Page 31 - Introduction to Information Optics

1. Entropy Information and Optics

H(Z) = H(Y/X) = - p(c) log 2p(c)dZ.
Jz

The channel capacity can be determined by maximizing the I(X; Y); that is,

C = max —~—, bits/time. (1.55)
T,p(*) T

Under the constraint of the signal mean-square fluctuation that cannot exceed
a specified value S,

2
\*\ p(a)dX ^ S. (1.56)
x
Since each of the vectors a, b, and c are represented by 2TAv continuous
variables, and each c l is statistically independent Gaussianly distributed with
zero mean, and has a variance equal to N 0/2T, we see that

2TAv
2TAv 2TAv
I(X; Y} = 7(y4 ; B ) - £ /(/I,; 5,). (1.57)

Thus, from Eq. (1.43), we have

H(Z) = 2TAvH(C ;), (1.58)
where

If we let N = ffc t = N 0Av, then H(Z) can be written as

ineN \
L
= TAvlog 2 -r \ (1-59)

In view of Eq. (1.42), we see that

H(B t) ^ Iog 2(27tgffj), (1.60)

where the equality holds if and only if /^ is Gaussianly distributed, with zero
mean and a variance equal to of, . Since b = a + c, />(b) is Gaussianly distrib-
uted if and only if p(a) is also Gaussianly distributed with zero mean. The