Page 217 - Schaum's Outlines - Probability, Random Variables And Random Processes

210 ANALYSIS AND PROCESSING OF RANDOM PROCESSES [CHAP 6

The m.s. derivative of X(t) exists if a2~,(t, s)/& as exists (Prob. 6.6). If X(t) has the m.s. derivative
X1(t), then its mean and autocorrelation function are given by

Equation (6.6) indicates that the operations of differentiation and expectation may be interchanged. If
X(t) is a normal random process for which the m.s. derivative X'(t) exists, then X'(t) is also a normal
random process (Prob. 6.10).

C. Stochastic Integrals:
A m.s. integral of a random process X(t) is defined by

whereto < t, < tand At, = ti+l -ti.
The m.s. integral of X(t) exists if the following integral exists (Prob. 6.1 1):

This implies that if X(t) is m.s. continuous, then its m.s. integral Y(t) exists (see Prob. 6.1). The mean
and the autocorrelation function of Y(t) are given by

Equation (6.1 0) indicates that the operations of integration and expectation may be interchanged. If
X(t) is a normal random process, then its integral Y(t) is also a normal random process. This follows
from the fact that Z, X(ti) Ati is a linear combination of the jointly normal r.v.'s. (see Prob. 5.60).

6.3 POWER SPECTRAL DENSITIES
In this section we assume that all random processes are WSS.

A. Autocorrelation Functions:
The autocorrelation function of a continuous-time random process X(t) is defined as [Eq. (5.7)]

Properties of RAT):