A Signal Theoretic Introduction to Random Processes

9THE AUTOCORRELATION FUNCTION

9.1 INTRODUCTION

As is evident from the previous chapter, a complete statistical specification of a random process, in general, is very difficult to achieve. It is usual to use partial statistical results, such as the autocorrelation function and power spectral density function, to obtain important information about a random process. This chapter is devoted to the autocorrelation function, and the autocorrelation function for the following random processes is detailed: a sinusoid with random phase, the random telegraph signal, generalized shot noise random processes, signalling random processes, jittered random processes, and the random walk.

9.2 NOTATION AND DEFINITIONS

The mean, variance, and autocorrelation function of a random process have been defined in Chapter 7. For notational convenience, in this chapter, the argument T is dropped from the autocorrelation functions, and a subscript of XX for a random process X is added, that is, R(T, t₁, t₂) is written as R_XX(t₁, t₂). The definitions and notation are as follows.

Definition: Mean, Variance, and Autocorrelation

For a random process defined on the interval [0, T], the mean, variance, and autocorrelation functions are, respectively, defined according to

(9.1)

9.2.1 Autocorrelation and ...