We analyze the behaviors of the FXLMS algorithm using a statistical-mechanical method. The cross-correlation between a primary path and an adaptive filter and the autocorrelation 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. The obtained equations are deterministic and closed-form. We analytically solve the equations to obtain the correlations and finally compute the mean-square error. The obtained theory can quantitatively predict the behaviors of computer simulations including the cases of both not only white but also nonwhite reference signals. The theory also gives the upper limit of the step size in the FXLMS algorithm.

It is shown that simple preprocessing on the reference signals in multichannel feedforward ANC system can improve the convergence performance of the adaptive ANC algorithm. A fast and efficient blind preprocessing algorithm in frequency domain is proposed to reduce the computational complexity even that the reference sensors are located far from the noise sources. The permutation problem at different frequency bin is also addressed and solved by an independent vector analysis algorithm. The basic principle and performance comparison are given to verify our conclusion.