Page 247 - Probability, Random Variables and Random Processes

CHAP. 61 ANALYSIS AND PROCESSING OF RANDOM PROCESSES

and A, = E(IXJ2) = 1.;

6.39. Find the Karhunen-Lokve expansion of the Wiener process X(t).

From Eq. (5.64),
.
<.

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

Differentiating Eq. (6.1 70) with respect to t, we get

Differentiating Eq. (6.1 71) with respect to t again, we obtain

A general solution of Eq. (6.1 72) is
t
+,(t) = a, sin o,, + bn cos con t con = a/fi
In order to determine the values of a,, b,, and A,, (or on), we need appropriate boundary conditions. From
Eq. (6.1 70), we see that c$,(O) = 0. This implies that b, = 0. From Eq. (6.1 71), we see that &(T) = 0. This
implies that
a (2n - 1)n (n - %)n
%I==-=- n = 1, 2, ...
A 2T T
Therefore the eigenvalues are given by
a2T2
I, = n = I., 2, ...
(n - &)'n2
The normalization requirement [Eq. (6.8311 implies that

a: T = -+an = 4
[(a, sin ant)' dt = I
2
Thus, the eigenfunctions are given by
sin - )
t = t o < t < T

and the Karhunen-Lokve expansion of the Wiener process X(t) is

t = f , , sin - i) i 0 c t < T
n= 1
where X, are given by

xn = 8 Fx(t) sin(. - i) t

and they are uncorrelated with variance in.