Analysis of adaptive filter algorithms in feedback systems

The paper presents some convergence results of an adaptive filter algorithm for a feedback-type active noise control (ANC) system with feedback path modeling using a dither signal and for feedback path modeling of hearing aids. In these systems, there is an adaptive filter in the feedback path so that the conventional method for analysis does not work. A frequency domain method is used for the analysis. It is based on the averaging method and the frequency domain expression of the adaptive algorithm with a causality constraint. The convergence conditions are derived for both problems and a bias expression of the weights for the latter problem is presented. The positive realness of the transfer function of the optimal predictor of the incoming signal is required for stability. Finally, the convergence condition and the bias are explicitly given for simple examples and their validities are shown by some simulations.