Random Processes: Filtering, Estimation, and Detection

5 Linear Systems: Random Processes

 

5.1 INTRODUCTION

In many scientific disciplines the description of input–output relationships for various types of systems is paramount to understanding the system. The inputs could be many different things. However, they are many times considered some form of excitation to the system, with the output representing the response of the system. System description can take many forms. For example, the input and output could be coupled through a differential equation or transfer function or impulse response. When the input to a system can be thought of as a realization from a particular random process, the output signal can be determined by using the system’s definition. Each realization of the process in turn generates an output signal. In this way there exists a mapping through the system from a set of outcomes governing the input random process and thus defining an output random process.

In this chapter we will explore the relationships that exist between input process characterizations and output process characterizations for a special class of systems called linear systems, while in the next chapter, we will explore the same problem for nonlinear systems. For example, the following question is the main theme: Knowing the mean, autocorrelation function, first-order density, and stationarity of the input process, what are the mean, autocorrelation function, first-order density, stationarity of the output process? It will be seen that a particular ...