The behavior of the modified FX-LMS algorithm with secondary path modeling errors

In active noise control there has been some research based in the modified filtered-X least mean square (LMS) algorithm (MFX-LMS). When the secondary path is perfectly modeled, this algorithm is able to perfectly eliminate it´s effect. It is also easily adapted to allow the use of fast algorithms such as the recursive least square, or algorithms with good tracking performance based on the Kalman filter. This letter presents the results of a frequency domain analysis about the behavior of the MFX-LMS in the presence of secondary path modeling errors and a comparison with the FX-LMS algorithm. Namely, it states that for small values of the secondary path delay both algorithms perform the same, but that the step-size of the FX-LMS algorithm decreases with increasing delay, while the MFX-LMS algorithm is stable for an arbitrary large value for the secondary path delay, as long as the real part of the ratio of the estimated to the actual path is greater than one half (Re{Sˆ_z/S_z}>1/2). This means that for the case of no phase errors the estimated amplitude should be greater than half the real one and for the case of no amplitude errors the phase error should be less than 60°. Analytical expressions for the limiting values for the step-size in the presence of modeling errors are given for both algorithms.