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

摘要

Gradient-type algorithms for adaptive infinite-impulse responsen(IIR) notch filters are very attractive in terms of performance andncomputation cost. However, it is generally quite difficult to assessntheir performances analytically. There are several trials to analyzenadaptive algorithms, such as the sign and the plain gradient algorithmsnfor some types of adaptive IIR notch filters, but the analysisntechniques used cannot be directly applied to different types ofnadaptive IIR notch filters. This brief presents closed form expressionsnfor steady-state estimation bias and mean square error (MSE) of a wellnknown plain gradient (LMS-like) second-order adaptive IIR notch filternwith constrained poles and zeros. First, theoretical expressions fornoutput signals of the notch filter and its corresponding gradient filternat their steady states are developed based on the Taylor seriesnexpansions of transfer functions of these two filters in the vicinity ofnthe sinusoidal signal frequency difference equations for convergences innthe mean and mean square are then established by using these outputnsignals, from which the steady-state bias and MSE of the algorithm arenderived. Stability bound of the algorithm is also investigated based onnthe difference equations. Extensive simulations are provided to supportnthe analytical findings