digital & analog communication systems (8th edition) – answer and script solutions manual
TRANSCRIPT
Chapter 1
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--=-~o:::oo!.oIl_
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1
INSTRUCTOR SOLUTIONS MANUAL (United States Edition)
Digital & Analog Comm. Systems 8th Edition, L. W. Couch, II
LET p--The probability for sending a binary 1, then the probability for sending a binary 0 is (l-p). From the entropy formula for H(p), we can draw the figure of R(p) ,and from this figure , we can find the maximum entropy and the p.
R(p)=(p*ln(P)+(1-p)*ln(1-p»j(-ln(2»
k := 0 •• 50 P k
:= k -50
HMAX := 1
h k
-1 :=-
In(2)
2
h ,HMAX I------+-----:=--+----=====--+------i k
o o P 1
k From the .above figure, we know the maximum entropy is 1 where the probability for sending 1 or 0 is 0.5.
INSTRUCTOR SOLUTIONS MANUAL (United States Edition)
Digital & Analog Comm. Systems 8th Edition, L. W. Couch, II
P I
::- o.L5 . ~l..:: ~~ ~ 0,15 .·\1-10 ) )
f 'f = ~ 5 "- P~ :.?) "Pg =- PC[ = D, 0 /
o. L 5" + 2 (O./S-) -I- cD (a. 0)) = 0 •cr I :. ~ :::. t).os ~;""te 2.
/0
p. == ',0'0 ~j ~ I
~ to
H = 2 p. ~~ ~ (t\ ~ [~J 2 p, ~ f· '-, J .)) 11 J j j- ..t:Ao'L ";'::'1
=[i-I"L] [. L5 .Lv. ,?. <; + Ll ) , ,,,. fr- ,/S
+(<0) ,oi t.,.. ,Of + .o~ Jr. ,o"l, ]
H ::: S, 0 g 4- b,'+~
I1-11 ~ ,) ?, = 0, ~ • P?:: 0 , '7
D,3bo.s +O,(~O,I
_~ L
o. gg I bl'ts
(h,) f! IIW>.x (oJ- P; = ~ = ~ = 0 S .i ~ I, 1M
II oM'" j. =- - L ( 6.5) J,.,.. 6," " 1 b,,'f "fI""c.'f-b"L
1 1-12 I M ~ /0 ~, :: -L '=:.. l lo
) 10 .) t
H = - I() (. J) ~ . I =- ~ . ~ (2.. h ;ts )ML
T~ li- :: S,5LL ~;+~ :; /,11 Sft, ::.T f!... 3 lA.; +~ 1st. c.
3
H =
INSTRUCTOR SOLUTIONS MANUAL (United States Edition)
Digital & Analog Comm. Systems 8th Edition, L. W. Couch, II
11 13 - I,~ LI~;r .,(11;-& /w...I 1>1=2.'1- ) ~ =~);r ~ ~ i e,,~fl j=tJt
U.. ~ 1J I,,~ -/'71..
=- f2"[ Ii} I] /11 (1)'1 -/- 3(t/'jyl 3(1)/1] - 1¥1l...
C~ecJ;. t'P: ~ l'l[U/~ J{i.F~ =9 J-J::. 1/. BII> Ms oJ"l i '=- 2. U ( ti(tJ'~:: /. 0
1 14 1 - I B -" 3~o U-z t~ )"8 "3tlJ B
c= ~ Jp~l.[l-l~)] ~ /;1. /11 [if ~j *~)J8/b I.,j-fr) (sIAlMB 3
-q- iJ:: ID J~ :::. (D':: II> Db
c.. =I:I>~ l~ [lDllQ - 1. 'l'1~ Ii)~ ;
1-15 1
(a) chars := 110 Number of characters available
log (chars) ]b := ceil Number of bits required to represent a character[ log(2)
====> b = 7 bits
(b) B:= 3200 Hz Channel bandwidth SNRdB := 20 dB Signal to noise ratio
SNRdB
10 SNR :;:;:: 10 ==;~===> SNR = 100 (Absolute power ratio)
C := B. [log (1 + SNR)] 4
==> C = 2.131·10 Channel capacity (bits/sec) log(2)
C 3 C :::::: ===> C = 3.044 10 Channel capacity (chars/sec)
b
(c) Assuming equally likely characters, information content of each character is:
1 P := ----- Probability of each character
chars
log[~] 1;=- =========> I = 6.781 bits
109(2)
4
INSTRUCTOR SOLUTIONS MANUAL (United States Edition)
Digital & Analog Comm. Systems 8th Edition, L. W. Couch, II
1 1-16 I R (S ) [()~ 1-' (I + J D~o/I~ ) :. EI~ I t i1 ::. 'B J,,~ /. t1.) :: 'B ~ Jt,'l4q)
(a) ~ J:Lt)O-3DlY~ crbD ~ R'::.(qOQ) (t4l/i/f) :: 13.1-5 k~/fJ"fr
(6] B::. 31t)~ -U;on-='/7oo:1 R:::.V7tJ~(j~91-q) '= 7-5~ 'H kl;cfs!r
(c) :B-=- 32. DO'" jDe>=' 2'tbTJ ~ ~~ (;l9~ (I'f. 9'f-Jl)=-~3. 35~ki!J.!r
[i -17=-J ~ t; ~ sL;ff. re1lsto- Ur\c.oJ~J Jt4+o. 1~
l!ti,ff' ,~ 1 b~-h ~t
(S~/ff ()lfi-
"-h~) .......,.--L..-,~-r--.L,--J.;,--L.-r-l
COd~J :l~t~ o~t
. . ~ ~',Is 4t (,. 1i;"e.)]r, f sf,,(j Y(3,rrey1-18
x .- 1 x .- 0 x .- 1 x .- 1 x .- 1 Input vector 0 1 2 3 4
ga .- 1 ga .- 0 ga .- 0 ga .- 1 Gain vector, mod2 adder 0 1 2 3
gb .- 1 gb .- 1 gb .- 1 gb .- 1 Gain vector, mod2 adder 0 1 2 3
k := 0 .. length(ga) - 2 v : = 0 k: = 0 .. length (x) - 1 v : = X
k k+length(ga)-l k k := length (x) + length (ga) - 1 .. length{x) + 2 length (ga) - 3
v := 0 i : = 0 .. length (v) - length (ga) k j := 0 .. length (ga) - 1
sa i
sb v:= V [9b ]i ~ length(gb)-j-l j+i
j
s := sa s := sb 2 i i 2 i+1 i
sb i
i := 0 .. 2 length (x) - 1
out : = S i i
For ====>
x T = (I 0 outT = (1
1 1
1 0
1) 1 1 0 0 Ill)
5
INSTRUCTOR SOLUTIONS MANUAL (United States Edition)
Digital & Analog Comm. Systems 8th Edition, L. W. Couch, II