2T
eaT
eaT
ea2T
−
+
+
−
a } | {z
a } |T{z
a2} |{z}
a
T a2
|{z}
| {z
C1

C2

C3

A1(F ) + C1(F )

= −

A2(F ) + C3(F )

=

A3(F ) + C5(F )
C2(F ) + c4(F )

C4

C5

ea3T /2
sin(πf T )
πf

ea3T /2
sin(πf T )
T aπf
eaT /2
= −
sin(πf T )
T aπf
ea3T /2
= −
sin(πf T )
πf

Then,
e−j2πf T
H(F ) =
T
(b)



sin(πf T )
πf

2

6.16
(a)
d(n)
E[d(n)]
E[d (n)] ≡ σd2
2

where ρx (1)

= x(n) − ax(n − 1)

= E[x(n)] − aE[x(n − 1)] = 0

= E [x(n) − ax(n − 1)]2

= σx2 + a2 σx2 − 2aE[x(n)x(n − 1)]
= σx2 + a2 σx2 − 2aγx (1)
= σx2 (1 + a2 − 2aρx (1))
γx (1)
=
σx2
γx (1)
≡
γx (0)
208

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T
H(F)
H
(F)

|H|

ideal

T/2

0
−6/T

−5/T

−4/T

−3/T

−2/T

−1/T

0
F

1/T

2/T

3/T

4/T

5/T

6/T

Figure 6.15-1:
(b)

d  2
σx (1 + a2 − 2aρx (1)) = 2a − 2ρx (1) = 0
da
a = ρx (1)
For this value of α we have
σd2

= σx2 [1 + ρ2x (1) − 2ρ2x (1)]
= σx2 [1 − ρ2x (1)]

(c) σd2 < σx2 is always true if |ρx (1)| > 0. Note also that |ρx (1)| ≤ 1.
(d)
d(n)
E[d (n)]
2

σd2

= x(n) − a1 x(n − 1) − a2 x(n − 2)

= E [x(n) − a1 x(n − 1) − a2 x(n − 2)]2

= σx2 (1 + a21 + a22 + 2a1 (a2 − 1)ρx (1) − 2a2 ρx (2))

d 2
σ
da1 d

=

0

⇒ a1

=

d 2
σ
da2 d

ρx (1)[1 − ρx (2)]
1 − ρ2x (1)

=

0

⇒ a2

=

Then, σd2 min

=

ρx (2) − ρ2x (1)
1 − ρ2x (1)
1 − 3ρ2x (1) − ρ2x (2) + 2ρ2x (1)ρx (2) + 2ρ4x (1) + ρ2x (1)ρ2x (2) − 2ρ4x (1)ρx (2)
[1 − ρ2x (1)]2

6.17
x(t) = Acos2πF t
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dx(t)
dt

= −A(2πF )sin2πF t

= −2πAF sin2πF t
△
dx(t)
|max = 2πAF ≤
dt
T
Hence, △ ≥ 2πAF T
2πAF
=
Fs
Refer to fig 6.17-1.

Figure 6.17-1:

6.18
Let Pd denote the power spectral density of the quantization noise. Then (a)
Pn

=

Z

B
Fs

− FBs

Pd df

2B
Pd
Fs
= σe2

=

SQNR

σx2
σe2
σ 2 Fs
= 10log10 x
2BPd
= 10log10

210

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

=

10log10

σx2 Fs
+ 10log10 Fs
2BPd

Thus, SQNR will increase by 3dB if Fs is doubled.
(b) The most efficient way to double the sampling frequency is to use a sigma-delta modulator.

6.19
(a)
Se (F )
|Hn (F )|
σn2

=

σe2
Fs

πF
|
2|sin
Fs
Z B
|Hn (F )|2 Se (F )dF
=
=

=

2

−B
Z B

4sin2 (

0

=
=
=

πF σe2
) dF
Fs Fs

Z
2πF
4σe2 B
(1 − cos
)dF
Fs 0
Fs
Fs
2πB
4σe2
[B −
sin
]
Fs
2π
Fs
2πB
2σe2 2πB
[
− sin
]
π Fs
Fs

(b)
2πB
Fs
2πB
sin
Fs

For

Therefore, σn2

<< 1,
≈
=
=

2πB
1 2πB 3
− (
)
Fs
6 Fs
2πB
1 2πB 3
2σe2 2πB
[
−
− (
) ]
π Fs
Fs
6 Fs
1 2 2 2B 3
)
π σe (
3
Fs

6.20
(a)
{[X(z) − Dq (z)]

1
z −1
− Dq (z)}
−1
1−z
1 − z −1
Dq (z)
Therefore, Hs (z)
and Hn (z)

= Dq (z) − E(z)
= z −1 X(z) + (1 − z −1 )2 E(z)
= z −1
= (1 − z −1 )2

(b)
|Hn (F )|

=
=

πF
)
Fs
2πF
))
2(1 − cos(
Fs

4sin2 (

211

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

(c)
σn2

=

Z

=

|Hn (F )|2

−B
Z B

σe2
dF
Fs

πF 2 2 σe2
) ]
dF
Fs
Fs
0
Z
32π 4 σe2 B 4
F dF
Fs5
0
1 4 2 2B 5
π σe (
)
5
Fs

≈ 2
=

B

[4(

6.21
(a)
x(n)
xa (t)

2π
n
N
= x(n)|n= Tt
= cos

2πt
NT
Fs
= cos2π( )t
N
Fs
=
N
= cos

Therefore, F0

(b) N analog sinusoids can be generated. There are N possible different starting phases.

6.22
(a)
h(t)

=
=

Z

∞

−∞
Z ∞

−∞

H(F )ej2πF t dF
[c(F − Fc ) + c∗ (−F − Fc )]ej2πF t dF

= c(t)ej2πFc t + c∗ (t)e−j2πFc t
=

2Re[c(t)ej2πFc t ]

(b)
H(F ) = C(F − Fc ) + C ∗ (−F − Fc )
1
X(F ) =
[U (F − Fc ) + U ∗ (−F − Fc )]
2
Y (F ) = X(F )H(F )
1
[C(F − Fc )U (F − Fc ) + U ∗ (−F − Fc )C ∗ (−F − Fc )]
=
2
1
+ [C(F − Fc )U ∗ (−F − Fc ) + U (F − Fc )C ∗ (−F − Fc )]
2
But C(F − Fc )U ∗ (−F − Fc ) = U (F − Fc )C ∗ (−F − Fc ) = 0
Z ∞
c(τ )u(t − τ )dτ ≡ v(t)
F −1 [C(F )U (F )] =
−∞

212

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Hence, y(t)

1
1
v(t)ej2πFc t + v ∗ (t)e−j2πFc t
2
2
= Re[v(t)ej2πFc t ]
=

6.23
(a) Refer to fig 6.23-1.
(b) Refer to fig 6.23-2.
First Order hold, N = 32 thd=0.1152
1
−−−> x(n)

−−−> x(n)

Zero Order hold: N = 32 thd = 0.1154
1
0.8
0.6

0.6
0.4
0
20
40
60
80
First Order hold, N = 64 thd=0.2329
1

−−−> x(n)

−−−> x(n)

0.4
0
20
40
60
80
Zero Order hold: N = 64 thd = 0.2331
1

0.8

0.5

0.5

0
0
50
100
150
First Order hold, N = 128 thd=0.4683
1
−−−> x(n)

−−−> x(n)

0
0
50
100
150
Zero Order hold: N = 128 thd = 0.4686
1

0.5

0
0

100

200

0.5

0
0

300

−−−> n

100

200

300

−−−> n

Figure 6.23-1:
(c) Refer to fig 6.23-3. The first order hold interpolator performs better than the zero order
interpolator because the frequency response of the first order hold is more closer to the ideal
interpolator than that of the zero order hold case.
(d) Refer to fig 6.23-4.
(e) Refer to fig 6.23-5. Higher order interpolators with more memory or cubic spline interpolators
would be a better choice.

213

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

First Order hold, N = 32 thd=0.1153
1.5
−−−> x(n)

−−−> x(n)

Zero Order hold: N = 32 thd = 0.1154
1.5
1
0.5

1
0.5

1
0.5

100

200

1
0.5
0
0
50
100
150
First Order hold, N = 128 thd=0.4687
1.5

−−−> x(n)

−−−> x(n)

0
0
50
100
150
Zero Order hold: N = 128 thd = 0.4689
1.5

0
0

0.5
0
0
20
40
60
80
First Order hold, N = 64 thd=0.2332
1.5

−−−> x(n)

−−−> x(n)

0
0
20
40
60
80
Zero Order hold: N = 64 thd = 0.2333
1.5

1

1
0.5
0
0

300

−−−> n

100

200

300

−−−> n

Figure 6.23-2:

214

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Zero Order Hold

First Order Hold

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

10

20

30

40

0

50

Zero Order Hold, filter spectrum
2

1.5

1.5

1

1

0.5

0.5

−0.5

0

0.5

20

30

40

50

First Order Hold, filter spectrum

2

0
−1

10

0
−1

1

−0.5

0

0.5

1

Figure 6.23-3:
Zero Order Hold, interpolated output
50

−−−−> |X(f)|

40
30
20
10
0
0

100

200

300

400

500

600

500

600

First Order Hold, Interpolated output
50

−−−−> |X(f)|

40
30
20
10
0
0

100

200

300
−−−−> n

400

Figure 6.23-4:
215

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Zero Order Hold, y(n)
0.2

0.15

0.15
−−−> y(n)

−−−> xi(n)

Zero Order Hold, xi(n)
0.2

0.1
0.05
0
0

0.1
0.05

10

20

30

0
0

40

0.2

0.15

0.15

0.1
0.05
0
0

20

30

40

First Order Hold, y(n)

0.2

−−−> y(n)

−−−> xi(n)

First Order Hold, xi(n)

10

0.1
0.05

10

20

30

0
0

40

10

20

30

40

Figure 6.23-5:

216

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

6.24
P∞
(a) xp (t) = n=−∞ xa (t − nTs ) is a periodic signal with period Ts . The fourier coefficients in a
fourier series representation are
ck

=

1
Ts

Z

=

1
Ts

Z

=

=
=
=
=

Ts
2

∞
X

− T2s

n=−∞

−j2πkt
Ts

dt

xa (t − nTs )e

−j2πkt
Ts

dt

∞ Z Ts
2
−j2πkt
1 X
xa (t − nTs )e Ts dt
Ts n=−∞ − T2s
∞ Z nTs + Ts
2
−j2πk(t′ +nTs )
1 X
Ts
xa (t′ )e
dt′
Ts n=−∞ nTs − T2s
Z ∞
−j2πkt′
1
xa (t′ )e Ts dt′
Ts −∞
1
k
Xa ( )
Ts
Ts
1
Xa (kδF )
Ts

(b) Let
w(t) =
If Ts ≥ 2τ,

xp (t)e

− T2s
Ts
2



1, − T2s ≤ t ≤
0, otherwise

Ts
2

xa (t) = xp (t)w(t)
Xa (F ) = Xp (F ) ∗ W (F )
" ∞
# 

X
sinπF Ts
k
Xa (F ) =
ck δ(F − ) ∗ Ts
Ts
πF Ts
k=−∞
∞
X

= Ts
=

ck

sinπ(F −

k=−∞
∞
X

π(F −

Xa (kδF )

k=−∞

k
Ts )Ts

k
Ts )Ts

sinπ(F −
π(F −

k
Ts )Ts

k
Ts )Ts

,

Ts =

1
δF

(c) If T < 2τ , there will be aliasing in every period of xp (t). Hence, xa (t) 6= xp (t)w(t) and
consequently, xa (t) cannot be recovered from xp (t).
(F −kδF )
P∞
sinπ
δF
(d) From (b) Xa (F ) = k=−∞ Xa (kδF ) (F −kδF
)
π

δF

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218

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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Chapter 7

7.1
Since x(n) is real, the real part of the DFT is even, imaginary part odd. Thus, the remaining
points are {0.125 + j0.0518, 0, 0.125 + j0.3018}

7.2
(a)
x̃2 (l)

= x2 (l),
0≤l ≤N −1
= x2 (l + N ),
− (N − 1) ≤ l ≤ −1
3π
= sin( l),
0≤l≤7
8
3π
− 7 ≤ l ≤ −1
= sin( (l + 8)),
8
3π
= sin( |l|),
|l| ≤ 7
8
3
X
x̃2 (n − m)
=

x̃2 (l)

Therefore, x1 (n)

8

x2 (n)

m=0

3π
3π
3π
|n|) + sin( |n − 1|) + . . . + sin( |n − 3|)
8
8
8
= {1.25, 2.55, 2.55, 1.25, 0.25, −1.06, −1.06, 0.25}

= sin(

(b)
x̃2 (n)

Therefore, x1 (n)

8

x2 (n)

3π
n),
0≤l≤7
8
3π
− 7 ≤ l ≤ −1
= −cos( n),
8
3π
= [2u(n) − 1] cos( n),
|n| ≤ 7
8


3
m
X
1
x̃2 (n − m)
=
4
m=0
= cos(

= {0.96, 0.62, −0.55, −1.06, −0.26, −0.86, 0.92, −0.15}

(c)
for (a) X1 (k)

=

7
X

π

x1 (n)e−j 4 kn

n=0

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= {4, 1 − j2.4142, 0, 1 − j0.4142, 0, 1 + j0.4142, 0, 1 + j2.4142}

similarly,
X2 (k)
DFT of x1 (n)

8

= {1.4966, 2.8478, −2.4142, −0.8478, −0.6682, −0.8478,
−2.4142, 2.8478}
= X1 (k)X2 (k)

x2 (n)

= {5.9864, 2.8478 − j6.8751, 0, −0.8478 + j0.3512, 0,
−0.8478 − j0.3512, 0, 2.8478 + j6.8751}

For sequences of part (b)
X1 (k)

= {1.3333, 1.1612 − j0.2493, 0.9412 − j0.2353, 0.8310 − j0.1248,
0.8, 0.8310 + j0.1248, 0.9412 + j0.2353, 1.1612 + j0.2493}
= {1.0, 1.0 + j2.1796, 1.0 − j2.6131, 1.0 − j0.6488, 1.0,

X2 (k)

1.0 + j0.6488, 1.0 + j2.6131, 1.0 − j2.1796}

Consequently,
DFT of x1 (n) 8 x2 (n)

= X1 (k)X2 (k)
= {1.3333, 1.7046 + j2.2815, 0.3263 − j2.6947, 0.75 − j0.664, 0.8,
0.75 + j0.664, 0.3263 + j2.6947, 1.7046 − j2.2815}

7.3
x̂(k) may be viewed as the product of X(k) with

1, 0 ≤ k ≤ kc , N − kc ≤ k ≤ N − 1
F (k) =
0, kc < k < N − kc

F (k) represents an ideal lowpass filter removing frequency components from (kc + 1) 2π
N to π.
Hence x̂(n) is a lowpass version of x(n).

7.4
(a)
x1 (n)

=

X1 (k)

=

also X2 (k)

=

So X3 (k)

=
=

and x3 (n)

=


2π
1  j 2π n
e N + e−j N n
2
N
[δ(k − 1) + δ(k + 1)]
2
N
[δ(k − 1) − δ(k + 1)]
2j
X1 (k)X2 (k)
N2
[δ(k − 1) − δ(k + 1)]
4j
2π
N
sin( n)
2
N

(b)
R̃xy (k)

⇒

r̃xy (n)

= X1 (k)X2∗ (k)
N2
[δ(k − 1) − δ(k + 1)]
=
4j
2π
N
= − sin( n)
2
N
220

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

(c)
R̃xx (k)

⇒

r̃xx (n)

= X1 (k)X1∗ (k)
N2
=
[δ(k − 1) + δ(k + 1)]
4
2π
N
cos( n)
=
2
N

(d)
R̃yy (k)

⇒

r̃yy (n)

= X2 (k)X2∗ (k)
N2
=
[δ(k − 1) + δ(k + 1)]
4
2π
N
cos( n)
=
2
N

7.5
(a)
N
−1
X

x1 (n)x∗2 (n)

=

n=0

=
=
=

N −1
2
2π
1 X  j 2π n
e N + e−j N n
4 n=0

N −1

4π
1 X  j 4π n
e N + e−j N n + 2
4 n=0

1
2N
4
N
2

(b)
N
−1
X

x1 (n)x∗2 (n)

= −

n=0

=
=
(c)

PN −1
n=0

N −1


2π
2π
2π
1 X  j 2π n
e N + e−j N n e−j N n − ej N n
4j n=0

N −1

4π
1 X  j 4π n
e N − e−j N n
4j n=0

0

x1 (n)x∗2 (n) = 1 + 1 = 2

7.6
w(n)

=

w(k)

=


 4π

 2π
2π
4π
0.42 − 0.25 ej N −1 n + e−j N −1 n + 0.04 ej N −1 n + e−j N −1 n
"N −1
#
N
−1
N
−1
X
X
X
2π
2π
2π
2π
2π
e−j N nk − 0.25
0.42
e−j N −1 n e−j N nk
ej N −1 n e−j N nk +
n=0

+0.04

"N −1
X
n=0

n=0

n=0

2π
j N4π
−1 n −j N nk

e

e

+

N
−1
X
n=0

2π
−j N4π
−1 n −j N nk

e

e

#

221

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

=

0.42N δ(k)
"
#
N
N
1 − ej2π[ N −1 −k]
1 − e−j2π[ N −1 +k]
−0.25
+
1
1
k
k
1 − ej2π[ N −1 − N ]
1 − e−j2π[ N −1 + N ]
"
#
2N
2N
1 − ej2π[ N −1 −k]
1 − e−j2π[ N −1 +k]
+0.04
+
2
2
k
k
1 − ej2π[ N −1 − N ]
1 − e−j2π[ N −1 + N ]
= 0.42N δ(k)
#
"
1
k
2πk
2πN
1 − cos( N
−1 ) − cos(2π( N −1 + N )) + cos( N )
−0.25
1 − cos(2π( N 1−1 + Nk ))
"
#
4πN
2
k
2πk
1 − cos( N
−1 ) − cos(2π( N −1 + N )) + cos( N )
+0.04
1 − cos(2π( N 2−1 + Nk ))

7.7

Xc (k)

=

N
−1
X
n=0

=
=
similarly, Xs (k)

=

 2πkn
 2πk0 n
2πk0 n
1
e− N
x(n) ej N + e−j N
2

N −1
N −1
2π(k−k0 )n
2π(k+k0 )n
1 X
1 X
N
N
+
x(n)e−j
x(n)e−j
2 n=0
2 n=0

1
1
X(k − k0 )modN + X(k + k0 )modN
2
2
1
1
X(k − k0 )modN − X(k + k0 )modN
2j
2j

7.8

y(n)

= x1 (n)
=

3
X

m=0

4

x2 (n)

x1 (m)mod4 x2 (n − m)mod4

= {17, 19, 22, 19}

7.9

X1 (k)
X2 (k)
⇒ X3 (k)

= {7, −2 − j, 1, −2 + j}
= {11, 2 − j, 1, 2 + j}

= X1 (k)X2 (k)
= {17, 19, 22, 19}
222

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

7.10

x(n)
x(n)x∗ (n)
E


2πkn
1  j 2πkn
e N + e−j N
2

4πkn
4πkn
1
2 + ej N + e−j N
=
4
N
−1
X
x(n)x∗ (n)
=
=

n=0

N −1

4πkn
4πkn
1 X
2 + ej N + e−j N
4 n=0

=

1
2N
4
N
2

=
=

7.11
(a)
x1 (n)
X1 (k)

= x(n − 5)mod8

= X(k)e−j

2π5k
8

= X(k)e−j

5πk
4

(b)
x2 (n)
X2 (k)

= x(n − 2)mod8

= X(k)e−j

2π2k
8

= X(k)e−j

πk
2

7.12
(a)
s(k)

= W2k X(k)
=

(−1)k X(k)
5

s(n)

=

1X
(−1)k X(k)WN−kn
6

N =6

k=0
5

=

1X
−k(n−3)
X(k)WN
6
k=0

s(n)

= x(n − 3)mod6
= {3, 4, 0, 0, 1, 2}

(b)
y(n)

=

IDFT



X(k) + X ∗ (k)
2


223

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

1
[IDFT {X(k)} + IDFT {X ∗ (k)}]
2

1
x(n) + x∗ (−n)modN
=
2


x(1) + x(5) x(2) + x(4)
x(4) + x(2) x(5) + x(1)
=
x(0),
,
, x(3),
,
2
2
2
2


1
1
=
0, , 3, 3, 3,
2
2
=

(c)
v(n)

=

IDFT

By similar means to (b)



X(k) − X ∗ (k)
2j





1
1
=
0, − j, j, 0, −j, j
2
2

v(n)

7.13
(a)
X1 (k)

=

N
−1
X

x(n)WNkn

n=0

X3 (k)

=

3N
−1
X

kn
x(n)W3N

n=0

=

N
−1
X

kn
x(n)W3N

+

n=0

=

N
−1
X

=

n=0

=

kn
x(n)W3N

x(n)WN +

N
−1
X

+

3N
−1
X

kn
x(n)W3N

n=2N

n=N

nk
3

nk
3

x(n)W3k WN +

n=0

n=0

N
−1
X

2N
−1
X

N
−1
X

nk

x(n)W32k WN 3

n=0

 nk

x(n) 1 + W3k + W32k WN 3

(1 + W3k + W32k )X1 (k)

(b)
X1 (k)

=

2 + W2k

X3 (k)

=

2 + W6k + 2W62k + W63k + 2W64k + W65k

=

(2 + W23 ) + W62k (2 + W23 ) + W64k (2 + W23 )
k
(1 + W3k + W32k )X1 ( )
3

k

=

k

k

7.14
(a)
y(n)

= x1 (n)

5

x2 (n)

= {4, 0, 1, 2, 3}
224

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

(b) Let x3 (n) = {x0 , x1 , . . . , x4 }. Then,

0 4 3
 1 0 4

 2 1 0

 3 2 1
4 3 2

2
3
4
0
1

1
2
3
4
0

Solving yields sequence

x3 (n) =
.








x0
x1
x2
x3
x4





 
 
=
 
 

1
0
0
0
0










−0.18, 0.22, 0.02, 0.02, 0.02
↑

7.15
△

Define H1 (z) = H −1 (z) and corresponding time signal h1 (n). The use of 64-pt DFTs of y(n)
and h1 (n) yields x(n) = y(n) 64 h1 (n) whereas x(n) requires linear convolution. However we
can simply recognize that
X(z) = Y (z)H1 (z)
so x(n)
with y(−1)

= Y (z) − 0.5Y (z)z −1
= y(n) − 0.5y(n − 1),
△

=

0 ≤ n ≤ 63

0

7.16

H(k)

=

N
−1
X

2π

h(n)e−j N kn

n=0

2π
1
1 + ( )e−j 4k0 k0 k
4
π
1
= 1 − e−j 2 k
4
1
=
H(k)
1
=
2π
1 − 41 e−j N k

 
2
1 −j π k
1 −j π k
= 1+
+
e 2
e 2
+ ...
4
4


4 16 − 4j 4 16 + 4j
,
, ,
, repeat k0 times
=
3
17
5
17

=

G(k)

g(n)

=

=

N −1
2π
1 X
G(k)ej N kn
N n=0

4k
0 −4
X
2π
1
4 j 4k
[
e 0 kn +
4k0
3
k=0,4,...

4k
0 −3
X

k=1,5,...



16 − 4j
17



2π

ej 4k0 kn

225

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.


4k
0 −1 
X
2π
2π
16 + 4j
4 j 4k
ej 4k0 kn ]
e 0 kn +
5
17
k=3,7,...
k=2,6,...

 X 
X  4 j 2π 2n X
2π
1
4
16 − 4j
n
j 4k
0
+
e
e 4k0
+
4k0
3
17
5


X
2π
16 + 4j
3n
j 4k
0
+
e
17

+
=

where
But

X

X

g(0)

△

=

4k
0 −2
X

kX
0 −1

2π

ej k0 ni

i=0

=
=
=

g(k0 ) =
=
g(2k0 ) =
=
g(3k0 ) =
=
and g(n)

=

Therefore, g(n) ∗ h(n)

=

1, yielding





4
1 4
16 − 4j
16 + 4j
+ +
+
4 3
17
5
17
256
255





4
16 − 4j
16 + 4j
1 4
− −j
+j
4 3
17
5
17
64
255





1 4
4
16 − 4j
16 + 4j
+ −
−
4 3
17
5
17
16
255





4
16 − 4j
16 + 4j
1 4
− +j
−j
4 3
17
5
17
4
255
0 for other n in [0, 4k0 ) .









 256
1
,0
, 0, 0, . . . , 0 , . . . , 0 , . . . , 0 , . . . , −

255
255 
↑
↑
↑




↑


k0
2k0
3k0
4k0

g(.) represents a close approximation to an inverse system, but not an exact one.

7.17

X(k)

=

7
X

x(n)e−j

2π
8 kn

n=0

|X(k)|
6

X(k)

= {6, −0.7071 − j1.7071, 1 − j, 0.7071 + j0.2929, 0, 0.7071 − j0.2929, 1 + j,
−0.7071 + j1.7071}
= {6, 1.8478, 1.4142, 0.7654, 0, 0.7654, 1.4142, 1.8478}


π
−π
, 0.3927, 0, −0.3927, , 1.9635
=
0, −1.9635,
4
4
226

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

7.18

x(n)

∞
X

=

i=−∞

y(n)

X

=

m

X

=

δ(n − iN )

h(m)x(n − m)
h(m)

m

X

=

i

"

X
i

#

δ(n − m − iN )

h(n − iN )

Therefore, y(.) is a periodic sequence with period N. So
Y (k)

N
−1
X

=

y(n)WNkn

n=0

= H(w)|w= 2π
N k
Y (k)

= H(

2πk
)
N

k = 0, 1, . . . , N − 1

7.19
Call the two real even sequences xe1 (.) and xe2 (.), and the odd ones xo1 (.) and xo2 (.) (a)
Let xc (n)
Then, Xc (k)

=
=
=

where Xe1 (k)

=

Xo1 (k)

=

Xe2 (k)

=

Xo2 (k)

=

[xe1 (n) + xo1 (n)] + j [xe2 (n) + xo2 (n)]
DFT {xe1 (n)} + DFT {xo1 (n)} + jDFT {xe2 (n)} + jDFT {xo2 (n)}

[Xe1 (k) + Xo1 (k)] + j [Xe2 (k) + Xo2 (k)]
Re[Xc (k)] + Re[Xc (−k)]
2
Re[Xc (k)] − Re[Xc (−k)]
2
Im[Xc (k)] + Im[Xc (−k)]
2
Im[Xc (k)] − Im[Xc (−k)]
2

(b)
si (0)
−si (N − n)

= xi (1) − xi (N − 1) = 0
= −xi (N − n + 1) + xi (N − n − 1)
= xi (n + 1) − xi (n − 1)

= si (n)
(c)
x(n)

=

[x1 (n) + s3 (n)] + j [x2 (n) + s4 (n)]

The DFT of the four sequences can be computed using the results of part (a)
For i = 3, 4, si (k)

=

N
−1
X

si (n)WNkn

n=0

227

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

N
−1
X

=

n=0
WN−k Xi (k)

=

=

X4 (k)

=

− WNk Xi (k)

2π
k)Xi (k)
N
s3 (k)
2jsin( 2π
N k)
s4 (k)
2jsin( 2π
N k)

=
Therefore, X3 (k)

[xi (n + 1) − xi (n − 1)] WNkn

2jsin(

(d) X3 (0) and X4 (0), because sin( 2π
N k) = 0.

7.20

X(k)

=

N
−1
X

x(n)WNkn

n=0

N
2

N
2

=

−1
X

x(n)WNkn

+

n=0
N
2

=

−1
X

x(n +

n=0

−1

N
k(n+ N
2 )
)WN
2

X

x(n) − x(n)W2k WNkn

n=0

If k is even, W2k = 1, and X(k) = 0
(b) If k is odd, W2k = −1, Therefore,
N
2

X(k)

=

−1
X

2x(n)WNkn

n=0
N
2

= 2

−1
X

nk

x(n)W N 2
2

n=0

For k

= 2l + 1,

l = 0, . . . ,

N
−1
2

N
2

X(2l + 1)

= 2

−1
X

ln
WNn
x(n)W N
2

n=0

=

N
− pt DFT of sequence 2x(n)WNn
2

7.21
(a) Fs ≡ FN = 2B = 6000 samples/sec
(b)
T

=
=

1
Fs
1
6000
228

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

1
LT

≤ 50

⇒L ≥
=
=
(c) LT =

1
6000

1
50T
6000
50
120 samples

× 120 = 0.02 seconds.

7.22
x(n)
X(k)

1 j 2π n 1 −j 2π n
e N + e N ,
2
2
N
−1
X
2π
x(n)e−j N kn
=
=

0 ≤ n ≤ N,

N = 10

n=0

=

N
−1
X
n=0

=

N −1
1 −j 2π (k−1)n X 1 −j 2π (k+1)n
+
e N
e N
2
2
n=0

5δ(k − 1) + 5δ(k − 9),

0≤k≤9

7.23
(a) X(k) =
(b)

PN −1
n=0

2π

δ(n)e−j N kn = 1,

X(k)

=

0≤k ≤N −1
N
−1
X
n=0

2π

δ(n − n0 )e−j N kn

2π

= e−j N kn0 ,

0≤k ≤N −1

(c)
X(k)

=

N
−1
X

2π

an e−j N kn

n=0

=

N
−1
X

2π

(ae−j N k )n

n=0

=

1 − aN

2π

1 − ae−j N k

(d)
N
2

X(k)

=

−1
X

2π

e−j N kn

n=0

=
=

2π N
2

1 − e−j N

k

2π
e−j N k

1−
1 − (−1)k
2π

1 − e−j N k
229

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

(e)
X(k)

N
−1
X

=

2π

2π

ej N nk0 e−j N kn

n=0

N
−1
X

=

2π

e−j N (k−k0 )n

n=0

= N δ(k − k0 )
(f)
x(n)

=

From (e) we obtain X(k)

=

1 j 2π nk0 1 −j 2π nk0
+ e N
e N
2
2
N
[δ(k − k0 ) + δ(k − N + k0 )]
2

(g)
x(n)

=

Hence X(k)

=

2π
1 j 2π nk0
1
− e−j N nk0
e N
2j
2j
N
[δ(k − k0 ) − δ(k − N + k0 )]
2j

(h)
X(k)

N
−1
X

=

2π

x(n)e−j N nk ( assume N odd )

n=0

2π

2π

2π

1 + e−j N 2k + e−j N 4k + . . . + e−j N (n−1)k

=

2π

1 − (e−j N 2k )

=

N +1
2

2π

1 − e−j N 2k
2π
1 − e−j N k

=

4π

1 − e−j N k
1

=

2π

1 − e−j N k

7.24
(a)
x(n)

=

N −1
2π
1 X
X(k)ej N nk
N
k=0

⇒

= N x(n)

k=0

=

4

j 3π
2

=

8

j3π

=

12

X(0) + X(1)e

jπ

X(0) + X(1)e

j 3π
2

X(0) + X(1)e
1
1
 1
j

 1 −1
1 −j

2π

X(k)ej N nk

X(0) + X(1) + X(2) + X(3)
jπ
2



N
−1
X

1
1
−1 −j
1 −1
−1
j

jπ

+ X(2)e

j2π

+ X(2)e

+ X(3)e

+ X(3)e

j 9π
2

j3π

+ X(2)e

X(0)
  X(1)

  X(2)
X(3)

+ X(3)e
 
4
  8
=
  12
4

= 4

X(0)
 X(1)

⇒
 X(2)

X(3)



7
  −2 − j 

=

  1
−2 + j




230

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(b)
X(k)

3
X

=

x(n)e−j

2π
4 nk

n=0

X(0)

3
X

=

x(n)

n=0

=

7
3
X

π

x(n)e−j 2 n

X(1)

=

X(2)

= −2 − j
3
X
x(n)e−jπn
=

n=0

n=0

=

X(3)

1
3
X

=

x(n)e−j

3π
2 n

n=0

= −2 + j

7.25
(a)
X(w)

∞
X

=

x(n)e−jwn

n=−∞
j2w

= e
+ 2ejw + 3 + 2e−jw + e−j2w
= 3 + 2cos(2w) + 4cos(4w)
(b)
V (k)

=

5
X

v(n)e−j

2π
6 nk

n=0

=
=

3 + 2e−j

2π
6 k

+ e−j

2π
6 2k

+ 0 + e−j
2π
π
3 + 4cos( k) + 2cos( k)
3
3

2π
6 4k

+ e−j

2π
6 5k

(c) V (k) = X(w)|w= 2πk = πk
6
3
This is apparent from the fact that v(n) is one period (0 ≤ n ≤ 7) of a periodic sequence
obtained by repeating x(n).

