Low Steady-state Misalignment Research Articles

Generalized Soft-Root-Sign Based Robust Sparsity-Aware Adaptive Filters

Robust adaptive filters utilizing hyperbolic cosine and correntropy functions have been successfully employed in non-Gaussian noisy environments. However, these filters suffer from high steady-state misalignment due to significant weight update in the presences of outliers. In addition, several practical systems exhibit sparse characteristics, which is not taken into account by these filters. In this paper, a generalized soft-root-sign (GSRS) function is proposed and the corresponding GSRS adaptive filter is designed. The proposed GSRS provides negligible weight update in the occurrence of large outliers and thereby results in lower steady-state misalignment. To further improve modelling performance for sparse systems and to achieve robustness, sparsity-aware GSRS algorithms are also developed in this paper. The bound on learning rate and the computational complexity of proposed algorithm is also investigated. Simulation studies confirmed the improved convergence characteristics achieved by the proposed algorithms over existing algorithms.

A mixing regularization parameter IPNLMS for underwater acoustic MIMO channel estimation

Less sensitivity to the sparsity of the underwater acoustic (UWA) channel is a drawback of the improved proportionate normalized least-mean-square (IPNLMS) algorithm. To address this problem, we propose an IPNLMS algorithm with a mixing regularization parameter (MRP-IPNLMS) for the multiple-input multiple-output (MIMO) UWA channel estimation. Firstly, the proposed MRP-IPNLMS exploits the sparsity of the UWA channel and integrates an approximate L0 norm into the cost function of the IPNLMS method, achieving the fast convergence in the sparse UWA channel estimation. Furthermore, we adopt a time-varying mixing regularization parameter to dynamically adjust the cost function. The proposed MRP-IPNLMS algorithm adaptively combines the fast convergence of the small regularization parameter with the low steady-state misalignment of the large regularization parameter. In addition, we evaluate the performance of the proposed algorithm in terms of its convergence, steady-state misalignment, and output signal-to-noise ratio (OSNR). Finally, simulations and real-data experiments are also conducted to demonstrate the effectiveness of the proposed algorithm.

Affine Projection Champernowne Algorithm for Robust Adaptive Filtering

The recently proposed affine projection Versoria (APV) algorithm has been widely used over other affine based algorithms due to its robustness against impulsive noises. However, the performance of the APV algorithm suffers from high steady state misalignment. In order to overcome this, we propose affine projection Champernowne adaptive filter (APCMAF) in which instead of taking Versoria function as a cost function we have used the probability density function of the Champernowne distribution as a cost function and data reuse technique. The proposed APCMAF algorithm provides low steady-state misalignment in impulsive noise environment. To verify the performance of the APCMAF algorithm, a set of simulation study has been done in system identification scenarios which confirms that the APCMAF provides better steady state performance with improved convergence performance over other existing algorithms in impulsive noise environments. Further, the bound of learning rate for stable convergence has been also derived and a detailed comparison of computational complexity is also presented.

M-estimate affine projection spline adaptive filtering algorithm: Analysis and implementation

This paper investigates the M-estimate affine projection spline adaptive filtering (MAPSAF) algorithm, which utilizes a modified Huber function with robustness against impulsive interference, and employs historical regression data to update the weight and the knot vector estimates for nonlinear filtering tasks. The detailed convergence and steady-state analyses of MAPSAF are also carried out in the mean and mean-square senses. In addition, an improved MAPSAF by exploiting the combined step sizes, called the CSS-MAPSAF algorithm, is derived to speed up the convergence on the premise of low steady-state misalignment. Numerical experiments in nonlinear system identification and nonlinear acoustic echo cancellation problems corroborate the theoretical performance analysis and the superiority of the proposed algorithms.

