A study on the step size of cascaded adaptive notch filter utilizing allpass filter

A cascaded second order notch filter implemented using an allpass filter, for elimination of multiple sinusoids, have been proposed. Various adaptive algorithms for the adaptive notch filters are used. As the one of them, an adaptive algorithm which has a reduced bias in comparison to the gradient based algorithm has been proposed. It has the problem that the convergence speed deteriorates when the number of the narrow band signal increases. Therefore small step size is required. However, the range of the step size for the adaptive notch filters to be stable has not been derived yet. The purpose of this paper is the derivation of it by using the principle of the contraction mapping. Finally, computer simulation results are presented to confirm the convergence characteristics.