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Summary of some new results on the performance of a well-known class of adaptive filters

The analog adaptive filters, nonlinear in general, which are considered here have been proposed in connection with the problem of identifying an unknown system with a particular linear structure from measurements of its input and output. We show first that under ideal conditions, and provided the input signal satisfies a nondegeneracy ("mixing") condition then the norm of the vector of differences between the actual and the estimated coefficients converges exponentially to zero. Both upper and lower bounds on the rate of convergence are derived and both bounds exhibit an identical and rather unexpected dependence on the gain of the filters' control loop. These results are subsequently used to bound the behavior of the filter for various perturbations from the ideal situation as when noise is present, the unknown system is time varying and the analog integrators have finite memory.