The mathematical background reviewed in Chapter 13 provides the basis for developing the statistical description of random functions (signals) known as random processes. Basically, we mathematically describe such signals by either a correlation function or a power spectral density, which are related through the Fourier transform. Among many other areas, the theory of random processes is essential to the field of statistical communication theory.

14.1 Introduction

In Chapter 13 we introduced the concept of random variable and its related probability distribution. A natural generalization of the random variable concept is that of random process. A random process, also called a stochastic process, is a collection of time fonctions and an associated probability description. The entire collection of such functions is called an ensemble. Ordinarily, we represent any particular member of the ensemble by simply x(t), called a sample function or realization. For a fixed value of time, say t₁, the quantity x₁ = x(t₁) can then be interpreted as a random variable.

A continuous random process is one in which the random variables x₁,x₂, …, can assume any value within a specified range of possible values. A discrete random process is one in which the random variables can assume only certain isolated values (possibly infinite in number).

One of the most common random process occurring in engineering applications is random noise, e.g., a “randomly” fluctuating voltage or current at the input to a receiver that interferes with the reception of a radio or radar signal, or the current through a photoelectric detector, and so on. Although we limit our discussion to random processes of time t, it is possible to readily extend these ideas to the notion of a random field, which in general is a fonction of both time t and space R = (x,y,z). Atmospheric wind velocity, temperature, and index of refraction fluctuations are typical examples of random fields.

14.2 Probabilistic Description of Random Process

If we imagine “sampling” the random process x(t) at a finite number of times t₁,t₂,…,t_n, then we obtain the collection of random variables x_k = x(t_k), k = 1,2,…,n. The probability measure associated with these random variables is described by the joint probability density function (PDF) of order n:

In principle, we can develop the theory of a continuous random process by describing the joint probability density fonction of all orders. However, this is generally an impossible task, so we usually settle for only first-order and/or second-order distributions. We denote the first-order PDF by f_x(x,t) and the second-order PDF by f_x(x₁,t₁;x₂,t₂).

DOI: 10.1117/3.467443.ch14

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