Simple adaptive IIR filtering

Several authors have suggested various methods of implementing an adaptive IIR filter. This paper suggests that hyperstability and associated positivity concepts can be used to determine the regions of assured convergence for adaptive IIR filter algorithms. This hyperstability interpretation is applied to the adaptive recursive LMS filter, for which it is demonstrated that convergence appears assured if the transfer function formed from the denominator (in z^-1) of the transfer function effectively being modelled and a unity numerator is strictly positive real. Examples show that this plausibly sufficient condition, though not necessary, loosely approximates the region of applicability, which seriously restricts the utility of this attractively simple adaptive synthesis method. Slight modifications, suggested by the hyperstability approach, can favorably alter the positive-real "convergence" region with subsequent dramatic improvement in applicability and convergence speed.