7.26

Let x(n)

=

∞
X

δ(n + lN )

l=−∞

Hence, x(n) is periodic with period N, i.e.
x(n)

= 1,

n = 0, ±N, ±2N, . . .
231

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

=
Then X(k)

=

0,

otherwise

N
−1
X

2π

x(n)e−j N nk = 1,

0≤k ≤N −1

n=0

and x(n)

N −1
2π
1 X
X(k)ej N nk
N

=

k=0

Hence,

∞
X

δ(n + lN )

1
N

=

l=−∞

N
−1
X

2π

ej N nk

k=0

7.27
(a)
Y (k)

=

M
−1
X

kn
y(n)WM

n=0

=

M
−1 X
X
n=0

Now X(w)

=

X

kn
x(n + lM )WM

l

x(n)e−jwn ,

n

2π
so X( k)
M

=

X

kn
WM

n

=

M
−1 X
X
n=0

=

l

M
−1 X
X
n=0

Therefore, Y (k)

k(n+lM )

x(n + lM )WM

kn
x(n + lM )WM

l

= Y (k)
= X(w)|w= 2π
M k

(b)
Y (k)

= X(w)|w= 2π
N k
2

k
Y ( ) = X(w)|w= 2π
N k
2
= X(k),
k = 2, 4, . . . , N − 2
(c)
X1 (k)
⇒ x1 (n)
Let y(n)

= X(k + 1)
2π

= x(n)e−j N n
= x(n)WNn
= x1 (n) + x1 (n +
=

Then X(k + 1)

0,

= X1 (k)
k
= Y ( ),
2

N
),
2

0≤n≤N −1

elsewhere

k = 0, 2, . . . , N − 2

232

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

where Y (k) is the

N
-pt DFT of y(n)
2

7.28
(a) Refer to fig 7.28-1.
(b)
X(w)
20

0.8

15
−−−> X(w)

−−−> x(n)

x(n)
1

0.6
0.4
0.2

10
5
0

0
0

10

20

−5
0

30

−−−> n

1

2
−−−> w

ck

xtilde(n)

3

4

0.6
1
−−−> xtilde(n)

0.4
0.2
0
−0.2
0

10

20

0.5

0
0

30

10

20

30

−−−> n

Figure 7.28-1:
∞
X

n=−∞

x(n)e−jwn

=

∞
X

a|n| e−jwn

n=−∞

= a+

1
X

a−n e−jwn +

= a+

an ejwn +

L
X

an e−jwn

1

1

= a+2

an e−jwn

1

−L

L
X

L
X

L
X

an cos(wn)

n=1

= x(0) + 2

L
X

x(n)cos(wn)

n=1

(c) Refer to fig 7.28-1.
(d) Refer to fig 7.28-1.
233

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

(e) Refer to fig 7.28-2.
(f) N=15. Refer to fig 7.28-3.
x(n)
1
0.9
0.8
0.7

−−−> x(n)

0.6
0.5
0.4
0.3
0.2
0.1
0
0

50

100

150

200

250

−−−> n

Figure 7.28-2:

234

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

X(w)
20

0.8

15
−−−> X(w)

−−−> x(n)

x(n)
1

0.6
0.4

10
5

0.2
0
−20

0
−10

0
−−−> n

10

−5
0

20

1

1.4

1

1.2

0.5
0
−0.5
0

5

3

4

5

10

xtilde(n)

1.5

−−−> xtilde(n)

−−−> ck

ck

2
−−−> w

10

1
0.8
0.6
−10

15

−5

−−−> w

0
−−−> n

Figure 7.28-3:

7.29
Refer to fig 7.29-1. The time domain aliasing is clearly evident when N=20.

7.30
Refer to fig 7.30-1.
(e)
xam (n)
Xam (w)

= x(n)cos(2πfc n)
=

N
−1
X

x(n)cos(2πfc n)e−j2πf n

n=0

=
Xam (w)

=

N −1
i
h
1 X
x(n) e−j2π(f −fc )n + e−j2π(f +fc )n
2 n=0

1
[X(w − wc ) + X(w + wc )]
2

7.31
2
1
(a) ck = { π2 , − π1 , 3π
, − 2π
. . .}
(b) Refer to fig 7.31-1. The DFT of x(n) with N = 128 has a better resolution compared to one
with N = 64.

235

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

X(w)

x(n)
1.5
−−> x(n)

−−> mag

10

5

0
0

2

4
6
X(w) with N=20

0
0

−−> mag

−−> x(n)

2
4
6
X(w) with N=100

3000

10
20
x(n) with N=100

30

1
0.5
0
0

8

1.5
−−> x(n)

10
−−> mag

1000
2000
x(n) with N=20

1.5

5

5

0
0

0.5

8

10

0
0

1

2

4
−−> w

6

1
0.5
0
0

8

50

100

150

−−> n
Figure 7.29-1:

7.32
(a)
1
P (jΩ) ∗ X(jΩ)
2π 

1
ΩT0 −j ΩT0
2
=
∗ [2πδ(Ω − Ω0 )]
T0 sin(
)e
2π
2
sin x
△
where sincx =
x


T0 (Ω − Ω0 ) −j T0 (Ω−Ω0 )
2
e
Y (jΩ) = T0 sinc
2
Y (jΩ)

=

(b) w0 P = 2πk for an integer k, or w0 =
(c)
Y (w)

=

2k
P π

N
−1
X

ejw0 n e−jwn

n=0

236

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

x(n)

xc(n)
1
xc(n)

x(n)

2

0

−2
0

100

xam(n)

200

−1
0

300

mag

mag

0

300

100
200
Xam(w) with N=100

50
100
Xam(w) with N=180

150

20

0
0

300

30

60

20

40

10
0
0

100
200
Xam(w) with N=128

40

mag

xam(n)

2

−2
0

0

20

50

100

0
0

150

100

200

300

Figure 7.30-1:
sin N2 (w − w0 ) −j N −1 (w−w0 )
2
e
0
sin w−w
2

=

(d)

Larger N ⇒ narrower main lobe of |Y (w)|. T0 in Y (jΩ) has the same effect.
Y (k)

|Y (k)|

= Y (w)|w= 2π
N k
=

sinπ(k − l)

=

|sinπ(k − l)|

sin π(k−l)
N

e−j

N −1
N π(k−l)

|sin π(k−l)
|
N
= N δ(k − l)

(e) The frequency samples 2π
N k fall on the zeros of Y (w). By increasing the sampling by a factor
of two, for example, we will obtain a frequency sample between the nulls.
2π
π
Y (w)|w= 2N
k= N
k,
k=0,1,...,2N −1

237

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

x(n)

DFT of x(n) with N=64

1.5

20

1
15
0.5
0

10

−0.5
5
−1
−1.5
0

200

400

600

0
0

800

20

40

60

80

DFT of x(n) with N=128
60
50
40
30
20
10
0
0

50

100

150

Figure 7.31-1:

238

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Chapter 8

8.1
2π
2π
2π
Since (ej N k )N = ej2πk = 1, ej N k satisfies the equation X N = 1. Hence ej N k is an N th root
PN −1 j 2π kn j 2π ln
of unity. Consider n=0 e N e N . If k 6= l, the terms in the sum represent the N equally
spaced roots in the unit circle which clearly add to zero. However, if k = l, the sum becomes
P
N −1
n=0 1 = N . see fig 8.1-1

j4 π
j2 π
12
e
e 12

z-plane

unit circle
Roots for N=12
Figure 8.1-1:

8.2
q(l−1)

2π

2π

2π

= e−j N q e−j N q(l−1) = e−j N ql = WNql
(a) WNq WN
q
(b) Let ŴN = WNq +δ where ŴNq is the truncated value of WNq . Now ŴNql = (WNq +δ)l ≈ WNql +lδ.
239

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Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Generally, single precision means a 32-bit length or δ = 5x10−10 ; while 4 significant digits means
δ = 5x10−5 . Thus the error in the final results would be 105 times larger.
(c) Since the error grows as lδ, after N iterations we have an error of N δ. If hWNqli is reset to -j

after every ql = N4 iterations, the error at the last step of the iteration is lδ =
error reduced by approximately a factor 4q.

N
4q

δ. Thus, the

8.3

X(k)

=

N
−1
X

x(n)WNkn

0≤k ≤N −1

n=0
N
2

−1
X

=

x(n)WNkn +

n=0

=

N
2

x(n)WNkn

+

n=0

−1
X

x(r +

r=0

′

′

′

′

=

′

k n
= WN
,

LetX (k )

x(n)WNkn

n= N
2

N
2

−1
X

N
−1
X

0 ≤ k′ ≤

= X(2k + 1),

N
(r+ N )k
)WN 2
2

N
−1
2

N
2

Then, X (k )
Using the fact that WN2k n


−1 
X
N
(n+ N )(2k′ +1)
(2k′ +1)n
x(n)WN
+ x(n + )WN 2
2
n=0
′

2

WNN = 1

N
2

′

′

X (k )


−1 
X
N
N
k′ n
n
k′ n
n
2
x(n)WN W N + x(n + )W N WN WN
2
2
2
n=0

=

N
2


−1 
X
N
k′ n
x(n) − x(n + ) WNn W N
2
2
n=0

=

8.4
Create three subsequences of 8-pts each

Y (k)

=

21
X

y(n)WNkn +

=

ki
y(3i)W N
+

i=0

△

y(n)WNkn +

3

7
X

ki
y(3i + 1)W N
WNk +
3

i=0

23
X

y(n)WNkn

n=2,5,...

n=1,4,7,...

n=0,3,6,...
7
X

22
X

= Y1 (k) + WNk Y2 (k) + WN2k Y3 (k)

7
X
i=0

ki
y(3i + 2)W N
WN2k
3

where Y1 , Y2 , Y3 represent the 8-pt DFTs of the subsequences.

8.5
X(z)

= 1 + z −1 + . . . + z −6
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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

X(k)

= X(z)|
=

z=ej
−j 2π
5

1+e

2π
5

+ e−j

4π
5

2π

x′ (n)
x′ (n)

+ . . . + e−j

4π

12π
5
8π

= 2 + 2e−j 5 + e−j 5 + . . . + e−j 5
= {2, 2, 1, 1, 1}
X
=
x(n + 7m),
n = 0, 1, . . . , 4
m

Temporal aliasing occurs in first two points of x′ (n) because X(z) is not sampled at sufficiently
small spacing on the unit circle.

8.6
(a) Zk = 0.8ej [
(b)

π
2πk
8 +8

] see fig 8.6-1

2π
8

z-plane
z1

z2

z0

z3

z4

π
8

z7
z6

z5
circle of radius 0.8

Figure 8.6-1:

X(k)

= X(z)|z=zk
7
i−n
h
X
π
2πk
x(n) 0.8ej [ 8 + 8 ]
=
n=0

s(n)

π

= x(n) 0.8e−j 8 n
241

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

8.7

Let M

=

N
,
2

L = 2. Then

N
2

F (0, q)

=

−1
X

mq
x(0, m)W N
2

n=0
N
2

F (1, q)

=

−1
X

mq
x(1, m)W N

n=0

2

which are the same as F1 (k) and F2 (k) in (8.1.26)
G(0, q)
G(1, q)

= F (0, q) = F1 (k)
= WNq F (1, q) = F2 (k)WNk

X(0, q) = x(k) = G(0, q) + G(1, q)W20
= F1 (k) + F2 (k)WNk
X1, q) = x(k) = G(0, q) + G(1, q)W21
= F1 (k) − F2 (k)WNk

8.8
1
W8 = √ (1 − j)
2
Refer to Fig.8.1.9. The first stage of butterflies produces (2, 2, 2, 2, 0, 0, 0, 0). The twiddle
factor multiplications do not change this sequence. The nex stage produces (4, 4, 0, 0, 0, 0, 0,
0) which again remains unchanged by the twiddle factors. The last stage produces (8, 0, 0, 0, 0,
0, 0, 0). The bit reversal to permute the sequence into proper order unscrambles only zeros so
the result remains (8, 0, 0, 0, 0, 0, 0, 0).

8.9
See Fig. 8.1.13.

8.10
Using (8.1.45), (8.1.46), and (8.1.47) the fig 8.10-1 is derived:

8.11
Using DIT following fig 8.1.6:
1st stage outputs
2nd stage outputs


1
1 1 1
, , ,...,
2 2 2
2


1
1
1
1
:
1, (1 + W82 ), 0, (1 − W82 ), 1, (1 + W82 ), 0, (1 − W82 )
2
2
2
2

:



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Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

x(0)
0
x(4)

0

x(0)
x(1)

0

x(8)

x(2)
0

x(12)
x(1)

x(3)
0

x(4)
0

x(5)

0

x(9)

0

1
2

x(5)
x(6)

3
x(13)

0

x(7)

0
x(2)
0

0
x(6)

2

x(8)
x(9)

0
4

x(10)
0
x(14)
x(3)
x(7)
x(11)

x(10)

6

0

x(11)

0
0

0
3
6

0

x(12)
x(13)
x(14)

9
x(15)

x(15)

Figure 8.10-1:
3rd stage outputs

:



1
1
2, (1 + W81 + W82 + W83 ), 0, (1 − W82 + W83 − W85 ), 0,
2
2

1
1
(1 − W81 + W82 − W83 ), 0, (1 − W82 − W83 + W85
2
2

Using DIF following fig 8.1.11:


1 1 1 1 1 1 1 1 1 2 1 3
, , , , , , W8 , W8 , W8
1st stage outputs :
2 2 2 2 2 2 2
2
2


1
1
1
1
2nd stage outputs :
1, 1, 0, 0, (1 + W82 ), 0, (W81 + W83 ), (1 − W82 ), (W83 − W85 )
2
2
2
2

1
1
3rd stage outputs :
2, 0, 0, 0, (1 + W81 + W82 + W83 ), (1 − W81 + W82 − W83 ),
2
2

1
1
(1 − W82 + W83 − W85 ), (1 − W82 − W83 + W85
2
2

8.12
Let




1
1
1
1
△  1 −j
−1
j 

A=
 1 −1
1 −1 
1
j −1 −j

T
△
x1 = x(0) x(4) x(8) x(12)
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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

△

x2 =
△

x3 =
△

x4 =






x(1)

x(5)

x(9)

x(13)

x(2)

x(6)

x(10)

x(14)

x(3)

x(7)

x(11)

x(15)





F (0)
4
 F (4) 
 0



 F (8)  = Ax1 =  0
F (12)
0









F (1)
0
 F (5) 
 0



 F (9)  = Ax2 =  0
F (13)
0

T







F (2)
−4
 F (6) 
 0



 F (10)  = Ax3 =  0
F (14)
0


0
F (3)
 0
 F (7) 



 F (11)  = Ax4 =  0
0
F (15)

T









T










As every F (i) = 0 except F (0) = −F (2) = 4,




x(0)
 x(7) 




 x(8)  = Ax4 
x(12)

 
0
F (0)
 8
F (1) 
=
F (2)   0
8
F (3)






which means that X(4) = X(12) = 8. X(k) = 0 for other K.

8.13
(a) ”gain” = W80 W80 (−1)W82 = −W82 = j
(b) Given a certain output sample, there is one path from every input leading to it. This is true
for every output.
(c) X(3) = x(0) + W83 x(1) − W82 x(2) + W82 W83 x(3) − W80 x(4) − W80 W83 x(5) + W80 W82 x(6) +
W80 W82 W83 x(7)

8.14
Flowgraph for DIF SRFFT algorithm for N=16 is given in fig 8.14-1. There are 20 real, non
trivial multiplications.
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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

x(0)

X(0)

x(1)

X(8)

-1
x(2)
-1
x(3)

w0

-1

-j

-1

-j

-1

x(10)

-1

x(11)

w2

-j

w3

+j

-1

x(13)
x(14)

w

-1

-1

X(5)
X(13)
X(3)

-1

X(11)

6

+j

-1

-j
+j

-1

w9

x(15)

X(9)

-1

3
w

+j

-1

X(1)
-1

0
w

-j

X(6)
X(14)

+j

w1

-j

-1

x(12)

-1

w0

-1

x(9)

-j

w6

+j

-1

x(8)

w0

+j

-1

X(10)

-1

-j

x(6)

X(2)

w2

x(5)

x(7)

X(12)

+j

-1

x(4)

X(4)

-j

-j
-1

+j

+j

X(7)
X(15)

Figure 8.14-1:

8.15
For the DIT FFT, we have
N
2

N
2

−1
X

X(k) =

nk

x(2n)W N +
2

n=0

−1
X

(2n+1)k

x(2n + 1)WN

n=0

The first term can be obtained from an N2 -point DFT without any additional multiplications.
Hence, we use a radix-2 FFT. For the second term, we use a radix-4 FFT. Thus, for N=8, the
DFT is decomposed into a 4-point, radix-2 DFT and a 4-point radix-4 DFT. The latter is
N
2

−1
X

x(2n +

(2n+1)k
1)WN

N
4

N
4

=

n=0

−1
X

x(4n +

k
1)WNk W N
4

n=0

−1
X

+

k
x(4n + 3)WN3k W N
4

n=0

N
The computation of X(k), X(k + N4 ), X(k + N2 ), X(k + 3N
4 ) for k = 0, 1, . . . , 4 − 1 are performed
from the following:
N
2

X(k)

=

−1
X

n=0

N
X(k + )
4
N
X(k + )
2

N
4

nk

x(2n)W N +
2

N
2

=

−1
X

n=0

=

n=0

N
4

x(4n +

n

2

2

−1
X

+

−1
X

−1
X

nk
x(4n + 3)WN3k W N
4

n=0

N
4

nk

x(4n + 1)(−j)W N +
4

n=0
N
4

x(2n)W N +

4

N
4

x(2n)W N (−1) +

nk

nk
1)WNk W N

n=0

nk

N
2

−1
X

−1
X

n=0
N
4

x(4n +

nk
1)(−1)WNk W N

n=0

4

+

−1
X

−1
X

n=0

nk
x(4n + 3)WN3k (j)W N
4

nk
x(4n + 3)(−1)WN3k W N
4

245

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

3N
X(k +
) =
4

N
4

N
2

−1
X

nk

n

x(2n)W N (−1) +
2

n=0

−1
X

N
4

x(4n +

nk
1)(j)WNn W N
4

n=0

+

−1
X

nk
x(4n + 3)(−j)WN3k W N

n=0

The basic butterfly is given in fig 8.15-1

X(k)
n
WN from the use of
x(2n)
from
x(2n+1)

X(k+N/4)

X

-1

Note that this is a mirror image of DIF-SRFFT butterfly.

X(k+N/2)

-j
from

X

X(k+3N/4)

j

-1

x(4n+3)
W3k
N

X(0)

x(0)
x(4)

X(1)

-1

x(6)

-1

This graph looks like the transpose of
X(3)

-1

W2

-1

W

1

an N-point DIF FFT. The twiddle factors
come before the second stage.

x(1)
x(5)

DIT/SRFFT

X(2)

-1

x(2)

-1

X(4)

-1

X(5)

-1

X(6)

-J

x(3)

J
-J

x(7)

-1

-1

J

W

X(7)

3

Figure 8.15-1:

8.16
x = xR + jxI
e
xR
xI
Total

= (a + jb)(c + jd)
= (a − b)d
1 add , 1 mult

= e + (c − d)a
2 adds 1 mult
= e + (c + d)b
2 adds 1 mult
5 adds 3 mult

8.17

X(z) =

N
−1
X

x(n)z −n

n=0

246

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

4

Hence, X(zk ) =

N
−1
X

2π

x(n)r−n e−j N kn

n=0

2π

where zk = re−j N k , k = 0, 1, . . . , N − 1 are the N sample points. It is clear that X(zk ), k =
0, 1, . . . , N − 1 is equivalent to the DFT (N-pt) of the sequence x(n)r−n , n ∈ [0, N − 1].

8.18

′

x (n)

=

=

=

=
Therefore Lx′ (Ln)

=

LN −1
1 X ′
−kn
X (k)WLN
LN
k=0
"k −1
#
LN
−1
0
X
X
1
−kn
−kn
′
′
X (k)WLN
X (k)WLN +
LN
k=LN −k0 +1
k=0
"k −1
#
LN
−1
0
X
X
1
−kn
−kn
X(k)WLN +
X(k + N − LN )WLN
LN
k=0
k=LN −k0 +1
"k −1
#
N
−1
0
X
X
1
−(k−N +LN )n
−kn
X(k)WLN
X(k)WLN +
LN
k=N −k0 −1
k=0
"k −1
#
N
−1
0
X
1 X
X(k)WN−kn
X(k)WN−kn +
N
k=N −k0 +1

k=0

= x(n)

L = 1 is a trivial case with no zeros inserted and


1 1
1
1
1
x′ (n) = x(n) =
, + j , 0, − j
2 2
2
2
2

8.19
X(k) =

N
−1
X

x(n)WNkn

n=0

Let F (t),

t = 0, 1, . . . , N − 1 be the DFT of the sequence on k X(k).
F (t)

=

N
−1
X

X(k)WNtk

k=0

=

N
−1
X
k=0

=

N
−1
X
n=0

=

N
−1
X

"N −1
X

x(n)WNkn

n=0

x(n)

"N −1
X

#

WNtk

k(n+t)
WN

k=0

#

x(n)δ(n + t)mod N

n=0

247

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

=

N
−1
X
n=0

x(n)δ(N − 1 − n − t)

t = 0, 1, . . . , N − 1

= {x(N − 1), x(N − 2), . . . , x(1), x(0)}

8.20

Y (k)

=

2N
−1
X

y(n)WNkn

n=0

2N
−1
X

=

n=0,n

=

N
−1
X

even

k = 0, 1, . . . , 2N − 1

kn
y(n)W2N

y(2m)WNkm

m=0

=

N
−1
X

x(m)WNkm

m=0

= X(k),
k ∈ [0, N − 1]
= X(k − N ),
k ∈ [N, 2N − 1]

8.21
(a)
1
2πn
(1 − cos
),
0≤n≤N −1
2
N −1
2πn
1 1 j N2πn
=
− (e −1 + e−j N −1 )
2 4
N
−1
X
w(n)z −n
W (z) =
w(n)

=

n=0

=

N
−1 
X
n=0


2πn
1 1 j N2πn
− (e −1 + e−j N −1 ) z −n
2 4
2π

1 1 − (z −1 ej N −1 )N
1 1 − z −N
−
2 1 − z −1
4 1 − z −1 ej N2π
−1

=

2π

−

1 1 − (z −1 e−j N −1 )N
4 1 − z −1 e−j N2π
−1

(b)
xw (n)
⇒ Xw (k)

= w(n)x(n)
= W (k)N X(k)

8.22
The standard DFT table stores N complex values WNk ,

k+ N
WN 2

=

−WNk ,

we need only store

WNk

k = 0, 1, . . . ,

N
2

k = 0, 1, . . . , N − 1. However, since
k+ N
4

− 1. Also, WN

= −jWNk which is

248

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

merely an interchange of real and imaginary parts of WNk and a sign reversal. Hence all essential
quantities are easily obtained from WNk
k = 0, 1, . . . , N4 − 1

8.23
The radix-2 FFT algorithm for computing a 2N-pt DFT requires 2N
N log2 2N = N + N log2 N
N
complex multiplications. The algorithm in (8.2.12) requires 2[ 2 log2 N + N2 ] = N2 +log2 N complex
multiplications.

8.24

since H(z)

=

2π
k)
N −1

=

H(

△

PM

k=0 bk z

−k

PN

1 + k=1 ak z −k
PM
kn
k=0 bk WN +1
PN
1 + k=1 ak WNkn+1

= H(k),

k = 0, . . . , N

Compute N + 1-pt DFTs of sequences {b0 , b1 , . . . , bM , 0, 0, . . . , 0} and {1, a1 , . . . , aN } (assumes
N > M ), say B(k) and A(k)
k = 0, . . . , N
H(k) =

B(k)
A(k)

8.25

Y (k)

=

8
X

y(n)W9nk

n=0

=

X

n=0,3,6

=

2
X

=

y(n)W9nk +

n=1,4,7

y(3m)W93km +

2
X

X

y(n)W9nk

n=2,5,8
(3m+1)k

y(3m + 1)W9

y(3m)W3km +

2
X

y(3m + 1)W3mk W9k +

2
X

(3m+2)k

y(3m + 2)W9

2
X

y(3m + 2)W3mk W92k

m=0

m=0

m=0

+

m=0

m=0

m=0
2
X

X

y(n)W9nk +

Total number of complex multiplies is 28 and the operations can be performed in-place. see
fig 8.25-1

8.26

X(k)

=

8
X

x(n)W9nk

n=0

249

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

x(0)

X(0)
1
W3
2
W3
2
W3
W 13

x(1)
x(2)

X(3)
X(6)

x(3)

X(1)
1
W3
2
W3

x(4)

1
W9
W

x(5)

W1
3

X(4)

2
9

X(7)

x(6)

X(2)

x(7)

2
W9

X(5)

x(8)

W1
9

X(8)

Figure 8.25-1:

=
=
x(3l)

=

2
X

=

n=3
2
X

=

x(n)W9kn +

n=0

2
X

2
X

2
X

2
X

2
X

n=0

x(n)W3nl +

x(n)W9nk

n=6
2
X

x(n + 6)W9nk W32k

n=0

x(n + 3)W3nl +

2
X

x(n + 6)W3nl

n=0

n=0

x(n)W3nl W9n +

8
X

x(n + 3)W9nk W3k +

n=0

n=0

x(3l + 2)

x(n)W9nk +

2
X

n=0

=

5
X

n=0

n=0

x(3l + 1)

x(n)W9kn +

2
X

x(n + 3)W3nl W9n W31 +

2
X

x(n + 6)W3nl W9n W32

n=0

n=0



W9n x(n) + W31 x(n + 3) + W32 x(n + 6) W3nl


W92n x(n) + W32 x(n + 3) + W31 x(n + 6) W3nl

The number of required complex multiplications is 28. The operations can be performed in-place.
see fig 8.26-1

8.27
(a)Refer to fig 8.27-1
(b)Refer to fig 8.27-2
250

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X(0)

x(0)
1
W
3
W32

x(1)
x(2)
x(3)
x(4)

W

1
W3

1
3

x(7)
x(8)

X(6)
X(1)

2
W3
1
W3
W2
3

W

1

1
W3

2
W3
2
W3
W31

9

2
W9

x(5)
x(6)

X(3)

2
W3
W 13

X(4)
X(7)
X(2)

2
W9

2 W1
W3 3

1
W3 2
2 W3
W
3 1
W3

4
W9

W1
3

X(5)
X(8)

Figure 8.26-1:
(c) DIF is preferable for computing all points. It is also better when only X(0), X(1), X(2), X(3)
are to be calculated. The rule is to compare the number of nontrivial complex multiplies and
choose the algorithm with the fewer.
(d) If M << N and L << N , the percentage of savings is
N
2

log2 N − M2L log2 N
N
2 log2 N

× 100% = (1 −

ML
N )

× 100%

251

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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

x(0)

X(0)
0
W16
X(8)
-1

x(1)
0
W16
4
W16

-1

0
W16

-1

X(4)
0
W16
X(12)
X(2)
0
W16
X(10)
X(6)
0
W16

4
W16
-1

X(14)
X(1)

0
W16
0
W16

-1

4
W16
-1

0
W16

-1

X(9)

X(5)
0
W16
X(13)
X(3)
0
W16
X(11)
X(7)
0
W16

4
W16
-1

X(15)

Figure 8.27-1:
x(0)

X(0)
X(1)
X(2)
X(3)
X(4)
X(5)
X(6)
X(7)
W0
16

x(1)

W1
16
2
W
16
W3
16
W4
16
W5
16
W6
16
W7
16

-1
-1

X(8)
X(9)

-1

X(10)

-1

X(11)

-1
-1

X(12)
X(13)

-1

X(14)

-1

X(15)

Figure 8.27-2:
252

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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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8.28
(a)Refer to fig 8.28-1. If data shuffling is not allowed, then X(0), . . . , X(3) should be computed
x(0)
x(1)
x(2)

-1

x(3)
x(4)
x(5)
x(6)
x(7)
0
W
16
-1
1
W
16
-1
W2
16
-1
W3
16
-1
W4
16
-1
W5
16
-1
W6
16
-1
W7
16
-1

x(8)
x(9)
x(10)
x(11)
x(12)
x(13)
x(14)
x(15)

-1

0
W
16
-1
2
W
16
-1
4
W
16
-1
W6
16
-1

0
W
16
4
W
16

-1

X(0)
0
W
16 X(8)

0
W
16
-1
0
W
16

0
W
16
-1
4
W
16
-1

0
W
16
-1
4
W
16
-1

0
W
16
-1
2
W
16
-1
4
W
16
-1
W6
16
-1

-1

-1

-1

X(4)
X(12)
X(2)
X(10)

X(6)
0
W
16 X(14)
X(1)
0
W
16 X(9)

0
W
16
-1

X(5)
X(13)
X(3)

0
W
16
0
W
16
-1
4
W
16
-1

-1
0
W
16
-1

X(11)
X(7)
X(15)

Figure 8.28-1:
by one DSP. Similarly for X(4), . . . , X(7) and X(8), . . . , X(11) and X(12), . . . , X(15). From the
flow diagram the output of every DSP requires all 16 inputs which must therefore be stored in
each DSP.
(b)Refer to fig 8.28-2
(c) The computations necessary for a general FFT are shown in the figure for part (a), Ng =
N
2 log2 N . Parallel computation of the DFTs requires
p−1

Np

=
=

XN 1
N
1N
log2
+
2M
M
2 2i
i=1

N
1
N
log2
+ N (1 −
)
2M
M
M

Complex operations, as is seen in the figure for (b). Thus
S

=

Ng
Np

=

N
2 log2 N
N
N
2M log2 M + N (1

=

1
−M
)
M log2 N
log2 N − log2 M + 2(M − 1)

253

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

x(0)
x(1)
x(2)
x(3)
x(4)
x(5)
x(6)
x(7)

W 0
16
2
W
16
-1
4
W
16
-1
W6
16
-1
-1

0
W
16
0
W
16
-1
4
W
16
-1

-1

-1

X(2)
X(10)

X(6)
0
W
16 X(14)

x(8)
x(9)
x(10)
x(11)
x(12)
x(13)
x(14)
x(15)

Figure 8.28-2:

8.29
Refer to fig 8.29-1

x(n)

=
=

=
=
x(n)

=

x(n + 4)

=

N −1
1 X
X(k)WN−kn
N
k=0
1 X
1 X
X(k)W8−kn +
X(k)W8−kn
8
8
k even
k odd

3
3
1 X
1 X
−mn
+
X(2m)W4
X(2m + 1)W4−mn W8−n
8 m=0
8 m=0

3

1 X
X(2m) + X(2m + 1)W8−n W4−mn
8 m=0
" 3
#
3
X
1 X
−n
−mn
−mn
+ W8
X(2m)W4
,
X(2m + 1)W4
8 m=0
m=0
" 3
#
3
X
1 X
−n
−mn
−mn
− W8
X(2m)W4
,
X(2m + 1)W4
8 m=0
m=0

0≤n≤3
0≤n≤3

This result can be obtained from the forward DIT FFT algorithm by conjugating each occurrence
of WNi → WN−i and multiplying each output by 81 (or 12 can be multiplied into the outputs of
each stage).
254

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
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1/8 x(0)

X(0)
-0
W
8
X(4)

-1

X(2)
-0
W
8
X(6)

-1

1/8 x(1)
-0
W
8
-2
W
8

-1

X(1)

-0
W
8
X(5)

-1

X(3)
-0
W
8
X(7)

-1

1/8 x(2)

-1

-0
W
8

1/8 x(3)
-0
W
8
-1
W
8
W -2
8

-1

-2
W
8

-1

W-3
8

-1
-1

1/8 x(4)
1/8 x(5)
1/8

-1
1/8
-1

x(6)
x(7)

Figure 8.29-1:

8.30

7

x(n)

=

1X
X(k)W8−kn
8
k=0
3

=

=

x(2l)

=

x(2l + 1)

=

7

1X
1X
X(k)W8−kn +
X(k)W8−kn
8
8
k=0
k=4
" 3
#
3
X
1 X
−kn
−kn
n
X(k)W8
+ (−1)
X(k + 4)W8
8
k=0
k=0
" 3
#
3
X
1 X
l = 0, 1, 2, 3
X(k + 4)W4−lk ,
X(k)W4−lk +
8
k=0
k=0
#
" 3
3
X
1 X
−k
−lk
−k
−lk
,
l = 0, 1, 2, 3
X(k + 4)W4 W8
X(k)W4 W8 −
8
k=0

k=0

Similar to the DIT case (prob. 8.29) result can be obtained by conjugating each WNi and scaling
by 18 . Refer to fig 8.30-1
255

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
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Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

X(0) 1/8

x(0)

X(1) 1/8
X(2)

-0
W
8

1/8
-1

X(3) 1/8
X(4)

1/8

X(5) 1/8
X(6) 1/8
X(7) 1/8

-1

-2
W
8

-1

-0
W
8
-1
-1
W
8
-1
W-2
8
-1
W-3
8
-1

-1

-0
W
8 x(4)
x(2)
-0
W
8 x(6)
x(1)

-0
W
8
-1

-1

-2
W
8

-1

-1

-0
W
8 x(5)
x(3)
-0
W
8 x(7)

Figure 8.30-1:

8.31
x(n)
IDFT(x∗ (n))

= x∗ (N − n)
=
=
=

N −1
1 X ∗
x (n)WN−kn
N n=0

N −1
1 X
x(N − n)WN−kn
N n=0

1
1 X
−k(N −m)
x(m)WN
N
m=N

=

N −1
′
1 X
x(N − m′ )WN−km
N ′
m =0

Since the IDFT of a Hermitian symmetric sequence is real, we may conjugate all terms in the
sum yielding
∗

IDFT(x (n))

=

N −1
′
1 X ∗
x (N − m′ )WNkm
N ′
m =0

=

N −1
1 X
x(n)WNkn
N n=0

256

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

=

1
X(k)
N

In general, the IDFT of an N-length sequence can be obtained by reversing the flow of a forward
FFT and introducing a scale factor N1 . Since the IDFT is apparently capable of producing the
(scaled) DFT for a Hermitian symmetric sequence, the reversed flow FFT will produce the desired
FFT.

