Page 232 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 63 ANALYSIS AND PROCESSING OF RANDOM PROCESSES

Now by Eq. (6.1 22),

Using assumptions 1,3, and 4 of the Wiener process (Sec. 5.7), and since s I a I t, we have

Finally, for 0 I B 5 s,

Substituting these results into Eq. (6.1 23), we get

Since R,(t, s) = RY(s, t), we obtain

POWER SPECTRAL DENSITY
6.13. Verify Eqs. (6.1 3) and (6.1 4).
From Eq. (6.12),

Setting t + T = s, we get

Next, we have
E{[X(t) + X(t + z)I2) 2 0
Expanding the square, we have
E[X2(t) + 2X(t)X(t + z) + x2(r + T)] 2 0
+
or ~[x~(t)] 2E[X(t)X(t + t)] + ~[Jr:'(t + T)] r 0
Thus 2Rx(0) +_ 2RX(z) 2 0
from which we obtain Eq. (6.14); that is,

6.14. Verify Eqs. (6.18) to (6.20).