Random Processes: Filtering, Estimation, and Detection

6 Nonlinear Systems: Random Processes

 

6.1 INTRODUCTION

In Chapter 5 the input–output characteristics were explored for a linear system excited by a random process. It was seen that the mean of the input process was sufficient to determine the mean of the output process, and correspondingly the autocorrelation function of the input was sufficient to determine the autocorrrelation function of the output process. However, there are many elements and systems, especially in communication theory, that are not linear, for example, devices like hard and soft limiters, rectifiers, modulators, and demodulators.

For these type of nonlinearities the autocorrelation functions are, in general, no longer sufficient to characterize the output autocorrelation functions, so it is necessary to introduce the concepts of higher-order correlation functions, cumulants, and higher-order spectrums. This chapter will also identify various classes of nonlinear systems and analyze them with respect to establishing input–output statistical relationships. The presentation will be guided more by what is mathematically tractable than what would be a complete and thorough investigation.

 

6.2 CLASSIFICATION OF NONLINEAR SYSTEMS

Currently no general theory exists that can handle all types of nonlinear systems. Therefore our presentation will contain special methods for analyzing certain classes of nonlinear systems. A hierarchical classification of nonlinear systems has been presented in Zadeh [12], and although ...