CHAPTER 5

Stationary Random Processes

This chapter discusses elementary and advanced concepts from stationary random processes theory to form a foundation for applications to analysis and measurement problems as contained in later chapters and in Refs 1–3. Material includes theoretical definitions for stationary random processes together with basic properties for correlation and spectral density functions. Results are stated for ergodic random processes, Gaussian random processes, and derivative random processes. Nonstationary random processes are covered in Chapter 12.

**5.1 BASIC CONCEPTS**

A *random process* {*x _{k}*(

*t*)}, −∞ <

*t*< ∞ (also called a

*time series*or

*stochastic process*), denoted by the symbol { }, is an ensemble of real-valued (or complex valued) functions that can be characterized through its probability structure. For convenience, the variable

*t*will be interpreted as time in the following discussion. Each particular function

*x*(

_{k}*t*), where

*t*is variable and

*k*is fixed, is called a

*sample function.*In practice, a sample function (or some time history record of finite length from a sample function) may be thought of as the observed result of a single experiment. The possible number of experiments represents a sample space of index

*k*, which may be countable or uncountable. For any number

*N*and any fixed times

*t*

_{1},

*t*

_{2},…,

*t*, the quantities

_{N}*x*(

_{k}*t*

_{1}),

*x*(

_{k}*t*

_{2}),…,

*x*(

_{k}*t*), represent

_{N}*N*random variables over the index

*k.*It is required that there exist a well-defined

*N*-dimensional probability ...