Page 248 - Probability, Random Variables and Random Processes

240 ANALYSIS AND PROCESSING OF RANDOM PROCESSES [CHAP 6

6.40. Find the Karhunen-Loeve expansion of the white normal (or white gaussian) noise W(t).
From Eq. (6.43),

Substituting the above expression into Eq. (6.86), we obtain

or [by Eq. (6.44)]

which indicates that all I, = o2 and &,(t) are arbitrary. Thus, any complete orthogonal set {4,(t)) with
corresponding eigenvalues A, = o2 can be used in the Karhunen-LoCve expansion of the white gaussian
noise.

FOURIER TRANSFORM OF RANDOM PROCESSES
6.41. Derive Eq. (6.94).

From Eq. (6.89),

Then

in view of Eq. (6.93).

6.42. Derive Eqs. (6.98) and (6.99).
Since X(t) is WSS, by Eq. (6.93), and letting t - s = z, we have

From the Fourier transform pair (Appendix B) 1 ++ 2nh(o), we have

Next, from Eq. (6.94) and the above result, we obtain