Steady-state analysis of a plain gradient algorithm for a second-order adaptive IIR notch filter with constrained poles and zeros

Gradient-type algorithms for adaptive infinite-impulse response (IIR) notch filters are very attractive in terms of performance and computation cost. However, it is generally quite difficult to assess their performances analytically. There are several trials to analyze adaptive algorithms, such as the sign and the plain gradient algorithms for some types of adaptive IIR notch filters, but the analysis techniques used cannot be directly applied to different types of adaptive IIR notch filters. This brief presents closed form expressions for steady-state estimation bias and mean square error (MSE) of a well known plain gradient (LMS-like) second-order adaptive IIR notch filter with constrained poles and zeros. First, theoretical expressions for output signals of the notch filter and its corresponding gradient filter at their steady states are developed based on the Taylor series expansions of transfer functions of these two filters in the vicinity of the sinusoidal signal frequency difference equations for convergences in the mean and mean square are then established by using these output signals, from which the steady-state bias and MSE of the algorithm are derived. Stability bound of the algorithm is also investigated based on the difference equations. Extensive simulations are provided to support the analytical findings.