A New Variable Regularized QR Decomposition-Based Recursive Least M-Estimate Algorithm—Performance Analysis and Acoustic Applications

This paper proposes a new variable regularized QR decompPosition (QRD)-based recursive least M-estimate (VR-QRRLM) adaptive filter and studies its convergence performance and acoustic applications. Firstly, variable L2 regularization is introduced to an efficient QRD-based implementation of the conventional RLM algorithm to reduce its variance and improve the numerical stability. Difference equations describing the convergence behavior of this algorithm in Gaussian inputs and additive contaminated Gaussian noises are derived, from which new expressions for the steady-state excess mean square error (EMSE) are obtained. They suggest that regularization can help to reduce the variance, especially when the input covariance matrix is ill-conditioned due to lacking of excitation, with slightly increased bias. Moreover, the advantage of the M-estimation algorithm over its least squares counterpart is analytically quantified. For white Gaussian inputs, a new formula for selecting the regularization parameter is derived from the MSE analysis, which leads to the proposed VR-QRRLM algorithm. Its application to acoustic path identification and active noise control (ANC) problems is then studied where a new filtered-x (FX) VR-QRRLM ANC algorithm is derived. Moreover, the performance of this new ANC algorithm under impulsive noises and regularization can be characterized by the proposed theoretical analysis. Simulation results show that the VR-QRRLM-based algorithms considerably outperform the traditional algorithms when the input signal level is low or in the presence of impulsive noises and the theoretical predictions are in good agreement with simulation results.