4.1 INTRODUCTION

In Chapter 3, we developed the concepts of continuous-time processes and described briefly the Markov process. In this chapter, we consider another class of random processes; namely, the discrete-time stochastic processes. A discrete random process may be a uniformly sampled version of a continuous-time process. A discrete random process is a correspondence that maps the sample space into a discrete-domain-functional space; that is, a functional space whose member functions are defined in a discrete set (time samples). Hence, it is a collection or an ensemble of real or complex discrete sequences of time, also called realizations, and denoted X(n). Many authors use the notation . In our case, we keep X( n) to be consistent with the notation X( t) of a continuous-time random process. Note that for the convenience of notation, we normalize the time with respect to the sampling period. Hence, for a fixed n, X( n) represents a random variable. One particular ensemble is the discrete-time series or just time series, where, for example, the sequence X( n), X( n ?1), , X( n ? M + 1), representing a time series, consists of the present observation X( n) and past ( M ?1) observation at times n ?1, n ?2, , n ?! M + 1. In fact, many discrete-time random processes are best approximated...