8.32

X(k)

N
−1
X

=

x(m)WNkm

m=0

N
−1
X

=

x(m)WNkm WN−kN since WN−kN = 1

m=0

N
−1
X

=

−k(N −m)

x(m)WN

m=0

This can be viewed as the convolution of the N-length sequence x(n) with the impusle response
of a linear filter.
△

= WNkn u(n), evaluated at time N
∞
X
WNkn z −n
Hk (z) =
hk (n)

n=0

=
=
yk (n)
yk (N )

1
1 − WNk z −1
Yu (z)
X(z)

= WNk yk (n − 1) + x(n),
= X(k)

yk (−1) = 0

8.33
(a) 11 frequency points must be calculated. Radix-2 FFT requires 1024
2 log2 1024 ≈ 5000 complex
multiplies or 20,000 real multiplies. FFT of radix-4 requires 0.75 × 5000 = 3, 750 complex
multiplies or 15,000 real multiplies. Choose Goertzel.
(b) In this case, direct evaluation requires 106 complex multiplies, chirp-z 22 × 103 comples
3
multiplies, and FFT 1000 + 5000
2 × 13 = 33 × 10 complex multiplies. Choose chirp-z.

8.34
In the DIF case, the number of butterflies affecting a given output is
the second, . . .. The total number is

N
2

in the first stage,

N
4

in

1
1 + 2 + . . . + 2ν−1 = 2−ν (1 − ( )ν ) = N − 1
2
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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

2

δ
Every butterfly requires 4 real multiplies, and the eror variance is 12
. Under the assumption that
the errors are uncorrelated, the variance of the total output quantization error is

σq2 = 4(N − 1)

δ2
N δ2
=
12
3

8.35
(a)
Re[Xn+1 (k)]

since |Xn (k)| <
since |Xl (k)| <

1
,
2

1
,
2

|Re[Xn (k)]|
|Re[WNm Xn (l)]|

so |Re[WNm Xn (l)]|
Therefore |Re[Xn+1 (k)]|

1 ∗
1
Xn+1 (k) + Xn+1
(k)
2
2
1
1
1
1
=
Xn (k) + WNm Xn (l) + Xn∗ (k) − WN−m Xn∗ (l)
2
2
2
2
= Re[Xn (k)] + Re[WNm Xn (l)]
1
<
2
1
<
2
1
<
2
≤ |Re[Xn (k)]| + |Re[WNm Xn (l)]| < 1
=

The other inequalities are verified similarly. (b)
Xn+1 (k)

Therefore, |Xn+1 (k)|

=

Re[Xn (k)] + jIm[Xn (k)]
2π
2π
[cos( m) − jsin( m)][Re[Xn (l)] + jIm[Xn (l)]]
N
N
= Re[Xn (k)] + cos(.)Re[Xn (l)] + sin(.)Im[Xn (l)]
+j {Im[Xn (k)] + cos(.)Im[Xn (l)] + sin(.)Re[Xn (l)]}
= |Xn (k)| + |Xn (l)| + A
△

where A =
also |Xn+1 (l)|2

Therefore, if A ≥ 0,
max[|Xn+1 (k)|, |Xn+1 (l)|]

2cos(.) {Re[Xn (k)]Re[Xn (l)] + Im[Xn (k)]Im[Xn (l)]}

+2sin(.) {Re[Xn (k)]Im[Xn (l)] − Im[Xn (k)]Re[Xn (l)]}
= |Xn (k)|2 + |Xn (l)|2 − A(∗)
= |Xn+1 (k)|

=
|Xn (k)|2 + |Xn (l)|2 + A

1
2

> max[|Xn (k)|, |Xn (l)|]

By similar means using (*), it can be shown that the same inequality holds if A < 0. Also,
from the pair of equations fro computing the butterfly outputs, we have
2Xn (k)

= Xn+1 (k) + Xn+1 (l)

2Xn (l)

= WN−m Xn+1 (k) − WN−m Xn+1 (l)

By a similar method to that employed above, it can be shown that
2max[|Xn (k)|, |Xn (l)|] ≥ max[|Xn+1 (k)|, |Xn+1 (l)|]
258

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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

(b) N=64 dc=8
8

15

6
magnitude

magnitude

(a) N=64 dc=16
20

10
5
0
0

4
2

20

40

60

0
0

80

800

15

600

10
5
0
0

40

60

80

(d) N=64 dc=7.664e−14

20

magnitude

magnitude

(c) N=128 dc=16

20

400
200

50

100

0
0

150

20

40

60

80

Figure 8.36-1:

8.36
Refer to fig 8.36-1.
(d) (1) The frequency interval between successive samples for the plots in parts (a), (b), (c) and
1
1
1
1
(d) are 64
, 64
, 128
and 64
respectively.
(2) The dc values computed theoretically and from the plots are given below:
theoretical
practical

part a
16
16

part b
8
8

part c
16
16

part d
0
8.203e − 14

Both theoretical and practical dc values match except in the last case because of the finite word
length effects the dc value is not a perfect zero.
(3) Frequency interval = Nπ1 .
(4) Resolution is better with N = 128.

8.37
(a) Refer to fig 8.37-1.
(b) Refer to fig 8.37-1.
(c) Refer to fig 8.37-1.
(d) Refer to fig 8.37-1.
(e) Refer to fig 8.37-2.

259

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

r=0.9, c=0.92, W(k)
25

5

20

4

magnitude

magnitude

r=0.9, Y(k)
6

3
2

15
10
5

1
0
0

50

100

0
0

150

50

r=0.5, Y(k)

100

150

r=0.5 , c=0.55, W(k)

1.4

6

1.2

magnitude

magnitude

5

1

4
3
2
1

0.8
0

50

100

0
0

150

50

100

150

Figure 8.37-1:

r = 0.5, Y(k)

W(k)

32

12

4

x 10

10
magnitude

magnitude

3
8
6
4

2
1

2
0
0

50

100

0
0

150

50

100

150

Figure 8.37-2:

260

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Chapter 9

9.1
(a) H(z) = 1 + 2z −1 + 3z −2 + 4z −3 + 3z −4 + 2z −5 + z −6 . Refer to fig 9.1-1
(b) H(z) = 1 + 2z −1 + 3z −2 + 3z −3 + 2z −4 + z −5 . Refer to fig 9.1-2

x(n)

z-1

z-1

z-1

z-1

z-1

z-1

4
+
3
+
2
+

y(n)

Figure 9.1-1:

9.2
Refer to fig 9.2-1
A4 (z) = H(z) =
B4 (z) =
Hence, K4

1 + 2.88z −1 + 3.4048z −2 + 1.74z −3 + 0.4z −4
0.4 + 1.74z −1 + 3.4048z −2 + 2.88z −3 + z −4

=

0.4
A4 (z) − k4 B4 (z)
A3 (z) =
1 − k42
=

1 + 2.6z −1 + 2.432z −2 + 0.7z −3
261

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

x(n)

z-1

z-1

z-1

z-1

z-1

+
3
+
2
+

y(n)

Figure 9.1-2:
B3 (z)
Hence, K3
A2 (z)

= 0.7 + 2.432z −1 + 2.6z −2 + z −3
= 0.7
A3 (z) − k3 B3 (z)
=
1 − k32

=
B2 (z)
Then, K2
A1 (z)

Therefore, K1

1 + 1.76z −1 + 1.2z −2

= 1.2 + 1.76z −1 + z −2
= 1.2
A2 (z) − k2 B2 (z)
=
1 − k22

=
=

1 + 0.8z −1
0.8

Since K2 > 1, the system is not minimum phase.

9.3

V (z)
v(n)
Y (z)
H(z)

h(n)

1
= X(z) + z −1 V (z)
2
1
= x(n) + v(n − 1)
2
= 2[3X(z) + V (z)] + 2z −1 V (z)
Y (z)
=
X(z)
8 − z −1
=
1 − 0.5z −1
= 8(0.5)n u(n) − (0.5)n−1 u(n − 1)
262

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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

z-1

x(n)

z-1

1

z-1

2.88

z-1
1.74

3.4048

+

+

+

0.4
+

y(n)

(a)

x(n)

f1(n)

+

f 2(n)

+

z-1

+

k2

k1
z-1

+

z-1

+

f (n)
3

+

f 4 (n) = y(n)

k
4

k3
+

z-1

+

(b)

Figure 9.2-1: (a) Direct form. (b) Lattice form

9.4
H(z) =
h(n)

=

1 + 2z 1
3z 1
1 −1 +
1 + 3z
1 − 21 z −1
1
1
1
5δ(n) + 3(− )n−1 u(n − 1) + ( )n u(n) + 2( )n−1 u(n − 1)
3
2
2
5+

9.5

H(z) =
=

6 + 29 z 1 − 35 z −2
(1 + 31 z −1 )(1 − 12 z −1 )

6 + 29 z 1 − 53 z −2
1 − 61 z 1 − 16 z −2

Refer to fig 9.5-1

9.6
1
1
+
1 − b1 z −1
1 − b2 z −1
1 − (b1 + b2 )z −1
H(z) =
(1 − b1 z −1 )(1 − b2 z −1 )
c0 + c1 z −1
For the second system, H(z) =
(1 − d1 z −1 )(1 − a2 z −1 )
clearly, c0 = 1
For the first system, H(z) =

263

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6

x(n)

y(n)

+

z-1

9/2

+

1/6

z-1

-5/3

+

1/6

Figure 9.5-1:
c1
d1
a2

= −(b1 + b2 )

= b1
= b2

9.7
(a)
y(n)
H(z)

= a1 y(n − 1) + a2 y(n − 2) + b0 x(n) + b1 x(n − 1) + b2 x(n − 2)
b0 + b1 z −1 + b2 z −2
=
1 + a1 z −1 + a2 z −2

(b)
1 + 2z −1 + z −2
1 + 1.5z −1 + 0.9z −2
= −1, −1
= −0.75 ± j0.58

H(z) =
Zeros at z
Poles at z

Since the poles are inside the unit circle, the system is stable.
1 + 2z −1 + z −2
1 + z −1 − 2z −2
= −1, −1
= 2, −1

H(z) =
Zeros at z
Poles at z

264

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The system is unstable.
(c)
x(n)

π
= cos( n)
3

H(z)

=

H(w)

=

π
H( )
3

=

100e−j 3

Hence, y(n)

=

π
π
100cos( n − )
3
3

1
1+
− 0.99z −2
1
1 + e−jw − 0.99e−j2w
z −1

π

9.8
y(n)

=

H(z)

=

1
y(n − 2) + x(n)
4
1
1 − 14 z −2

(a)
h(n)

=

H(z)

=



1 1 n
1
( ) + (− )n u(n)
2 2
2
1−

1
2
1 −1
2z

+

1+

1
2
1 −1
2z

(b)
x(n)

=

X(z) =
X(z) =
Y (z) =



1 n
1 n
( ) + (− ) u(n)
2
2
1
1
+
1 − 21 z −1
1 + 12 z −1
2
1 − 41 z −2
X(z)H(z)

1 −1
− 21 z −1
2z
+
1 −1
1 −1
1 −1 2
1+ z
1 − 2z
(1 − 2 z )
(1 + 21 z −1 )2

 2
1
1
1
1
=
( )n + (− )n − n( )n + n(− )n u(n)
2
2
2
2

=
y(n)

1

1

+

+

(c)Refer to fig 9.8-1
(d)
H(w)

=
=

1
1−
√

1 −j2w
4e

sin2w
4
6 − tan−1
4 − cos2w
17 − 8cos2w

Refer to fig 9.8-2.

265

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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Direct form 2

cascade form

+

x(n)

y(n)
x(n)

+

+

y(n)

D
D
1/2

D
-1/2

D
1/4
Parallel form
1/2

+

+

y(n)

D
1/2
x(n)
1/2

+
D
-1/2

Figure 9.8-1:
Magnitude of H(w)

−−−> |H(w)|

1.4

1.2

1

0.8
0

0.5

1

1.5

2

2.5

3

3.5

2.5

3

3.5

−−−> w

−−−> angle of H(w)

Phase of H(w)
0.2

0

−0.2
0

0.5

1

1.5

2
−−−> w

Figure 9.8-2:
266

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

9.9
(a)
1 + 31 z −1
1 − 43 z −1 + 18 z −2

H(z) =

1 + 13 z −1
(1 − 21 z −1 )(1 − 14 z −1 )

=
=

1

10
3
− 21 z −1

+

− 73
1 − 41 z −1

Refer to fig 9.9-1
(b)

Direct form II:

Direct form I:

x(n)

+

+

y(n)

+

x(n)

+

z-1
z-1

+

3/4

1/3

3/4

+
z-1

1/3
z-1

-1/8

-1/8

Cascade:
x(n)

y(n)

z-1

Parallel:

+

+

+

y(n)
+

z-1
1/2

z-1
1/3

z-1
x(n)

1/4

10/3

1/2
+

y(n)

1/4
z-1

-7/3

+

Figure 9.9-1:

H(z)

Refer to fig 9.9-2
(c)
H(z)

0.7(1 − 0.36z −2 )
1 + 0.1z −1 − 0.72z −2
0.7(1 − 0.6z −1 )(1 + 0.6z −1 )
=
(1 + 0.9z −1 )(1 − 0.8z −1 )
0.1853
0.1647
−
= 0.35 −
1 + 0.9z −1
1 − 0.8z −1
=

=

3(1 + 1.2z −1 + 0.2z −2 )
1 + 0.1z −1 − 0.2z −2
267

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Direct form II:

Direct form I:

x(n)

+

+

+

y(n) x(n)

+

z-1
z-1

-0.1

+
z-1

z-1

z-1

0.72

-0.36

0.72

-0.36

Parallel:

Cascade:
x(n)

y(n)

z-1

-0.1

+

0.7

+

+

+

+

0.7

y(n)
+

z-1
-0.9

z-1
0.8

-0.6

-0.1853

z-1
0.8

x(n)

0.6

0.35

+

y(n)

-0.9
z-1

0.1647

+

Figure 9.9-2:
3(1 + 0.2z −1 )(1 + z −1 )
(1 + 0.5z −1 )(1 − 0.4z −1 )
7
1
= −3 +
−
1 − 0.4z −1
1 + 0.5z −1
=

Refer to fig 9.9-3
(d)
H(z) =
=
=
Refer to fig 9.9-4
(e)

√
2(1 − z −1 )(1 + 2z −1 + z −2 )
(1 + 0.5z −1 )(1 − 0.9z −1 + 0.8z −2 )
√
√
2 + (2 2 − 2)z −1 + (2 − 2 2)z −2 − 2z −3 )
1 − 0.4z −1 + 0.36z −2 + 0.405z −3
B + Cz −1
A
+
1 + 0.5z −1
1 − 0.9z −1 + 0.8z −1

H(z)

=
=
=

1 + z −1
1−
− 41 z −2
1 −1
2z

1 + z −1
(1 − 0.81z −1 )(1 + 0.31z −1 )
−0.62
1.62
+
1 − 0.81z −1
1 + 0.31z −1

Refer to fig 9.9-5
1−z −1 +z −2
(f) H(z) = 1−z
−1 +0.5z −2 ⇒ Complex valued poles and zeros.Refer to fig 9.9-6 All the above
268

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Direct form II:

Direct form I:

x(n)

+

3

+

+

y(n) x(n)

+

z-1
z-1
+
1.2
z-1

y(n)

z-1

-0.1

-0.1

+

1.2

+

+
z-1

z-1

0.2

0.2

0.2

0.2

Parallel:

Cascade:
x(n)

3

+

+

+

+

z-1
-0.5

3

y(n)

7

+

z-1
0.2

0.4

z-1
x(n)

1

0.4

-3

+

y(n)

-0.5
z-1
-1

+

Figure 9.9-3:
systems are stable.

269

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Direct form II:

Direct form I:
x(n)

+

3

+

x(n)

y(n)

+

0.4

+

y(n)

z-1

z-1
z-1

3

+

2 -1

0.4

+

+

+
2 -1
z-1
1- 2

z-1

z-1
+

+

z-1

z-1

z-1

1- 2

-0.36

+

-0.36

+

-0.405

-1

-0.405

-1

Parallel:

x(n) 2

A

+

Cascade:

z-1
+

+

+

+

y(n)

x(n)

-0.5

-3

y(n)

+
-0.8

z-1
-0.5

z-1
-1

+

z-1

0.9 1.414

+

+

0.9

z-1
-0.81

C
z-1
B

+

1

+

Figure 9.9-4:

Direct form II:

Direct form I:

x(n)

+

+

+

y(n) x(n)

+

z-1
z-1

+

z-1

1/2

1/2

+
z-1

1
z-1

1/4

1/4

Cascade:
x(n)

y(n)

Parallel:

+

+

+

y(n)
1.62

+
z-1
0.81

z-1
1

z-1
0.81

x(n)

-0.31

+

y(n)

-0.3
z-1
+
-0.62

Figure 9.9-5:
270

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
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Direct form I:

Direct form II, cascade, parallel:
y(n)

x(n)

+

y(n) x(n)

+

+

+

z-1
z-1
-1

+

z-1

1

+

1

+

+
z-1

z-1
1

-1
z-1

-1/2

-0.5

1

Figure 9.9-6:

271

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

9.10
Refer to fig 9.10-1

x(n)

+

v(n)

+

w(n)

r sin w0

z-1 w(n-1)

+

y(n)

Figure 9.10-1:
1
1 − 2rcosw0 z −1 + r2 z −2
V (z) = X(z) − rsinw0 z −1 Y (z)

H(z) =
(1)
(2)
(3)

W (z) = V (z) − rcosw0 z −1 W (z)
Y (z) = rcosw0 z −1 Y (z) − rsinw0 z −1 W (z)

By combining (1) and (2) we obtain
(4)

W (z)

=

rsinw0 z −1
1
X(z) −
Y (z)
−1
1 − rcosw0 z
1 − rcosw0 z −1

Use (4) to eliminate W (z) in (3). Thus,
Y (z)[(1 − rcosw0 z −1 )2 + r2 sin2 w0 z −2 ] = X(z)
Y (z)[1 − 2rcosw0 z −1 + (r2 cos2 w0 + r2 sin2 w0 )z −2 ] = X(z)
1
Y (z)
=
X(z)
1 − 2rcosw0 z −1 + r2 z −2

9.11
A0 (z) = B0 (z) = 1
A1 (z) = A0 (z) + k1 B0 (z)z −1
1
= 1 + z −1
2
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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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A2 (z)

1
+ z −1
2
= A1 (z) + k2 B1 (z)

B2 (z)

= 1 + 0.3z −1 + 0.6z −2
= 0.6 + 0.3z −1 + z −2

B1 (z)

=

A3 (z)

= A2 (z) + k3 B2 (z)
= 1 − 0.12z −1 + 0.39z −2 − 0.7z −3
= −0.7 + 0.39z −1 − 0.12z −2 + z −3

B3 (z)
A4 (z)

= A3 (z) + k4 B3 (z)
53 −1
1
= 1−
z + 0.52z −2 − 0.74z −3 + z −4
150
3
1
53 −1
−2
−3
z + 0.52z − 0.74z + z −4 )
= C(1 −
150
3

Therefore, H(z)
where C is a constant

9.12
Refer to fig 9.12-1
b
0k

x (n)
k

+

y (n)
k

z-1
w1k (n)

b1k

-a
1k

+

z-1
w2k(n)
-a 2k

b2k

+

x(n) =
x1(n)

H (z)
1

xN(n)

H2(z)

H (z)
N

y(n)=y (n)
N

Figure 9.12-1:
b0k + b1k z −1 + b2k z −2
1 + a1k z −1 + a2k z −2
= b0k xk (n) + w1k (n − 1)

Hk (z) =
yk (n)

273

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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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w1k (n)
w2k (n)

= b1k x(n) − a1k yk (n) + w2k (n − 1)
= b2k x(n) − a2k yk (n)

9.13
YJM1 = G * XIN
DO 20 J=1,K
YJ=B(J,0) * XIN + W1(J)
W1(J) = B(J,1)*XIN - A(J,1)*YJ + W2(J)
W2(J) = B(J,2)*XIN - A(J,2)*YJ
YJM1 = YJM1 + YJ
20
CONTINUE
YOUT = YJM1
RETURN

9.14
YJM1 = XIN
DO 20 J=1,K
W = -A(J,1) * WOLD1 - A(J,2) * WOLD2 + YMJ1
YJ = W + B(J,1)*WOLD1 + B(J,2)*WOLD2
WOLD2 = WOLD1
WOLD1 = W
YJM1 = YJ
20
CONTINUE
YOUT = YJ
RETURN

9.15
1
1 + 2z −1 + z −2
3
1
−1
+ 2z + z −2
=
3
1
=
3
A2 (z) − k2 B2 (z)
=
1 − k22
3
= 1 + z −1
2
3
=
2

H(z) = A2 (z)

=

B2 (z)
k2
A1 (z)

k1

9.16
(a)




A2 (z)
B2 (z)



A1 (z)
B1 (z)

=





1
− 13

=



1
k1

− 13
1





z −1



=



1 + 12 z −1
1
−1
2 +z

A1 (z)
z −1 B1 (z)



=



1 + 31 z −1 − 13 z −2
− 31 + 13 z −1 + z −2

k1
1

1




274

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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.



A3 (z)
B3 (z)



=



1
1

1
1



A2 (z)
z −1 B2 (z)



H1 (z) = A3 (z) = 1 + z −3 ⇒

π

zeros at z = −1, e±j 3
(b)

H2 (z)

The zeros are z

= A2 (z) − z −1 B2 (z)
2
2
= 1 + z −1 − z −2 − z −3
3
3
√
−5 ± j 11
= 1,
6

(c) If the magnitude of the last coefficient |kN | = 1, i.e., kN = ±1, all the zeros lie on the unit
circle.
(d) Refer to fig 9.16-1. We observe that the filters are linear phase filters with phase jumps at

−−−> phase of H1(w)

2
1
0
−1
−2
0

0.05

0.1

0.15

0.2
0.25
0.3
−−−> freq(Hz)

0.35

0.4

0.45

0.5

0.05

0.1

0.15

0.2
0.25
0.3
−−−> freq(Hz)

0.35

0.4

0.45

0.5

−−−> phase of H2(w)

4
2
0
−2
−4
0

Figure 9.16-1:
the zeros of H(z).

9.17
(a) Refer to fig 9.17-1

275

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
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Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

x(n)

f1(n)

+

z-1

+

+

f (n) = y(n)
3

0.8

-0.34

0.65
z-1

f 2(n)

+

z-1

+

g (n)
1

+

g (n)
2

g (n)
3

Figure 9.17-1:
x(n)
f1 (n)
g1 (n)
f2 (n)
g2 (n)
h(n) = f3 (n)

= δ(n)
= δ(n) + 0.65δ(n − 1)

= 0.65δ(n) + δ(n − 1)
= f1 (n) − 0.34g1 (n − 1)

= δ(n) + 0.429δ(n − 1) − 0.34δ(n − 2)
= −0.34f1 (n) + g1 (n − 1)

= −0.34δ(n) + 0.429δ(n − 1) + δ(n − 2)
= f2 (n) + 0.8g2 (n − 1)
= δ(n) + 0.157δ(n − 1) + 0.0032δ(n − 2) + 0.8δ(n − 3)

(b) H(z) = 1 + 0.157z −1 + 0.0032z −2 + 0.8z −3 . Refer to fig 9.17-2

x(n)

+

y(n)

z-1
0.157

+

z-1
0.0032

+

z-1
0.8

Figure 9.17-2:

276

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

9.18
(a)

H(z) =

C3 (z)
A3 (z)

A3 (z) =

1 + 0.9z −1 − 0.8z −2 + 0.5z −3

B3 (z) = 0.5 − 0.8z −1 + 0.9z −2 + z −3
k3 = 0.5
A3 (z) − k3 B3 (z)
A2 (z) =
1 − k32
= 1 + 1.73z −1 − 1.67z −2
B2 (z) = −1.67 + 1.73z −1 + z −2
k2

= −1.67
A2 (z) − k2 B2 (z)
A1 (z) =
1 − k22
=

1 + 1.62z −1

B1 (z) = 1.62 + z −1
k1 = 1.62
C3 (z) = 1 + 2z −1 + 3z −2 + 2z −3
D3 (z) = 2 + 3z −1 + 2z −2 + z −3
k3 = 2
C3 (z) − k3 D3 (z)
C2 (z) =
1 − k32
4
1
= 1 + z −1 + z −2
3
3
1 4 −1
+ z + z −2
D2 (z) =
3 3
1
k2 =
3
C2 (z) − k2 D2 (z)
C1 (z) =
1 − k22
3
= 1 + z −1
4
3
+ z −1
D1 (z) =
4
3
k1 =
4
C3 (z) = v0 + v1 D1 (z) + v2 D2 (z) + v3 D3 (z)
=

1 + 2z −1 + 3z −2 + 2z −3

From the equations, we obtain
v0
v1
v2
v3

107
48
13
= −
4
= −1

= −

=

2

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The equivalent lattice-ladder structure is: Refer to fig 9.18-1
(b) A3 (z) = 1 + 0.9z −1 − 0.8z −2 + 0.5z −3 ,
|k1 | > 1 and |k2 | > 1 ⇒ the system is unstable.

x(n)

+

+
0.5

1.62

-1.67
z -1

+

+

z -1

+

v3 = 2

v =-1
2

v = -13/4
1

+

z -1

+

v

0 =-107/48

+

+

y(n)

Figure 9.18-1:

9.19
Refer to fig 9.19-1
[rsinΘX(z) + rcosΘY (z) − rsinΘC(z)] z −1

Y (z)

=

C(z)

[−rcosΘX(z) + rsinΘY (z) + rcosΘC(z)] z −1
Y (z)
=
X(z)
rsinΘz −1
=
1 − 2rcosΘz −1 + r2 z −2
= rn sin(Θn)u(n)
=

H(z)

Hence, h(n)

= rsinΘx(n − 1) + 2rcosΘy(n − 1) − r2 y(n − 2)

and y(n)

The system has a zero at z = 0 and poles at z = re±jΘ .

9.20
H(z) =
=
S(z) =

1
1 − 2rcosw0 z −1 + r2 z −2
0
0
rcosw0 − j rcos2w
rcosw0 + j rcos2w
2sinw0
2sinw0
1+
+
z − (rcosw0 + jrsinw0 ) z − (rcosw0 − jrsinw0 )
0
rcosw0 − j rcos2w
2sinw0

z − (rcosw0 + jrsinw0 )

X(z)

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r cos θ

r sin θ

y(n)

z-1

+

r sin θ
x(n)
-r sin θ
c(n)

z-1

+
-r cos θ

r cos θ

Figure 9.19-1:

s(n)

= v1 (n) + jv2 (n)

p
⇒ α1

= α1 + jα2
= rcosw0

⇒ q1

= rcosw0
−rcosw0
=
2sinw0
= α1 v1 (n) − α2 v2 (n) + q1 x(n)
= rcosw0 v1 (n) − rsinw0 v2 (n) + rcosw0 x(n)
= α2 v1 (n) + α1 v2 (n) + q2 x(n)
−rcosw0
= rsinw0 v1 (n) + rcosw0 v2 (n) +
x(n)
2sinw0

α2
A
q2

v1 (n + 1)
v2 (n)

= rsinw0
= q1 + jq2

or, equivalently,

v(n + 1) =



rcosw0
rsinw0

−rsinw0
rcosw0



v(n) +



rcosw0
rcosw0
2sinw0



x(n)

y(n) = s(n) + s∗ (n) + x(n)
=

2v1 (n) + x(n)

or, equivalently,
y(n) =

[2 0]v(n) + x(n)

where
v(n) =



v1 (n)
v2 (n)



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9.21
(a)
k1

=

0.6
1 + 0.6z −1
0.6 + z −1

A1 (z) =
B1 (z) =

A2 (z) = A1 (z) + k2 B1 (z)z −1
= 1 + 0.78z −1 + 0.3z −2
B2 (z) = 0.3 + 0.78z −1 + z −2
A3 (z) = A2 (z) + k3 B2 (z)z −1
= 1 + 0.93z −1 + 0.69z −2 + 0.5z −3
B3 (z) = 0.5 + 0.69z −1 + 0.93z −2 + z −3
H(z) = A4 (z) = A3 (z) + k4 B3 (z)z −1
1 + 1.38z −1 + 1.311z −2 + 1.337z −3 + 0.9z −4

=

(b) Refer to fig 9.21-1

Direct form:

z-1

x(n)

z-1

1

z-1

z-1

1.311

1.38

+

+

+

0.9

1.337

y(n)
Lattice form:

x(n)

+

+

-

+

z-1

+

-

0.3

0.6
z-1

+

-

-

0.5
+

z-1

y(n)

0.9
+

z-1

+

Figure 9.21-1:

9.22
(a)
From (9.3.38) we have
y(n)
But, y(n)
Hence, k2
and, k1 (1 + k2 )
k1

= −k1 (1 + k2 )y(n − 1) − k2 y(n − 2) + x(n)
= 2rcosw0 y(n − 1) − r2 y(n − 2) + x(n)
= r2
= −2rcosw0
2rcosw0
+ −
1 + r2
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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
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Refer to fig 9.22-1
(b) When r = 1, the system becomes an oscillator.

x(n)

+

+
-

y(n)
-

k2

k1
z -1

+

z -1

+

Figure 9.22-1:

9.23

H(z)

=
=

For the all-pole system

1
, we have
A(z)
k1 (1 + k2 )
k2

⇒ k1
k2
For the all-zero system, C2 (z)
A2 (z)
B2 (z)
k2
A1 (z)

= 0.1
= −0.72

= 0.357
= 0.72
= 1 − 0.8z −1 + 0.15z −2
=
=

=

A0 (z)

1 − 0.8z −1 + 0.15z −2
0.15 − 0.8z −1 + z −2

0.15
A2 (z) − k2 B2 (z)
=
1 − k22

=
B1 (z)
k1

1 − 0.8z −1 + 0.15z −2
1 + 0.1z −1 − 0.72z −2
B(z)
A(z)

1 − 0.696z −1

= −0.696 + z −1
= −0.696
= B0 (z) = 1

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
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C2 (z) =

2
X

vm Bm (z)

m=0

The solution is:
v2
v1 − 0.18v2
v0 − 0.696v1 + 0.15v2
⇒ v0

v1
v2

= v0 + v1 B1 (z) + v2 B2 (z)
= 1 − 0.8z −1 + 0.15z −2
=

0.15

= −0.8
= 1
= 1.5
= −0.68
= 0.15

Thus the lattice-ladder structure is: Refer to fig 9.23-1

x(n)

+

+
-

0.15

+

-0.696
z -1

z -1

+
v = -0.68
1

v = 0.15
2
+

v = 1.5
0
+

y(n)

Figure 9.23-1:

9.24
√

H(z) =

1− 22 z −1 +0.25z −2
1−0.8z −1 +0.64z −2 .

