Page 244 - Probability, Random Variables and Random Processes

ANALYSIS AND PROCESSING OF RANDOM PROCESSES [CHAP 6

Again taking the inverse Fourier transform of both sides and using the result of part (a), we have
d d
RX*(z) = - - RXx,(z) = - -
dz' R~(z)
dz
(c) From Eq. (6.63),
Sxr(o) = ( H(o) (2Sx(o) = (jo (2SX(o) = o2SX(w)
Note that Eqs. (6.159) and (6.160) were proved in Prob. 6.8 by a different method.

FOURIER SERIES AND KARHUNEN-LOEVE EXPANSIONS
6.33. Verify Eqs. (6.80) and (6.81).

From Eq. (6.78),

Since X(t) is WSS, E[X(t)] = p,, and we have

Again using Eq. (6.78), we have

Now

Letting t - s = z, and using Eq. (6.76), we obtain

Thus

6.34. Let z(t) be the Fourier series representation of X(t) shown in Eq. (6.77). Verify Eq. (6.79).