Existence of stationary points for reduced-order hyperstable adaptive IIR filters

We establish the existence of asymptotic stationary points for a class of adaptive IIR filtering algorithms, including (S)HARF, the Feintuch (1996) algorithm, and Landau´s (1976) algorithm, for reduced-order cases. We show first that the nonlinear equations characterizing a stationary point admit a solution giving rise to a stable transfer function, when the input is white noise. We then show that an analytic procedure to construct the solution may be reduced to the Nevanlinna-Pick interpolation problem. The white noise assumption on the input simplifies the mathematics of an already difficult problem, although the existence proof appears extendable to correlated inputs as well