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

8.8 SYSTEMS WITH RANDOM INPUTS

Although many systems have multiple inputs and multiple outputs, in this chapter we focus on single-input single-output (SISO) systems as depicted in Figure 8.9(a). Let the input be random process X(t) which has known properties such as its first and second moments, possibly its pdf, and whether or not it is uncorrelated. Typically, the goal in the analysis or design of a system is to modify X(t) in order to generate an output Y(t) that has some desired characteristics. For example, X(t) might contain a communication signal embedded in additive noise, and the goal of the system is to filter X(t) such that Y(t) has reduced noise, and without adversely affecting the underlying signal of interest. Figure 8.9(b) shows the configuration for an equalizer which is designed to compensate for the distorting effects of the channel H(z) through which the transmitted sequence X[k] propagates. Figure 8.9(c) shows a system identification configuration where G(z) is a model of H(z), obtained from measurements of the input and output sequences X[k] and Y[k]. Of course, in an actual implementation, the various filters operate on realizations of the random sequences. As was done in previous chapters, a probabilistic representation will be used so that we can describe how the ensemble of realizations are processed.

Figure 8.9 LTI systems. (a) SISO system. (b) Channel equalization. (c) System identification.

Generally, systems can be categorized as being (i) linear ...