Refer to fig 9.24-1

9.25
H(z) =
H(z) =

1+z −1
√ 1
1−z −1 . 1−0.8 2z −1 +0.64z −2
−1
2.31
−1.31+2.96z
√
+ 1−0.8
1− 12 z −1
2z −1 +0.64z −2

Refer to fig 9.25-1

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
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Direct form II :

x(n)

Transposed form :

+

+

x(n)

y(n)

y(n)

+

D
+

a

b

D

+

D
d

b

c

+

a

where a=0.8, b = - 2 /2, c = 0.25 and d = -0.64
D

d

c

+

Figure 9.24-1:

9.26
(a)
For positive numbers, range is
01. 00
. . . 0} ×21001
| {z

− 01. |11 {z
. . . 1} ×20111

negitive numbers
10. 11
. . . 1} ×21001
| {z

− 10. |00 {z
. . . 0} ×20111

11

11

or 7.8125 × 10−3

− 2.5596875 × 102

11

11

−3

or − 7.8163 × 10

−

−2.56 × 102

(b)
For positive numbers, range is
01. 00
. . . 0} ×210000001
| {z

− 01. 11
. . . 1} ×201111111
| {z

10. 11
. . . 1} ×210000001
| {z

− 10. 00
. . . 0} ×201111111
| {z

23

−39

or 5.8774717 × 10
negitive numbers
23

−39

or − 5.8774724 × 10

23

− 3.4028234 × 1038

23

−

−3.4028236 × 1038

283

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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Parallel:

x(n)

2.31

+

Cascade:
+

+

z-1

y(n)

+

0.5

x(n)

y(n)
+

-0.64
z-1
1/2

z-1
1

+

z-1

0.8 2

+

2.96

0.8 2
z-1

z-1
-0.64

-1.31

+

+

Figure 9.25-1:

9.27
(a) Refer to fig 9.27-1

x(n)

+

y(n)

+
-a 2

-a 1

z-1

z -1

Figure 9.27-1:

if a1 ≥ 0,

HR (z)

=

poles zp1,2

=

for stability
(i)a21 − 4a2
p
−a1 − a21 − 4a2
2
q
⇒

a21 − 4a2

⇒ a1 ≤ 2 and a1 − a2

(1 + a1 z −1 + a2 z −2 )−1
p
−a1 ± a21 − 4a2
2

≥ 0
≥

−1

≤ 2 − a1

≤ 1

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

−a1 −

if a1 < 0,

⇒

p

a21 − 4a2
2
q

a21 − 4a2

≤ 1
≤ 2 + a1

⇒ a1 ≥ −2 and a1 + a2 ≥ −1
p
4a2 − a21
a1
) ≤ 1
(ii)(− )2 + (
2
2
a2 ≤ 1

Refer to fig 9.27-2. The region of stability in the a1 − a2 plane is shaded in the figure. There are

The stable area of (a , a )
1 2
2

a2

1

-2

-1

1

2

a1

-1

Figure 9.27-2:
nine integer pairs (a1 , a2 ) which satisfy the stability conditions. These are (with corresponding
system functions):
(0, −1)

HR1 (z) =

(1 − z −2 )−1

(0, 0)
(0, 1)

HR2 (z) =
HR3 (z) =

1
(1 + z −2 )−1

(1, 0)
(1, 1)
(2, 1)

HR4 (z) =
HR5 (z) =
HR6 (z) =

(1 + z −1 )−1
(1 + z −1 + z −2 )−1
(1 + 2z −1 + z −2 )−1

(−1, 0)
(−1, 1)

HR7 (z) =
HR8 (z) =

(−2, 1)

HR9 (z) =

(1 − z −1 )−1
(1 − z −1 + z −2 )−1

(1 − 2z −1 + z −2 )−1

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(b)
HR1 (z) = HR4 (z)HR7 (z)
HR6 (z) = HR4 (z)HR4 (z)
HR9 (z) = HR7 (z)HR7 (z)
(c) Only the following cases can make h(n) FIR:
(i)
hR (n)

= δ(n)
N
X

Then H(z) =

z −i

i=0

y(n)

N
X

=

i=0

x(n − i)

(ii)
hR (n) ∗ hF (n)

= δ(n)

Then H(z) = 1
y(n) = x(n)

(d) see above.

9.28
Refer to fig 9.28-1
Note that 4 multiplications and 3 additions are required to implement H1 (z). The advantage

x(n)
+
+
+

z-1

z-1

b
0

b1

b2

z-1
b

3

Figure 9.28-1: Structure of H1 (z)
of Horner’s method is in evaluating H1 (z) for a specific z0 . Thus, if
H1 (z)

= b0 + b0 b1 z −1 + b0 b1 b2 z −2 + b0 b1 b2 b3 z −3
= b0 + z −1 (b0 b1 + z −1 (b0 b1 b2 + z −1 b0 b1 b2 b3 ))
286

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

the 3 multiplications and 3 additions are required for the evaluation of 9.1 in the field of z.
If the various powers of z are prestored, then Horner’s scheme has no advantage over the direct
evaluation of 9.1. Refer to fig 9.28-2
This requires 4 multiplications and 3 additions. The linear-phase system is written as

z-1

z-1

z-1

z-1

z-1

+

b

3

+

+

b
0

b1

b2

z-1

Figure 9.28-2: Structure of H(z) = b0 z −3 + b0 b1 z −2 + b0 b1 b2 z −1 + b0 b1 b2 b3
H(z) = z 2 a3 + za2 + a1 + z −1 a0 + z −2 a1 + z −3 a2 + z −4 a3
By applying Horner’s scheme, we can rewrite this as
H(z) = z 3 (a3 + z −1 (a2 + z −1 (a1 + z −1 (a0 + z −1 (a1 + z −1 (a2 + z −1 a3 ))))))
Assuming that z −1 and z are given, a direct evaluation of H(z) at z = z0 requires 8 multiplications
and 6 additions. Using Horner’s scheme based on 9.28, requires the same number of operations
as direct evaluation of H(z). Hence, Horner’s scheme does not offer any savings in computation.

9.29
(a) When x1 and x2 are positive, the result is obvious. If x1 and x2 are negative, let
x1
x2
x3

= −0 n1 n2 . . . nb
= −0 n1 n2 . . . nb + 0 0 0 . . . 0 1

= −0 m1 m2 . . . mb
= −1 m1 m2 . . . mb + 0 0 0 . . . 0 1
= x1 + x2

= −0 n1 0 . . . 0 + 0 m1 0 . . . 0 + c
where c = 0 0 n2 . . . nb + 0 0 m2 . . . mb + 0 0 0 . . . 0 1 0
If the sign changes, there are two possibilities
(i)

n 1 = m1

=

0

287

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

⇒ n1 = m1

=

⇒ |x1 |

>

⇒ |x3 |
(ii)(n1 = 1, m1 = 0, c = 0)
1
⇒ (|0 n1 0 . . . 0)10 | >
2

>
or
or

and |c10 |

>

⇒ |x3 |

>

1
1
1
,
|x2 | >
2
2
1, overflow
(n1 = 0, m1 = 1, c = 0)
1
(|0 m1 0 . . . 0)10 | >
2
1
2
1, overflow

(b)
x1
x2
x3
x1 + x2
x1 + x2 + x3

=

0100

= 0110
= −0 1 1 0 = 1 0 1 0
= 1 0 1 0, overflow
=

0 1 0 0, correct result

9.30
(a)
H(z)
|H(ejw )|2

−a + z −1
1 − az −1
−a + e−jw 2
= |
|
1 − ae−jw
(−a + cosw)2 + (−sinw)2
=
(1 − acosw)2 + (asinw)2
a2 − 2acosw + 1
=
= 1 ∀w
1 − 2acosw + a2
=

(b) Refer to fig 9.30-1
(c) If |â| = | − â|, where â means the quantized value of a, then the filter remains all-pass.

x(n)

-a

+

+

y(n)

z-1
a

1

Figure 9.30-1:
(d) Refer to fig 9.30-2
(e) Yes, it is still all-pass.
288

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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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z-1

x(n)

+

+

y(n)

z-1

a
-

+

+

Figure 9.30-2:

9.31


(a) y(n) = 2( 12 )n − ( 41 )n u(n)
(b) Quantization table
x
31
≥x
32
29
≥x
32
1
≥x
32

>
>
>
...
>
...

x

<

Therefore x(n)

=

y(n)

=

y(n)

=

1−

1
32

29
32
27
32
1
32

x=1

15
16
14
x=
16
x=

x=

14
16

1
−1 +
x = −1
32


4 1
1, , , 0, . . . , 0
↑ 16 16
8
y(n − 1) + x(n)
16


12 7 3 1
1, , , , , 0, 0, . . .
↑ 16 16 16 16

(c)
y(n)
y(n)



3 7 15 31 63
,
,...
=
1, , , ,
↑ 4 16 64 256 1024


3 7 12 16
, 0, 0, . . .
=
1, , , ,
↑ 4 16 64 256

Errors occur when number becomes small.
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9.32
y(n)

=

0.999y(n − 1) + e(n)


e(n) is white noise, uniformly distributed in the interval − 219 , 219



E y 2 (n)
= 0.9992 E y 2 (n − 1) + E e2 (n)


(1 − 0.9992 )E y 2 (n)
= E e2 (n)
Z △2
1 2
e de
=
△
−△
2
=

Therefore, E y 2 (n)

=
=

△2
where △ = 2−8
12
1
1 1 2
( 8)
12 2
1 − 0.9992
6.361x10−4

9.33
(a) poles zp1 = 0.695,
(b) Truncation

x(n)

zp2 = 0.180 Refer to fig 9.33-1

+

+

y(n)

D

D

0.695

0.18

Figure 9.33-1:

poles z p1 = 0.625,

0.695

→

0.180

→

5
= 0.625
8
1
= 0.125
8

z p2 = 0.125

(c) Rounding

poles z p1 = 0.75,

0.695

→

0.180

→

6
= 0.75
8
1
= 0.125
8

z p2 = 0.125
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(d)
|0.75 − 0.695|
Rounding is better

< |0.695 − 0.625|
1

|Ha (w)|

=

[(1.483 + 1.39cosw)(1.0324 + 0.36cosw)]− 2

=

[(1.391 + 1.25cosw)(1.0156 + 0.25cosw)]− 2

|Hc (w)|

=

[(1.563 + 1.5cosw)(1.0156 + 0.25cosw)]− 2

|Hb (w)|

1

1

9.34
(a)
H1 (z) =
h1 (n)

=

H2 (z) =
h2 (n)

=

H3 (z) =
h3 (n)

=

1
1 − z −1
 2 
1
1, −
2
1
(1 − z −1 )−1
4
1
( )n u(n)
4
1
(1 + z −1 )−1
4
1
(− )n u(n)
4

Refer to fig 9.34-1 Cascade the three systems in six possible permutations to obtain six realiza-

H1 (z)
+

H (z)
2

H (z)
3

+

+

z -1
-1/2

z -1
1/4

z -1
-1/4

Figure 9.34-1:
tions.
(b) Error sequence ei (n) is uniformly distributed over interval ( 12 2−b , 12 2−b ). So σe2i =
any i (call it σe2 )

2−2b
12

for

291

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+

+

+

z -1

z -1

z -1

-1/2

+

+

e (n)
1

+

1/4

-1/4

e (n)
2

e (n)
3

Figure 9.34-2:
(c) consider cascade H1 - H2 - H3 Refer to fig 9.34-2
h4 (n)

σq2

= h2 (n) ∗ h3 (n)


1
1
=
1, 0, , 0, ( )2 , 0, . . .
16
16
#
" ∞
∞
X
X
= σe2 2
h23 (n)
h24 (n) +
n=0



n=0

1
2
1 2 +
1 − ( 16
1 − ( 14 )2
)

=

σe2

=

3.0745σe2

σq2

=

3.0745σe2

σq2

=

3.3882σe2

σq2

=

3.2588σe2

σq2

=

3.2627σe2

σq2

=

3.3216σe2



using similar methods:
H1 − H2 − H3

H2 − H1 − H3

H2 − H3 − H1

H3 − H1 − H2

H3 − H2 − H1

9.35
y(n) = Q[0.1δ(n)] + Q[αy(n − 1)]
(a)
y(n)
y(0)

= Q[0.1δ(n)] + Q[0.5y(n − 1)]
1
= Q[0.1] =
8
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Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

y(1) = Q[
y(2) = y(3) = y(4) =

1
]=0
16

0

no limit cycle
(b)
y(n)

= Q[0.1δ(n)] + Q[0.75y(n − 1)]
1
= Q[0.1] =
8
1
3
= Q[ ] =
32
8
3
1
= Q[ ] =
32
8
1
=
8

y(0)
y(1)
y(2)
y(3) = y(4)
limit cycle occurs

9.36
(a) σx2 = rxx (0) = 3 ⇒ Ax =
(b)

√1
3

2−6
△2
12

△ =
σe2

=

1
12 × 212
1
10log10 2
σe

=
so SNR

=

10log10 (12 × 212 )
46.91dB

=
=
(c) left-justified.
(d)
σq2

=

σe2

∞
X

n=0
2
Now σe1
X
and
h2 (n)
n

so σq2

and SNR

2

h (n) +

2
σe1

∞
X

h2 (n)

n=0

1 1 2
( )
12 28
1
16
=
=
,
2
1 − 0.75
7


1 1 2
16 1 1 2
( ) + ( 6)
=
7 12 28
12 2
17
=
344, 064
1
= 10log10 2
σq
= 43.06dB
=

293

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

9.37
△

△

Define ρc = rcosθ, ρs = rsinθ for convenience, (a)
−ρs y(n − 1) + e1 (n) + x(n) + ρc v(n − 1) + e2 (n)
ρs v(n − 1) + e3 (n) + ρc y(n − 1) + e4 (n)

= v(n)
= y(n)

(b)
−ρs z −1 Y (z) + E1 (z) + X(z) + ρc z −1 V (z) + E2 (z) = V (z)
ρs z −1 V (z) + E3 (z) + ρc z −1 Y (z) + E4 (z) = Y (z)
ρs z −1
[X(z) + E1 (z) + E2 (z)]
1 − 2ρc z −1 + r2 z −2
1 − ρc z −1
[E3 (z) + E4 (z)]
+
1 − 2ρc z −1 + r2 z −2
= H1 (z)X(z) + H1 (z)[E1 (z) + E2 (z)]
+H2 (z)[E3 (z) + E4 (z)]

Y (z) =

when H1 (z) and H2 (z) are as defined in the problem statement
h1 (n)

h2 (n)

1 n−1
r
sinθnu(n − 1)
sinθ
= rn sin(nθ)u(n − 1)
= ρs

= rn sin(nθ)u(n)
1 n
1 n−1
=
r sin(n + 1)θu(n) + ρc
r
sin(θn)u(n − 1)
sinθ
sinθ
n
r
[sin(n + 1)θ − cosθsin(nθ)]u(n − 1)
= δ(n) +
sinθ
n
= δ(n) + r cos(nθ)u(n − 1)
= rn cos(nθ)u(n)

(c)
2
2
2
2
σe2 = σe1
= σe2
= σe3
= σe4

=
=
=

σq2

=

△2
12
1 −b 2
(2 )
12
2−2b
12
∞
∞
X
X
2σe2
h21 (n) + 2σe2
h22 (n)

=

2σe2

=

2σe2

n=0
∞
X

n=0

[r2n sin2 nθ + r2n cos2 nθ]

n=0

=

1
1 − r2n
−2b
2
1
6 1 − r2n

9.38
(a)
h1 (n)

1
= ( )n u(n)
2
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h2 (n)
h(n)
σq2

1
( )n u(n)
4
1
1
= [2( )n − ( )n ]u(n)
2
4
∞
∞
X
X
2
2
= 2σe1
h21 (n) + 2σe2
h22 (n)
=

n=0

=

n=0

16
64 2
σ + σ2
35 e1 15 e2

(b)
σq2

2
= σe1

X

2
h2 (n) + σe2

n

=

X

h21 (n)

n

64 2
4
σ + σ2
35 e1 3 e2

9.39
Refer to fig 9.39-1

x(n)

1

+

y(n)

+

e (n)
1

+

e

+

e

-1
z
a1

-1
z

aM-2

M-2

(n)

-1
z
a

M-1

M-1

(n)

Figure 9.39-1:
σe2i

=

σq2

=
=

1 −2b
2
∀i
12
(M − 1)σe2i

(M − 1) −2b
2
12

295

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

9.40

H(z)

B(z)
A(z)

π
π
(1 − 0.8ej 4 )(1 − 0.8e−j 4 )
=
G1
(1 − 0.5z −1 )(1 + 13 z −1 )

(1 + 0.25z −1 )(1 − 58 z −1 )
G2
π
π
(1 − 0.8ej 3 )(1 − 0.8e−j 3 )
= H1 (z)H2 (z)
=

(a)
z −1
At w = 0, z −1
H1 (w)|w=0
π
π
(1 − 0.8ej 4 )(1 − 0.8e−j 4 )
G1
(1 − 0.5)(1 + 13 )
G1
5
(1 + 0.25)(1 − 8 )
G2
π
π
(1 − 0.8ej 3 )(1 − 0.8e−j 3 )
G2

= e−jw
= 1
= 1
= 1
= 1.1381
= 1
= 1.7920

(b) Refer to fig 9.40-1.
(c) Refer to fig 9.40-2.
Refer to fig 9.40-3.
Refer to fig 9.40-4.

296

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Direct for m I:
x(n)

1

+

+

y(n)

G

-1
z

-1
z
-3/8

+

1/6

+

-1
z

-1
z
-5/32

1/6

Direct form II and cascade structure:

x(n)

+

+

y(n)

G

-1
z
+

-3/8

1/6

+

-1
z
-5/32

1/6

Figure 9.40-1:
Direct form I, impulse response
1.5

−−−> mag

1
0.5
0
−0.5
−1
0

10

20

30

40

50

60

70

80

90

100

70

80

90

100

Direct form I, step response
1.6

−−−> mag

1.4
1.2
1
0.8
0

10

20

30

40

50
−−−> n

60

Figure 9.40-2:
297

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Direct form II, impulse response
1.5

−−−> mag

1
0.5
0
−0.5
0

10

20

30

40

50

60

70

80

90

100

70

80

90

100

70

80

90

100

70

80

90

100

Direct form II, step response
1.5

−−−> mag

1.4
1.3
1.2
1.1
1
0

10

20

30

40

50
−−−> n

60

Figure 9.40-3:
Cascade form , impulse response
1.5

−−−> mag

1
0.5
0
−0.5
0

10

20

30

40

50

60

Cascade form, step response
1.5

−−−> mag

1.4
1.3
1.2
1.1
1
0

10

20

30

40

50
−−−> n

60

Figure 9.40-4:
298

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

9.41
(a)
k1

−3
8
27
32

=

4
9
5
= −
32

= −
k2
Refer to fig 9.41-1a.
(b)

x(n) f (n)
0

f1(n)

+

f 2 (n) = y(n)

+
k2

k1
z-1

z-1

+

z-1

+

(a)
Forward
x(n)

f1(n)

+
-

f (n)
2

f (n)
0

+

y(n)

k2

k1

g (n)
2
z-1

+

z-1

+

g (n)
0

Reverse
(b)

Figure 9.41-1:

A(z)

=
=

k2

1
(1 −

1−
1
= −
6

0.5z −1 )(1

+ 31 z −1 )

1

1 −1
6z

− 61 z −2

299

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

1
6
1
= −
5

k1 (1 + k2 ) = −
k1

Refer to fig 9.41-1b.
(c) Refer to fig 9.41-2.
(e) Refer to fig 9.41-3.

x(n)

+

+
-

-1/6

+

-1/5
z -1

z -1

+
v =0.4336
1

v =-0.1563
2

v =0.7829
0

+

+

y(n)

Figure 9.41-2:
(f) Finite word length effects are visible in h(n) for part f.

9.42
Refer to fig 9.42-1.

c =
H1 (z)

=

H2 (z)

=

15
16
1
1

9
10
− 21 z −1
83
80
+ 31 z −1

300

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

IR for part a

IR for part b

0.5

0.5
h(n)

1

h(n)

1

0

−0.5
0

0

50

−0.5
0

100

IR for part c

50

100

IR for part f

0.8

1

0.5

0.4

h(n)

h(n)

0.6

0.2

0

0
−0.2
0

50
−−> n

100

−0.5
0

50
−−> n

100

Figure 9.41-3:

301

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Parallel form structure:
c

x(n)

y(n)

H (z)
1

H (z)
2
Parallel form structure using 2nd-order coupled-form state-space sections
1
1/2

x(n)

A

B

+

z

+

z

-1

+

y(n)

-1

-1/3

Figure 9.42-1:

302

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Chapter 10

10.1
(a) To obtain the desired length of 25, a delay of
Hd (w)

hd (n)

=

1e−j12w ,

=

0,

=

1
2π

= 12 is incorporated into Hd (w). Hence,
0 ≤ |w| ≤

π
6

otherwise
Z

π
6

−π
6
π
sin 6 (n

Hd (w)e−jwn dw

− 12)
π(n − 12)
= hd (n)w(n)

=
Then, h(n)

25−1
2

where w(n) is a rectangular window of length N = 25.
P24
(b)H(w) = n=0 h(n)e−jwn ⇒ plot |H(w)| and 6 H(w). Refer to fig 10.1-1.
(c) Hamming window:
w(n)

=

h(n)

= hd (n)w(n)

nπ
)
12
for 0 ≤ n ≤ 24

(0.54 − 0.46cos

Refer to fig 10.1-2.

303

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

−−−> mag(dB)

50
0
−50
−100
−150
0

0.05

0.1

0.15

0.2
0.25
0.3
−−−> Freq(Hz)

0.35

0.4

0.45

0.5

0.05

0.1

0.15

0.2
0.25
0.3
−−−> Freq(Hz)

0.35

0.4

0.45

0.5

−−−> phase

4
2
0
−2
−4
0

Figure 10.1-1:

−−−> mag(dB)

50
0
−50
−100
−150
0

0.05

0.1

0.15

0.2
0.25
0.3
−−−> Freq(Hz)

0.35

0.4

0.45

0.5

0.05

0.1

0.15

0.2
0.25
0.3
−−−> Freq(Hz)

0.35

0.4

0.45

0.5

−−−> phase

4
2
0
−2
−4
0

Figure 10.1-2:
304

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

(d) Bartlett window:
w(n)

=

1−

2(n − 12)
24

0 ≤ n ≤ 24

Refer to fig 10.1-3.

−−−> mag(dB)

0
−10
−20
−30
−40
−50
0

0.05

0.1

0.15

0.2
0.25
0.3
−−−> Freq(Hz)

0.35

0.4

0.45

0.5

0.05

0.1

0.15

0.2
0.25
0.3
−−−> Freq(Hz)

0.35

0.4

0.45

0.5

−−−> phase

4
2
0
−2
−4
0

Figure 10.1-3:

10.2
(a)
Hd (w)

hd (n)

=

1e−j12w ,

=

0,

=

1
2π

|w| ≤

π
π
≤ |w| ≤
6Z
3

π
,
6

π
≤ |w| ≤ π
3

π

Hd (w)e−jwn dw

−π

= δ(n) −

sin π3 (n − 12) sin π6 (n − 12)
+
π(n − 12)
π(n − 12)

(b) Rectangular window:
w(n)

=

1,

=

0, otherwise

0 ≤ n ≤ 24

305

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Refer to fig 10.2-1.
(c) Hamming window:
10
−−−> mag(dB)

0
−10
−20
−30
−40
−50
0

0.05

0.1

0.15

0.2
0.25
0.3
−−−> Freq(Hz)

0.35

0.4

0.45

0.5

0.05

0.1

0.15

0.2
0.25
0.3
−−−> Freq(Hz)

0.35

0.4

0.45

0.5

−−−> phase

4
2
0
−2
−4
0

Figure 10.2-1:

w(n)

=

h(n)

= hd (n)w(n)

H(w)

=

(0.54 − 0.46cos
24
X

nπ
)
12

h(n)e−jwn

n=0

Refer to fig 10.2-2.
(d) Bartlett window:
w(n)

=

1−

(n − 12)
,
12

0 ≤ n ≤ 24

Refer to fig 10.2-3.
Note that the magnitude responses in (c) and (d) are poor because the transition region is
wide. To obtain sharper cut-off, we must increase the length N of the filter.

306

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

−−−> mag(dB)

5
0
−5
−10
−15
−20
0

0.05

0.1

0.15

0.2
0.25
0.3
−−−> Freq(Hz)

0.35

0.4

0.45

0.5

0.05

0.1

0.15

0.2
0.25
0.3
−−−> Freq(Hz)

0.35

0.4

0.45

0.5

−−−> phase

4
2
0
−2
−4
0

Figure 10.2-2:

−−−> mag(dB)

0

−5

−10

−15
0

0.05

0.1

0.15

0.2
0.25
0.3
−−−> Freq(Hz)

0.35

0.4

0.45

0.5

0.05

0.1

0.15

0.2
0.25
0.3
−−−> Freq(Hz)

0.35

0.4

0.45

0.5

−−−> phase

4
2
0
−2
−4
0

Figure 10.2-3:
307

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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10.3
(a) Hanning window: w(n) = 12 (1 − cos πn
0 ≤ n ≤ 24. Refer to fig 10.3-1.
12 ),
πn
(b) Blackman window: w(n) = 0.42 − 0.5cos πn
12 + 0.08cos 6 . Refer to fig 10.3-2.

−−−> mag(dB)

50
0
−50
−100
−150
0

0.05

0.1

0.15

0.2
0.25
0.3
−−−> Freq(Hz)

0.35

0.4

0.45

0.5

0.05

0.1

0.15

0.2
0.25
0.3
−−−> Freq(Hz)

0.35

0.4

0.45

0.5

−−−> phase

4
2
0
−2
−4
0

Figure 10.3-1:

10.4
(a) Hanning window: Refer to fig 10.4-1.
(b) Blackman window: Refer to fig 10.4-2.
The results are still relatively poor for these window functions.

10.5

M

=

4,

Hr (0) = 1,
M
2

Hr (w)

=

2

−1
X

n=0

=

2

h(n)cos[w(

1
π
Hr ( ) =
2
2
M −1
− n)]
2

1
X

3
h(n)cos[w( − n)]
2
n=0

308

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−−−> mag(dB)

0

−50

−100

−150
0

0.05

0.1

0.15

0.2
0.25
0.3
−−−> Freq(Hz)

0.35

0.4

0.45

0.5

0.05

0.1

0.15

0.2
0.25
0.3
−−−> Freq(Hz)

0.35

0.4

0.45

0.5

−−−> phase

4
2
0
−2
−4
0

Figure 10.3-2:

At w = 0, Hr (0) = 1

= 2

1
X

h(n)cos[0]

n=0

2[h(0) + h(1)]
At w =

π
1
π
, Hr ( ) =
2
2
2

−h(0) + h(1)
Solving (1) and (2), we get

= 1
= 2

(1)

1
X

π 3
h(n)cos[ ( − n)]
2 2
n=0

= 0.354

h(0)
h(1)

= 0.073 and
= 0.427

h(2)
h(3)

= h(1)
= h(0)

Hence, h(n)

(2)

= {0.073, 0.427, 0.427, 0.073}

309

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−−−> mag(dB)

5
0
−5
−10
−15
0

0.05

0.1

0.15

0.2
0.25
0.3
−−−> Freq(Hz)

0.35

0.4

0.45

0.5

0.05

0.1

0.15

0.2
0.25
0.3
−−−> Freq(Hz)

0.35

0.4

0.45

0.5

−−−> phase

4
2
0
−2
−4
0

Figure 10.4-1:

−−−> mag(dB)

5
0
−5
−10
−15
0

0.05

0.1

0.15

0.2
0.25
0.3
−−−> Freq(Hz)

0.35

0.4

0.45

0.5

0.05

0.1

0.15

0.2
0.25
0.3
−−−> Freq(Hz)

0.35

0.4

0.45

0.5

−−−> phase

4
2
0
−2
−4
0

Figure 10.4-2:
310

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10.6
M = 15.Hr (

2πk
)=
15



1,
0,

k = 0, 1, 2, 3
k = 4, 5, 6, 7

M −3

Hr (w)
h(n)
h(n)
Hr (w)

2
X
M −1
M −1
h(n)cosw(
)+2
− n)
= h(
2
2
n=0

= h(M − 1 − n)
= h(14 − n)

= h(7) + 2

6
X

n=0

Solving the above eqn yields,
h(n)

h(n)cosw(7 − n)

= {0.3189, 0.0341, −0.1079, −0.0365, 0.0667, 0.0412, −0.0498, 0.4667
0.4667, −0.0498, 0.0412, 0.0667, −0.0365, −0.1079, 0.0341, 0.3189}

10.7

 1,
2πk
0.4,
)=
M = 15.Hr (

15
0,

k = 0, 1, 2, 3
k=4
k = 5, 6, 7

M −3

Hr (w)
h(n)
h(n)
Hr (w)

2
X
M −1
M −1
= h(
)+2
− n)
h(n)cosw(
2
2
n=0

= h(M − 1 − n)
= h(14 − n)

= h(7) + 2

6
X

n=0

Solving the above eqn yields,
h(n)

h(n)cosw(7 − n)

= {0.3133, −0.0181, −0.0914, 0.0122, 0.0400, −0.0019, −0.0141, 0.52,
0.52, −0.0141, −0.0019, 0.0400, 0.0122, −0.0914, −0.0181, 0.3133}

10.8
(a)
ya (t)

Hence, H(F )

dxa (t)
dt
d j2πF t
[e
]
=
dt
= j2πF ej2πF t
=

= j2πF
311

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
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(b)
|H(F )|
6

=

2πF
π
H(F ) =
,
F >0
2
π
F <0
= − ,
2

Refer to fig 10.8-1.
(c)
B=pi/6

−−> magnitude

0.6

0.4

0.2

0
−0.1

−0.08

−0.06

−0.04

−0.02
0
0.02
−−> Freq(Hz)

0.04

0.06

0.08

0.1

−0.08

−0.06

−0.04

−0.02
0
0.02
−−> Freq(Hz)

0.04

0.06

0.08

0.1

2

−−> phase

1
0
−1
−2
−0.1

Figure 10.8-1:
H(w)
|H(w)|
6

H(w)

= jw,

|w| ≤ π

= |w|
π
,
w>0
=
2
π
= − ,
w<0
2

Refer to fig 10.8-2.
we note that the digital differentiator has a frequency response that resembles the response
of the analog differentiator.
(d)
y(n) = x(n) − x(n − 1)
H(z) = 1 − z −1

H(w)

=

1 − e−jw

312

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−−> magnitude

4
3
2
1
0
−4

−3

−2

−1

0
−−> w

1

2

3

4

−3

−2

−1

0
−−> w

1

2

3

4

2

−−> phase

1
0
−1
−2
−4

Figure 10.8-2:

|H(w)|
6

H(w)

w
w
= e−j 2 (2jsin )
2
w
= 2|sin |
2
π w
=
−
2
2

Refer to fig 10.8-3.
w
Note that for small w, sin w2 ≈ w2 and H(w) ≈ jwe−j 2 , which is a suitable approximation
to the differentiator in (c).
(e) The value H(w0 ) is obtained from (d) above. Then y(n) = A|H(w0 )|cos(w0 n + θ + π2 − w20 )

10.9
Hd (w)
hd (n)

hd (n)

= we−j10w ,
0≤w≤π
−j10w
= −we
,
−π ≤w ≤0
Z π
1
=
Hd (w)e−jwn dw
2π −π
cosπ(n − 10)
=
,
n 6= 10
(n − 10)
= 0,
n = 10
cosπ(n − 10)
,
0 ≤ n ≤ 20, n 6= 10
=
(n − 10)
313

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−−> magnitude

2
1.5
1
0.5
0
−4

−3

−2

−1

0
−−> w

1

2

3

4

−3

−2

−1

0
−−> w

1

2

3

4

2

−−> phase

1
0
−1
−2
−4

Figure 10.8-3:
=

0,

n = 10

With a Hamming window, we obtain the following frequency response: Refer to fig 10.9-1.