A Novel Family of Sparsity-Aware Robust Adaptive Filters Based on a Logistic Distance Metric

In the recent past, logarithmic hyperbolic cosine based-cost function has been widely applied in adaptive filters as it offers robust performance against outliers. However, the performance of such adaptive filters suffers from high steady-state misalignment due to its significant weight update, even in the presence of outliers. This paper proposes a logistic distance metric-based novel robust cost function, and the corresponding logistic distance metric adaptive filter (LDMAF) has been developed. The proposed LDMAF provides negligible weight update when the desired signals are affected by significant outliers, resulting in low steady-state misalignment. The bound on learning rate has been estimated, and computational complexity comparison of the proposed and other existing algorithms has also been carried out. To further exploit the system's sparse nature and robustness against the outliers, zero attraction-based LDMAF (ZA-LDMAF) and re-weighted zero attraction-based LDMAF (RZA-LDMAF) algorithms have also been developed in this paper. In addition, a new sparse penalty function based on a generalized multivariate Geman-McClure function has been introduced, which provides smooth <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$l_{0}$</tex-math></inline-formula> -norm approximation over other existing functions. Based on this new sparse penalty function, this paper has also developed a novel sparsity-aware robust adaptive filter called generalized Geman-McClure LDMAF (GGM-LDMAF). Simulation studies confirmed the improved convergence behaviour achieved by the proposed algorithms over other existing algorithms for system identification and acoustic echo cancellation scenarios.

An Improved Least Stochastic Entropy Algorithm for Strong Noncircular Inputs and Noise

When the noise is second-order (SO) noncircular, the least stochastic entropy (LSE) algorithm can obtain a low steady-state misalignment compared to the complex-valued least mean-square (CLMS) algorithm. However, the convergence rate of LSE will decrease if the input of the adaptive filter is SO noncircular. This letter analyzes the cause of this problem and proposes an improved LSE (ILSE), which uses a combination strategy to accelerate its convergence rate. To predict its stochastic behavior, the performance of ILSE is analyzed. Simulation results are provided to verify the effectiveness of ILSE and the accuracy of the theoretical analysis.

Normalized Combinations of Proportionate Affine Projection Sign Subband Adaptive Filter

The proportionate affine projection sign subband adaptive filter (PAP-SSAF) has a better performance than the affine projection sign subband adaptive filter (AP-SSAF) when we eliminate the echoes. Still, the robustness of the PAP-SSAF algorithm is insufficient under unknown environmental conditions. Besides, the best balance remains to be found between low steady-state misalignment and fast convergence rate. In order to solve this problem, we propose a normalized combination of PAP-SSAF (NCPAP-SSAF) based on the normalized adaption schema. In this paper, a power normalization adaptive rule for mixing parameters is proposed to further improve the performance of the NCPAP-SSAF algorithm. By using Nesterov’s accelerated gradient (NAG) method, the mixing parameter of the control combination can be obtained with less time consumed when we take the l1-norm of the subband error as the cost function. We also test the algorithmic complexity and memory requirements to illustrate the rationality of our method. In brief, our study contributes a novel adaptive filter algorithm, accelerating the convergence speed, reducing the steady-state error, and improving the robustness. Thus, the proposed method can be utilized to improve the performance of echo cancellation. We will optimize the combination structure and simplify unnecessary calculations to reduce the algorithm’s computational complexity in future research.

Variable Step-Size Sparsity-Induced Augmented Complex-Valued NLMS Algorithm

The widely linear model has attracted much attention due to its good features for non-circular adaptive signal processing in recent years. In this paper, a sparsity-induced augmented complex-valued NLMS algorithm is proposed to promote the performance of the adaptive filter for estimating sparse systems, which is established by incorporating the $$l_0$$ -norm regularization into the squared error normalized by the input vector. To address the problem of trade-off between fast convergence rate and low steady-state misalignment, we minimize the variance of the a posteriori error to derive an optimal step-size and then some practical problems are considered. Simulation results are provided to verify the superior performance of the proposed algorithm.

