Page 216 - Schaum's Outlines - Probability, Random Variables And Random Processes

Chapter 6

Analysis and Processing of Random Processes

6.1 INTRODUCTION
In this chapter, we introduce the methods for analysis and processing of random processes. First,
we introduce the definitions of stochastic continuity, stochastic derivatives, and stochastic integrals of
random processes. Next, the notion of power spectral density is introduced. This concept enables us
to study wide-sense stationary processes in the frequency domain and define a white noise process.
The response of linear systems to random processes is then studied. Finally, orthogonal and spectral
representations of random processes are presented.

6.2 CONTINUITY, DIFFERENTIATION, INTEGRATION
In this section, we shall consider only the continuous-time random processes.

A. Stochastic Continuity:
A random process X(t) is said to be continuous in mean square or mean square (m.s.) continuous if
lim E{[X(t + E) - X(t)I2) = 0 (6.1)
8-0
The random process X(t) is m.s. continuous if and only if its autocorrelation function is continuous
(Prob. 6.1). 1fx(t) is WSS, then it is mas. continuous if and only if its autocorrelation function Rx(r) is
continuous at z = 0. If X(t) is m.s. continuous, then its mean is continuous; that is,
lirn p,(t + E) = pX(t) (6.2)
E+O
which can be written as
lirn E[X(t + E)] = E[lim X(t + E)]
8-0 &-'O
Hence, if X(t) is m.s. continuous, then we may interchange the ordering of the operations of expecta-
tion and limiting. Note that m.s. continuity of X(t) does not imply that the sample functions of X(t)
are continuous. For instance, the Poisson process is m.s. continuous (Prob. 6.46), but sample func-
tions of the Poisson process have a countably infinite number of discontinuities (see Fig. 5-2).

B. Stochastic Derivatives:
A random process X(t) is said to have a m.s. derivative X1(t) if

where 1.i.m. denotes limit in the mean (square); that is,