10.10
H(s) has two zeros at z1 = −0.1 and z2 = ∞ and two poles p1,2 = −0.1 ± j3. The matched
z-transform maps these into:
z̃1
z̃2

= e−0.1T = e−0.01 = 0.99
= e−∞T = 0

p̃1
p̃2

= e(−0.1+j3)T = 0.99ej0.3
= 0.99e−j0.3
1 − rz −1
, w0 = 0.3 r = 0.99
=
1 − 2rcosw0 z −1 + r2 z −2

Hence, H(z)

From the impulse invariance method we obtain


1
1
1
H(s) =
+
2 s + 0.1 − j3 s + 0.1 + j3


1
1
1
H(z) =
+
2 1 − e−0.1T ej3T z −1
1 − e−0.1T e−j3T z −1
1 − rcosw0 z −1
=
1 − 2rcosw0 z −1 + r2 z −2
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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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3
−−−> mag(dB)

2.5
2
1.5
1
0.5
0
0

0.05

0.1

0.15

0.2
0.25
0.3
−−−> Freq(Hz)

0.35

0.4

0.45

0.5

0.05

0.1

0.15

0.2
0.25
0.3
−−−> Freq(Hz)

0.35

0.4

0.45

0.5

−−−> phase

4
2
0
−2
−4
0

Figure 10.9-1:
The poles are the same, but the zero is different.

10.11

Ha (s) =
s =
H(z) =
=
where α

=

(Ωu − Ωl )s
− (Ωu − Ωl )s + Ωu Ωl
2 1 − z −1
T 1 + z −1
s2

(Ωu − Ωl )

2
T

(1 − z −1 )(1 + z −1 )
( T2 )2 (1 − z −1 )2 + (Ωu − Ωl )( T2 )(1 − z −1 )(1 + z −1 ) + Ωu Ωl (1 + z −1 )2

2(α − β)(1 − z −2 )
[4 + 2(α − β) + αβ] − 2(4 − αβ)z −1 + [4 − 2(α − β) + αβ]z −2
Ωu T,
β = Ωl T

In order to compare the result with example 10.4.2, let
wu
wl
Then, H(z)
In our case, we have α = Ωu T
β = Ωl T

3π
5
2π
= Ωl T =
5
0.245(1 − z −2 )
=
1 + 0.509z −2
wu
= 2.753
= 2tan
2
wl
= 1.453
= 2tan
2
=

Ωu T =

( example 8.3.2 )

315

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By substituting into the equation above, we obtain
2.599(1 − z −2 )
10.599 + 5.401z −2
0.245(1 − z −2 )
1 + 0.509z −2

H(z) =
=

10.12
Let T = 2
(a) H(z) =
(b)

1+z −1
1−z −1

⇒ y(n) = y(n − 1) + x(n) + x(n − 1)
Ha (Ω) =

1
|Ω|
6

H(Ω) =

(c)
w
|H(w)| = |cot |
2

H(w) =
6





− π2 ,
π
2,

Ω≥0
Ω<0

− π2 , 0 ≤ w ≤ π
π
−π < w < 0
2,

(d) The digital integrator closely matches the magnitude characteristics of the analog integrator.
The two phase characteristics are identical.
(e) The integrator has a pole at w = 0. To avoid overflow problems, we would have E[x(n)] = 0,
i.e., a signal with no dc component.

10.13
(a)
H(z) = A
= A
H(z)|z=1

=

⇒A =

(1 −

(1 + z −1 )3
− 12 z −1 + 41 z −2 )

1 −1
)(1
2z
−1

(1 + z )(1 + 2z −1 + z −2 )
(1 − 21 z −1 )(1 − 12 z −1 + 41 z −2 )

1
3
1
1
1
, b1 = 2, b2 = 1, a1 = 1, c1 = − , d1 = − , d2 =
64
2
2
4

(b) Refer to fig 10.13-1

10.14
(a) There are only zeros, thus H(z) is FIR.
(b)
Zeros: z1
z2
z3,4
z5,6

4
= − ,
3
3
= − ,
4
3 ±j π
e 3
=
4
4 ±j π
e 3
=
3
316

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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

x(n)

3/64

+

+

y(n)

z-1
+

-1

3

+

z-1
+

1/2

3

+

z-1
-1/8

x(n)

3/64

1

+

+

+

+
z-1

z-1
-0.5

y(n)

1

-1/2

+

2

+

z-1
1/4

1

Figure 10.13-1:
z7

=

Hence, z2

=

z4

=

z5

=

z6

=

z1

=

and H(z) =

1
1
z1∗
z3∗
1
z3∗
z5∗
1
=1
z7
z −6 H(z −1 )

Therefore, H(w) is linear phase.
(c) Refer to fig 10.14-1

10.15
From the design specifications we obtain
ǫ =

0.509

δ

=

fp

=

fs

=

99.995
4
1
=
24
6
1
6
=
24
4
317

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

+

+

y(n)

z-1
x(n)

+

z-1

25/12

+

z-1

+

z-1

z-1
-4/3 - 3/4 = -25/12
+
+

z-1

z-1

2
o
(4/3)2 + (3/4) + 4 cos 2 60 = 481/144

Figure 10.14-1:

Assume t = 1. Then, Ωp

=
and Ωs

wp
2
2tanπfp = 1.155
ws
2tan
2
2tanπfs = 2
δ
= 196.5
ǫ
Ωs
= 1.732
Ωp
logη
= 9.613 ⇒ N = 10
logk
cosh−1 η
= 5.212 ⇒ N = 6
cosh−1 k
1
= 0.577 ⇒ α = 35.3o
k
1
= 0.577 ⇒ β = 0.3o
η
k(sinα) k(cosβ)
.
= 3.78 ⇒ N = 4
k(cosα) k(sinβ)

= 2tan

=
=

η

=

k

=

Butterworth filter: Nmin

≥

Chebyshev filter: Nmin

≥

Elliptic filter: sinα

=

sinβ

=

Nmin

≥

318

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Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

10.16
From the design specifications we have
ǫ =
δ =
fp

=

fs

=

Ωp

=

Ωs

=

η

=

k

=

Butterworth filter: Nmin

≥

Chebyshev filter: Nmin

≥

Elliptic filter: Nmin

≥

0.349
99.995
1.2
= 0.15
8
2
= 0.25
8
wp
2tan
= 1.019
2
ws
2tan
=2
2
δ
= 286.5
ǫ
Ωs
= 1.963
Ωp
logη
= 8.393 ⇒ N = 9
logk
cosh−1 η
= 4.90 ⇒ N = 5
cosh−1 k
q

k( 1 − η12 )
k( k1 )
q
.
⇒ N =4
k( η1 )
k( 1 − k12 )

10.17
Passband ripple = 1dB ⇒ ǫ = 0.509
Stopband attenuation = 60dB ⇒ δ = 1000
wp
ws

=
=

Ωp

=

Ωs

=

η

=

k

=

Nmin

≥

0.3π
0.35π
wp
= 1.019
2tan
2
ws
= 1.226
2tan
2
δ
= 1965.226
ǫ
Ωs
= 1.203
Ωp
8.277
cosh−1 η
=
= 13.2 ⇒ N = 14
−1
cosh k
0.627

Special software package, such as MATLAB or PC-DSP may be used to obtain the filter coefficients. Hand computation of these coefficients for N = 14 is very tedious.
319

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10.18
Passband ripple = 0.5dB ⇒ ǫ = 0.349
Stopband attenuation = 50dB
wp
ws

=
=

Ωp

=

Ωs

=

η

=

k

=

Nmin

≥

0.24π
0.35π
wp
2tan
= 0.792
2
ws
= 1.226
2tan
2
δ
= 906.1
ǫ
Ωs
= 1.547
Ωp
7.502
cosh−1 η
=
= 7.48 ⇒ N = 8
−1
cosh k
1.003

Use a computer software package to determine the filter coefficients.

10.19
(a) MATLAB is used to design the FIR filter using the Remez algorithm. We find that a filter
of length M = 37 meets the specifications. We note that in MATLAB, the frequency scale is
normalized to 12 of the sampling frequency. Refer to fig 10.19-1.
20
15
(b)δ1 = 0.02, δ2 = 0.01, △f = 100
− 100
= 0.05
1.2

1

|H(w)|

0.8

0.6

0.4

0.2

0
0

0.1

0.2

0.3

0.4

0.5
−−> f

0.6

0.7

0.8

0.9

1

Figure 10.19-1:
With equation (10.2.94) we obtain
M̂

=

√
−20log10 ( δ1 δ2 ) − 13
+ 1 ≈ 34
14.6△f

320

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With equation (10.2.95) we obtain
D∞ (δ1 δ2 )

=

1.7371

f (δ1 δ2 )

=

and M̂

=

11.166
D∞ (δ1 δ2 ) − f (δ1 δ2 )(△f )2
+ 1 ≈ 36
△f

Note (10.2.95) is a better approximation of M .
(c) Refer to fig 10.19-2.
Note that this filter does not satisfy the specifications.
1
0.9
0.8
0.7

|H(w)|

0.6
0.5
0.4
0.3
0.2
0.1
0
0

0.1

0.2

0.3

0.4

0.5
−−> f

0.6

0.7

0.8

0.9

1

Figure 10.19-2: M=37 FIR filter designed by window method with Hamming window
(d)The elliptic filter satisfies the specifications. Refer to
(e)
FIR
order
37
storage
19
No. of mult. 19

fig 10.19-3.
IIR
5
16
16

10.20
(a)
h(n)



=
0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, . . .
↑

10
X

h(n)z −n

H(z)

=

H(w)

= z −1 + 2z −2 + 3z −3 + 4z −4 + 5z −5 + 4z −6 + 3z −7 + 2z −8 + z −9
= e−j9w [2cos4w + 4cos3w + 6cos2w + 8cosw + 5]

n=0

(b)|H(w)| = |2cos4w + 4cos3w + 6cos2w + 8cosw + 5|. Refer to fig 10.20-1.
321

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1
0.9
0.8
0.7

|H(w)|

0.6
0.5
0.4
0.3
0.2
0.1
0
0

0.05

0.1

0.15

0.2

0.25
−−> f

0.3

0.35

0.4

0.45

0.5

Figure 10.19-3:

10.21
(a)
dc gain: Ha (0)
3dB frequency: |Ha (jΩ)|2
or

α2
α2 + Ω2c
⇒ Ωc

For all Ω, only H(j∞)
ha (τ )
⇒ e−αt
⇒τ

=

1
1
=
2
1
=
2
= α
=

0
1
1
=
ha (0) =
e
e
= e−1
1
=
α

(b)
h(n)
H(z)
H(w)
H(0)

3dB frequency: |H(wc )|2
(1 − αT α coswc )2 + (e−αT sinwc )2

= ha (nT )
= e−αnT u(n)
1
=
1 − e−αT z −1
1
=
1 − e−αT e−jw
= H(w)|w=0
1
=
1 − e−αT
1
=
|H(0)|2
2
= 2(1 − e−αT )2

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25

−−> magnitude

20

15

10

5

0
0

0.05

0.1

0.15

0.2

0.25
0.3
−−> Freq(Hz)

0.35

0.4

0.45

0.5

Figure 10.20-1:

it oscillates between

Hence, wc

=

Since |H(w)|2

=

1
1
and
(1 − e−αT )2
(1 + e−αT )2
but never reaches zero
h(τ )
⇒τ

τ is the smallest integer that is larger than

αT
)
2
1
1 − 2e−αT cosw + e−2αT

2sin−1 (sinh

= e−ατ T = e−1
1
≥
αT

1
T

(c)
H(z)

=
=
=

DC Gain: H(z)|z=1

=

At z = −1(w = π), H(z)

=

α
2 1−z −1
T 1+z −1

+α

αT (1 + z −1 )
2(1 − z −1 ) + αT (1 + z −1 )
αT (1 + z −1 )
2 + αT + (αT − 2)z −1
1

0
1
since |Ha (jΩc )|2 =
, we have Ωc = α
2
Ωc
wc = 2tan−1 T
2
= 2tan−1 αT 2
2 − αT
Let a =
2 + αT
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Then H(z) =
h(n)

=

h(0)

=

h(n)
h(0)

=

⇒ (1 + a)an−1

=

n =
=



1−a
(1 + a)z −1
1+
2
1 − az −1


1−a
δ(n) + (1 + a)an−1 u(n − 1)
2
1−a
2
1
e
1
e
a
ln 1+a
−1
lna
ln( 2−αT
4 )−1
2−αT
ln( 2+αT )

10.22
(a)
hd (n)

=
=
=

T
2π
T
2π

Z

π
T

π
−T

"Z

Hd (w)ejwn dw

− 0.4π
T

ejwn dw +

π
− 2T

Z

0.5π
T

ejwn dw

0.4π
T

#



πn
2πn
T
sin
− sin
nπ
2T
5T

(b)
Let hs (n)
Then, h(n)
(c)

= hd (n)w(n),
− 100 ≤ n ≤ 100(M = 101)
= hs (n − 100) will be the impulse of the filter for 0 ≤ n ≤ 200

0,
0 ≤ w ≤ 0.4π

T


0.5π
−j100w

, 0.4π
 e
T ≤w ≤ T
0.5π
1.5π
0,
Hd (w) =
T  N , it is not possible to determine the {ak } and the order p.

10.28
(1) The set of linear equations are:
M
−1
X
k=0

h(k)rxx (k − l)
where rxx (l)
ryx (l)

= ryx (l),
=
=

∞
X

n=0
∞
X

n=0

E

=

l = 0, 1, . . . , M − 1

x(n)x(n − l) and
y(n)x(n − l)

∞
X

[y(n) −

n=−∞

M
−1
X
k=0

h(k)x(n − k)]2

(2) Refer to fig 10.28-1.
(3) Refer to fig 10.28-2.
Total squared error
8.28
8.26
8.24
8.22

error

8.2
8.18
8.16
8.14
8.12
8.1
8.08
8

9

10

11
filter order

12

13

14

Figure 10.28-1:
(4) v(n) = y(n) + 0.01w(n). Refer to fig 10.28-3.
328

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fft of original h(n)

M=11E=8.093
0.4

1
−−>|H(w)|

−−>|H(w)|

0.3

0.5

0.2
0.1

0
0

1000

2000

0
0

3000

1000

0.5

0.4

0.4

0.3
0.2
0.1
0
0

3000

M=13E=8.101

0.5

−−>|H(w)|

−−>|H(w)|

M=12E=8.084

2000

0.3
0.2
0.1

1000

2000

0
0

3000

1000

2000

3000

Figure 10.28-2:

10.29
(a) Since δ(n − k) = 0 except for n = k, equation (1) reduces to
h(n) = −a1 h(n − 1) − a2 h(n − 2) − . . . − aN h(n − N ) + bn , 0 ≤ n ≤ M
(b) Since δ(n − k) = 0 except for n = k, equation (1) reduces to
h(n) = −a1 h(n − 1) − a2 h(n − 2) − . . . − aN h(n − N ), 0n > M
(c) We use the linear equation given in (b) to solve for the filter parameters {ak }. Then we use
values for the {ak } in the linear equation fiven in (a) and solve for the parameters {bk }.

10.30
Hd (z) =

2
1 − 21 z −1

We can see that by setting M = 0 and N = 1 in
PM
−1
k=0 bk z
H(z) =
PN
1 + k=1 ak z −1
we can provide a perfect match to Hd (z) as given in
H(z) =

b0
.
1 + a1 z −1
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fft of original h(n)

M=8 E=8.42

M=9 E=8.305

−−>|H(w)|

1.5

0.4
0.5

1
0.2
0.5
0
0

1000 2000 3000
M=10 E=8.228

0.4

0
0

1000 2000 3000
M=11 E=8.228

0
0

1000 2000 3000
M=12 E=8.221

0.4
0.5

0.2

0
0

0.2

1000 2000 3000
M=13 E=8.238

0.5

0
0

0
0

1000 2000 3000
M=14 E=8.24

0
0

1000

2000

3000

0.5

1000

2000

3000

0
0

1000

2000

3000

Figure 10.28-3:
With δ(n) as the input to H(z), we obtain the output
h(n) = −a1 h(n − 1) + b0 δ(n).
For n > M = 1, we have
h(n) = −a1 h(n − 1)
or, equivalently,
hd (n) = −a1 hd (n − 1).
Substituting for hd (n), we obtain a1 = − 21 . To solve for b0 , we use the equation given in 10.29(a)
with h(n) = hd (n),
1
hd (n) = hd (n − 1) + b0 δ(n).
2
For n = 0 this equation yields b0 = 2. Thus
H(z) =

2
1 − 12 z −1

.

(b)
Hd (z) = H(z)
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10.31
(a)
h(n) = −

N
X

k=1

ak h(n − k) +

M
X

k=0

bk δ(n − k)

n≥0

(b) Based on (a)
ĥd (n) = −

ε1

=

N
X

k=1

∞ h
X

M +1

=

∞
X

M +1

ak hd (n − k)

n>M

i2
hd (n) − ĥd (n)

"

hd (n) −

N
X

k=1

#2

ak hd (n − k)

By differentiating with respect to the parameters {ak }, we obtain the set of linear equations of
the form
N
X
ak rhh (k, l) = −rhh (l, 0)
l = 1, 2, . . . , N
k=1

where,
rhh (k, l)

=
=

∞
X

n=1
∞
X

n=0

hd (n − k)hd (n − l)
hd (n)hd (n + k − l) = rhh (k − l)

The solution of these linear equations yield to the filter parameters {ak }.
(c) We can find the least-squares solution for {bk } from the minimization of
ε2 =

∞
X

n=0

"

ĥd (n) −

M
X

k=0

#2

bk v(n − k)

Thus we obtain a set of linear equations for the parameters {bk }, in the form
M
X

bk rvv (k, l) = rhv (l)

l = 0, 1, . . . , M

k=0

where
rvv (k, l)
rhv (k)

=
=

∞
X

n=0
∞
X

n=0

v(n − k)v(n − l)
h(n)v(n − k)

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10.32
(a)
y(n)

=

1.5198y(n − 1) − 0.9778y(n − 2) + 0.2090y(n − 3)
+0.0812x(n) + 0.0536x(n − 1) + 0.0536x(n − 2) + 0.0812x(n − 3)

hd (n) can be found by substituting x(n) = δ(n). Fig 10.32-1 shows the hd (n).
0.3

0.25

0.2

d

h (n)

0.15

0.1

0.05

0

−0.05

−0.1

0

5

10

15

20

25
n

30

35

40

45

50

Figure 10.32-1:
(b) The poles and zeros obtained using Shanks’ method are listed in Table 10.32. The magnitude
response for each case together with the desired response is shown in Fig. 10.32-2. The frequency
response characteristics illustrate that Shanks’ method yields very good designs when the number
of poles and zeros equals or exceeds the number of poles and zeros in the actual filter. Thus the
inclusion of zeros in the approximation has a significant effect in the resulting design.
Filter
Order
N=3
M=2
N=3
M=3
N=4
M=3

Poles
0.5348
0.6646 ± j0.4306
0.3881
0.5659 ± j0.4671
-0.00014
0.388
0.566 ± j0.4671

Zeros
−0.2437 ± j0.5918
-1
0.1738 ± j0.9848
-1
0.1738 ± j0.9848

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Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

0

−10
Desired response
N=3, M=3
N=4, M=3

−20

Magnitude (dB)

−30

−40
N=3, M=2
−50

−60

−70

−80

−90

0

0.1

0.2

0.3
0.4
0.5
0.6
0.7
Normalized Frequency (xπ rad/sample

0.8

0.9

1

Figure 10.32-2:

333

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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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334

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Chapter 11

11.1
(a) Let the corresponding baseband spectrum be called Xb (Ω). Then
1
[Xb (Ω − 2000π) + Xb (Ω + 2000π)]
2

Xa (Ω) =

With frequencies normalized to Fx ,
w′ =

Ω
Fx

. The sequence x(n) has DTFT
X(w′ ) =
=

∞
X

q=−∞
∞
X

Xa (w′ − 2πq)

[Xa (w′ − 0.8π − 2πq) + Xb (w′ + 0.8π − 2πq)]

q=−∞

modulation by cos(0.8π) causes shifts up and down by 0.8π (and scaling by

|X(w’)|

1
2)

of each

Assumes peak of X (.) normalized to unity
b

0.5
X (w’) shifted to 0.8π
b

period 2π
−π

−0.8π

0

0.8π

π

w’

β=0.16π

Figure 11.1-1:
component in the spectrum. Refer to fig 11.1-1. Ideal LPF preserves only the baseband spectrum
(of each period). Refer to fig 11.1
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Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

|W(w’)|

1

H(w’)

0.5

−π

period 2 π

0.5

−0.4π

0.1π

0.4π

π

w

β

Figure 11.1-2:

′
The downsampling produces the figure in fig 11.1, where w′′ = FΩy = ΩD
Fx = 10w . Note that
′′
there is no aliasing in the spectrum |Y (w )| because the decimated sample rate, in terms of w′ ,
is 2π
10 > 0.04π.
(b) The assumed spectral amplitude normalization in fig 11.1-1 implies that the analog FT
(magnitude spectrum) of xa (t) is (refer to fig 11.1-4).

The given sample rate is identical to Fy above, Fy = 250Hz. The DTFT of samples taken
P
at this rate is Ỹ (Ω) = T1y q Xa (Ω − qΩy ) where Ωy = 2πFy . On a scaled frequency axis
P
w′′ = ΩTy = FΩy , Ỹ (w′′ ) = T1y q Xa (w′′ − q2π). Consequently ỹ(n) = y(n).

|V(w’)|
period 2 π

0.08π

π

w’

π

w’’

|Y(w’’)| = 0.1 |V(0.1w’’)|
0.1
period 2 π

0.8π

Figure 11.1-3:
336

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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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|Xa ( Ω) |
Tx /2

Ωc
400π

−Ωc

Ω

Figure 11.1-4:

11.2
(a) X(w) =

1
(1−ae−jw )

′

(b) After decimation Y (w′ ) = 12 X( w2 ) =

1
2(1−ae−

jw′
2

)

(c)
DTFT {x(2n)}

=

X

x(2n)e−jw2n

n

=

X

′

x(2n)e−jw n

n

= Y (w′ )

11.3
(a)Refer to fig 11.3-1
(b)

x(n)
y(m)
F
x

-1
z

1/2

Fy = Fx

+

Figure 11.3-1:
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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Ω
,
Fx
X

Ω
w′
=
Fy
2

Let w′

=

Y (w′′ )

X 1 n−1
′′
n
n + 1 −jw′′ n
x( )e−jw n +
[x(
) + x(
)]e
2
2
2
2
n even
n odd
X
′′
1X
−jw′′ 2p
[x(q) + x(q + 1)]e−jw (2q+1)
=
x(p)e
+
2 q
p
=

w′′ =

′′
′′
1
= X(2w′′ ) + e−jw [X(2w′′ ) + ej2w X(2w′′ )]
2
= X(2w′′ )[1 + cosw′′ ]

X(w′ ) =



′′

X(2w ) =

=

Y (w′′ ) =





0 ≤ |w′ | ≤ 0.2π
otherwise

1,
0,


1,
0,

1,
0,

0 ≤ |2w′′ | ≤ 0.2π
otherwise

0 ≤ |w′′ | ≤ 0.1π
otherwise

1 + cosw′′ ,
0,

0 ≤ |w′′ | ≤ 0.1π
otherwise

(c) Refer to fig 11.3-2

X(.)

0
0

0.7π
0.35 π

0.9π π
0.45 π

1.1π
0.55 π

1.3π
0.65 π

2π
π

w’
w ’’

Figure 11.3-2:

11.4


 1 + cosw′′ , 0.35π ≤ |w′′ | ≤ 0.45π
′′
or 0.55π ≤ |w′′ | ≤ 0.65π
Y (w ) =

0,
otherwise

w′′ = ΩD
(a) Let w′ = FΩx ,
Fx . Refer to fig 11.4-1
′′
Let x (n) be the downsampled sequence.
338

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|X(w ’)|
1

-w m
’

wm’

π

w’

Dw’
m

π

w ’’

|X’’(w’’)|
1/D

-Dw’
m

Figure 11.4-1:
x′′ (n)
X ′′ (w′′ )

= x(nD)
w′′
1
X( )
=
D
D

′
As long as Dwm
≤ π, X(w′ ) [hence x(n)] can be recovered from X ′′ (w′′ )[x′′ (n) = x(Dn)]
using interpolation by a factor D:

X(w′ ) = DX ′′ (Dw′ )
2π
′
′
′
The given sampling frequency is ws′ = 2π
D . The condition Dwm ≤ π → 2wm ≤ D = ws
(b) Let xa (t) be the ral analog signal from which samples x(n) were taken at rate Fx . There exists
a signal, say x′a (t′ ), such that x′a (t′ ) = Xa ( Ttx ). x(n) may be considered to be the samples of x′ (t′ )
1
.
taken at rate fx = 1. Likewise x′′ (n) = x(nD) are samples of x′ (t′ ) taken at rate fx′′ = fDx = D
′ ′
′′
From sampling theory, we know that x (t ) can be reconstructed from its samples x (n) as long
1
π
as it is bandlimited to fm ≤ 2D
, or wm ≤ D
, which is the case here. The reconstruction formula
is
X
x′ (t′ ) =
x′′ (k)hr (t′ − kD)
k

where
hr (t′ ) =

π ′
t)
sin( D
π ′
(Dt )

Refer to fig 11.4-2
′
Actually the bandwidth of the reconstruction filter may be made as small as wm
, or as large
2π
′
as D − wm , so hr may be
sin(wc′ t′ )
hr (t′ ) =
(wc′ t′ )
′
where wm
≤ wc′ ≤

2π
D

′
− wm
. In particular x(n) = x′ (t′ = n) so

x(n) =

X
k

x(kD)hr (n − kD)
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|X(w ’)| period 2 π/D

Hr (w ’)

w’
m

0

π/D

2 π/D - w ’
m

2 π/D

w’

Figure 11.4-2:

(c) Clearly if we define
v(p) =



x(p),
0,

if p is an integer multiple of D
other p

then, we may write 11.4 as
x(n) =

X
p

v(p)hr (n − p)

so x(n) is reconstructed as (see fig 11.4-3)

x’’(n)=x(kn)

D

v(n)

x(n)
h r(n)

Figure 11.4-3:

340

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11.5
(a)
Let w
Xs (w)

Ω
2Ω
,
w′′ =
Fx
Fx
X
−jwn
=
xs (n)e

=

n

=

X

x(2m)e−jw2m

m

=
=

2π
1X
q)
X(w −
2 q
2
1X
X(w − πq)
2 q

To recover x(n) from xs (n): see fig 11.5-1
(b)

X (w)
s
period π

1/2

−π

x (n)
s

−2π/3

v(n)

2

−π/3

π/3

2π/3

π

w

x(n)
h (n)
r

where
H (w)
r
1

−π/2

π/2

w

Figure 11.5-1:
Recall w′
′

Xd (w )

=
=

2w
X

′

xd (n)e−jw n

n

=
=

X

xs (n)e−jw

′n
2

n even
X
′n
xs (n)e−jw 2
n

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since xs (n) = 0 when n odd
= Xs (

w′
)
2

see fig 11.5-2
No information is lost since the decimated sample rate still exceeds twice the bandlimit of

period 2 π

1/2

−π

2π/3

π

w’

Figure 11.5-2:
the original signal.

11.6
A filter of length 30 meets the specification. The cutoff frequency is wc =
are given below:
h(1)
h(2)
h(3)
h(4)
h(5)

and the coefficients

= h(30) = 0.006399
= h(29) = −0.01476
= h(28) = −0.001089

= h(27) = −0.002871
= h(26) = 0.01049

h(6)
h(7)

= h(25) = 0.02148
= h(24) = 0.01948

h(8)
h(9)

= h(23) = −0.0003107
= h(22) = −0.03005

h(10)
h(11)
h(12)

π
5

= h(21) = −0.04988
= h(20) = −0.03737
= h(19) = 0.01848

h(13)
h(14)

= h(18) = 0.1075
= h(17) = 0.1995

h(15)

= h(16) = 0.2579
342

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pk (n)

= h(n + k),

k = 0, 1, . . .

corresponding polyphase filter structure (see fig 11.6-1)

p (n)
0

+

p (n)
1

+

y(n)

x(n)

p (n)
4

F
x

F = Fx /D
y

Figure 11.6-1:

11.7
A filter of length 30 meets the specification. The cutoff frequency is wc =
are given below:
h(1)

= h(30) = 0.006026

h(2)
h(3)
h(4)

= h(29) = −0.01282
= h(28) = −0.002858
= h(27) = 0.01366

h(5)
h(6)

= h(26) = −0.004669
= h(25) = −0.01970

h(7)
h(8)
h(9)
h(10)
h(11)
h(12)
h(13)
h(14)
h(15)
pk (n)

π
2

and the coefficients

= h(24) = 0.01598
= h(23) = 0.02138

= h(22) = −0.03498
= h(21) = −0.01562

= h(20) = 0.06401
= h(19) = −0.007345
= h(18) = −0.1187
= h(17) = 0.09805
= h(16) = 0.4923

= h(2n + k),

k = 0, 1; n = 0, 1, . . . , 14
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corresponding polyphase filter structure (see fig 11.7-1)

p (n)
0

x(n)

y(n)
p (n)
1

F
x

F = Fx /D
y

Figure 11.7-1:

11.8
The FIR filter that meets the specifications of this problem is exactly the same as that in Problem
11.6. Its bandwidth is π5 . Its coefficients are
g(n, m)

g(0, m)
g(1, m)

g(14, m)

= h(nI + (mD)I )
mD
= h(nI + mD − [
]I)
I
5m
])
= h(2n + 5m − 2[
2
= {h(0), h(1)}
= {h(2), h(3)}
..
.
= {h(28), h(29)}

A polyphase filter would employ two subfilters, each of length 15
p0 (n)
p1 (n)

= {h(0), h(2), . . . , h(28)}
= {h(1), h(3), . . . , h(29)}

11.9
(a)
x(n)
D = I = 2. Decimation first

= {x0 , x1 , x2 , . . .}

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z2 (n)
y2 (n)
Interpolation first
z1 (n)
y1 (n)
so y2 (n)

= {x0 , x2 , x4 , . . .}
= {x0 , 0, x2 , 0, x4 , 0, . . .}
= {x0 , 0, x1 , 0, x2 , 0, . . .}
= {x0 , x1 , x2 , . . .}

6= y1 (n)

(b) suppose D = dk and I = ik and d, i are relatively prime.
x(n)
Decimation first
z2 (n)
y2 (n)

= {x0 , x1 , x2 , . . .}
= {x0 , xdk , x2dk , . . .}






x0 , 0, . . . , 0, xdk , 0, . . . , 0, x2dk , . . .
=
| {z }


 | {z }

ik−1

Interpolation first
z1 (n)

y1 (n)

ik−1







x0 , 0, . . . , 0, x1 , 0, . . . , 0, x2 , 0, . . . , 0, . . .
=
| {z }
| {z }



 | {z }
ik−1
ik−1
ik−1






x0 , 0, . . . , 0, xd , 0, . . . , 0, . . .
=
| {z }


 | {z }

d−1

d−1

Thus y2 (n) = y1 (n) iff d = dk or k = 1 which means that D and I are relatively prime.

