Page 226 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 63 ANALYSIS AND PROCESSING OF RANDOM PROCESSES

Similarly, the inverse Fourier transform

should also be interpreted in the m.s. sense. Note that z(Q + 2n) = z(Q) and the properties of
discrete-time Fourier transforms (Appendix B) also hold for discrete-time random signals. For
instance, if Y(n) is the output of a discrete-time LTI system with input X(n), then

where H(i2) is the frequency response of the system.
Let &al, Q,) be the two-dimensional Fourier transform of Rx(n, m):

Then the autocorrelation function of R(Q) is given by (Prob. 6.44)

If X(n) is a WSS random process with autocorrelation function Rx(n, m) = R,(n - m) = R,(k) and
power spectral density Sx(Q), then

Equation (6.106) shows that the Fourier transform of a discrete-time WSS random process is nonsta-
tionary white noise.

Solved Problems

CONTINUITY, DIFFERENTIATION, INTEGRATION
6.1. Show that the random process X(t) is m.s. continuous if and only if its autocorrelation function
Rx(t, s) is continuous.
We can write

Thus, if Rx(t, s) is continuous, then
lim E{[X(t + E) - X(t)I2) = lim {Rx(t + E, t + E) - 2Rx(t + E, t) + Rx(t, t)} = 0
E-0 &+O
and X(t) is m-s. continuous. Next, consider
Rx(t + El, t + E2) - RX(t, t) = E{[X(t + El) - X(t)][X(t + E2) - X(t)])
+ E([X(t + 8,) - X(t)lX(t)) + E([X(t + 6,) - X(t)]X(t))
Applying Cauchy-Schwarz inequality (3.97) (Prob. 3.33, we obtain
Rx(t + El, t + E2) - Rx(t, t) 2 (E{[X(t + E,) - X(t)12) E{[x(~ + c2) - ~(t)]~))"~
+
+
(E{[x(~
+ iE(Cx(t + 8,) - ~(t)]~)~[x~(t)])~~~ - ~(t)]~}~[x~(t)])l/~
Thus if X(t) is m.s. continuous, then by Eq. (6.1) we have
lim Rx(t + el, t + E2) - Rx(t, t) = o
El. EZ-'~
that is, R,(t, s) is continuous. This completes the proof.