Page 238 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 61 ANALYSIS AND PROCESSING OF RANDOM PROCESSES

(b) Find the power spectral density S,(Q of Y(n).
(a) The mean of Y(n) is
E[Y(n)] = E[X(n)] + E[W(n)] = E(A) + E[W(n)] = 0
The autocorrelation function of Y(n) is
Ry(n, n + k) = E{[X(n) + W(n)][X(n + k) + W(n + k)])
= E[X(n)x(n + k)] + ECX(n)]E[W(n + k)] + E[W(n)]E[X(n + k)] + E[W(n) W(n + k)]
= E(A2) + Rw(k) = a,' + a26(k) = Ry(k) (6.1 46)
Thus Y(n) is WSS.
(b) Taking the Fourier transform of Eq. (6.146), we obtain

RESPONSE OF LINEAR SYSTEMS TO RANDOM INPUTS
6.24. Derive Eq. (6.58).
Using Eq. (6.56), we have

6.25. Derive Eq. (6.63).
From Eq. (6.62), we have

Taking the Fourier transform of RAT), we obtain

Letting r + a - #? = 1, we get

6.26. A WSS random process X(t) with autocorrelation function