Signal Processing and Integrated Circuits

Chapter 7

Stochastic Signals and Power Spectra

7.1 Introduction

So far in our discussion of signals and systems, it has been tacitly assumed that the signals are defined by analytic expressions, differential equations, difference equations or even arbitrary graphs. Such signals are called deterministic. The same is true for deterministic systems described by differential equations, difference equations or any functional. However, most signals are random, or at best contain random components due to factors such as noise introduced by the generating sources or by the channel over which the signals are transmitted. Such signals require the use of statistical methods for their description; this consideration leads to the area of stochastic signal processing [11, 12]. This chapter gives an introduction to the concepts and techniques suitable for the description of stochastic (random) signals. The discussion encompasses both analog and digital signals. However the systems which perform the processing of these signals are themselves deterministic.

7.2 Random Variables

Consider performing a certain experiment a number of times, and each time an outcome ζ_i(i = 1, 2, …) results. Thus, we obtain a set Λ of possible outcomes ζ₁, ζ₂, … which can be finite or (theoretically) infinite in number. We then assign a number f(ζ) to each ζ according to some rule. This way we construct a function f(ζ) or simply f whose domain is the set Λ and whose range is a set of numbers f(ζ₁), f(ζ₂) …. This function ...