Page 241 - Schaum's Outlines - Probability, Random Variables And Random Processes

ANALYSIS AND PROCESSING OF RANDOM PROCESSES [CHAP 6

Now, by Eq. (6.48),

and by Eq. (6.72), the power spectral density of Y(n) is

Taking the inverse Fourier transform of Eq. (6.151), we obtain

Thus, by Eq. (6.33), the average power of Y(n) is

6.30. Let Y(t) be the output of an LTI system with impulse response h(t), when X(t) is applied as input.
Show that

(a) Using Eq. (6.56), we have

(b) Similarly,

6.31. Let Y(t) be the output of an LTI system with impulse response h(t) when a WSS random process
X(t) is applied as input. Show that

(a) If X(t) is WSS, then Eq. (6.152) of Prob. 6.30 becomes

which indicates that Rxy(t, s) is a function of the time difference z = s - t only. Hence