Page 224 - Probability, Random Variables and Random Processes

216 ANALYSIS AND PROCESSING OF RANDOM PROCESSES [CHAP 6

When X(n) is WSS, then from Eq. (6.66),

where H(0) = H(Q)I,=, and H(Q) is the frequency response of the system defined by the Fourier
transform of h(n) :

The autocorrelation function of Y(n) is, from Eq. (6.67),

Setting m = n + k, we get

From Eqs. (6.68) and (6.71), we see that the output Y(n) is also WSS. Taking the Fourier transform of
Eq. (6.71), the power spectral density of Y(n) is given by (Prob. 6.28)

which is the same as Eq. (6.63).

6.6 FOURIER SERIES AND KARHUNEN-LOEVE EXPANSIONS

A. Stochastic Periodicity :
A continuous-time random process X(t) is said to be m.s. periodic with period T if
E([X(t + T) - X(t)I2) = 0 (6.73)
If X(t) is WSS, then X(t) is m.s. periodic if and only if its autocorrelation function is periodic with
period T; that is,

RX(z + T) = RX(z) (6.74)

B. Fourier Series :
Let X(t) be a WSS random process with periodic RX(z) having period T. Expanding RX(z) into a
Fourier series, we obtain

where

Let T(t) be expressed as