Page 228 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 61 ANALYSIS AND PROCESSING OF RANDOM PROCESSES

Let
By the Cauchy criterion (see the note at the end of this solution), the m.s. derivative X1(t) exists if

lim E{[Y(t; E,) - Y(t; &,)I2) = 0
El, &2+0

Thus lim E[Y(~; c2)Y(t; E,)] =
El, ~2 -0
provided d2Rx(t, s)/at as exists at s = t. Setting E, = E, in Eq. (6.1 12), we get
lim E[Y2(t; E,)] = lim E[Y2(t; &,)I = R2
El -0 ~2-0
and by Eq. (6.1 1 I), we obtain

Thus, we conclude that X(t) has a m.s. derivative X1(t) if a2Rx(t, s)/at ds exists at s = t. If X(t) is WSS, then
the above conclusion is equivalent to the existence of a2 RX(z)/Z22 at 7 = 0.

Note: In real analysis, a function g(&) of some parameter e converges to a finite value if

This is known as the Cauchy criterion.

6.7. Suppose a random process X(t) has a m.s. derivative X'(t).
(a) Find E[X'(t)].
(b) Find the cross-correlation function of X(t) and X1(t).
(c) Find the autocorrelation function of X1(t).
(u) We have

= lim E I
c+O