Time-varying performance surfaces for adaptive IIR filters: geometric properties and implications for filter stability

We present a new framework for understanding the performance of adaptive IIR filters which enhances the understanding of filter stability during on-line operation. This new understanding arises from examining geometric properties of time-varying performance surfaces which are defined by the data, rather than the standard steady-state error surfaces as defined by statistics of the data. Data-dependent descent directions used in adaptive algorithms to update filter coefficients typically are viewed as functions of gradients defined on fixed performance surfaces, and data is used to approximate these gradients at each iteration. In contrast, we view data-dependent descent directions at each iteration as functions of exact gradients on time-varying performance surfaces. By examining the shape of these time-varying performance surfaces near filter stability boundaries, we are able to identify the origin of on-line stability problems associated with existing adaptive IIR filtering formulations, and suggest corrective measures. Specifically, by using exact z-domain methods, we define time-varying performance surfaces which geometrically enforce filter stability, and maintain the geometric and physical properties of the “true” error surface at each iteration. Development of adaptive algorithms based on this measure is expected to result in adaptive filters having improved stability performance during on-line operation