Digital Signal and Image Processing Using MATLAB

Chapter 8

Random Processes

8.1   Introduction

At the beginning of Chapter 7, we pointed out that the concept of random variable was needed to describe with a model the variability of certain phenomena said to be random. Speech signal observed at a microphone’s output is an example. There is no use to try and describe it with a deterministic expression such as x(t) = A cos(2πf₀t), which is relevant however when describing electrical voltage, hence the idea of using random variables for describing the phenomenon at every instant. This leads us to the following definition.

Definition 8.1 A random process is a set of time-indexed random variables X(t) defined in the same probability space. If the possible values for t belong to , the process is called a continuous-time random process. If the possible values fort belong to , then we are dealing with a discrete-time random process¹.

The definition implies that a random process associates a real value called a realization with every instant t and every outcome ω. A random process can therefore be interpreted as two different perspectives (Figure 8.1):

1. either as a set of functions of time, also called trajectories, each one associated with an outcome;
2. or as a set of random variables, each one associated with a given time.

In MATLAB^®, the ...