Saving complexity of modified filtered-X-LMS and delayed update LMS algorithms

In some applications, like in active noise control, the error signal cannot be obtained directly but only a filtered version of it. A gradient adaptive algorithm that solves the identification problem under this condition is the well known Filtered-x Least-Mean-Squares (FxLMS) algorithm. If only one coefficient of this error-filter function is nonzero, a special case of the FxLMS algorithm, the Delayed-update Least-Mean-Squares (DLMS) algorithm is obtained. The drawback of these algorithms is the increased dynamic order which, in turn, decreases the convergence rate. Recently, some modifications for these algorithms have been proposed, overcoming the drawbacks by additional computations of the same filter order as the filter length M. In this contribution, an improvement is shown yielding reduced complexity if the error path filter order P is much smaller than the filter order M, which is the case for many applications. Especially for the DLMS algorithm a strong saving can be obtained.