Statistical-mechanics approach to the filtered-X LMS algorithm

The learning curves of the filtered-X least-mean-square (LMS) algorithm are theoretically obtained using a statistical-mechanics approach. The direction cosines among the 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, simultaneous differential equations are obtained that describe the dynamical behaviours of the macroscopic variables in a deterministic form. The equations are solved analytically and show that the obtained theory quantitatively agrees with computer simulations. In the analysis, neither the independence assumption nor the few-taps assumption is used.