Page 233 - Schaum's Outlines - Probability, Random Variables And Random Processes

ANALYSIS AND PROCESSING OF RANDOM PROCESSES [CHAP 6

By Eq. (6.1 7),
Rxy(-r) = E[X(t)Y(t - T)]
Setting t - z = s, we get
Rxu(-r) = E[X(s + ~)Y(s)] = E[Y(s)X(s + T)] = RYx(z)
Next, from the Cauchy-Schwarz inequality, Eq. (3.97) (Prob. 3.35), it follows that
+
{E[X(t)Y(t + T)]}~ E[X2(t)]~[Y2(t T)]
5
or CRx Az)l 5 Rx(O)R AO)
from which we obtain Eq. (6.19); that is,
I RXYW I 5 JKKKiW
Now E{[X(t) - Y(t + z)I2) 2 0
Expanding the square, we have
-
E[x~(~) 2X(t) Y(t + z) + y2(t + z)] 2 0
or E[X2(t)] - 2E[X(t)Y(t + 2)] + E[Y2(t + z)] 2 0
Thus Rx(0) - 2Rxy(z) + Ry(0) 2 0
from which we obtain Eq. (6.20); that is,
RXY(4 l 3CRx(O) + RY(0)I

6.15. Two random processes X(t) and Y(t) are given by

where A and o are constants and O is a uniform r.v. over (0, 274. Find the cross-correlation
function of X(t) and Y(t) and verify Eq. (6.18).
From Eq. (6.1 7), the cross-correlation function of X(t) and Y(t) is

A2
-
-- sin oz = RXy(z)
2
Similarly,

A2
= -- sin wr = RYx(z)
2
From Eqs. (6.1 25) and (6.1 26), we see that
A2 A2
Rxy(-2) = - sin a(-r) = - - sin oz = R,,(T)
2 2
which verifies Eq. (6.1 8).

6.16. Show that the power spectrum of a (real) random process X(t) is real and verify Eq. (6.26).