Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

6.1 INTRODUCTION

A random process is a collection of random variables that are indexed by time. Let be the set of time instants over which the random process is defined. This set could be continuous time such as or , or it might be discrete time with or . A discrete-time random process is also called a random sequence or time series, and the time instants are usually equally spaced, though it is not necessary. In this chapter, we often use the term random process in a general sense, to include continuous-time processes as well as discrete-time sequences. When it is necessary to restrict our discussion specifically to discrete time, we will use random sequence. A random process is also known as a stochastic process (stochastic from Greek stokhastikos means “capable of guessing”).

Figure 6.1 shows an example realization of a Gaussian random process. We are interested in exploring various characterizations of such a random process in order to have some understanding of its properties. This ...