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What can go wrong

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