11.10
(a) Refer to fig 11.10-1
y1 (n)

=
=
=

h(n) ∗ w1 (n)
h(n) ∗ x(nD)
∞
X
h(k)x[(n − k)D]

k=0

H(z D )

=

. . . h(0)z 0 + h(1)z D + h(2)z 2D + . . .

H(z D ) ↔

h̃(n)




h0 , 0, . . . , 0, H1 , 0, . . . , 0, h(2), . . .
| {z }
 | {z }


=

D−1

so w2 (n)

=
=
=

nD−1
X
k=0
n
X

k=0
n
X

k=0

D−1

h̃(k)x(n − k)

h̃(kD)x(n − kD)

h(k)x(n − kD)
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w1(n)

x(n)

H(z)

D

x(n)

H(zD )

w2(n)

D

y (n)
1

y (n)
2

Figure 11.10-1:
y2 (n)

=
=
=

w2 (nD)
n
X
h(k)x(nD − kD)

k=0
n
X

k=0

So y1 (n)

=

h(k)x[(n − k)D]

y2 (n)

(b)
w1 (n)

=

∞
X

k=0

h(k)x(n − k)

y1 (n)

= w1 (p),
n = pI(p an integer )
= 0,
other n

w2 (n)

= x(p),
n = pI
= 0,
other n

Let h̃(n) be the IR corresponding to H(z I )
y2 (n)

=
=
=

∞
X

k=0
∞
X

k=0
∞
X

k=0

for n
y2 (n)

h̃(k)w2 (n − k)
h̃(kI)w2 (n − kI)
h(k)w2 (n − kI)

= pI
∞
X
h(k)w2 ((p − k)I)
=
k=0

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Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

∞
X

=

h(k)x(p − k)

k=0

= w1 (p)( see above )
for n 6= pI
∞
X
h(k).0 = 0
y2 (n) =
k=0

so we conclude y1 (n)

= y2 (n)

11.11
(a)
H(z) =

X

h(2n)z −2n +

n

=

X

X

h(2n + 1)z −2n−1

n

2 −n

h(2n)(z )

+ z −1

n

X

h(2n + 1)(z 2 )−n

n

= H0 (z 2 ) + z −1 H1 (z 2 )
X
Therefore H0 (z) =
h(2n)z −n
n

H1 (z) =

X

h(2n + 1)z −n

n

(b)
H(z)

=

X

h(nD)z −nD +

n

+

D−1
X
k=0

Therefore Hk (z)

=

X

h(nD + 1)z −nD−1 + . . .

n

X
n

=

X

h(nD + D − 1)z −nD−D+1

z −k

X

h(nD + k)(z D )−n

n

h(nD + k)z −n

n

(c)
H(z) =
=
H0 (z) =

1
1 − az −1
∞
X
an z −n

n=0
∞
X

a2n z −n

n=0

=
H1 (z) =

1
1 − a2 z −1
∞
X
a2n+1 z −n

n=0

=

a
1 − a2 z −1

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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

11.12
The output of the upsampler is X(z 2 ). Thus, we have
Y1 (z)



X(z)H1 (z 1/2 ) + X(z)H1 (z 1/2 W 1/2 )


= 12 H1 (z 1/2 ) + H1 (z 1/2 W 1/2 ) X(z)
= H2 (z)X(z)
1
2

=

11.13
(a) Refer to Fig. 11.13-1 for I/D = 5/3.

DTFT[x(n)]

5Fx

Fx

Filter

5Fx

Fx
DTFT[y(m)]

(1/2)min(Fx,Fy)

Fy

2Fy

3Fy

Figure 11.13-1:
(b) Refer to Fig. 11.13-2 for I/D = 3/5.

11.14
(a) The desired implementation is given in Fig. 11.14-1
(b) The polyphase decomposition is given by
Hk (z) = (1 + z −1 )5
= 1 + 5z −1 + 10z −2 + 10z −3 + 5z −4 + z −5
= 1 + 10z −2 + 5z −4 + (5 + 10z −2 + z −4 )z −1
= P0 (z) + P1 (z)z −1
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DTFT[x(n)]

3Fx

Fx
Filter

(1/2)min(Fx,Fy)

3Fx

DTFT[y(m)]

5Fy

Fy

Figure 11.13-2:

11.15
(a)
H(z) =

N
−1
X

z −n Pn (z N )

n=0

where
Pn (z) =

∞
X

h(kN + n)z −k

k=−∞

Let m = N − 1 − n. Then
H(z) =

N
−1
X
n=0

=

N
−1
X

z −(N −1−m) PN −1−m (z N )
z −(N −1−m) Qm (z N )

n=0

(b)

(1 + z −1 )5

2

2

(1 + z −1 )5

(1 + z −1 )5

2

Figure 11.14-1:
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Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

x(n)

P0 (zN)

+

y(n)

x(n)

P0 (zN)

z−1

+

y(n)

z−1
P1 (zN)

+

P1 (zN)

+

PN−2(zN )

+

PN−2(zN )

+

z−1

z−1
PN−1(zN )

PN−1(zN )

Figure 11.15-1: Type 1 Polyphase Decomposition

x(n)

Q 0 (zN )
z−1
Q 1 (zN )

+

QN−2(zN )

+
z−1

QN−1(zN )

+

y(n)

Figure 11.15-2: Type 2 Polyphase Decomposition

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11.16
x(n)

Q 0 (zN )

3
z−1

Q 1 (zN )

+

3

z−1
Q2 (zN)

+

3

y(n)

Figure 11.16-1:

11.17

D1

=

25,

D2 = 4

F0

=

10 kHz ,

=

F1
= 100 Hz
D2

F1 =

F0
= 400 Hz
D1

Passband 0 ≤ F ≤ 50

Transition band 50 < F ≤ 345
Stopband 345 < F ≤ 5000
F2

Passband 0 ≤ F ≤ 50

Transition band 50 < F ≤ 55
Stopband 55 < F ≤ 200
For filter 1, δ1

=

△f

=

M̂1

=

For filter 2, δ1

=

△f

=

M̂2

=

0.1
= 0.05,
δ2 = 10−3
2
345 − 50
= 2.95x10−2
10, 000
−10logδ1 δ2 − 13
+ 1 = 71
14.6△f
0.05,
δ2 = 10−3
55 − 50
= 7.5x10−3
400
−10logδ1 δ2 − 13
+ 1 ≈ 275
14.6△f

The coefficients of the two filters can be obtained using a number of DSP software packages.
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11.18
To avoid aliasing Fsc ≤

Fx
2D .

Thus D = I = 50.
Single stage
δ1

=

△f

=

M̂1

=

Two stages
D1

=

stage 1:F1

=

0.1,
δ2 = 10−3
65 − 60
= 5x10−4
10, 000
−10logδ1 δ2 − 13
+ 1 ≈ 3700
14.6△f
25,
D2 = 2
10, 000
= 400
25

Passband 0 ≤ F ≤ 60
Transition band 60 < F ≤ 335
Stopband 335 < F ≤ 5000
δ1

=

△f

=

stage 2:F2

0.1,

δ2 =

2.75x10−2
400
=
= 200
2

I1 = 2,

I2 = 25

10−3
4
M̂1 = 84

Passband 0 ≤ F ≤ 60

Transition band 60 < F ≤ 65
Stopband 65 < F ≤ 100
δ1

=

0.1,

△f

=

0.1875

10−3
4
M̂2 = 13

δ2 =

Use DSP software to obtain filter coefficients.

11.19
b+ (n) is nonzero for 0 ≤ n ≤ 2N − 2 with N even. Let c(n) = b+ [n − (N − 1)]. So c(n) is nonzero
for −(N − 1) ≤ n ≤ N − 1. From (11.11.35)
B+ (w) + (−1)N −1 B+ (w − π) = αe−jw(N −1)
or B+ (z) + (−1)N −1 B+ (−z) = αz −(N −1)
Therefore, C(z)z −(N −1) + (−1)N −1 C(−z)(−z)−(N −1)
or C(z) + C(−z)
c(n) + c(−n)
when n 6= 0c(n)

when n is odd c(n)
when n is even but n 6= 0, c(n)
(half-band filter)
when n = 0, c(n)

= αz −(N −1)
= α
= αδ(n)
= −c(−n)

= −c(−n)
= 0
=

α
2

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11.20
one stage:

M̂ =

δ1

=

△f

=

−10logδ1 δ2 − 13
+ 1 ≈ 2536
14.6△f
two stages: F0
I1

=
=

F1

=

Passband 0 ≤ F ≤ 90
Transition band 90 < F ≤ 19, 900

M̂1 =

Therefore △f

=

and δ11

=

0.01,
δ2 = 10−3
100 − 90
= 10−3
10, 000

2 × 105 Hz
1,
I2 = 2
F0
= 2 × 104 Hz
I1

19, 900 − 90
= 0.09905
2 × 105
δ1
,
δ12 = δ2
2

−10logδ1 δ2 − 13
+ 1 ≈ 29
14.6△f
F2

=

F1
= 1 × 104 Hz
I2

Passband 0 ≤ F ≤ 90

Transition band 90 < F ≤ 9, 900

M̂2 =

Therefore △f

=

and δ21

=

9, 900 − 90
= 0.4905
2 × 104
δ1
,
δ22 = δ2
2

−10logδ1 δ2 − 13
+1≈7
14.6△f

11.21
Suppose the output of the analysis section is xa0 (m) and xa1 (m). After interpolation by 2, they
become y0 (m) and y1 (m). Thus

xak ( m
2 ), m even k = 0, 1
yk (m) =
0,
m odd
The final output is
z(m)
when m is even, say m
z(m) = z(2j)

= y0 (m) ∗ 2h(m) + y1 (m) ∗ [−2(−1)m h(m)]

= 2j,
= 2y0 (m) ∗ h(m) − 2y1 (m) ∗ h(m)
X
X
= 2
y0 (k)h(m − k) − 2
y1 (k)h(m − k)
k

=

2

X
l

k

y0 (2l)h(2j − 2l) − 2

X
l

y1 (2l)h(2j − 2l)

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=

2

X
l

=
In the same manner, it can be shown that
z(2j + 1)

xa0 (l)h[2(j − l)] − 2

2[xa0 (j) − xa1 (j)] ∗ h(2j)

=

2[xa0 (j) − xa1 (j)] ∗ p0 (j)

=

2[xa0 (j) + xa1 (j)] ∗ p1 (j)

X
l

xa1 (l)h[2(j − l)]

11.22
Refer to fig 11.22-1, where hi (n) is a lowpass filter with cutoff freq.

I
1

h (n)
1

π
Ii .

After transposition (refer

I
L

Interpolator 1

h(n)
L
Interpolator L

Figure 11.22-1: I = I1 I2 . . . IL L-stage interpolator
to fig 11.22-2). As D = I, let Di = IL+1−i , then D = D1 D2 . . . DL . Refer to fig 11.22-3
Obviously, this is equivalent to the transposed form above.

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h(n)
L

I
L

h (n)
1

I
1

Figure 11.22-2:

h (n)
L

D
L

h (n)
1

D1

Figure 11.22-3: L-stage decimator

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11.23
Suppose that output is y(n). Then Ty = kI Tx . Fy =
filter is h(n) of length M = kI (see fig 11.23-4)

1
Ty

=

I 1
k Tx

= kI Fx . Assume that the lowpass

coefficient storage
x(n)

F

x

input
buffer

g(n,0)

n=0,1, ..., K-1

g(n,1)

n=0,1, ..., K-1

g(n,I-1)

n=0,1, ..., K-1

length K

buffer

1
2
+

length K
K
K-1
n=0

output
buffer
length I

y(n)
F = ( I/k) Fx
y

Figure 11.23-4:

11.24
(a)
for any n = lI + j
I−1
X

k=0

pk (n − k) =

I−1
X

k=0

(0 ≤ j ≤ I − 1)

pk (lI + j − k)

= pj (lI)
= pj (l)
= h(j + lI)
= h(n)

Therefore, h(n)

=

I−1
X

k=0

pk (n − k)

356

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Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

(b) z-transform both sides
H(z) =

I−1
X

z −k pk (z)

k=0

(c)
I−1

2πl(n−k)
n−k
1 XX
h(n)ej I z − I
I n

l=0

I−1

1 XX
h(k + mI)ej2πlm z −m
I m
l=0
X
=
h(k + mI)z −m

=

m

X

=

pk (m)z −m

m

= pk (z)

11.25
(a) Refer to fig 11.25-1.
(b)

spectrum of x(n)

spectrum of y(n)
0.8

0.8

−−> magnitude

−−> magnitude

1

0.6
0.4
0.2
0
0

2

4
−−> w

6

0.7
0.6
0.5
0.4
0

8

2

4
−−> w

6

8

Figure 11.25-1:
Bandwidth
cut off freq
sampling freq of x(n)
sampling freq for the desired band of frequencies
Therefore, D

π
3
π
=
2
= 2π
2π
=
=π
2
2π
=2
=
2
=

(c) Refer to fig 11.25-2.
(d) Refer to fig 11.25-3.
357

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x(n)

|X(w)|

40

1200
1000
−−> magnitude

−−>x(n)

30
20
10

800
600
400
200

0
0

500

1000
−−−> n

0
0

1500

500

1000

Figure 11.25-2:

spectrum of s(n)
1000
900
800

−−> magnitude

700
600
500
400
300
200
100
0
0

200

400

600

800

1000

1200

Figure 11.25-3:

358

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1500

11.26
x(n)

Q 0 (zN )

I
z−1

Q 1 (zN )

I

+

QN−2(zN )

I

+
z−1

QN−1(zN )

+

I

y(n)

Figure 11.26-1:

11.27
H0 (z) =

N
−1
X

z −n Pn (z N )

n=0

where
Pn (z) =

∞
X

h0 (kN + n)z −k ,

k=0

Then,

0≤k ≤N −1

k
Hk (z) = H0 (ze−j2πk/N ) = H0 (zwN
)

where wN = e−j2π/N .
(a)
Hk (z)

=

N
−1
X

−kl
kN
z −l wN
Pl (z N wN
)

l=0

=

N
−1
X

−kl
z −l wN
Pl (z N ),

l=0

k = 0, 1, . . . , N − 1

Therefore, Hk (z), 0 ≤ k ≤ N − 1 can be expressed in matrix form as

P0 (z N )
−1

i z P1 (z N )
h

−(N −1)k
−2k
−k
Hk (z) = 1 wN

. . . wN
wN
..

.

z −1 P1 (z N )







359

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(b) From part (a), we have



1
H0 (z)
 1
 H1 (z) 



 =  ..

..
 .


.
HN −1 (z)
1

1
−1
wN
..
.

1
−2
wN
..
.

−(N −1)

wN


−2(N −1)

wN

P0 (z N )
 z −1 P1 (z N )

= N W −1 
..

.
z −1 P1 (z N )

where W id the DFT matrix.
(c)

x(n)



···
···
···

1
−(N −1)

wN

..
.

−(N −1)(N −1)

wN







P0 (z N )
−1
z P1 (z N )
..
.
z −1 P1 (z N )












y0 (n)

P0 (zN)

z−1
y1 (n)

P1 (zN)
N−point
IDFT

yN−2(n)

PN−2(zN )
z−1

yN−1(n)

PN−1(zN )

Figure 11.27-1:
(d)
y0 (n)

P0 (zN)
z−1

y1 (n)

P1 (zN)

+

PN−2(zN )

+

N−point
DFT
yN−2(n)

z−1
yN−1(n)

v(n)
PN−1(zN )

+

Figure 11.27-2:

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
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11.28
H0 (z) = 1 + z −1 + 3z −2 + 4z −3
(a)
Hk (z) = H0 (zw4k ),

1≤k≤3

−j2πk/4

= H0 (ze

)

Then,
H1 (z)

=

H2 (z)
H3 (z)

=
=

1 + jz −1 − 3z −2 + j4z −3

1 − jz −1 + 3z −2 − 4z −3
1 − jz −1 − 3z −2 + j4z −3

Note that the impulse response hk (n) are complex-valued, in general. Consequently, |Hk (w)| is
not symmetric with respect to w = 0.
(b) Let us use the polyphase implementation of the uniform filter bank. We have
Pl (z) =

∞
X

h0 (l + 3n)z −n ,

l = 0, 1, 2, 3

n=0

This yields P0 (z) = 1, P1 (z) = 1, P2 (z) = 3, and P3 (z) = 4. By using the results in Problem 11.27, we have the equation for the synthesis filter bank as


1 1
H0 (z)

 H1 (z) 
 =  1 j

 1 −1
 H2 (z) 
1 −j
H3 (z)

1 1
 1 j
= 
 1 −1
1 −j




P0 (z 4 )
1
1
4 
 −1
−1 −j 
  z P1 (z ) 
1 −1   z −2 P2 (z 4 ) 
z −3 P3 (z 4 )
−1 j



1
1
1
1
−1
 z −1 

−1 −j 
z
 = 4W −1 

 3z −2
1 −1   3z −2 
4z −3
−1 j
4z −3






where W denotes the DFT matrix. Thus, we have the analysis filter bank given in fig 11.28-1.
(c) The synthesis filter bank in fig. 11.28-2

11.29
H(z) = −3 + 19z −2 + 32z −3 + 19z −4 − 3z −6
(a)
H(z −1 ) = −3 + 19z 2 + 32z 3 + 19z 4 − 3z 6

z −6 H(z −1 ) = −3z −6 + 19z −4 + 32z −3 + 19z −2 − 3
= H(z)
Therefore, H(z −1 ) and H(z) hve roots that are symmetric, such that if zi is not a root, then
1/zi is also a root. This implies that H(z) has linera phase.
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4
x(n)

y0 (n)

+

P0 (z4 )
z−1

y1 (n)

P1 (z 4)
4−point
IDFT

z−1
P2 (z 4)

y2 (n)

z−1
y3 (n)

P3 (z 4 )

Figure 11.28-1:
y0 (n)

P0(z N)
z−1

y1 (n)

y2 (n)

P1(z N)
4−point
DFT

+
z−1

P2(z N)

+
z−1
v(n)

y3 (n)

P3(z N)

+

Figure 11.28-2:
(b) We may express H(z) as:


H(z) = z −3 −3z 3 + 19z 1 + 32 + 19z −1 − 3z −3

Thus, we have the coefficients:

h(2n) =



32,
0,

n=0
n 6=

Therefore, H(z) is a half-band filter.

362

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
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Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

(c)
70
60

|H(w)|

50
40
30
20
10
0
−4

−3

−2

−1

0
w

1

2

3

4

−3

−2

−1

0
w

1

2

3

4

10

angle(H(w))

5

0

−5

−10
−4

Figure 11.29-1:

11.30
H0 (z) = 1 + z −1
(a)
Pl (z)
P0 (z)
P1 (z)

=
=
=

∞
X

h0 (l + 2n)z −n

n=0
∞
X

n=0
∞
X

h0 (2n)z −n = 1
h0 (l + 2n)z −n = 1

n=0

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(b)
H1 (z) = P0 (z 2 ) − z −1 P1 (z 2 )
= 1 − z −1

x(n)
1

+

2

1

+

2

z−1

Figure 11.30-1: Anaylsis section
(c)
G0 (z) = P0 (z 2 ) + z −1 P1 (z 2 ) = 1 + z −1


G1 (z) = − P0 (z 2 ) − z −1 P1 (z) = − 1 + z −1
x(n)
2

+

1

+

1

2
z−1

z−1
2

−+

1

−+

1

2

+

^
x(n)

Figure 11.30-2: QMF in a polyphase realization
(d) For perfect reconstruction,
Q(z) =

1
[H0 (z)G0 (z) + H1 (z)G1 (z)] = Cz −k
2

where C is a constant. We have
Q(z) =


1
(1 + z −1 )2 − (1 − z −1 )2 = 2z −1
2

11.31
(a)


 
1 + z −1 + z −2
H0 (z)
H(z) =  H1 (z)  =  1 − z −1 + z −2  = P (z 3 )a(z)
1 − z −2
H2 (z)





1
where a(z) =  z −1 . Then
z −2

 
P00 (z 3 )
1 + z −1 + z −2
 1 − z −1 + z −2  =  P10 (z 3 )
P20 (z 3 )
1 − z −2




1
P01 (z 3 ) P02 (z 3 )
P11 (z 3 ) P12 (z 3 )   z −1 
z −2
P21 (z 3 ) P22 (z 3 )

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

1
Clearly, P (z 3 ) =  1
1


1
1
−1 1 
0 −1

(b) The synthesis filters are given as
G(z) = z −3 Qt (z 3 )a(z −1 )
where Q(z) = Cz −k [P (z)]

−1

. But

[P (z)]

−1

By selecting C = 4 and k = 1, we have



1
1
=  2
4
1


1
2
−2 0 
1 −2



1 1
Q(z) = z  2 −2
1 1

Therefore,



2
0 
−2



G0 (z)
1 1
2
 G1 (z)  = z −2  2 −2 0
G2 (z)
1 1 −2

1 + 2z −1 + z −2

1 − 2z −1 + z −2
=
−2 + 2z −1

(c)
x(n)

3



1



  z −1 
z −2



3
z−1

z−1
3

P(z)

Q(z)

3

+
z−1

z−1
3

3

+

v(n)

Figure 11.31-1:

365

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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366

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Chapter 12

12.1
(a)
Γxx (z)

=

H(z)

=

2
σw

=

and

so x(n)

25
(1 −
1−

z −1

+

1 −2
)(1
2z

1
+ 12 z −2

− z −1 + 12 z −2 )

z −1

25

1
= x(n − 1) − x(n − 2) + w(n)
2

(b) The whitening filter is H −1 (z) = 1 − z −1 + 12 z −2

12.2
(a) Γxx (z) =

1
1
27 (1− 3 z 1 )(1− 3 z)
2 (1− 12 z 1 )(1− 12 z)

For a stable filter, denominator (1 − 21 z 1 ) must be chose. However, either numerator factor
(1 − 13 z 1 )
(1− 31 z)
may be used.
H(z) =
1 1 or (1− 21 z)
(1 − z )
| {z2 }
[min.pk.]

(b) Must invert the min. pk. filter to obtain a stable whitening filter.
H −1 (z) =

(1 − 12 z 1 )
(1 − 31 z 1 )

12.3
(a)
1 + 0.9z −1
1 − 1.6z −1 + 0.63z −2
1 − 1.6z −1 + 0.63z −2
whitening filter, H −1 (z) =
1 + 0.9z −1
zeros: z = 0.7 and 0.9
H(z) =

pole: z

= −0.9
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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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(b)
2
= σw
H(w)H(−w)
|1 + 0.9e−jw |2
2
= σw
|1 − 1.6e−jw + 0.63e−2jw |2

Γxx (w)

12.4
A(z) =
k3

=

B3 (z) =
k3

=

B2 (z) =
A1 (z) =
=
k1

=

1+

13 −1 5 −2 1 −3
z + z + z
24
8
3

1
3
1 5 −1 13 −2
+ z + z + z −3
3 8
24
1
2
1 3 −1
+ z + z −2
2 8
A2 (z) − k2 B2 (z)
1 − k22
1
1 + z −1
4
1
4

12.5
1
1 + 2z −1 + z −2
3
1
−1
B2 (z) =
+ 2z + z −2
3
1
k2 =
3
A2 (z) − k2 B2 (z)
A1 (z) =
1 − k22
3
= 1 + z −1
2
3
k1 =
2
A2 (z) =

12.6
(a)
1
1 + z −1
2
1
+ z −1
B1 (z) =
2
A2 (z) = A1 (z) + k2 B1 (z)z −1
A1 (z) =

368

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1
1
1 + z −1 − z −2
3
3
1 1 −1
B2 (z) = − + z + z −2
3 3
H(z) = A3 (z) = A2 (z) + k3 B2 (z)z −1
= 1 + z −3
=

The zeros are at z

π

= −1, e±j 3

Refer to fig 12.6-1

1

Figure 12.6-1:

(b)
If k3 = −1, we have
H(z) = A3 (z) = A2 (z) − B2 (z)z −1
2
2
= 1 + z −1 − z −2 − z −3
3
3
√
11
5
The zeros are at z = −1, − ± j
6
6
(c) If |kp | = 1, the zeros of H(z) = Ap (z) are on the unit circle. Refer to fig 12.6-2.
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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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unit circle

Figure 12.6-2:

12.7
A1 (z) =
B1 (z) =

1 + 0.6z −1
0.6 + z −1

A2 (z) = A1 (z) + k2 B1 (z)z −1
= 1 + 0.78z −1 + 0.3z −2
B2 (z) = 0.3 + 0.78z −1 + z −2
A3 (z) = A2 (z) + 0.52B2 (z)z −1
= 1 + 0.93z −1 + 0.69z −2 + 0.5z −3
B3 (z) = 0.5 + 0.69z −1 + 0.93z −2 + z −3
H3 (z) = A3 (z) + 0.9B3 (z)z −1
1 + 1.38z −1 + 1.311z −2 + 1.337z −3 + 0.9z −4


=
1, 1.38, 1.311, 1.337, 0.9, 0, . . .

=
h(n)

↑

12.8
Let y(m) = x(2n − p − m). Then, the backward prediction of x(n − p) becomes the forward
prediction of y(n). Hence, its linear prediction error filter is just the noise whitening filter of the
corresponding anticausal AR(p) process.

12.9

x̂(n + m) = −

p
X

k=1

ap (k)x(n − k)

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e(n)

= x(n + m) − x̂(n + m)
p
X
= x(n + m) +
ap (k)x(n − k)
k=1

E[e(n)x∗ (n − l)] = 0,
l = 1, 2, . . . , p
p
X
⇒
ap (k)γxx (k − l) = −γxx (l + m),
l = 1, 2, . . . , p
k=1

The minimum error is

E{|e(n)|2 } = E[e(n)x∗ (n + m)]
p
X
= γxx (0) +
ap (k)γxx (m + k)
k=1

Refer to fig 12.9-1.

x(n+m)

+

e(n)
-

forward
z -m-1

x(n+m)

linear
predictor

Figure 12.9-1:

12.10

x̂(n − p − m) = −
e(n)

p−1
X

k=0

bp (k)x(n − k)

= x(n − p − m) − x̂(n − p − m)
= x(n − p − m) +

E[e(n)x∗ (n − l)]

⇒

p−1
X

k=0

=

0,

p−1
X

k=0

bp (k)x(n − k)

l = 0, 2, . . . , p − 1

bp (k)γxx (l − k) = −γxx (l − p − m),

l = 0, 2, . . . , p − 1

371

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The minimum error is
E{|e(n)|2 }

= E[e(n)x∗ (n − p − m)]
= γxx (0) +

p−1
X

k=0

bp (k)γxx (p + m − k)

Refer to fig 12.10-1.

Backward
linear
predictor

x(n)

z

x(n-p-m)

-p-m
x(n-p-m)

+

e(n)

Figure 12.10-1:

12.11
The Levinson-Durbin algorithm for the forward filter coefficients is
t

am (m) ≡ km
am (k)
but bm (k)
or am (k)
Therefore, b∗m (0) ≡ km
b∗m (m − k)
∗
Equivalently, bm (0) = km

bm (k)

= −

γxx (m) + γ bm−1 am−1

f
Em
= am−1 (k) + km a∗m−1 (m − k),
k = 1, 2, . . . , m − 1; m = 1, 2, . . . , p
∗
= am (m − k),
k = 0, 2, . . . , m

= b∗m (m − k)
γxx (m) + γ tm−1 b∗m−1
= −
b
Em
∗
= bm−1 (m − 1 − k) + km bm−1 (k)
=

∗
(m) + γ ∗m−1 btm−1
γxx

b
Em
∗ ∗
= bm−1 (k − 1) + km
bm−1 (m − k)

This is the Levinson-Durbin algorithm for the backward filter.
372

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Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

12.12
Let
bm =
Then,

"

Γm−1
γ bt
m−1

∗

γ bm−1
γxx (0)



#

bm =



+

bm−1
0



bm−1
0





dm−1
bm (m)

+





dm−1
bm (m)



=



cm−1
cm (m)



Hence,
∗

Γm−1 bm−1 + Γm−1 dm−1 + bm (m)γ bm−1

= cm−1

b
d
+ γ bt
+ bm (m)γxx (0)
γ bt
m−1 m−1
m−1 m−1

= cm (m)

But Γm−1 bm−1
⇒ Γm−1 dm−1
Hence, dm−1
∗

b
Also, Γ−1
m−1 γ m−1
∗

Therefore, bm (m)γ bt
+ bm (m)γxx (0)
ab
m−1 m−1
solving for bm (m), we obtain
bm (m)

= cm−1
∗

= −bm (m)γ bm−1
∗

= abm−1

= cm (m) − γ bt
b
m−1 m−1
=
=

we also obtain the recursion
bm (k)

∗

b
= −bm (m)Γ−1
m−1 γ m−1

b
cm (m) − γ bt
m−1 m−1
∗

ab
γxx (0) + γ bt
m−1 m−1
b
cm (m) − γ bt
m−1 m−1
f
Em−1

= bm−1 (k) + bm (m)a∗m−1 (m − k),
k = 1, 2, . . . , m − 1

12.13
Equations for the forward linear predictor:
Γm am = cm
where the elements of cm are γxx (l + m),

l = 1, 2, . . . , p. The solution of am is

am (m) =

cm (m) − γ bt
a
m−1 m−1

=

a
cm (m) − γ bt
m−1 m−1

am (k)
where αm is the solution to Γm αm

∗

γxx (0) + γ bt
ab
m−1 m−1

f
Em−1
∗
= am−1 (k) + am (m)αm−1
(m − k),

k = 1, 2, . . . , m − 1;
= γm

m = 1, 2, . . . , p

The coefficients for the m-step backward predictor are bm = abm .
373

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

12.14
(a)
x̂(n)
But x(n)

(b)

= −a1 x(n − 1) − a2 x(n − 2) − a3 x(n − 3)
14
9
1
=
x(n − 1) + x(n − 2) − x(n − 3) + w(n)
24
24
24

9
14
, a2 = − 24
, a3 =
E{[x(n) − x̂(n)]2 } is minimized by selecting the coefficients as a1 = − 24

γxx (m)

= −
= −

3
X

k=1
p
X

k=1

ak γxx (m − k),

1
24

m>0

2
ak γxx (m − k) + σw
,

m=0

Since we know the {ak } we can solve for γxx (m), m = 0, 1, 2, 3. Then we can obtain γxx (m)
for m > 3, by the above recursion. Thus,
γxx (0)
γxx (1)

=
=

4.93
4.32

γxx (2)
γxx (3)

=
=

4.2
3.85

γxx (4)
γxx (5)

=
=

3.65
3.46

(c)
A3 (z)

=

k3

=

B3 (z)

=

A2 (z)

=

k2
B2 (z)
A1 (z)

9
1
14 −1
z − z −2 + z −3
24
24
24

1
24
9
14
1
− z −1 − z −2 + z −3
24 24
24
A3 (z) − k3 B3 (z)
1 − k32

= 1 − 0.569z −1 − 0.351z −2
= −0.351

= −0.351 − 0.569z −1 + z −2
A2 (z) − k2 B2 (z)
=
1 − k22

=
k1

1−

1 − 0.877z −1

= −0.877

12.15
(a)
Γxx (z) =

2
4σw
(2 − z −1 )(2 − z)
9 (3 − z −1 )(3 − z)

2
= σw
H(z)H(z −1 )

374

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
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Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

The minimum-phase system function H(z) is
H(z) =
=

2 2 − z −1
3 3 − z −1
4 1 − 21 z −1
9 1 − 31 z −1

(b) The mixed-phase stable system has a system function
H(z) =
=

2 1 − 2z −1
3 3 − z −1
2 1 − 2z −1
9 1 − 31 z −1

12.16
(a)
1 − 2rcosΘz −1 + r2 z −2

A2 (z)

