THEORETICAL DISCUSSION OF THE FILTERED-X LMS ALGORITHM BASED ON STATISTICAL MECHANICAL ANALYSIS

We theoretically obtain the learning curves of the FXLMS algorithm on the basis of statistical mechanical analysis. Cosines of angles between the coefficient vectors of an adaptive filter, its shifted filters, and an unknown system are treated as macroscopic variables. Assuming that the tapped-delay line is sufficiently long and exactly calculating the correlations between the past tap input vectors and the coefficient vector of the adaptive filter, we obtain simultaneous differential equations that describe the dynamical behaviors of the macroscopic variables in a deterministic form. We analytically solve the equations and show that the obtained theory quantitatively agrees with computer simulations. In the analysis, neither the independence assumption, the small step-size condition, nor the few-taps assumption is used.