Statistical performance of the memoryless nonlinear gradient algorithm for the constrained adaptive IIR notch filter

Gradient-type algorithms for the adaptive infinite-impulse response (IIR) notch filters are very attractive in terms of both performance and computational requirements for various real-life applications. This paper presents, in detail, a statistical analysis of the memoryless nonlinear gradient (MNG) algorithm applied to the well-known second-order adaptive IIR notch filter with constrained poles and zeros. This analysis is based on a proper use of Taylor series expansion and nonlinearization of output signals of the notch and gradient filters. Two difference equations are derived first for the convergence in the mean and mean square senses, respectively. Two closed-form expressions, one for the steady-state estimation bias and the other for the mean-square error, are then derived based on the difference equations, with the former valid for both fast and slow adaptations and the latter valid for slow adaptation only. A closed-form coarse stability bound for the step size parameter of the algorithm is also derived. Extensive simulations are performed to reveal the validity and limitations of the analytical findings. Comparisons between the MNG and the conventional plain gradient algorithm are also made.