Solutions Manual to accompany

Communication Systems

An Introduction to Signals and Noise in Electrical Communication

Fourth Edition

A. Bruce Carlson Rensselaer Polytechnic Institute

Paul B. Crilly

University of Tennessee

Janet C. Rutledge University of Maryland at Baltimore

Solutions Manual to accompany COMMUNICATION SYSTEMS: AN INTRODUCTION TO SIGNALS AND NOISE IN ELECTRICAL COMMUNICATION, FOURTH EDITION A. BRUCE CARLSON, PAUL B. CRILLY, AND JANET C. RUTLEDGE Published by McGraw-Hill Higher Education, an imprint of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020. Copyright © The McGraw-Hill Companies, Inc., 2002, 1986, 1975, 1968. All rights reserved. The contents, or parts thereof, may be reproduced in print form solely for classroom use with COMMUNICATION SYSTEMS: AN INTRODUCTION TO SIGNALS AND NOISE IN ELECTRICAL COMMUNICATION, provided such reproductions bear copyright notice, but may not be reproduced in any other form or for any other purpose without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning. www.mhhe.com

2-1

Chapter 2 2.1-1

0 0

0

/ 2 2 ( )

/ 20

sinc( )0 otherwise

jj T j m n f t jn T

Ae n mAec e dt Ae m n

T

φφπ φ−

−

== = − =

∫

2.1-2

0 0

0

0

/ 4 / 2

0 / 40 0 0

( ) 0

2 2 2 2cos ( )cos sin

2

T T

n T

c v t

nt nt A nc A dt A dt

T T T nπ π π

π

=

= + − =∫ ∫

n 0 1 2 3 4 5 6 7

nc 0 2 /A π 0 2 / 3A π 0 2 / 5A π 0 2 / 7A π

arg nc 0 180± ° 0 180± ° 2.1-3

0

0

/2

200 0 0

( ) / 2

2 2 2cos sin (cos 1)

( )T

n

c v t A

At nt A Ac A dt n n

T T T n nπ

π ππ π

= =

= − = − −

∫

n 0 1 2 3 4 5 6

nc 0.5A 0.2A 0 0.02A 0 0.01A 0

arg nc 0 0 0 0 2.1-4

0 / 2

0 00 0

2 2cos 0

T tc A

T Tπ

= =∫ (cont.)

2-2

( ) ( )

[ ]

0

0

/ 2/ 2 0 0

00 0 0 0 0 0 0

sin 2 / sin 2 /2 2 2 2cos cos

4( ) / 4( ) /

/ 2 1sinc(1 ) sinc(1 )

0 otherwise2

TT

n

n t T n t Tt nt Ac A dt

T T T T n T n T

A nAn n

π π π ππ ππ π π π

− + = = + − +

= ±= − + + =

∫

2.1-5

0

0

/ 2

00 0

( ) 0

2 2sin (1 cos )

T

n

c v t

nt Ac j A dt j n

T T nπ

ππ

= =

= − = − −∫

n 1 2 3 4 5

nc 2 /A π 0 2 / 3A π 2 / 5A π

arg nc 90− ° 90− ° 90− ° 2.1-6

0 ( ) 0c v t= =

( ) ( )

[ ]

0

0

/ 2/ 2 0 0

00 0 0 0 0 0 0

sin 2 / sin 2 /2 2 2 2sin sin

4( ) / 4( ) /

/ 2 1sinc(1 ) sinc(1 )

0 otherwise2

TT

n

n t T n t Tt nt Ac j A dt j

T T T T n T n T

jA nAj n n

π π π ππ ππ π π π

− + = − = − − − +

= ±= − − − + =

∫m

2.1-7

]0 00 0

0

/ 2

0 / 20

1( ) ( )

T Tjn t jn tn T

c v t e dt v t e dtT

ω ω− −= +∫ ∫

0 00 0 0 0

0

00

/ 2 / 20/ 2 0

/ 2

0

where ( ) ( /2)

( )

