B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

766 INTRODUCTION TO INFORMATION THEORY
We have assumed that the mean square value of the signal x(t) is constrained to have a value
S, and the mean square value of the noise is N. We shall also assume that the signal x(t) and
the noise n(t) are independent. In such a case, the mean square value of y will be the sum of
the mean square values of x and n. Hence,
y 2 = S +N
For a given noise [given H(n)], /(x; y) is maximum when H(y) is maximum. We have seen
that for a given mean square value of y (y 2 = S + N), H (y) will be maximum if y is Gaussian,
and the maximum entropy Hmax(Y) is then given by
1
Hmax (Y) = log [2ne(S + N)]
2
(13.61)
Because
y=x+ n
and n is Gaussian, y will be Gaussian only if x is Gaussian. As the mean square value of x is
S, this implies that
and
lmax (x; y) = Hmax (y) - H (n)
1
= log [2ne(S + N)] - H (n)
2
For a white Gaussian noise with mean square value N,
1
H(n) = 2
Iog 2neN
N=NB
and
(13.62a)
(13.62b)
The channel capacity per second will be the maximum information that can be transmitted
per second. Equations (13.62) represent the maximum information transmitted per sample.
If all the samples are statistically independent, the total information transmitted per second
will be 2B times C s . If the samples are not independent, then the total information will be
less than 2BC s . Because the channel capacity C represents the maximum possible information
transmitted per second,
C = 2B [ log ( 1 + ! ) ]
= Blog ( 1 + ! ) bit/s (13.63)