Convergence analysis of linear and nonlinear filtered-X LMS algorithms for active control of multitonal noise

In the presence of tonal noise generated by periodic noise source like rotating machines, the filtered-X LMS algorithm is used for active control of such noises. However, the algorithm is derived under the assumption of slow adaptation limit and the exact analysis of the algorithm is restricted to the case of one real sinusoid in the literature. In this paper for the general case of arbitrary number of sources, the characteristic polynomial of the equivalent linear system describing the filtered-X LMS algorithm is derived and a method for calculating the stability limit is presented. Also, a new nonlinear algorithm free from the above assumption is proposed. Simulation results show that in the early stage of adaptation the nonlinear algorithm gives faster decay of errors.

I. Introduction

In practical active noise control (ANC) systems, the inputs to the adaptive system are often tonal signals generated by periodic noise source like rotating machines. In the case where a pair of a sinusoid and a cosinusoid is used as reference signals, i.e., in the notch filtering, the analysis of the LMS algorithm using z-transfonn is well known [1]. Also, a new method using the eigen decomposition of the deterministic correlation matrix has been proposed for the twin-reference case [2]. If the effect of the secondary path is significant, the filtered-X LMS (FXLMS) algorithm must be used. In [3], the FXLMS algorithm is treated for the case of one real sinusoid to analyze the effect of the secondary path to the passband characteristic of the ANC system.