Page 225 - Schaum's Outlines - Probability, Random Variables And Random Processes

2 18 ANALYSIS AND PROCESSING OF RANDOM PROCESSES [CHAP 6

6.7 FOURIER TRANSFORM OF RANDOM PROCESSES
A. Continuous-Time Random Processes:
The Fourier transform of a continuous-time random process X(t) is a random process x(.(o) given
by
X(W) = J_bX(ne-jmt dt (6.89)

which is the stochastic integral, and the integral is interpreted as an m.s. limit; that is,

Note that g(w) is a complex random process. Similarly, the inverse Fourier transform

is also a stochastic integral and should also be interpreted in the m.s. sense. The properties of
continuous-time Fourier transforms (Appendix B) also hold for random processes (or random
signals). For instance, if Y(t) is the output of a continuous-time LTI system with input X(t), then

where H(o) is the frequency response of the system.
Let bdq, 03 be the two-dimensional Fourier transform of Rx(t, s); that is,

Then the autocorrelation function of z(w) is given by (Prob. 6.41)

If X(t) is a WSS random process with autocorrelation function Rx(t, s) = Rx(t - s) = R,(z) and power
spectral density SAW), then (Prob. 6.42)

Equation (6.99) shows that the Fourier transform of a WSS random process is nonstationary white
noise.

B. Discrete-Time Random Processes:
The Fourier transform of a discrete-time random process X(n) is a random process X(0) given by
(in m.s. sense)