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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 highperformance 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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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 quasilinearization 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 timehonored procedure for dealing in a practical way with nonlinear
systems is to construct a linear model to approximate the nonlinearities.
Thus, for example, a smallsignal 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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 Spring '04
 EricFeron
 The Land

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