This paper proposes a new class of local polynomial modeling (LPM)-based variable forgetting factor (VFF) recursive least squares (RLS) algorithms called the LPM-based VFF RLS (LVFF-RLS) algorithms. It models the time-varying channel coefficients as local polynomials so as to obtain the expressions of the bias and variance terms in the mean square error (MSE) of the RLS algorithm. A new locally optimal VFF (LOVFF) is then derived by minimizing the resulting MSE and the theoretical analysis is found to be in good agreement with experimental results. Methods for estimating the parameters involved in this LOVFF are also developed, resulting in an improved RLS algorithm with VFF. The algorithm is further extended to include variable regularization and a QR decomposition (QRD) version which is numerically more stable and amenable to multiplier-less implementation using coordinate rotation digital computer (CORDIC) algorithm. Applications of these algorithms to frequency estimation and adaptive beamforming in time-varying speech and audio signals are also presented to illustrate the effectiveness of the proposed algorithms. Simulations show that the convergence and tracking performance of the proposed algorithms compare favorably with conventional algorithms.

This paper proposes a new regularized transform domain normalized LMS (R-TDNLMS) algorithm and studies its mean and mean square convergence performances. The proposed algorithm extends the conventional TDNLMS algorithm by imposing a regularization term on the filter coefficients to reduce the variance of estimators due to the lacking of excitation in a certain frequency band or in the presence of modeling errors. Difference equations describing the mean and mean square convergence behaviors of this algorithm are derived so as to characterize its convergence condition and steady-state excess mean square error (MSE). It shows that regularization can help to reduce the MSE by trading slight bias for variance. Based on this analysis, a new formula to select the regularization parameter for white Gaussian inputs is proposed, which leads to a new variable regularized TDNLMS (VR-TDNLMS) algorithm. Computer simulations are conducted to examine the improved convergence performance, steady-state MSE and robustness to power-varying inputs of the proposed algorithm and verify the effectiveness of the theoretical analysis. Furthermore, the application of the proposed VR-TDNLMS algorithm to the design and implementation of acoustic system identification and active noise control (ANC) systems show that they considerably outperforms traditional TDNLMS algorithms at low excitation or in the presence of modeling errors. Moreover, the theoretical analysis provides simple design formulas for achieving a given excess MSE (EMSE) and step-size bound for stable operation.

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, variableL2regularization 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.

This paper proposes a new state-regularized (SR) and QR decomposition based recursive least squares (QRRLS) algorithm with variable forgetting factor (VFF) for recursive coefficient estimation of time-varying autoregressive (AR) models. It employs the estimated coefficients as prior information to minimize the exponentially weighted observation error, which leads to reduced variance and bias over traditional regularized RLS algorithm. It also increases the tracking speed by introducing a new measure of convergence status to control the FF. Simulations using synthetic and real speech signals show that the proposed method has improved tracking performance and reduced estimation error variance than conventional TVAR modeling methods during rapid changing of AR coefficients.

This paper studies the convergence behaviors of the noise-constrained normalized least mean squares (NCNLMS) algorithm recently proposed in the work of Chan et al. (2008). Like its LMS counterpart, the NCNLMS algorithm employs the prior knowledge of the additive noise to adjust its step-size. Following (Wei et al., 2001), the convergence behaviors of the NCLMS under the noise mismatch cases are firstly derived. Using a novel transformation approach and the small step-size properties of the NCNLMS algorithm at convergence, the mean and mean squares behaviors of this algorithm are derived. The validity of the proposed analysis is verified well by computer simulations and the relative merits of the NCLMS and NCNLMS algorithms are also compared.