Page 219 - Schaum's Outlines - Probability, Random Variables And Random Processes

212 ANALYSIS AND PROCESSING OF RANDOM PROCESSES [CHAP 6

Similarly, the power spectral density Sx(Q) of a discrete-time random process X(n) is defined as the
Fourier transform of Rx(k):

Thus, taking the inverse Fourier transform of Sx(Q), we obtain

Properties of S#):
1. Sx(Q + 271) = Sx(R)
2. Sx(Q) is real and Sx(Q) 2 0.
3. sx( - n) = s,(q

Note that property 1 [Eq. (6.30)] follows from the fact that e-jm is periodic with period 271. Hence it
is sufficient to define SAR) only in the range (-n, n).

D. Cross Power Spectral Densities:
The cross power spectral density (or cross power spectrum) Sxy(w) of two continuous-time random
processes X(t) and Y(t) is defined as the Fourier transform of RXy(z):

Thus, taking the inverse Fourier transform of Sxdo), we get

Properties of S,(o) :
Unlike Sx(o), which is a real-valued function of o, Sxy(o), in general, is a complex-valued func-
tion.

Similarly, the cross power spectral density S,dQ) of two discrete-time random processes X(n) and
Y(n) is defined as the Fourier transform of Rxy(k):

Thus, taking the inverse Fourier transform of Sxy(R), we get