Analysis of adaptation rate of the FXLMS algorithm

We analyze the behaviors of active noise control using a statistical-mechanical method. The principal assumption used in the analysis is that the impulse responses of the primary path and adaptive filter are sufficiently long. In particular, in this paper we analyze the adaptation rate of the mean square error (MSE) using two measures. The first measure is the MSE initial decreasing rate. The second measure is an adaptation constant. This is defined by the negative of the maximum eigenvalue of the coefficient matrix of differential equations that describe the dynamical behaviors of the macroscopic variables. Introducing these two measures, we theoretically show that the optimal step size depends on whether we focus on the rate of decrease in the MSE at the initial stage or the MSE after sufficient adaptation time.