T Tjn t jn jn T

T

T jn tjn

v t e dt v T e e d

e v t e dt

ω ω λ ω

ωπ

λ λ− − −

−

= +

= −

∫ ∫

∫

since 1 for even , 0 for even jnne n c nπ = =

2-3

2.1-8

2 2 2 2 2 20 0 0 0 0 0 0 0

1

0

22 2 2 2

22 2 2 2 2 2

2 2 sinc 2 sinc2 2 sinc3

1where 4

1 1 1 31 2sinc 2sinc 2sinc 0.23

16 4 2 4

2 1 1 3 5 3 71 2sinc 2sinc 2sinc 2sinc 2sinc 2sinc

16 4 2 4 4 2 4

nn

P c c Af Af f Af f Af f

f

Af P A

Af P

τ τ τ τ τ τ τ

τ

τ

τ

∞

=

= + = + + + +

=

> = + + + = > = + + + + + +

∑ L

2

22 2 2

0.24

1 1 11 2sinc 2sinc 0.21

2 16 4 2

A

Af P A

τ

= > = + + =

2.1-9

0 0

0

2

2 2/ 2 / 2

/ 2 00 0 0 0

2 2 2

2 2 2

0 02 2 2

0 even

2 odd

n

41 2 4 1a) 1 1

3

4 4 42 2 2 0.332 so / 99.6%

9 258 8 8

b) ( ) cos cos3 cos59 25

n

T T

T

nc

n

t tP dt dt

T T T T

P P P

v t t t

π

π π π

ω ω ωπ π π

−

=

= − = − =

′ ′= + + = =

′ = + +

∫ ∫

0t

2.1-10

( )0

0

2 2 2/ 2 2

/ 20

0 even2

odd

1 2 2 2a) 1 1 2 0.933 so / 93.3%

3 5

n

T

T

nc j

nn

P dt P P PT

π

π π π−

= −

′ ′= = = + + = =

∫

(cont.)

2-4

( ) ( ) ( )

( ) ( ) ( )

0 0 0

0 0 0

4 4 4b) ( ) cos 90 cos 3 90 cos 5 90

3 54 4 4

sin sin 3 sin 53 5

v t t t t

t t t

ω ω ωπ π π

ω ω ωπ π π

′ = − ° + − ° + − °

= + +

2.1-11

0

2

00 0

1/2 01 11/2 03

T

n

ntP dt c

n nT T π=

= = = ≠ ∫

4 4

4 4 4 odd

2 2 1 1 1 12 2

1 3 5 3n

Pnπ π

∞ = = + + + =

∑ L

2 2

2 2 2

1 1 1 4 1 1Thus,

1 2 3 2 3 4 6π π + + + = − =

L

2.1-12

0

2/ 2

200 0

0 even2 4 11

(2/ ) odd3T

n

ntP dt c

n nT T π

= − = =

∫

2 2

2 2 2 21

1 1 1 2 1 1 1 12

2 2 4 4 1 2 3 3n

Pnπ π

∞

=

= + = + + + + =

∑ L

4 4

4 4 4 4

1 1 1 1Thus,

1 3 5 2 2 3 96π π

+ + + = =⋅

L

2.2-1

( )( )

( )( ) [ ]

/ 2

0( ) 2 cos cos2

sin 2 sin 22 22 sinc( 1/2) sinc( 1/2)

22 2 2 2

tV f A ftdt

f f AA f f

f f

τ

π πτ τπ πτ τ

ππ

ττ τ

π π ττ τ

π π

+

+

=

− = + = − + +

−

∫

(cont.)

2-5

2.2-2

( )( )

( )( ) [ ]

/ 2

0

2 2

2 2

2( ) 2 sin cos2

sin 2 sin 22 22 sinc( 1) sinc( 1)

22 2 2 2

tV f j A ftdt

f f Aj A j f f

f f

τ

π πτ τπ πτ τ

ππ

ττ τ

π π ττ τ

π π

+

+

= −

− = − − = − − − +

−

∫

2.2-3

2 220

2( ) 2 cos 2sin 1 1 sinc

( ) 2t A

V f A A tdt A fτ τ ωτ

ω τ ττ ωτ

= − = − + = ∫

2.2-4

20

2( ) 2 sin (sin cos )

( )

(sinc2 cos2 )

t AV f j A tdt j

Aj f f

f

τ τω ωτ ωτ ωτ

τ ωτ

τ π τπ

= − = − −

= − −

∫

2.2-5

22

2

1( ) sinc2

2 2

1 1 1sinc2

2 2 4 2

fv t Wt

W W

fWt dt df df

W W W W∞ ∞ ∞

−∞ −∞ −∞

= ↔ Π

= Π = = ∫ ∫ ∫

2-6

2.2-6

( )2 2 2

2

2 20 0

22 arctan

2 (2 )

