A frequency domain model for `filtered´ LMS algorithms-stability analysis, design, and elimination of the training mode

A frequency domain model of the filtered LMS algorithm is presented for analyzing the behavior of the weights during adaptation. In particular, expressions for stable operation of the algorithm are derived as a function of the algorithm step size, the input signal power, and the transfer functions of the linear filters. The expressions show that algorithm stability can be achieved over a frequency band of interest by inserting an appropriately chosen delay in the reference input to the LMS algorithm weight update equation. This result implies that it is not necessary to use a training mode to estimate the loop transfer functions before or during adaptation if the input is limited to a band of frequencies. It is only necessary to know the approximate delay introduced by the transfer functions in the band. The single delay parameter can be estimated much more easily than the entire transfer function. Simulations of the time domain algorithm are presented to support the theoretical predictions of the frequency domain model