=

⇒ k2
B2 (z)

= r2
= r2 − 2rcosΘz −1 + z −2
A2 (z) − k2 B2 (z)
=
1 − k22
2rcosΘ −1
z
= 1−
1 + r2
2rcosΘ
= −
1 + r2

A1 (z)

Hence, k1
(b) As r → 1, k2 → 1 and k1 → −cosΘ

12.17
(a)
a1 (1)

= −1.25, a2 (2) = 1.25, a3 (3) = −1

Hence, A3 (z) =

1 − 1.25z −1 + 1.25z −2 − z −3

First, we determine the reflection coefficients. Clearly, k3 = −1, whcih implies that the roots
of A3 (z) are on the unit circle. We may factor out one root. Thus,
A3 (z)

=
=

where α

=

1
(1 − z −1 )(1 − z −1 + z −2 )
4
(1 − z −1 )(1 − αz −1 )(1 − α∗ z −1 )
√
1 + j 63
8

Hence, the roots of A3 (z) are z = 1, α, and α∗ .
(b) The autocorrelation function satisfies the equations
γxx (m) +

3
X

k=1

a3 (k)γxx (m − k) =



2
σw
,
0,

m=0
1≤m≤3

375

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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

γxx (0)
 γxx (1)

 γxx (2)
γxx (3)

γxx (1)
γxx (0)
γxx (1)
γxx (2)


1
γxx (3)
 −1.25
γxx (2) 

γxx (1)   1.25
−1
γxx (0)

γxx (2)
γxx (1)
γxx (0)
γxx (1)


2
σw
  0 

=
  0 
0




f
f
= Em−1
(1 − |km |2 ) implies that E3f = 0. This
(c) Note that since k3 = −1, the recursion Em
2
=0
implies that the 4x4 correlation matrix Γxx is singular. Since E3f = 0, then σw

12.18

γxx (0)
γxx (1)
γxx (2)
γxx (3)
Use the Levinson-Durbin algorithm
a1 (1)

= 1
= −0.5

= 0.625
= −0.6875

A1 (z) =
⇒ k1 =

γxx (1)
1
=
γxx (0)
2
1
1 + z −1
2

= −

1
2

3
4
γxx (2) + a1 (1)γxx (1)
1
a2 (2) = −
=−
E1
2
1
a2 (1) = a1 (1) + a2 (2)a1 (1) =
4
1 −1 1 −2
Therefore,A2 (z) = 1 + z − z
4
2
1
⇒ k2 = −
2
9
E2 = (1 − a22 (2))E1 =
16
1
γxx (3) + a2 (1)γxx (2) + a2 (2)γxx (1)
=
a3 (3) = −
E2
2
3
a3 (2) = a2 (2) + a3 (3)a2 (1) = −
8
a3 (1) = a2 (1) + a3 (3)a2 (2) = 0
3
1
Therefore,A3 (z) = 1 − z −2 + z −3
8
2
1
⇒ k3 =
2
27
E3 = (1 − a23 (3))E2 =
64
E1

=

(1 − a21 (1))γxx (0) =

376

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

12.19
(a)
∞
X

Γxx (z) =

γxx (m)z −m

−∞

∞
−1
X
X
1
1
( )m z −m
( )−m z −m +
4
4
−∞
0

=
=

1

=

1
4z
− 41 z

+

1
1 − 14 z −1

15
16

(1 − 41 z)(1 − 14 z −1 )

since Γxx (z) = σ 2 H(z)H(z −1 ),
0.968
H(z) =
1 − 41 z −1

is the minimum-phase solution. The difference equation is
x(n) =

1
x(n − 1) + 0.968w(n)
4

where w(n) is a white noise sequence with zero mean and unit variance.
(b) If we choose
H(z)

=
=

1
1 − 41 z
z −1
z −1 −

4z
1 − 4z −1
4x(n − 1) − 4 × 0.968w(n − 1)

= −
then, x(n)

=

1
4
−1

12.20
γxx (0)
γxx (1)

=
=

1
0

= −a2
= 0
γxx (1)
a1 (1) = −
=0
γxx (0)
A1 (z) = 1
⇒ k1 = 0
γxx (2)
γxx (3)

E1

a2 (2)
a2 (1)

(1 − a21 (1))γxx (0) = 1
γxx (2) + a1 (1)γxx (1)
= −
= a2
E1
= a1 (1) + a2 (2)a1 (1) = 0

=

377

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Therefore,A2 (z)
⇒ k2 = a2
E2

a3 (3)
a3 (2)
a3 (1)
Therefore,A3 (z) = A2 (z)
⇒ k3 = 0
E 3 = E2

1 + a2 z −2

=

(1 − a22 (2))E1 = 1 − a4
γxx (3) + a2 (1)γxx (2) + a2 (2)γxx (1)
= −
=0
E2
= a2 (2) + a3 (3)a2 (1) = a2
= a2 (1) + a3 (3)a2 (2) = 0

=

=

1 + a2 z −2

=

1 − a4

12.21
Ap (z) = Ap−1 (z) + kp Bp−1 (z)z −1
where Bp−1 (z) is the reverse polynomial of Ap−1 (z).
For |kp | < 1, we have all the roots inside the unit circle as previously shown.
For |kp | = 1, Ap (z) is symmetric, which implies that all the roots are on the unit circle.
For |kp | > 1, Ap (z) = As (z) + ǫBp−1 (z)z −1 , where As (z) is the symmetric polynomial with all
the roots on the unit circle and Bp−1 (z) has all the roots outside the unit circle. Therefore, Ap (z)
will have all its roots outside the unit circle.

12.22


1
∗
km

km
1



Vm =

V m JV

=



1
∗
km

t∗
m

−km
−1

=





1
∗
km

1
∗
km

= (1 − |km |2 )



km
1


=

1 0
0 −1

km
1

1
0





0
−1



1 − |km |2
0

1
∗
km

km
1



0
−(1 − |km |2 )



= (1 − |km |2 )J

12.23
(a)
E[fm (n)x(n − i)]

= E[

m
X

k=0

=

am (k)x(n − k)x(n − i)]

0, by the orthogonality property
378

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

(b)
E[gm (n)x(n − i)]

=
=

m
X

k=0
m
X

k=0

=

0,

a∗m (k)E[x(n − m + k)x(n − i)]
a∗m (k)γxx (k − m + i)
i = 0, 1, . . . , m − 1

(c)
E[fm (n)x(n)]

= E{fm (n)[fm (n) −
= E{|fm (n)|2 }

m
X

k=1

am (k)x(n − k)]}

= Em

E[gm (n)x(n − m)]

= E{gm (n)[gm (n) −
= E{|gm (n)|2 }
= Em

m−1
X
k=0

bm (k)x(n − k)]}

(d)
E[fi (n)fj (n)]

= E{fi (n)[x(n) +

j
X

k=1

= E{fi (n)x(n)}

aj (k)x(n − k)]}

= Ei
= Emax (i, j)
where i > j has been assumed
(e)
E[fi (n)fj (n − t)] = E{fi (n)[x(n − t) +

j
X

k=1

aj (k)x(n − t − k)]}

when 0 ≤ t ≤ i − j, x(n − t − 1), x(n − t − 2), . . . , x(n − t − j) are just a subset of x(n − 1), x(n −
2), . . . , x(n − i) Hence, from the orthogonality principle,
E[fi (n)fj (n − t)] = 0
Also, when −1 ≥ t ≥ i − j holds, via the same method we have
E[fi (n)fj (n − t)] = 0
(f)
E[gi (n)gj (n − t)] = E{gi (n)[x(n − t − j) +

j−1
X

k=0

bj (k)x(n − t − k)]}

when 0 ≤ t ≤ i − j, {x(n − t), x(n − t − 1), . . . , x(n − t − j)} is a subset of {x(n), . . . , x(n − i + 1)}
Hence, from the orthogonality principle,
E[gi (n)gj (n − t)] = 0
379

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
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Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Also, when 0 ≥ t ≥ i − j + 1 we obtain the same result (g)
for i = j, E{fi (n + i)fj (n + j)}

= E{fi2 (n + i)}
= Ei

for i 6= j, suppose that i > j. Then
E{fi (n + i)fj (n + j)}

= E{fi (n + i)[x(n + j) +

j
X

k=1

=

0

aj (k)x(n + j − k)]}

(h)
suppose i > j
E{gi (n + i)gj (n + j)}

= E{gi (n + i)[x(n) +

j−1
X

k=0

= E[gi (n + i)x(n)]
= Ei

bj (k)x(n + j − k)]}

(i)
for i ≥ j
E{fi (n)gj (n)}

= E{fi (n)[x(n − j) +
= E{fi (n)[bj (0)x(n)]}
= kj E[fi (n)x(n)]

j−1
X

k=0

bj (k)x(n − k)]}

= kj Ei
for i < j,
E{fi (n)gj (n)}

= E{gj (n)[x(n) +

i
X

k=1

=

0

ai (k)x(n − k)]}

(j)
E{fi (n)gi (n − 1)}

= E{fi (n)[x(n − 1 − j) +
= E[fi (n)x(n − 1 − i)]
= E{fi (n)[gi+1 (n) −

i
X

k=0

i−1
X

k=0

bi (k)x(n − 1 − k)]}

bi+1 (k)x(n − k)]}

= −E[fi (n)bi+1 (0)x(n)]
= −ki+1 Ei
(k)
E{gi (n − 1)x(n)}

= E{gi (n − 1)[fi+1 (n) −

i+1
X

k=1

ai+1 (k)x(n − k)]}

= −E[gi (n − 1)ai+1 (i + 1)x(n − 1 − i)]
= −ki+1 Ei

E{fi (n + 1)x(n − i)}

= E{fi (n + 1)[fi (n − i) −

i
X

k=1

ai (k)x(n − i − k)]}

380

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

(l)
suppose i > j
E{fi (n)gj (n − 1)}

= E{fi (n)[x(n − 1 − j) +
=

0

Now, let i ≤ j. then
E{fi (n)gj (n − 1)}

= E{gj (n − 1)[x(n) +
= E{gj (n − 1)x(n)}

= −kj+1 Ej

i
X

k=1

j−1
X

k=0

bj (k)x(n − 1 − k)]}

ai (k)x(n − k)]}

from (d)

12.24
(a) E[fm (n)x∗ (n − i)] = 0,
1≤i≤m
(b) E[gm (n)x∗ (n − i)] = 0,
0≤i≤m−1
(c) E[fm (n)x∗ (n)] = E[gm (n)x∗ (n − m)] = Em
(d) E[fi (n)fj∗ (n)] = Emax (i, j)
(e)

1 ≤ t ≤ i − j,
∗
E[fi (n)fj (n − t)] = 0, for
−1 ≥ t ≥ i − j,
(f)
E[gi (n)gj∗ (n − t)] = 0, for



0 ≤ t ≤ i − j,
i>j
0 ≥ t ≥ i − j + 1, i < j

(g)
E[fi (n +

i)fj∗ (n

+ j)] =

(h) E[gi (n + i)gj∗ (n + j)] = Emax (i, j)
(i)



Ei ,
0,

i=j
i 6= j

=



kj∗ Ei ,
0,

E[fi (n)gj∗ (n − 1)] =



0,
ij
i 0.

In this version we need 5 extra multiplications for the calculation of

fm−1 (n)
|fm−1 (n)|2
,
ãm−1 (n−1) , wE f
(n−1)
m−1

gm−1 (n)
em (n)
gm−1 (n)c̃mm (n), ãm−1
(n−1) , ãm (n) and we save m multiplications from the estimation of K̃m−1 (n).
FAST RLS algorithm: Version B (a-posteriori version)

fm−1 (n)
gm−1 (n)
am−1 (n)
f˜m−1 (n, n)
f
Em−1
(n)

K̃m (n)
K̃m−1 (n)
ãm (n)

= x(n) + atm−1 (n − 1)Xm−1 (n − 1)

= x(n − M + 1) + btm−1 (n − 1)Xm−1 (n)
fm−1 (n)
= am−1 (n − 1) − K̃m−1 (n − 1)
ãm−1 (n − 1)
fm−1 (n)
=
ãm−1 (n − 1)
|fm−1 (n)|2
f
= wEm−1
(n − 1) +
ãm−1 (n − 1)




 
∗
fm−1
(n)
0
1
C̃m−1 (n)
=
+
=
f
K̃m−1 (n − 1)
c̃mm (n)
wEm−1
(n − 1) am−1 (n − 1)
= C̃m−1 (n) − bm−1 (n − 1)c̃mm (n)
|fm−1 (n)|2
= ãm−1 (n − 1) +
f
wEm−1
(n − 1)

ãm−1 (n)

= ãm (n) − gm−1 (n)c̃mm (n)

bm−1 (n)

= bm−1 (n − 1) − K̃m−1 (n − 1)

ˆ
d(n)
em (n)
hm (n)

gm−1 (n)
ãm−1 (n)

= htm (n − 1)Xm (n)
ˆ
= d(n) − d(n)
= hm (n − 1) +

K̃m (n)em (n)
ãm (n)

Initialization:
f
am−1 (−1) = bm−1 (−1) = 0, K̃m−1 (−1) = 0, hm−1 (−1) = 0, Em−1
(−1) = E > 0, ãm−1 (−1) = 1.

In this version we need 3 extra multiplications for the calculation of

fm−1 (n)
gm−1 (n)
em (n)
ãm−1 (n−1) , ãm−1 (n−1) , ãm (n)

and we save m multiplications from the estimation of K̃m−1 (n).

13.12


E =E g−

M
−1
X
n=0

!2 

h(n)x(n)



400

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Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

∂E
=0
∂h(k)

"

=⇒

E 2 g−

Thus,
E[gx(k)] = E

"M −1
X

M
−1
X

!

h(n)x(n) x(k) = 0,

n=0

#

h(n)x(n)x(k) ,

n=0

E[gx(k)]

E

"M −1
X

#

h(n)x(n)x(k)

n=0

#

k = 0, · · · , M − 1.

k = 0, · · · , M − 1.

= E[g(gv(k) + w(k))] = E[g 2 ]v(k) + E[gw(k)]
= Gv(k)
(ifg, w(k)areuncorrelated)
=

M
−1
X

h(n)E[x(n)x(k)]

n=0

=

M
−1
X

h(n)E[(gv(n) + w(n))(gv(k) + w(k))]

n=0

=

M
−1
X

h(n)E[g 2 v(n)v(k) + gv(n)w(k) + gv(k)w(n) + w(n)w(k)]

n=0

= G

M
−1
X

2
h(n)v(k)v(n) + σw
h(k)

n=0

Hence,
Gv(k) = Gv(k)

M
−1
X

2
h(n)v(n) + σw
h(k)

n=0

or

2
(GvvT + σw
I)h = Gv

where
v = [v(0), · · · , v(M − 1)]T ,

h = [h(0), · · · , h(M − 1)]T .

13.13
Let
H(z) =

M
−1
X

hk z −k

and

Hn = H(z = ej2πn/M ) =

M
−1
X

hk e−j2πnk/M .

k=0

k=0

The sequence {hk } is related to the sequence {Hn } by the inverse discrete Fourier transform
hk =

M −1
1 X
Hn ej2πn/M ,
M n=0

k = 0, · · · , M − 1.

When hk , given above is substituted in the expression for H(z) the double sum that results can
be simplified to yield
M −1
Hk
1 − z −M X
.
H(z) =
j2πk/M
M
1−e
z −1
k=0

The filter structure is shown in Fig. 13.13-1.

1. Let yk (n) be the output at time t = nT of the filter with transfer function
1 − z −M
1
.
M
1 − ej2πk/M z −1
401

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
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Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

y

1

0

−1

X

1−z

H0
y

1

d(n)
1

1 − z e j2 π /M
−1

−M

1−z

X
+

^

M

H1

d(n)

+

+
−

y

1
1 − z e j2 π
−1

e(n)
M−1

(M−1)/M

X
HM−1

Figure 13.13-1:
Then the response of the recursive filter at t = nT is
ˆ =
d(n)

M
−1
X

Hk (n)yk (n).

k=0

ˆ
where {Hk (n)} are the filter coefficients at t = nT . If e(n) = d − d(n)
then, an algorithm
for adjusting the coefficients Hk (n) is given by
Hk (n + 1) = Hk (n) + △e(n)yk (n)k = 0, · · · , M − 1.
2. The cascade of the comb filter
with frequency response

1−z −M
M

Hk (f ) =
Thus,
|Hk (f )|

=
=

with each of the single-pole filter forms a system
1 − ej2πf /M
.
M (1 − ej2π(k/M −f ) )

ej2πM f − e−j2πM f
1
e−j2πM f
·
M ej2π(k/M −f )
e−j2π(k/M −f ) − ej2π(k/M −f )
1
1
2j sin(πM f )
sin(πM f )
=
.
M −2j sin(π(k/M − f ))
M sin(π(k/M − f ))

We observe that |Hk (f )| = 0 at the frequencies f = n/M , n 6= k and |Hk (f )| = 1 at
f = k/M .
Thus, the kth system has a resonant frequency at f = k/M , and it is zero at the resonant
frequencies of all the other systems. This means that if the desired signal is
d(n) =

M
−1
X

Ak cos(ωk n), ωk =

k=0

2πk
,
M

the coefficient of each single-pole filter can be adjusted independently without any interaction from the other filters.
402

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

13.14
∂J
= 2h(n) − 40
∂h(n)
Thus,
h(n + 1) = h(n) − △h(n) + 20△ = h(n)(1 − △) + 20△.
1. For an overdamped system,
|1 − △| < 1 =⇒ 0 < △ < 2.
2. Fig. 13.14-1 contains a plot of J(n) vs. n. The step △ was set to 0.5 and the initial value
of h was set to 0. In Fig. 13.14-2 we have plotted J(h(n)) vs. h(n). As it is observed from
the figures the minimum value of J which is −372, is reached within 5 iterations of the
algorithm.

50

0

−50

−100

J(n)

−150

−200

−250

−300

−350

−400

0

5

10

15

20

25
n

30

35

40

45

50

Figure 13.14-1:

13.15
Normal Equations:
M
−1
X
k=0

a(k)rvv (l − k) = ryv (l)

l = 0, 1, · · · , M − 1

403

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

50

0

−50

J(h(n))

−100

−150

−200

−250

−300

−350

−400

0

2

4

6

8

10
h(n)

12

14

16

18

20

Figure 13.14-2:
rvv (l − k) = rw3 w3 (l − k) + rv2 v2 (l − k)
Power spectral density of v2 (n):
2
2
|H(f )|2 = σw
Γv2 v2 (f ) = σw

1
0.75
σ2
= w
.
−j2πf
2
|1 − 0.5e
|
0.75 1.25 − cos(2πf )

Thus,
rv2 v2 (m) =

2
σw
(0.5)|m| .
0.75

Hence,
2
rvv (l − k) = σw
δ(l − k) +

2
σw
(0.5)|l−k| .
0.75

Assuming that x(n), w1 (n), w2 (n), w3 (n) are mutually uncorrelated, it follows that
"

E[y(n)v(n − l)] = E[w2 v2 (n − l)] = E w2

∞
X

k=0

#

h(k)w2 (n − l − k) ,

where h(k) = 0.5k . Thus,
E[y(n)v(n − l)] =

∞
X

k=0

h(k)E [w2 (n)w2 (n − l − k)] =

∞
X

2
2
h(k)σw
δ(l + k) = σw
δ(l).

k=0

404

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

w3 (n)
w2(n)

v2(n)

1
1−0.5 z−1

v(n)

+

−

A(z)

+

e(n)

y(n)=x(n) + w1 (n) + w2(n)

Figure 13.15-1:
The normal equations take the form






2
σw
+

2
σw
0.75

2
0.5σw

0.75
2
0.25σw
0.75

=⇒

2
0.5σw
0.75

2
σw
+

2
σw

2
0.5σw

0.75

2
0.5σw
0.75

a(0) =



2
0.25σw
0.75

0.75
2
σw

15
,
32

+

2
σw
0.75

a(1) = −


  2 
σw
 a(0)

 a(1)  =  0 
 a(2)
0
4
,
32

a(2) = −

1
.
32

13.16

e(n) = x(n) − a1 x(n − 1) − a2 x(n − 2)
E = E[e2 (n)]
=⇒
∂E
= E[(x(n) − a1 x(n − 1) − a2 x(n − 2))x(n − 1)] = 0
∂a1
∂E
= E[(x(n) − a1 x(n − 1) − a2 x(n − 2))x(n − 2)] = 0
∂a2
=⇒
E[x(n)x(n − 1)] − a1 E[x(n − 1)x(n − 1)] − a2 E[x(n − 2)x(n − 1)] = 0
E[x(n)x(n − 2)] − a1 E[x(n − 1)x(n − 2)] − a2 E[x(n − 2)x(n − 2)] = 0

But,
E[x(n)x(n − 1)] = E[x(n − 2)x(n − 1)] = a
E[x(n − 1)x(n − 1)] = E[x(n − 2)x(n − 2)] = 1

E[x(n)x(n − 2)] = a2
Thus, we obtain the system


1
a

a
1



a1
a2



=



a
a2



with solution a1 = a, a2 = 0.
405

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

13.17
The optimum linear predictor in Prob. 13.16 is a first order filter with transfer function
A(z) = 1 − az −1 .
Thus, the corresponding lattice has one stage with the forward and backward errors given by
f (n) = f0 (n) + Kb0 (n − 1)

b(n) = b0 (n − 1) + Kf0 (n)

Since f0 (n) = b0 (n) = x(n), we obtain
f (n) = x(n) + Kx(n − 1)

b(n) = x(n − 1) + Kx(n).

Comparing with the prediction error:
e(n) = x(n) − ax(n − 1)
we identify K as −a.

f(n)=e(n)

+
−a

x(n)

−a
z

−1

g(n)

+
Figure 13.17-1:

13.18
1
X

k=0

bk ryy (l − k) = rdy (l) = rxy (l),

l = 0, 1

where y(n) is the input of the adaptive FIR filter B(z)
2
ryy (l − k) = rss (l − k) + rww (l − k) = rss (l − k) + σw
δ(l − k)

where s(n) is the output of the system C(z).
If x(n) is white with variance σx2 then,
rss (l − k) =

σx2
σx2
(−0.9)|l−k| =
(−0.9)|l−k|
2
1 − 0.9
1 − 0.19

rxy (l) = E[x(n)y ∗ (n − l)] = E[x(n)(s∗ (n − l) + w∗ (n − l))].
If x(n) and w(n) are uncorrelated then,
rxy (l) = E[x(n)s∗ (n − l)] = σx2 δ(l).
406

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Thus, we obtain the system:



2
σx
0.19

2
+ σw

σ2

σ2

x
− 0.19
(0.9)

x
− 0.19
(0.9)

2
σx
0.19

2
+ σw

2
With σx2 and σw
known, we can determine b0 , b1 .






b0
b1



=



σx2
0



.

13.19
(a)
fm (n)
gm (n)

= fm−1 (n) − km gm−1 (n − 1)
∗
= gm−1 (n − 1) − km
fm−1 (n)

εLS
m =

n
X
l=0

dεLS
m
∗
dkm
n
X
l=0

= −2

n
X
l=0

h
i
2
2
wn−l |fm (l)| + |gm (l)|

 ∗

∗
wn−l gm−1
(l − 1)fm (l) + fm−1 (l)gm
(l) = 0

 ∗
 ∗

wn−l gm−1
(l − 1) [fm−1 (l) − km gm−1 (l − 1)] + fm−1 (l) gm−1
(l − 1) − km fm−1 (l)

Solving for kM , we obtain
Pn

∗
wn−l fm−1 (l)gm−1
(l − 1)
u (n)
i= m
h
2
2
vm (n)
n−l
|fm−1 (l)| + |gm−1 (l − 1)|
l=0 w

km (n) = P
n

(b)

km (n) =

2

l=0

∗
wum (n − 1) + 2fm−1 (n)gm−1
(n − 1)
2

wvm (n − 1) + |fm−1 (n)| + |gm−1 (n − 1)|

fm−1 (n)gm−1 (n − 1)

fm−1 (n)gm−1 (n − 1)
Therefore,
where

2

 ∗

∗
= fm−1 (n) gm
(n) + km (n)fm−1
(n)

∗
= fm−1 (n)gm
(n) + km (n) |fm−1 (n)|

2

= gm−1 (n) [fm (n) + km (n)gm−1 (n − 1)]

= gm−1 (n − 1)fm (n) + km (n) |gm−1 (n − 1)|

2

i
h
2
2
∗
2fm−1 (n)gm−1
(n − 1) = km (n) |fm−1 (n)| + |gm−1 (n)| + z(n)
∗
z(n) = fm−1 (n)gm (n) + fm (n)gm−1
(n − 1)

Now,
∗
2fm−1 (n)gm−1
(n − 1)

i
h
2
2
= z(n) + km (n) wvm (n) + |fm−1 (n)| + |gm−1 (n)|
−km (n)wvm (n − 1)

= z(n) + km (n)vm (n) − km (n)wvm (n − 1)
407

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Then,
2

n−1
X
l=0

∗
∗
wn−l fm−1 (l)gm−1
(l − 1) + 2fm−1 (n)gm−1
(n − 1)

= wum (n − 1)
+z(n) + km (n)vm (n)
−km (n)wvm (n − 1)

But km (n) = um (n)/vm (n). Therefore,
km (n)wm (n) = z(n) + wum (n − 1) + km (n)vm (n) − km wvm (n − 1)
and, then
km (n)
km (n)

wum (n − 1)
z(n)
+
wvm (n − 1)
wvm (n − 1)
z(n)
= km (n − 1) +
wvm (n − 1)
=

408

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Chapter 14

14.1
(a)
"

1
limT0 →∞ E
|
2T0
=
=
=
=
=

Z

T0

−j2πF t

x(t)e
−T0

2

dt|

#

"

#
Z T0
Z T0
1
∗
j2πF τ
−j2πF t
limT0 →∞ E
x (τ )e
dτ
x(t)e
dt
2T0 −T0
−T0
Z T0 Z T0
1
E[x(t)x∗ (τ )]e−j2πF (t−τ ) dtdτ
limT0 →∞
2T0 −T0 −T0
Z T0 Z T0
1
limT0 →∞
γxx (t − τ )e−j2πF (t−τ ) dtdτ
2T0 −T0 −T0
Z t+T0 Z T0
1
limT0 →∞
γxx (α)e−j2πF (α) dtdα
2T0 t−T0 −T0
Z ∞
γxx (α)e−j2πF (α) dα
−∞

= γxx (F )
(b)

N
X

γxx (m)

=

γxx (m)e−j2πf m

=

N −1
1 X
x(n + m)x∗ (n)
N n=0
N
X

m=−N

m=−N

=

N
−1
X
n=0

=

=

1
N

1
N

N −1
1 X
x(n + m)x∗ (n)e−j2πf m
N n=0
n+N
X

x(l)x∗ (n)e−j2πf (l−n)

l=n−N

−1
N
−1 N
X
X

x(l)x∗ (n)e−j2πf l ej2πf n

n=0 l=0

N −1
1 X
x(n)e−j2πf n |2
|
N n=0

409

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

14.2

2

E[|γxx (m)| ]

=
=

1
N2

N −|m|−1 N −|m|−1

X

n=0

X

E[x∗ (n)x(n + m)x(n′ )x∗ (n′ + m)]

n′ =0

1 XX
{E[x∗ (n)x(n + m)]E[x(n′ )x∗ (n′ + m)]
N2 n ′
n

+E[x∗ (n)x(n′ )]E[x∗ (n′ + m)x(n + m)]
=

+E[x∗ (n)x∗ (n′ + m)]E[x(n′ )x(n + m)]}
1 XX 2
2
[γ (m) + γxx
(n − n′ )
N 2 n ′ xx
n

Let p
E[|γxx (m)|2 ]

Therefore,
var[γxx (m)]

∗
+γxx
(n′ + m − n)γxx (n + m − n′ )]
= n − n′ . Then
2

1 XX 2
N − |m|
∗
2
+ 2
[γ (p)γxx
(p − m)γxx (p + m)]
= γxx (m)
N
N n p xx
1 XX 2
∗
= |E[γxx (m)]|2 + 2
[γ (p)γxx
(p − m)γxx (p + m)]
N n p xx

=
≈

1 XX 2
∗
[γ (p)γxx
(p − m)γxx (p + m)]
N 2 n p xx
∞
1 X 2
∗
[γ (p)γxx
(p − m)γxx (p + m)]
N p=−∞ xx

14.3
(a)
∗
E[γxx (m)γxx
(m′ )]



 1 N −|m|−1
X
= E 
x∗ (n)x(n + m) .
 N
n=0


N −|m|−1

X
1
x(n′ )x∗ (n′ + m′ )

N
′
n =0

=
=

1 XX
E{x∗ (n)x(n + m)x(n′ )x∗ (n′ + m′ )}
N2 n ′
n
1 XX
{E[x∗ (n)x(n + m)]E[x(n′ )x∗ (n′ + m′ )]
N2 n ′
n

+E[x∗ (n)x(n′ )]E[x∗ (n′ + m′ )x(n + m)]
+E[x∗ (n)x∗ (n′ + m′ )]E[x(n′ )x(n + m)]}
σx4 X X
[δ(m)δ(m′ ) + δ(n − n′ )δ(m − m′ )
=
N2 n ′
n

′

+δ(n + m′ − n)δ(n + m − n′ )]

Hence, E[pxx (f1 )pxx (f2 )]

=

N
−1
X

N
−1
X

′

E[γxx (m)γxx (m′ )]e−j2πmf1 e−j2πm f2

m=−(N −1) m′ =−(N −1)

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=

σx4 X X X X
[δ(m)δ(m′ ) + δ(n − n′ )δ(m − m′ )
N2 m ′ n ′
n

m

′

+δ(n′ + m′ − n)δ(n + m − n′ )]e−j2πmf1 e−j2πm f2
(

2 
2 )
sinπ(f1 + f2 )N
sinπ(f1 − f2 )N
4
= σx 1 +
+
N sinπ(f1 + f2 )
N sinπ(f1 − f2 )
(b)
cov[pxx (f1 )pxx (f2 )]

= E[pxx (f1 )pxx (f2 )] − E[pxx (f1 )]E[pxx (f2 )]

= E[pxx (f1 )pxx (f2 )] − σx4
(
2 
2 )
sinπ(f1 − f2 )N
sinπ(f1 + f2 )N
4
+
= σx
N sinπ(f1 + f2 )
N sinπ(f1 − f2 )
(c)
var[pxx (f )]

=

cov[pxx (f1 )pxx (f2 )]|f1 =f2 =f
"

2 #
sin2πf N
4
= σx 1 +
N sin2πf

14.4
Assume that x(n) is the output of a linear system excited by white noise input w(n), where
σx2 = 1. Then pxx (f ) = Γxx (f )pww (f ). From prob. 12.3, (a), (b) and (c), we have
E[pxx (f1 )pxx (f2 )]

=
=

cov[pxx (f1 )pxx (f2 )]

=
=

var[pxx (f )]

=
=

Γxx (f1 )Γxx (f2 )E[pww (f1 )pww (f2 )]
(
2 
2 )

sinπ(f1 + f2 )N
sinπ(f1 − f2 )N
Γxx (f1 )Γxx (f2 ) 1 +
+
N sinπ(f1 + f2 )
N sinπ(f1 − f2 )
Γxx (f1 )Γxx (f2 )cov[pww (f1 )pww (f2 )]
(
2 
2 )
sinπ(f1 − f2 )N
sinπ(f1 + f2 )N
+
Γxx (f1 )Γxx (f2 )
N sinπ(f1 + f2 )
N sinπ(f1 − f2 )
cov[pxx (f1 )pxx (f2 )]|f1 =f2 =f
"

2 #
sin2πf N
f
Γxx 1 +
N sin2πf

14.5
Let yk (n)

= x(n) ∗ hk (n)
=

N
−1
X

x(m)e

j2πk(n−m)
N

m=0

= e

j2πkn
N

N
−1
X

x(m)e

−j2πkm
N

m=0

yk (n)|n=N

=

N
−1
X

x(m)e

−j2πkm
N

m=0

= X(k)
411

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Note that this is just the Goertzel algorithm for computing the DFT. Then,
|yk (n)|2 = |X(k)|2

= |

N
−1
X

x(m)e

−j2πkm
N

m=0

|2

14.6
From (14.2.18) we have

W (f )

=
=

Z

1
2

W (f )df

=

− 12

=

=
by the definition of U in (14.2.12)

M −1
1 X
w(n)e−j2πf n |2
|
M U n=0

M −1 M −1
′
1 X X
w(n)w∗ (n′ )e−j2πf (n−n )
M U n=0 ′
n =0
Z 21
X
X
′
1
∗ ′
e−j2πf (n−n ) df
w(n)w (n )
1
MU n ′
−2
n
1 XX
w(n)w∗ (n′ )δ(n − n′ )
MU n ′
n
" M −1
#
1 1 X
2
|w(n)| = 1
U M n=0

14.7
(a) (1) Divide x(n) into subsequences of length M
2 and overlapped by 50% to produce 4k subseM
quences. Each subsequence is padded with 2 zeros.
(2) Compute the M-point DFT of each frame or subsequence.
(3) Compute the magnitude square of each DFT.
(4) Average the 4k M-point DFT’s.
(5) Perform the IDFT to obtain an estimate of the autocorrelation sequence.
(b)
X3 (k)

=

M
−1
X

x3 (m)e−

j2πkm
M

x1 (m)e−

j2πkm
M

m=0
M
2

=

−1
X

+

m=0

=

M
−1
X

m= M
2
− j2πkm
M

x1 (m)e

−jπk

+e

x2 (m −
M
−1
X

M − j2πkm
)e M
2

x2 (m′ )e−

j2πkm′
M

m′ =0

m=0

X3 (k)

M
−1
X

−jπk

= X1 (k) + e

X2 (k)

(c) Instead of zero-padding, we can combine two subsequences to produce a single M-point
subsequence and thus reduce the number of sequences form 4k to 2k. Then, we use the relation
in (b) for the DFT.
412

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14.8
0.9
= 90.
(a) Obviously, △f = 0.01. From (12.2.52), M = △f
(b) From (14.2.53), the quality factor is QB = 1.1N △f . This expression does not depend on M ;
hence, there is no advantage to increasing the value of M beyond 90.

