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

8.1 INTRODUCTION

In this chapter, we consider systems that filter a random process to achieve a particular goal, such as removing additive noise or compensating for channel distortion in communications. Generally, we are interested in linear time-invariant (LTI) systems that are described by an impulse-response function h(t) and the corresponding transfer function H(ω). Such systems can be represented by a linear differential equation (DE) with constant coefficients, so they include a combination of integration and differentiation of the signals. In Chapter 7, we considered mean-square (MS) definitions of derivatives and integrals of a random process, which led to an evaluation of the autocorrelation function RXX(t1, t2). Table 8.1 summarizes the results for MS continuity, derivatives, and integrals; the wide-sense stationary case, where the autocorrelation function RXX(τ) depends only on the time lag , is also included. In this chapter, we focus on wide-sense stationary processes such that

(8.1)

for all . Techniques are developed for investigating how a random process is modified when it is processed by an LTI system, in terms of its time response as well as its frequency response. ...