Page 28 - Introduction to Information Optics

1.3. Communication Channel $ 3

the mean-square fluctuation of the output signal. Thus, for a given mean-
square value of of, one can show that for the corresponding entropy, derived
from p(b), there exists an upper bound:

2
H(B)^ilo g2 (27re<r c ), (1,42)
where the equality holds if and only if the probability density p(b) is Gaussianly
distributed with zero mean and the variance equals to of.
From the additivity property of the channel noise H(B/A\ which is
dependent solely on p(c), we see that

2
H(B/A) ^ i Iog 2(27te<i ), ( 1 .43)

where the equality holds when p(c] is Gaussianly distributed, with zero mean
and the variance equal to of.
Thus, if the additive noise in a memoryless continuous channel has a
Gaussian distribution, with zero mean and the variance equal to N, where N
is the average noise power, then the average mutual information is bounded
by the following inequality:

I(A; B) ^ ilog 2(27E«tf) - $\og 2(2neN) ^ ilog 2 ^. (1.44)

Since the input signal and the channel noise are assumed to be statistically
independent,

at = a% + at = oi + W, (1.45)

we have

/( fl ;&)^log 2 t^ . (1.46)

By maximizing the preceding equation, we have the channel capacity as written
by

(1.47)

where the input signal is Gaussianly distributed, with zero mean and a variance
equal to S.
Let's now evaluate one of the most useful channels; namely, a memoryless,
time-continuous, band-limited, continuous channel. The channel is assumed to