Performance analysis of the sign algorithm for a constrained adaptive IIR notch filter

Adaptive infinite impulse response (IIR) notch filters are very attractive in terms of their reasonable performances and low computational requirements. Generally, it is very difficult to assess their performances analytically due to their IIR nature. This paper analyzes in detail the steady-state performance of the sign algorithm (SA) for a well-known adaptive IIR notch filter with constrained poles and zeros. Slow adaptation and Gaussianity of the notch filter output are assumed for the sake of analysis. Two difference equations are first established for the convergences in the mean and mean square in the vicinity of the steady state of the algorithm. Steady-state estimation error or bias and mean square error (MSE) of the SA are then derived in closed forms. A coarse stability bound is also derived for the algorithm. Theory-based comparison between the algorithm and the plain gradient (PG) algorithm is done in some detail. Extensive simulations are conducted to demonstrate the validity of the analytical results for both slow and relatively fast adaptations.