Page 29 - Introduction to Information Optics

14 1. Entropy Information and Optics

be disturbed by an additive white Gaussian noise, and a band-limited time-
continuous signal, with an average power not to exceed a given value S, is
applied at the input end of the channel.
Note that, if a random process is said to be a stationary Gaussian process,
then the corresponding joint probability density distribution, assumed by a
time function at any finite time interval, is independent of the time origin, and
it has a Gaussian distribution. If a stationary Gaussian process is said to be
white, then the power spectral density must be uniform (constant) over the
entire range of the frequency variable.
Let us denote c(i) be a white Gaussian noise. By the Karhunen-Loeve
expansion theorem [1.5, 1.6], c(t) can be written over a time interval — 77
2 < t < 7/2:

C(t) == £ c/0,-(0, (1-48)
i = — oo

where the <^(0's are orthonormal functions that can be represented by

CTI2 ^ - = ^

J - r/2 (A J ^ A

and c { are real coefficients commonly known as orthogonal expansion coeffi-
cients. Furthermore, the c/s are statistically independent, and the individual
probability densities have a stationary Gaussian distribution, with zero mean
and the variance equal to N 0/2T, where N 0 is the corresponding power spectral
density.
Now we consider an input time function a(t) as applied to the communica-
tion channel, where the frequency spectrum is limited by the channel band-
width Av. Since the channel noise is assumed an additive white Gaussian noise,
the output response of the channel is

b(t) = a(t) + c(t). (1.50)

Such a channel is known as a band-limited channel with additive white
Gaussian noise.
Again by the Karhunen-Loeve expansion theorem, the input and output
time functions can be expanded:

a(t) =