Steady-state properties of the sign algorithm for the constrained adaptive IIR notch filter

Many algorithms have been proposed for the constrained adaptive IIR notch filter for frequency estimation. The sign algorithm (SA) is a good option in terms of low computational cost and robustness against additive noise of impulsive nature. However, unlike most of the other algorithms, the performance of the SA has not been reported on. This is because of the difficulty due to the presence of the sign function. To overcome this difficulty, we, here, present an effective approach where relatively slow adaptation and Gaussianity of the notch filter output are assumed. Two difference equations are first established for the convergences in the mean and in the mean square, respectively. Steady-state estimation error and mean square error (MSE) of the SA are then derived in closed forms. Theory-based comparison between the SA and the plain gradient (PG) algorithm is done in some detail. Extensive simulations demonstrate the validity of our analytical results not only for the slow adaptation cases but also for cases of relatively fast adaptation.