Frequency-Domain Adaptive Filtering Revisited, Generalization to the Multi-Channel Case, and Application to Acoustic Echo Cancellation

We drive a new frequency-domain adaptive algorithm by using a frequency-domain recursive least squares criterion, minimizing an error signal in the frequency-domain. A similar criterion was proposed by Mansour and Gray by using mathematical expectations. Here, however, we propose to go one step further and derive an exact adaptive algorithm from the so-called normal equation. It is shown that the obtained algorithm is complex to implement, and to reduce the complexity, we need to remove a constraint resulting in the unconstrained frequency-domain LMS (UFLMS) algorithm. We also give the optimal adaptation step size for the UFLMS. Most importantly, we generalize all this to the multi-channel case, thereby exploiting the cross-power spectra among all the channels which is very important (for a fast convergence rate) in multi-channel acoustic echo cancellation (AEC), where the input signals are highly correlated.