Wbt A A A WE Ae dt E df

b b f b bπ

π π

∞ − ′= = = =+∫ ∫

50% / 22 2arctan

84% 2 /W bE WW bE b

ππππ

=′ = = =

2.2-7

( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

j t

j t

v t w t dt v t W f e df dt

W f v t e dt df W f V f df

ω

ω

∞ ∞ ∞

−∞ −∞ −∞

∞ ∞ ∞− −

−∞ −∞ −∞

= = = −

∫ ∫ ∫

∫ ∫ ∫

22( ) * ( ) when ( ) is real, so ( ) ( ) * ( ) ( )V f V f v t v t dt V f V f df V f df∞ ∞ ∞

−∞ −∞ −∞− = = =∫ ∫ ∫

2.2-8

2 2 2 ( )( ) ( ) ( ) ( )

Let ( ) ( ) so ( ) ( ) and ( ) ( )

Hence ( ) ( ) ( ) ( )

j ft j ft j f tw t e dt w t e dt w t e dt W f

z t w t Z f W f W f Z f

v t z t dt V f Z f df

π π π∗ ∗∞ ∞ ∞∗ − − − ∗

−∞ −∞ −∞

∗ ∗ ∗

∞ ∞

−∞ −∞

= = = = = − = −

= −

∫ ∫ ∫

∫ ∫

2.2-9

1sinc so sinc

2 2( ) sinc ( ) for

2 2

t fA Af At

A A A

t fv t V f A

τ ττ τ

Π ↔ ↔ Π

= ↔ = Π =

2.2-10

[ ]

[ ]

[ ]

cos sinc( 1/2) sinc( 1/2)2

( )so sinc( 1/2) sinc( 1/2) cos cos

2

Let and 2 ( ) sinc(2 1/2) sinc(2 1/2)

t t BB f f

B f f f ft t B B

B A W z t AW Wt Wt

π ττ τ

τ ττ π π

τ ττ τ τ τ

τ

Π ↔ − + +

− − − + + ↔ Π = Π

= = ⇒ = − + +

2.2-11

[ ]

[ ]

[ ]

2sin sinc( 1) sinc( 1)

22 ( ) 2

so sinc( 1) sinc( 1) sin sin2

Let and 2 ( ) sinc(2 1) sinc(2 1)

t t BB j f f

B f f f fj t t B B

B jA W z t AW Wt Wt

π ττ τ

τ ττ π π

τ ττ τ τ τ

τ

Π ↔ − − + +

− − − − + + ↔ Π = − Π

= − = ⇒ = − + +

2-7

2.2-12

( )( )

( )

22 2 2 2 2 2

222

22 2 0 2 2

2

2 30 2 2

2 4 /(2 ) (2 ) (2 )

1 /2

2

1 1Thus,

2 2 4

b t a t

a t

b a ae e

b f a f a f

a a dfe dt df

a a f a f

dxa a aa x

π

π

π ππ π π

ππ π

π ππ

− −

∞ ∞ ∞−

−∞ −∞

∞

↔ ⇒ ↔ =+ + +

= = = + +

= = +

∫ ∫ ∫

∫

2.3-1

( ) ( ) ( ) where v( ) ( / ) sinc

so Z( ) ( ) ( ) 2 sinc cos2j T j T

z t v t T v t T t A t A f

f V f e V f e A f fTω ω

τ τ τ

τ τ π−

= − + + = Π ↔

= + =

2.3-2

2 2

( ) ( 2 ) 2 ( ) ( 2 ) where v( ) ( / ) sinc

( ) ( ) ( ) ( ) 2 (sinc )(1 cos4 )j T j T

z t v t T v t v t T t a t A f

Z f V f e V f V f e A f fTω ω

τ τ τ

τ τ π−

= − + + + = Π ↔

= + + = +

2.3-3

2 2

( ) ( 2 ) 2 ( ) ( 2 ) where ( ) ( / ) sinc

( ) ( ) 2 ( ) ( ) 2 (sinc )(cos4 1)j T j T

z t v t T v t v t T v t a t A f

Z f V f e V f V f e A f fTω ω

τ τ τ

τ τ π−

= − − + + = Π ↔

= − + = −

2.3-4

/ 2

/ 2( ) ( )

