A theory of the FXLMS algorithm based on statistical-mechanical method

We analyze the learning curves of the FXLMS algorithm using a statistical-mechanical method. Cross-correlations between the element of a primary path and that of an adaptive filter and 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 conditions in which the tapped-delay line is long. We analytically solve the equations to obtain the correlations and finally compute the mean-square error. Introducing the correlation function of the input signal, the theory can treat not only the white but also the nonwhite signal. The obtained theory quantitatively agrees with the results of computer simulations.