Page 246 - Probability, Random Variables and Random Processes

238 ANALYSIS AND PROCESSING OF RANDOM PROCESSES [CHAP 6

Then

since Xn's are orthogonal; that is, E(Xm X,*) = 0 if m # n. But by Eq. (6.84),

Thus, equating Eqs. (6.165) and (6.166), we obtain

where An = E( I Xn 12).

6.37. Let z(t) be the Karhunen-Lokve expansion of X(t) shown in Eq. (6.82). Verify Eq. (6.88).
From Eqs. (6.166) and (6.86), we have

Now by Eqs. (6.83), (6.84), and (6.1 67) we obtain

6.38. Let z(t) be the Karhunen-Lobe expansion of X(t) shown in Eq. (6.82). Verify Eq. (6.85).
From Eq. (6.82), we have

Using Eqs. (6.1 67) and (6.168), we have

since by Mercer's theorem [Eq. (6.87)]