2

( ) 2 sinc2 ( ) sincj T j T

t T t Tv t A B A

T T

V f AT fTe B A T fTeω ω− −

− − = Π + − Π

= + −

2-8

2.3-5

2 2

2 2( ) ( )

4 2

( ) 4 sinc4 2( ) sinc2j T j T

t T t Tv t A B A

T T

V f AT fTe B A T fTeω ω− −

− − = Π + − Π

= + −

2.3-6

/ /

1Let ( ) ( ) ( ) ( / )

1Then ( ) [ ( / )] ( / ) so ( ) ( ) ( / )d dj t a j t a

d d

w t v at W f V f aa

z t v a t t a w t t a Z f W f e V f a ea

ω ω− −

= ↔ =

= − = − = =

2.3-7

2 ( )( ) ( ) ( ) ( )c c cj t j t j f f tj tcv t e v t e e dt v t e dt V f fω ω πω∞ ∞ − −−

−∞ −∞ = = = − ∫ ∫F

2.3-8

[ ]

( ) ( / )cos with 2 /

( ) sinc( ) sinc( ) sinc( 1/2) sinc( 1/2)2 2 2

c c c

c c

v t A t t f

A A AV f f f f f f f

τ ω ω π π τ

τ τ ττ τ τ τ

= Π = =

= − + + = − + +

2.3-9

[ ]

/ 2 / 2

( ) ( / )cos( /2) with 2 2 /

( ) sinc( ) sinc( )2 2

sinc( 1) sinc( 1)2

c c c

j j

c c

v t A t t f

e eV f A f f A f f

Aj f f

π π

τ ω π ω π π τ

τ τ τ τ

ττ τ

−

= Π − = =

= − + +

= − − − +

2.3-10

2

2 2 2 2

2( ) ( )cos ( )

1 (2 )1 1

( ) ( ) ( )2 2 1 4 ( ) 1 4 ( )

tc

c cc c

Az t v t t v t Ae

fA A

Z f V f f V f ff f f f

ωπ

π π

−= = ↔+

= − + + = ++ − + +

2.3-11

/ 2 / 2

( ) ()cos( /2) ( ) for 01 2

/ 2 / 2( ) ( ) ( )

2 2 1 2 ( ) 1 2 ( )/ 2 / 2

2 ( ) 2 ( )

tc

j j

c cc c

c c

Az t v t t v t Ae t

j f

e e jA jAZ f V f f V f f

j f f j f fA A

j f f j f f

π π

ω ππ

π π

π π

−

−

= − = ≥ ↔+

−= − + + = +

+ − + +

= −− − − +

2-9

2.3-12

( )

22

( ) ( ) ( ) 2 sinc2

sin2 2( ) 2 (2 ) cos2 2 sin2

2 (2 )

1( ) ( ) sinc2 cos2

2

A tv t t z t z t A f

d d f AZ f A f f f

df df f f

d jAV f Z f f f

j df f

ττ τ

π τπτ π τ πτ π τ

π τ π τ

τ π τπ π

= = Π ↔

= = −

−= = −

−

2.3-13

2 2

22 2 2 2

2( ) ( ) ( )

(2 )

1 2 2( )

2 (2 ) (2 )

b t Abz t tv t v t Ae

b f

d Ab j AbfZ f

j df b f b f

π

π π π

−= = ↔+

= = − + +

2.3-14

( ) [ ]

2

2 3

( ) ( ) ( ) for 02

1 2( )

22 2

t Az t t v t v t Ae t

b j f

d A AZ f

df b j fj f b j f

π

ππ π

−= = ≥ ↔+

= = +− +

2.3-15

2 2

2 2

2 2

2 2

( ) ( / )

2 ( ) ( / )

( ) ( / )

( ) ( / )

1( ) ( )

2( ) ( ) 2

1( ) ( )

2

Both results are equivalent to

bt f b

bt f b

bt f b

bt f b

v t e V f eb

d j fa v t b te e

dt bd f

b te V f ej df jb

bte jf e

π π

π π

π π

π π

ππ

π

− −

− −

− −

− −

= ↔ =

= − ↔

↔ =−

↔ −

2.4-1

2

0

2

0

( ) 0 0

0 22

2 2

t

y t t

AtA d t

A d A t

λ λ

λ λ

= <

= = < <

= = >

∫

∫