Page 224 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 61 ANALYSIS AND PROCESSING OF RANDOM PROCESSES

where X, are r.v.'s given by

Note that, in general, Xn are complex-valued r.v.'s. For complex-valued r.v.'s, the correlation between
two r.v.'s X and Y is defined by E(XY*). Then 2(t) is called the m.s. Fourier series of X(t) such that
(Prob. 6.34)

Furthermore, we have (Prob. 6.33)

C. Karhunen-Ldve Expansion
Consider a random process X(t) which is not periodic. Let *(t) be expressed as

where a set of functions {+,(t)) is orthonormal on an interval (0, T) such that

and X, are r.v.'s given by

Then %(t) is called the Karhunen-Lokve expansion of X(t) such that (Prob. 6.38)

Let Rdt, s) be the autocorrelation function of X(t), and consider the following integral equation :

where A, and 4,(t) are called the eigenvalues and the corresponding eigenfunctions of the integral
equation (6.86). It is known from the theory of integral equations that if RJt, s) is continuous, then
4,(t) of Eq. (6.86) are orthonormal as in Eq. (6.83), and they satisfy the following identity:

which is known as Mercer's theorem.
With the above results, we can show that Eq. (6.85) is satisfied and the coeficient X, are orthog-
onal r.v.'s (Prob. 6.37); that is,