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7 DESCRIBING FUNCTIONS FOR NONLINEAR SYSTEMS WITH RANDOM INPUTS 7.0 INTRODUCTION The preceding chapters have dealt with approximate descriptions of non- linearities having inputs consisting of the sums of two commonly considered signal forms, sinusoids and constants. We wish now to add a third form to this repertory of input signals, a random process. The study of nonlinear systems with random inputs is of consequence both because the design of many high-performance systems is significantly influenced by the presence of undesired random noise and because the signal inputs expected in many operational circumstances do not permit description in deterministic terms. Realistic input signals often can be characterized only as members of an ensemble of possible input functions, the ensemble having certain statistical properties which are known or can be estimated. Actually, the general theory of describing functions developed in Chap. 1 was couched in statistical terms at the outset. This was done so most signal forms of interest at the nonlinearity input could be included in a single format. To this end, sinusoidal and bias signals have been treated as special
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366 DESCRIBING FUNCTIONS FOR NONLINEAR SYSTEMS WITH RANDOM INPUTS cases of random processes. This led to describing functions for these signals which have the more familiar interpretation in terms of harmonic analysis. In this chapter we consider, in addition to constants and sinusoids, random signals of the ordinary sort which do not have characteristic waveshapes. In this case a statistical approach is clearly essential, and no alternative deterministic interpretation seems possible. There is a considerable litera- ture on the subject of quasi-linearization of nonlinear elements driven by random signals. Much of this is related to random signals having finite power density spectra, which rules out consideration of biases and sinusoids. This literature is briefly reviewed in the following section, after which the calculation and application of describing functions for systems with random inputs is discussed. A review of probability theory, random variables, and random processes appears in Appendix H for convenient reference. 7.1 STATISTICAL LINEARIZATION A time-honored procedure for dealing in a practical way with nonlinear systems is to construct a linear model to approximate the nonlinearities. Thus, for example, a small-signal linearization consists of expanding the nonlinear function in a Taylor series about some operating point and retaining only the linear terms in the analysis. If the signals at the input to the nonlinearity are not small enough to permit this simple form of lineariza- tion, one can do better by allowing the linear approximator to depend on the input.
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