SHARF Convergence Properties

A class of stable algorithms for adapting infinite impulse response (IIR) digital filters based on the concepts of nonlinear stability theory prominent in the control literature is emerging. While this class of adaptive filters offers much promise in practical applications, little has been done toward providing a characterization that would guide selection of design parameters such as adaptation constants and error smoothing coefficients. This paper focuses on the simplest well-behaved member of this class of adaptive recursive filters, SHARF. Progression from a local linearization of the nonlinear parameter estimate convergence behavior, through an idealized eigenvalue/eigenvector analysis of the parameter estimate time-varying recursion, to Lyapunov function establishment for the full output and parameter error system reveals the exponential, local, nongradient descent convergence character of SHARF and provides initial insight into the effects of adaptation constants and error smoothing coefficients on these characteristics.