Page 245 - Probability, Random Variables and Random Processes

CHAP. 61 ANALYSIS AND PROCESSING OF RANDOM PROCESSES

From Eq. (6.77), we have

Now, by Eqs. (6.81) and (6.162), we have

Using these results, finally we obtain

since each sum above equals Rx(0) [see Eq. (6.75)].

6.35. Let X(t) be m.s. periodic and represented by the Fourier series [Eq. (6.77)]

Show that

From Eq. (6.81), we have

Setting z = 0 in Eq. (6.75), we obtain

Equation (6.1 63) is known as Parseval's theorem for the Fourier series.

6.36. If a random process X(t) is represented by a Karhunen-Loeve expansion [Eq. (6.82)]

and Xn's are orthogonal, show that 4,(t) must satisfy integral equation (6.86); that is,

Consider