Page 30 - Introduction to Information Optics

1.3. Communication Channel

and

Thus, we see that

6,. = <*, + £,.. (1.51)

Since the input function a(t) is band-limited by Av, only 2TAv coefficients a,,
/ = 1, 2, ..., 2TAv, within the passband are considered. In other words, the
input signal ensemble can be represented by a 2rAv-order product ensemble
2TAv
over a; that is, ^ . This is similarly true for the output ensemble over b; that
is, B 2TAv . Thus, the average mutual information between the input and output
ensembles is

It is also clear that a, b, and c each form a 2 TAv -dimensional vector space, for
which we write

b = a + c. (1.53)

If we let />(a) and p(c) be the probability density distribution of a and c
respectively, then the transitional probability density of p(b/si) is

where a and c are statistically independent. For simplicity, we let X = A 2TAiV
be the vector space (the product space) of a. The probability density distribu-
tion of b can be determined by

where the integral is over the entire vector space X.
Similarly, 7 4 5 2TAv and Z 4 C 2TAv represent the vector space of b and c,
respectively. The average mutual information can therefore be written by

I(X; 7) = H(Y) - H(Z\ (1.54)
where

H(Y)=-{ p(b)\oB 2p(b)dY,
JY