Statistical-mechanical analysis of the FXLMS algorithm with actual primary path

A theory that predicts the behaviors of the Filtered-X LMS algorithm was derived by using a statistical-mechanical method. In this paper, the theory is generalized to explain the system behaviors in the case of an actual primary path. In the theory, 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. Simultaneous differential equations that describe the dynamical behaviors of the macroscopic variables are obtained under conditions in which the tapped-delay line is sufficiently long. The equations are analytically solved to obtain the correlations and finally compute the mean-square error. In order to generalize the theory to the case of an actual primary path, the correlations of the elements of the primary path are absorbed. The generalized theory quantitatively predict the behaviors in the case of an actual primary path.