Page 48 - Introduction to Information Optics

1.5. Signal Analysis

mean-square error can be written as

u(r - a) - 2
J — or,
(1.113)

where RU(T) is the autocorrelation function of the input signal, R id(r) is the
cross-correlation of the input with the desired-output function, and R dd(ty is
the mean-square value of the desired-output function, as given by

U*(t)~\dt. (1.114)

It is trivial to see that the mean-square error is a function of h(t), which is the
central problem to determine h(t') by minimizing the error. The minimization
of function is subject to calculus of variation, which leads to the Wiener-Hopf
integral equation, as given by

- a}da - R id(i] = 0. (1.115)

We note that by following the realizable constraint of a physical filter, strictly
speaking the preceding result should be written as

- RM < '° r T < . (1.116)
[ = 0, for T ^ 0

In view of the Wiener-Hopf equation (1.115) as written by,

i opt(r)R it(r - ff)dff, (1.117)

by which the optimum filter h 0 f (t) is needed to solve.