14.9
(a) From table 14.1, we have
QB
⇒ △f
Qw
⇒ △f
QBT
⇒ △f

=

1.11N △f
QB
=
=
1.11N
= 1.39N △f
Qw
=
=
1.39N
= 2.34N △f
QBT
=
=
2.34N

1
111
1
139
1
234

(b)
For the Bartlett estimate,
QB

=

⇒M

=

N
M
N
= 100
QB

For the Welch estimate with 50% overlap,
Qw

=

⇒M

=

16N
M
16N
= 178
Qw

For the Blackman-Tukey estimate,
QBT

=

⇒M

=

1.5N
M
1.5N
= 150
QBT

14.10
(i)

(a) Suppose PB (f ) is the periodogram based on the Bartlett method. Then,
(i)
PB (f )

=

(0)
Pxx
(f )

=

(1)
Pxx
(f )

=
=

(2)
Pxx
(f )

M −1
1 X
xi (m)e−j2πf n |2 ,
|
M n=0

i = 0, 1, . . . , k − 1

0

M −1
1−w X
x1 (m)e−j2πf n |2
|
M
n=0
(1)

(1 − w)PB (f )

(1)

(1)
= wPxx
(f ) + (1 − w)PB (f )

413

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(1)

=

(2)

(1 − w)[wPB (f ) + PB (f )]
X
(k)
= (1 − w)
mwm−k PB (f )

(m)
Pxx
(f )

k=1

Therefore,

(M )
E{Pxx
(f )}

=

=
=
(M )
var{Pxx
(f )}

(1 − w)

X

(k)

M wm−k E[PB (f )]

k=1


2
Z 21
1 − wM 1
sinπ(f − α)M
(1 − w)
Γxx (α)
dα
1 − w M − 12
sinπ(f − α)

2
Z 21
1
sinπ(f − α)M
Γxx (α)
(1 − wM )
dα
M − 21
sinπ(f − α)

(M )
(M )
= E{[Pxx
(f )]2 } − [E{Pxx
(f )}]2
X
(k)
= E{[(1 − w)
M wm−k PB (f )]2 }

(M )
var{Pxx
(f )}

k=1

−{E[(1 − w)

=
=
=
=

2

(1 − w)

"

X

(k)

M wm−k PB (f )]}2

k=1

X

Mw

2(M −k)

(k)
E{PB (f )}2

k=1

(1 − w)2

X

−

(k)
{E[PB (f )]}2

#

(k)

M w2(M −k) var[PB (f )]

k=1

"
2 #

2M
sin2πf
M
1
−
w
Γ2 (f ) 1 +
(1 − w)2
1 − w2 xx
M sin2πf
"
2 #

sin2πf M
2w 1 − w 2
Γ (f ) 1 +
(1 − w )
1 + w xx
M sin2πf

(b)
(M )
E{Pxx
(f )}

(w)
= E{Pxx
(f )}
Z 21
Γxx (α)W (f − α)dα
=
− 12

where W (f )

=

(M )
var[Pxx
(f )]

=
=

M −1
1 X
w(n)e−j2πf n |2
|
M U n=0

(1 − w)2
(1 − w

M
X

(i)
w2(M −k) var[P̃xx
(f )]

k=1

2M

)



1−w
1+w



Γ2xx (f )

14.11
(i)

Let Rxx be defined as follows:

(i)
Rxx



(i)

rxx (0)
 (i)

r
1  xx (−1)
=
M


(i)

rxx (1)
(i)
rxx (0)
..
.



...
...
(i)

rxx (0)







414

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Then,
(i)
E ∗t (f )Rxx
E(f )

=

−1
M
−1 M
X
X
k=0 k′ =0

=

′
1 (i)
rxx (k − k ′ )e−j2π(k−k )f
M

M −1
1 X
M

X

(i)
rxx
(m)e−j2πmf

k=0 m=k−(M −1)

=

(M −1)

X

−(M −1)

(M| m|) (i)
rxx (m)e−j2πmf
M

(i)
= Pxx
(f )
(B)
Therefore, Pxx
(f )

=

K
1 X ∗t
(k)
E (f )Rxx
E(f )
K
k=1

14.12
To prove the recursive relation in (12.3.19) we make use of the following relations:
Êm

=

N
−1
X

[|fm (n)|2 + |gm (n − 1)|2 ]

(1)

n=m

where fm (n)
gm (n)
and Êm−1

= fm−1 (n) + km gm−1 (n − 1)

∗
= k̂m
fm−1 (n) + gm−1 (n − 1)
N
−1
X

=

n=m−1

(2)

[|fm−1 (n)|2 + |gm−1 (n − 1)|2 ]

= |fm−1 (m − 1)|2 + |gm−1 (m − 2)|2
+

N
−1
X

n=m

Also,

N
−1
X

∗
[fm−1 (n) + gm−1
(n − 1)]

n=m

[|fm−1 (n)|2 + |gm−1 (n − 1)|2 ]

1
= − k̂m Êm−1
2

We substitute for fm (n) and gm (n − 1) from (2) into (1), and we expand the expressions. Then,
use the relations for Êm−1 and k̂m to reduce the result.

14.13
x(n)

=

E[x(n)]

=

since E[w(n)]
To determine the autocorrelation, we have

=

h(0)

=

h(1)

=

1
x(n − 1) + w(n) − w(n − 1)
2
1
E[x(n − 1)] + E[w(n)] − E[w(n − 1)]
2
0, it follows that E[x(n)] = 0
1
h(−1) + δ(0) − δ(−1) = −1
2
1
1
h(0) + δ(1) − δ(0) = −
2
2

415

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Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

p = q = 1,

1
a=− ,
2

b0 = 1,

b1 = −1

Hence, γxx (0)

=

γxx (1)

=

and γxx (0)

=

γxx (1)

=

γxx (m)

=
=

γxx (m)

=
=

1
1
2
γxx (1) + σw
(1 + )
2
2
1
2
γxx (0) + σw (−1)
2
4 2
σ
3 w
1 2
− σw
3
−a1 γxx (m − 1)
1 1
2
− ( )m−1 σw
,
m>1
3 2
γxx (−m)
1 1
2
,
m<0
− ( )−m+1 σw
3 2

14.14
x(n)
E[x(n)]
γxx (m)

= w(n) − 2w(n − 1) + w(n − 2)
=

0 since E[w(n)] = 0
q
X
2
bk bk+m ,
0≤m≤q
= σw
k=0

where q = 2,

b0 = 1,

b1 = −2,

b2 = 1

Hence, γxx (0)

2
= σw

2
X

2
b2k = bσw

k=0

γxx (1)

2
= σw

2
X

k=0

γxx (2)

=

2
σw

γxx (m)

=

0,

2
X

2
bk bk+1 = −4σw
2
bk bk+2 = σw

k=0

γxx (−m)

|m| ≥ 3,

= γxx (m)

14.15
(a)
Γxx (z)
√

√
1±j 3 1±j 3
The four zeros are
,
2
2
The minimum-phase system is
H(z)
Hence, H(z)

=
=

X

γxx (m)z −m

m
−2

2z

(z 4 − 2z 3 + 3z 2 − 2z + 1)

= G(1 − z −1 + z −2 ), where G =
√
=
2(1 − z −1 + z −2 )

√

2

416

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

(b) The solution is unique.

14.16
(a)
Γxx (z) =

∞
X

γxx (m)z −m

m=−∞
2

z
(6 − 35z −1 + 62z −2 − 35z −3 + 6z −4 )
62
1
1
z2
(1 − 3z −1 )(1 − 2z −1 )(1 − z −1 )(1 − z −1 )
=
62
2
3
1 1
The four zeros are z = 3, 2, ,
3 2
6
1
1
The minimum phase system is H(z) = √ (1 − z −1 )(1 − z −1 )
2
3
62
1
= √ (6 − 5z −1 + z −2 )
62
=

(b) The maximum phase system is H(z) = √162 (1 − 5z −1 + 6z −2 )
(c) There are two possible mixed-phase systems: H1 (z) = √162 (3 − 7z −1 + 2z −2 )
√1 (2 − 7z −1 + 3z −2 )
62

H2 (z) =

14.17
(a)
H(z) =
Γhh (f )

=
=
=

γxx (m)

=

⇒ Γxx (f )

=
=

Γyy (f )

=
=

1 + z −1
1 − 0.8z −1
H(z)H(z −1 )|z=ej2πf
1 + e−j2πf
1 + ej2πf
1 − 0.8e−j2πf 1 − 0.8ej2πf
cos2 πf
4
1.64 − 1.6cos2πf
1 |m|
( )
2
∞
X
1
( )|m| e−j2πf m
2
m=−∞

0.75
1.25 − cos2πf
Γxx (f )Γhh (f )
3cos2 πf
(1.64 − 1.6cos2πf )(1.25 − cos2πf )

(b)
Γyy (f )

=

75
54
2
−
1.64 − 1.6cos2πf
1.25 − cos2πf

417

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Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

=
γyy (m)

9
25

150

− 50

1.64 − 1.6cos2πf
1
150(0.8)|m| − 50( )|m|
2

=

3
4

1.25 − cos2πf

2
= γxx (0) = 150 − 50 = 100
(c) σw

14.18
proof is by contradiction.
(a) Assume the |km | > 1. Since Em = (1 − |km |2 )Em−1 , this implies that either Em < 0 or
2
Em−1 < 0. Hence, σw
< 0, and
 2 
σw
 0 


at Γxx a = at  .  ⇒ Γxx
 .. 
0

is not positive definite.
(b) From the Schur-Cohn test, Ap (z) is stable if |km | < 1. Hence, the roots of Ap (z) are inside
the unit circle.

14.19
(a)



γxx (0)
 γxx (−1)
γxx (−2)

γxx (1)
γxx (0)
γxx (−1)


  2 
γxx (2)
σw
1
= 0 
γxx (1)  
0
0
γxx (0)
−0.81

γxx (m)
γxx (m)
Hence,
2
σw
The values of the parameters dm

=

0.81γxx (m − 2),

m≥3

= {2.91, 0, 2.36, 0, 1.91, 0, 1.55, 0, . . .}
=

q
X

bk bk+m are as follows:

k=0

M A(2) : dm
M A(4) : dm
M A(8) : dm

= {2.91, 0, 2, 36}

= {2.91, 0, 2, 36, 0, 1.91}
= {2.91, 0, 2, 36, 0, 1.91, 0, 1.55, 0}

(b) The M A(2), M A(4) and M A(8) models have spectra that contain negative values. On the
other hand, the spectrum of the AR process is shown below. Clearly, the MA models do not
provide good approximations to the AR process. Refer to fig 14.19-1.

14.20

2
2
γxx (m) = 1.656σw
, 0, 0.81σw
, 0, . . . .
For AR(2) process:
418

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−−−> magnitude

6
5
4
3
2
1
0
0

0.05

0.1

0.15

0.2
0.25
0.3
−−−> frequency(Hz)

0.35

0.4

0.45

0.5

Figure 14.19-1:


The solution is

2
1.656σw

0
2
0.81σw

0
2
1.656σw
0

g
a1
a2

 


2
2
gσw
1
0.81σw
  a1  =  0 
0
2
0
a2
1.656σw
=

1.12

= 0
= −0.489

For the AR(4) process, we obtain g = 1.07 and
a = {1, 0, −0.643, 0, 0.314}

For the AR(8) process, we obtain g = 1.024 and
a = {1, 0, −0.75, 0, 0.536, 0, −0.345, 0, 0.169}

Refer to fig 14.20-1.

14.21
(a) (1)
H(w)
Γxx (w)
Γxx (w)

1 − e−jw
1 + 0.81e−jw
2
= |H(w)|2 σw
1 − e−jw 2 2
= |
| σ
1 + 0.81e−jw w
=

(2)
H(w)
Γxx (w)

=

(1 − e−j2w )

2
= |H(w)|2 σw
2
= 4σw
sin2 w

(3)
H(w)

=

1
1 − 0.81e−jw
419

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MA(2)

AR(2)
2.5
−−−> magnitude

−−−> magnitude

2
1.5
1
0.5
0
0

0.2
0.4
−−−> frequency(Hz)

2
1.5
1
0.5
0

0.6

0.2
0.4
−−−> frequency(Hz)

AR(4)

AR(8)
2
−−−> magnitude

−−−> magnitude

2

1.5

1

0.5
0

0.6

0.2
0.4
−−−> frequency(Hz)

1.5
1
0.5
0
0

0.6

0.2
0.4
−−−> frequency(Hz)

0.6

Figure 14.20-1:

Γxx (w)
(b) Refer to fig 14.21-1.
(c) For (2),

=

2
σw
1.6561 − 1.62cosw


P3
2
 σw
k=0 bk bk+m , 0 ≤ m ≤ 2
γxx (m) =
0,
m>2
 ∗
m<0
γxx (−m),

since b0
γxx (0)
γxx (2)
γxx (−2)
γxx (m)

= 1,
b1 = 0 and b2 = −1, we have
2
= 2σw
2
= −σw
2
= −σw
= 0,
m 6= 0, ±2

For (3), the AR process has coefficients a0 = 1, a1 = 0 and a2 = 0.81.


  2 
1
0
0.81
σw
γxx (0)
 0
1.81
0   γxx (1)  =  0 
0
0.81
0
1
γxx (2)
γxx (0)

=

2
2.9σw

420

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(1)

(2)
2
−−−> magnitude

−−−> magnitude

8
6
4
2
0
0

0.2
0.4
−−−> frequency(Hz)

0.6

1.5
1
0.5
0
0

0.2
0.4
−−−> frequency(Hz)

0.6

(3)

−−−> magnitude

6
5
4
3
2
1
0
0

0.2
0.4
−−−> frequency(Hz)

0.6

Figure 14.21-1:

γxx (m) =
γxx (m) =

0,
m odd
2
2.9(0.9)|m| σw
,

m even

14.22

(a) For the Bartlett estimate,
M

=
=

(b)M

=

(c)for (a), QB

=
=

for (b), QB

=
=

0.9
△f
0.9
= 90
0.01
0.9
= 45
0.02
N
M
2400
= 26.67
90
N
M
2400
= 53.33
45

421

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14.23
2
Γxx (f ) = σw

|ej2πf

|ej2πf − 0.9|2
− j0.9|2 |ej2πf + j0.9|2

(a)
z − 0.9 z −1 − 0.9
z 2 + 0.81 z −2 + 0.81
z − 0.9
z 2 + 0.81
z −1 (1 − 0.9z −1 )
1 + 0.81z −2

2
= σw

Γxx (z)
Therefore, H(z)

=
=

(b) The inverse system is
1 + 0.81z −2
1
= −1
H(z)
z (1 − 0.9z −1 )
This is a stable system.

14.24
N
−1
X

X(k) =

x(n)e

−j2πnk
N

n=0

(a)
E[X(k)]

=

X

E[x(n)]e

−j2πnk
N

=0

n

E[|X(k)|2 ]

=

XX
n

=

n

=

−j2πk(n−m)
N

m

XX

= σx2

E[x(n)x∗ (m)]e

m

N
−1
X

σx2 δ(n − m)e

−j2πk(n−m)
N

1

n=0
N σx2

(b)
E{X(k)X ∗ (k − m)}

=

XX
n

=

σx2

−j2πkn
N

e

j2πn′ (k−n)
N

n′

XX
n

=
=
=

E[x(n)x∗ (n′ )]e

n′

δ(n − n′ )e

−j2πmn′
N

e

−j2πk(n−n′ )
N

j2πmn
σx2 e N
N σx2 ,
m

0,

= pN
otherwise

p = 0, ±1, ±2, . . .

422

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14.25
γvv (m)

= E[v ∗ (n)v(n + m)]
q X
q
X
=
b∗k bk′ E[w∗ (n − k)w(n + m − k ′ )]
=
=
=

Then, Γvv (f )

=

k′ =0 k=0
q X
q
X
2
σw
b∗k bk′ δ(m
′
k =0 k=0
q
X
2
σw
b∗k bk+m

+ k − k′ )

k=0
2
σ w dm
q
X
2
σw
dm e−j2πf m
m=−q

14.26
γxx (m)

= E[x∗ (n)x(n + m)]
= A2 E{cos(w1 n + φ)cos[w1 (n + m) + φ]}
A2
E{cosw1 m + cos[w1 (2n + m) + 2φ]}
=
2
A2
cosw1 n
=
2

14.27
(a)
x(n)
y(n)
⇒ x(n)
y(n) − v(n)

Therefore, y(n)
so that y(n) is an ARMA(2,2) process

= 0.81x(n − 2) + w(n)
= x(n) + v(n)
= y(n) − v(n)
= 0.81y(n − 2) − 0.81v(n − 2) + w(n)

=

0.81y(n − 2) + v(n) − 0.81v(n − 2) + w(n)

(b)
x(n)
y(n)
⇒ x(n)
y(n) − v(n)
y(n) +

p
X

k=1

ak y(n − k)

= −

p
X

k=1

ak x(n − k) + w(n)

= x(n) + v(n)
= y(n) − v(n)
p
X
= −
ak [y(n − k) − v(n − k)] + w(n)
k=1

= v(n) +

p
X

k=1

ak v(n − k) + w(n)

423

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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Hence, y(n) is an ARMA(p,p) process
p
X
Note that X(z)[1 +
ak z −k ] = W (z)
k=1

H(z) =
=

1
Pp
1 + k=1 ak z −k
1
Ap (z)

2
Γxx (z) = σw
H(z)H(z −1 )
2
and Γyy (z) = σw
H(z)H(z −1 ) + σv2
2
σw
=
+ σv2
Ap (z)Ap (z −1 )
2
+ σv2 Ap (z)Ap (z −1 )
σw
=
Ap (z)Ap (z −1 )

14.28
(a)
γxx (m)

= E{[

K
X

Ak cos(wk n + φk ) + w(n)][

XX
k

=

Ak Ak′ E{cos(wk n + φk )cos(wk′ (n + m) + φk′ )} + E[w(n)w(n + m)]

k′

K
X
A2

k=1

Ak′ cos(wk′ (n + m) + φk′ ) + w(n + m)]}

k′ =1

k=1

=

K
X

2

2
cos(wk n) + σw
δ(m)

(b)
Γxx (w)

=

∞
X

γxx (m)e−jwm

m=−∞

=

K
∞
X
A2 X

k=1

=

K
X

k=1

=

4

2
(ejwk + e−jwk )e−jwn + σw

m=−∞

A2
2
[2πδ(w − wk − 2πm) + 2πδ(w + wk − 2πm)] + σw
4

K
πX 2
2
Ak [δ(w − wk − 2πm) + 2πδ(w + wk − 2πm)] + σw
2
k=1

14.29
T

T

E = a∗ Γyy a + λ(1 − a∗ a)
dE
= 0
da
⇒ Γyy a − λa = 0
or Γyy a = λa

424

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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
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Thus, a is an eigenvector corresponding to the eigenvalue λ. Substitute Γyy a = λa into E. Then,
2
E = λ. To minimize E, we select th smallest eigenvalue, namely, σw
.

14.30
(a)
γxx (0)
γxx (1)
γxx (2)
By the Levinson-Durbin algorithm,
a1 (1)

2
= P + σw
= P cos2πf1

= P cos4πf

=
k1
E1

=
=
=

a2 (2)

=
=

a2 (1)

=
=

(b) k2 = a2 (2)
(c)

γxx (1)
γxx (0)
P cos2πf1
−
2
P + σw
a1 (1)
(1 − k12 )γxx (0)
4
2
+ σw
P 2 sin2 2πf1 + 2P σw
2
P + σw
γxx (2) + a1 (1)γxx (1)
−
E1
2
P σw
cos4πf1 − P 2 sin2 2πf1
− 2 2
2 + σ4
P sin 2πf1 + 2P σw
w
a1 (1) + a2 (2)a1 (1)


2
P cos2πf1
P 2 sin2 2πf1 − P σw
cos4πf1
−
1
+
2
2 + σ4
P + σw
P 2 sin2 2πf1 + 2P σw
w

= −

k1 = a1 (1) as given above.
2
If σw
a2 (1)

→
=

a2 (2)

=
=

−2cos2πf1
1

k2
k1

=
=

1
−cos2πf1

0, we have
−(cos2πf1 )(1 + 1)

14.31
ε(h) = hH Γxx h + µ(1 − E H (f )h) + µ∗ (1 − hH E(f ))

(a) To determine the optimum filter that minimizes σy2 subject to the constraint, we differentiate
ε(h) with respect to hH (compute the complex gradient):
ε(h)
= Γxx h − µ∗ E(f ) = 0
hH
Thus,

−1
hopt = µ∗ Γxx
E(f )

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.
(b) To solve for the Langrange multipliers using the constraint, we have
E H (f )hopt = µ∗ E H (f )Γ−1
xx E(f ) = 1
Thus,
µ∗ =

1
E (f )Γ−1
xx E(f )
H

By substituting for µ∗ in the result given in (a) we obtain the optimum filter as
hopt =

Γ−1
xx E(f )
H
E (f )Γ−1
xx E(f )

14.32
The periodogram spectral estimate is
PXX (f ) =
where
X(f ) =

1
1
2
|X(f )| = X(f )X ∗ (f )
N
N

N
−1
X

x(n)e−j2πf n = E H (f )X(n)

n=0

By substituting X(f ) into Pxx (f ), we obtain
Pxx (f ) =

1 H
E (f )X(n)X(n)H E(f )
N

Then,
E [Pxx (f )]

=
=



1 H
E (f )E X(n)X(n)H E(f )
N
1 H
E (f )Γxx E(f )
N

14.33
We use the Pisasenko decomposition method. First, we compute the eigqnvalues of the correlation
matrix.
g(λ) =
=

3−λ
0
−2

0
3−λ
0

−2
0
3−λ

0
−2
3−λ


(3 − λ)3 − 2(2)(3 − λ) = (3 − λ) (3 − λ)2 − 4 = 0
= (3 − λ)

3−λ
0

0
3−λ

3−λ
0

Thus, λ = 5, 3, 1 and the noise varinace is λmin = 1. The corresponding eigenvector is


  


1
0
1
2 0 −2
 0 2 0   a1  =  0  ⇒ a2 = 1, a1 = 0 ⇒  0 
1
0
a2
−2 0 2

The frequency is found from the equation 1 + z −2 = 0 ⇒ z = ±j. Therefore, ejw = ±j yields
w = ±π/2 and the power is P = 2.
426

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Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

14.34
The eigenvalues are found from
g(λ) =

2−λ
−j
−1
j
2−λ
−j
−1
j
2−λ

and the normalized eigenvectors are
√ 

−j/√ 3
v 1 =  1/√3 
j/ 3

⇒ λ1 = 1, λ2 = 1, λ3 = 4.

 p


2/3
√
v 3 =  −j/√ 6 
1/ 6


0√
v 2 =  j/√2 
1/ 2


By computing the denominator of (14.5.28), we find that the frequency is ω = π/2 or f = 1/4. We
may also find the frequency by using the eigenvectors v 2 and v 3 to construct the two polynomials
(Boot Music Method):
V2 (z)
V3 (z)

1
j
√ z − √ z −2
2
2
r
1
1
2
=
− √ z −1 + √ z −2
3
6
6

=

Then, we form the polynomials
V2 (z)V2∗ (1/z ∗ ) + V3 (z)V3∗ (1/z ∗ ) =

1 2 2
2
1
z + jz + 2 − jz −1 + z −2
3
3
3
3

It is easily verified that the polynomial has a double root at z = j or, equivalently, at ω = π/2.
The other two roots are spurious roots that are neglected.
Finally, the power of the exponential signal is P1 = 1.

14.35
1

PM U SIC (f ) = PM

2

k=p+1

The denominator can be expressed as
M
X

sH (f )v k

2

=

k=p+1

M
X

|sH (f )vk |

sH (f )v k v H
k s(f )

k=p+1



= sH (f ) 

M
X

k=p+1



 s(f )
vk vH
k

14.36
PM −1
(a) Vk (z) = n=0 vk (n+1)z −n and Vk (f ) = Vk (z) |z=ej2πf Then, the denominator in PM U SIC (f )
may be expressed as
M
X

= sH (f )v k

2

=

X

M Vk (f )Vk∗ (f )

k=p+1

k=p+1

=

X

k=p+1

M Vk (z)Vk∗ (1/z ∗ ) |z=ej2πf

427

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

(b) For the roots of Q(z), we consruct (from Problem 14.34) Q(z) as
Q(z) = V2 (z)V2∗ (1/z ∗ ) + V3 (z)V3∗ (1/z ∗ )
1 2 2
2
1
=
z + jz + 2 − jz −1 + z −2
3
3
3
3
Thus polynomial has a double root at z = j and two spurious roots. Therefore, the desired
frequency is ω = π/2.

14.37
(a)
γxy (n0 )

=

N
−1
X
n=0

E[γxy (n0 )]

=

N
−1
X
n=0

=

N
−1
X
n=1

=
var[γxy (n0 )]

y(n − n0 )[y(n − n0 ) + w(n)]
E[y 2 (n − n0 )]
E[A2 cos2 w0 (n − n0 )]

M A2
2

2
= E[γxy
(n0 )](

=

XX
n

=

0≤n≤M −1

n′
2

M A2 2
)
2

E{y(n − n0 )[y(n − n0 ) + w(n)]y(n′ − n0 )[y(n′ − n0 ) + w(n′ )]} − (

M A2 2
)
2

MA 2
σw
2

(b)
SNR

=

{E[γxy (n0 )]}2
var[γxy (n0 )]
2

=
=

( M2A )2
M A2 2
2 σw
2

MA
2
2σw

(c) As M increases, the SNR increases.

14.38
Refer to fig 14.38-1.

14.39
Refer to fig 14.39-1.

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writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

autocor of w(n)

periodogram Pxx(f)

3

80

2.8
60
2.6
2.4

40

2.2
20
2
1.8
−20

−10

0

10

0
0

20

200

400

600

avg periodogram Pxx(f)
80
60
40
20
0
0

200

400

600

Figure 14.38-1:
theoretical psd with M = 100

Bartlett with M = 50

60

50

40

40

20

30

0

20

−20

10

−40
0

1

2

3

0
0

4

1

2

3

4

Blackman−Tukey psd with lag=25
120
100
80
60
40
20
0

1

2

3

4

Figure 14.39-1:
429

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as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in
writing from the publisher. For the exclusive use of adopters of the book Digital Signal Processing, Fourth Edition, by John G.
Proakis and Dimitris G. Manolakis. ISBN 0-13-187374-1.

Co rrections to Digital Signal Processing, 4 t h Edition
by
John G . Proakis and Dimitri s G . Manolakis
1. Page 18, two lines below equation (1.3.18)
sk(n) should be sk(n)
2. Page 34, Figure 1.4.8
The quantized value of the signal between 2T and 3T should be 4
3. Page 66, line below equation (2.2.43)
“is relaxed” should be “is non-relaxed”
4. Page 101, last term of equation (2.4.24)
n
n

should be

N

5. Page 147, last sentence above Section 3.1
Move this sentence to line above, just before the word “Finally, “
6. Page 161, figure 5.2.1
The mapping is w = a-1z
7. Page 237, line 2 from the top of page
“radian” should be “radial”
8. Page 321, Figure 5.2.3, magnitude plot
Scale on the ordinate should be multiplied by 5

9. Page 387, line 8 below equation (6.1.15)
X(Fs) should be X(F)
10. Page 390, Figure 6.1.3(b)
X(F/Fs) should be X(F)
11. Page 391, Figure 6.1.5 upper right-hand part of the figure
X(F/Xf) should be X(F)
12. Page 396, Figure 6.2.3, graph of Y(F)
For F<0, the Fs on the abscissa should be -Fs
13. Page 424, two lines below equation (6.4.68)
The word “envelop” should be “envelope”
14. Page 454, equation on line above Section 7.1.2

e-j2 kN should be e-j2 k/N
15.Page 463, line below equation (7.1.39)
(7.1.38) should be (7.1.39)
16.Page 506, problem 7.23(e)
The exponent should be j(2 /N) kon
17. Page 526, Figure 8.1.10
Delete the factor of 2 in the expression for B
18. Page 582, line 4 from the top
B2(z) = 1/2+3/8 z-1+z-2

19. Page 646, Problem 9.22
In the denominator of H(z), the term r2 should be r2
20. Page 672, two lines below equation (10.2.35)
G(k+x) should be ((k+ )
21. Page 679, line above equation (10.2.52) and in equation (10.2.52)
Add the term

˜
˜
b (1) = 2b(1) -2 b(0); Then, in (10.2.52), k = 2,3,…,M/2 -2
22. Page 680, line above Case 4:
The equation should be

˜
˜
c(0) – ½ c(2) = c(1)

23. Page 725, Figure 10.3.14, graph on left
The value of 1 is the peak value
24. Page 742, problem 10.2.3, lines 4 and 6
Add subscripts l and u on the expressions for
H(s) should b Ha(s)
25. Page 809, equation (11.12.15)
t

Q(zM) should be Q (zM)

26. Page 811, in Solution of example 11.12.1
The matrix for G0(z), G1(z) and G2(z) should be transposed
Thus,
G0(z) = 1-z-1 + z-2, G1(z) = -1-z-1+3z-2, G2(z)=1+3z-1-5z-2
27. Page 818, problem 11.16
Change the statement of the problem to the following:
Use the result in Problem 11.15 to determine the type II form of the I=3
interpolator in Figure 11.5.12(b)
28. Page 821, third line from bottom of page
Should be f0 = 1/6 and f = 1/3
29. Page 958, problem 13.19
In the expression for the least squares error,
f(m)n should be fm(l) and gm(n) should be gm(l)
30. Page 962, equations (14.1.6), (14.1.7) and (14.1.8)
X(F/X(F)) should be X(F)
31. Page 964, in Solution of Example 14.1.1, line 2
Figure 10.2.2(a) should be Figure 10.2.2
32. Page 1038, problem 14.35
In the denominator of the equation, v kv k should be v kv kH



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