The document appears to discuss topics related to digital and analog communication systems, specifically focusing on entropy in information theory and probabilities for signal transmission. It covers calculations related to channel capacity and information content, including maximum entropy at a probability of 0.5 for binary systems. The content seems to be from an instructor solutions manual for the 8th edition of 'Digital & Analog Communication Systems' by L.W. Couch II.

1. Chapter 1
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[ 1-2 J U.rin, (1-/), d. ~
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1
INSTRUCTOR SOLUTIONS MANUAL
(United States Edition)
Digital & Analog Comm. Systems
8th Edition, L. W. Couch, II

2. 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

3. PI
::- o.L5 . ~l..:: ~~ ~ 0,15 .·1-10 ) )
f'f = ~5 "- P~ :.?) "Pg =- PC[ = D, 0 /
o. L5" + 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
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_~ L
o. gg I bl'ts
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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
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)ML
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f!... 3 lA.;+~ 1st.c.
3
H =
INSTRUCTOR SOLUTIONS MANUAL
(United States Edition)
Digital & Analog Comm. Systems
8th Edition, L. W. Couch, II

4. 11 13
- I,~ LI~;r .,(11;-&/w...I 1>1=2.'1- ) ~=~);r~ ~i e,,~flj=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
1 141 - I B -"3~o U-z t~)"8 "3tlJB
c= ~ Jp~l.[l-l~)] ~ /;1. /11 [if ~j *~)J8/bI.,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

5. 1 1-16 I R (S ) [()~1-' (I + JD~o/I~)
:. EI~ I t i1 ::. 'B J,,~/. t1.) :: 'B ~Jt,'l4q)
(a) ~ J:Lt)O-3DlY~ crbD ~ R'::.(qOQ)(t4l/i/f) :: 13.1-5k~/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
- Urc.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 I l l )
5
INSTRUCTOR SOLUTIONS MANUAL
(United States Edition)
Digital & Analog Comm. Systems
8th Edition, L. W. Couch, II