The Behavior of the Modified FX-LMS Algorithm With Secondary Path Modeling Errors

Abstract

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^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.Index Terms-Active noise control, filtered-X least mean square (FX-LMS), Kalman, least mean square (LMS), modeling, recursive least square (RLS), secondary path.

Recent advances on active noise control: open issues and innovative applications

“…Other analyses on the convergence behaviors of the ANC systems include the presence of a feedback path [145], the ANC dealing with moving noise sources [146], the modified FXLMS algorithm [147], the leaky FXLMS algorithm [148,149], and the filtered-x adaptive filter with an averaging algorithm [150].…”

Adaptive Repetitive Control Design With Online Secondary Path Modeling and Application to Bit-Patterned Media Recording

“…When the step size is small, a sufficient condition for robustness to the secondary path modeling mismatch is that the phase difference between these two systems should be <90° [6]-i.e., Real(R(e j ω ))/R(e j ω ))) > 0 for all values of ω.…”

“…Nevertheless, the MFX-LMS algorithm will diverge if the internal model, used in the secondary path, does not match the actual closed-loop dynamics very well. This divergence behavior happens when the phase mismatch between the nominal model and real closed-loop system exceeds 90°at the excitation frequency range [6]. We propose a self-tuning secondary path identification scheme in Section IV to compensate for these dynamics mismatches in an online fashion, to prevent the divergence of the adaptive algorithm.…”

FxLMS and MFxLMS Stability Constrains when used in Active Noise Control

“…Por otro lado de comparar (3) y (4) queda de manifiesto la ventaja del algoritmo MFxLMS en cuanto a la velocidad de adaptación máxima relativa al retraso del camino secundario. En [6] se analizan los límites teóricos de la diferencia entre Sˆ(z) y S(z) en cuanto a la amplitud y fase para ambos modelos. …”