Statistical-mechanical analysis of the FXLMS algorithm with nonwhite reference signals

We analyze the learning curves of the FXLMS algorithm using a statistical-mechanical method when the reference signal is not necessarily white. We treat the nonwhite reference signal by introducing the correlation function of the signal to the method proposed in our previous study. 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 meansquare error. The obtained theory quantitatively agrees with the results of computer simulations. The theory also gives the upper limit of the step size in the FXLMS algorithm.