Convergence analysis of filtered-x least mean squares algorithm for active control of repetitive impact noise

Abstract

The prevalent adaptive active noise control (ANC) algorithm, namely, the filtered-x least mean squares (FXLMS), exhibits a critical challenge for treating transient impact noise. This is because the FXLMS algorithm requires certain adaptation time to converge satisfactorily. However, the FXLMS algorithm may have its learning capacity when the transient noise shows certain repeatability. In this paper, a distinctive theoretical analysis of the convergence behavior of ANC system with the standard FXLMS algorithm is conducted for repetitive impulse-induced transient noise control. To simplify the derivation, the secondary path is assumed to be a pure delay model. Through this analysis, a step size bound condition is derived, and an optimal step size that leads to the fastest convergence is determined. To validate the analysis, extensive numerical simulations are performed considering various pure delay secondary path models. Calculations are in very good agreement with the theoretical analysis. Finally, a more general secondary path is considered to further demonstrate the effectiveness of the FXLMS algorithm for repetitive impulse noise control. The results indicate that ANC system with the FXLMS algorithm can be a very promising technique for repetitive transient noise typically seen in industrial facilities such as punching machines.