Statistical-mechanical analysis of the FXLMS algorithm with time-varying primary path

We analyze the learning curves of the active noise control with a time-varying primary path using a statistical-mechanical method. The cross-correlation between the element of a primary path and that of the adaptive filter and the autocorrelations of the elements of the adaptive filter are treated as macroscopic variables. We obtain simultaneous differential equations that describe the dynamical behaviors of the macroscopic variables under the condition that the tapped-delay line is sufficiently long. We analyze the case where the primary path has the Markovian property. As a result, we show that an optimal step size exists.