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

7.1 INTRODUCTION

In this chapter, we consider additional properties of random processes that further characterize their behavior. These include convergence of a random sequence X[k] as , and its counterpart for a random process X(t): continuity at , where is a continuous set of time instants. Derivatives and integrals of a random process are also defined, as well as differential equations (DEs). These are useful because in Chapter 8, we consider signals and linear systems, which are modeled by DEs with constant coefficients. With this background, we can examine how a linear system alters probabilistic features of a random process, especially its correlation. Finally, we describe how a random signal is decomposed into additive components that separately provide greater insight about the original process.

Consider the realization of a Poisson process shown in Figure 7.1. In this chapter, we are interested in different types of continuity for a process that may or may not apply to each realization. Even though the realization shown in the figure is not continuous, there are stochastic definitions of continuity where the Poisson process is considered to be continuous. Various definitions ...