150132750 45150908 Communication Systems 4th Edition 2002 Carlson Solution Manual PDF
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Transcript of 150132750 45150908 Communication Systems 4th Edition 2002 Carlson Solution Manual PDF
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
= <
= = < <
= = >
∫
∫