A Variable Step-Size Partial-Update Normalized Least Mean Square Algorithm for Second-Order Adaptive Volterra Filters

Partial-update (PU) algorithms offer reduced computational complexity to adaptive second-order Volterra filters (SOV) in nonlinear systems while retaining acceptable performance. In this paper, a new selective partial-update technique for the normalized LMS (NLMS) SOV algorithm is proposed, which requires lesser number of comparison operations per iteration than existing methods while providing close performance to the standard M-Max NLMS-SOV algorithm. Convergence properties of the proposed algorithm are enhanced by making the algorithm step-size time varying based on the natural logarithm function. Simulation experiments compare the proposed algorithm with existing PU and variable step-size NLMS-SOV algorithms, which illustrate the advantageous properties of the new algorithm. The proposed algorithm achieves both lower steady-state misalignment and faster convergence speed when compared with the fixed step-size full-update NLMS-SOV algorithm. Simulations also show that comparison operations overhead of the proposed algorithm is reduced significantly compared to that of the standard M-Max NLMS-SOV algorithm.

Robust Generalized Maximum Correntropy Criterion Algorithms for Active Noise Control

As a robust nonlinear similarity measure, the maximum correntropy criterion (MCC) has been successfully applied to active noise control (ANC) for impulsive noise. The default kernel function of the filtered-x maximum correntropy criterion (FxMCC) algorithm is the Gaussian kernel, which is desirable in many cases for its smooth and strict positive-definite. However, it is not always the best choice. In this study, a filtered-x generalized maximum correntropy criterion (FxGMCC) algorithm is proposed, which adopts the generalized Gaussian density (GGD) function as its kernel. The FxGMCC algorithm has greater robust ability against non-Gaussian environments, but, it still adopts a single error norm which exhibits poor convergence rate and noise reduction performance. To surmount this problem, an improved FxGMCC (IFxGMCC) algorithm with continuous mixed Lp- norm is proposed. Moreover, to make a trade-off between fast convergence rate and low steady-state misalignment, a convexly combined IFxGMCC (C-IFxGMCC) algorithm is further developed. The stability mechanism and computational complexity of the proposed algorithms are analyzed. Simulation results in the context of different impulsive noises as well as the real noise signals verify that the proposed algorithms are superior to most of the existing robust adaptive algorithms.

L₁/L₂-Mode Switching Adaptive Filter Algorithm Based on Novel Mean Square Deviation Analysis

This paper proposes an $\mathcal {L}_{1}/\mathcal {L}_{2}$ -mode switching adaptive filter algorithm by comparing the performances in $\mathcal {L}_{1}$ and $\mathcal {L}_{2}$ modes. In the $\mathcal {L}_{1}$ or $\mathcal {L}_{2}$ mode, the proposed algorithm adopts, as the update equation, that of the normalized sign (NS) algorithm or that of the normalized least mean square (NLMS) algorithm, respectively. By analyzing the mean square deviations (MSDs) of the NS algorithm as well as of the NLMS algorithm, the algorithm selects the better mode in the sense that the next MSD of the algorithm in the selected mode becomes smaller than that in the other mode. The algorithm mainly operates in the $\mathcal {L}_{1}$ mode when the impulsive noises occur but in the $\mathcal {L}_{2}$ mode otherwise, which leads to robustness against the impulsive noises like the NS algorithm and also leads to a low steady-state misalignment like the NLMS algorithm. Furthermore, the proposed algorithm is faster than the NS and the NLMS algorithms in terms of the convergence rate. A modified reset algorithm is also applied to maintain performance when the unknown system is abruptly changed. Simulations conducted in various system identification scenarios show that the proposed algorithm outperforms the conventional algorithms in terms of the convergence rate and the steady-state misalignment, whenever impulsive noises exist or not.

Adaptive feedback cancellation with prediction error method and howling suppression in train public address system

Train public address (PA) system is an important component of Passenger Information System. Acoustic feedback is a well-known problem in public address systems. It will cause howling under certain conditions, which can severely degrade sound quality and limit the maximum allowable amplification of the PA systems. The general adaptive feedback cancellation (AFC) algorithms lead to a biased estimation because of the correlation between source and feedback signals. In order to solve the problem and achieve both fast convergence and low steady-state misalignment, a novel variable step-size improved proportionate affine projection algorithm (IPAPA) based on mixed error cost function with prediction error method (PEM) and howling suppression is proposed in this paper. The experimental results demonstrate the effectiveness of the proposed algorithm.

