Stability of adaptive recursive filters

It is shown that the stochastic error gradient cannot be computed exactly for adaptive recursive filters because of parameter time variations. The usual pseudogradient, (G), which recursively filters the increment, is based on the assumption of slow variations, which is false when the stability boundary is crossed. In the example of adaptive prediction with narrowband inputs, it is shown that (G) may eventually yield an unbounded output. The recursive LMS suppresses the increment filtering and ensures a bounded output, due to the self-stabilization phenomenon, without external stabilization procedures although the adaptive poles may momentarily go outside the unit circle