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When you think about it, adaptive filters are a
little scary. They process a signal and then decide to adjust
themselves in order to alter the signals characteristics.
How can you be sure that the filter makes the right decision and
doesn't make things worse ? Mathematically, what we are concerned
with here is the *stability* of the filter. The question of
the filters stability arises because of two related properties of
adaptive filters. First, the feedback of an error term makes the
filter behave as a differential equation. It is perfectly valid
for a differential equation to have an unstable solution. If we
have happened to design an adaptive filter whose underlying
differential equation has an unstable solution, then we are in
trouble. Second, since we are implementing our filter in the
discrete time domain the filter implementation becomes a *finite
difference approximation* to the differential equation.
Finite difference equations have their own stability constraints
such that it is possible to have numerically unstable solutions
even when the analytic solution is stable.
Analyzing the stability of an adaptive filter tends to be a deep
exercise in mathematics (involving such things as Laplace transforms,
and numerical analysis). Having a mathematical software package
such as `Mathematica` or `Macsyma` helps a great deal,
but often in the real world such filters are just empirically
tested. Hopefully, a proper analysis is used for filters that will
be installed into safety critical systems.

** Next:** General principles
** Up:** number9
** Previous:** Introduction
Skip Carter
2008-08-20