Performance Analysis of Shrinkage Linear Complex-Valued LMS Algorithm

The shrinkage linear complex-valued least mean squares (SL-CLMS) algorithm with a variable step size overcomes the conflicting issue between fast convergence and low steady-state misalignment. To the best of our knowledge, the theoretical performance analysis of the SL-CLMS algorithm has not been presented yet. This letter focuses on the theoretical analysis of the excess mean square error transient and steady-state performance of the SL-CLMS algorithm. Simulation results obtained for identification scenarios show a good match with the analytical results.

Variable step-size widely linear complex-valued NLMS algorithm and its performance analysis

The shrinkage widely linear complex-valued least mean square (SWL-CLMS) algorithm with a variable step-size (VSS) overcomes the tradeoff between fast convergence and low steady-state misalignment, but meanwhile suffers from instability for highly correlated input signals because of the gradient noise amplification problem. To obtain a VSS that is also applicable to the case of highly correlated input signals, in this paper, we propose the VSS widely linear complex-valued normalized least mean square (VSS-WL-CNLMS) algorithm, where the VSS is derived by minimizing the mean-square deviation (MSD). Owing to the normalization, the VSS-WL-CNLMS algorithm is convergent in the mean square sense. By using the Rayleigh distribution, we calculate the mean step-size, which is then combined with the approximate uncorrelating transform to analyze the transient and steady-state mean square error (MSE) behaviors. Simulations for system identification scenario show that the proposed VSS-WL-CNLMS algorithm outperforms some well-known techniques and verify the accuracy of the theoretical analysis.

A Normalized Subband Adaptive Filter with Combined Regularization Parameter

Proposed is a novel variable regularization algorithm for normalized subband adaptive filter (NSAF) by employing the mixing parameter to combine a large regularization parameter with a small regularization parameter. The mixing parameter is derived by minimizing the energy of noise-free a posteriori subband error. The combined regularization parameter can vary from a small value to a large value. Therefore, the proposed algorithm exhibits fast convergence rate meanwhile obtains low steady-state misalignment. Experimental simulations demonstrate the effectiveness of our work.

Polynomial variable scaling factor improved least sum of exponentials algorithm with maximum correntropy criterion

In this paper, a polynomial variable scaling factor improved least sum of exponentials algorithm with maximum correntropy criterion is proposed for sparse system identification. Sparse system estimation problem is increasing important topics in broadband wireless communications systems. Sparse learning algorithms for system identification achieved a better performance under Gaussian assumption, such as the zero-attracting least mean square (ZA-LMS). However, in non-Gaussian environments the existing algorithms suffer from performance degradation due to random impulsive noises. To further improve the robustness of the zero-attracting algorithms, an attempt has been made to design an improved sum of error exponentials that utilize the maximum correntropy criterion. In addition, a polynomial zero attractor is introduced to enhance the capability of sparse system identification. The test on sparse system identifications under an impulsive noise environment demonstrates that the proposed algorithm has a low steady-state misalignment compared with the others.

Bias-compensated Adaptive Filter Algorithm under Minimum Error Entropy Criterion

This paper proposes a bias-compensated adaptive filtering algorithm under minimum error entropy criterion, which outperforms with low steady-state misalignment for signal processing with noisy input in an environment containing impulsive output noise. In previous studies, much works use the minimum error entropy criterion, which is called MEE, to develop adaptive filter under the assumption of no input noise. However, pure input signal without any noise is nonexistent in the real-world environment. To address above issue, we introduce a bias-compensated vector into the traditional MEE algorithm and propose a bias-compensated adaptive filtering algorithm under minimum error entropy criterion named BCMEE, which has stronger robustness and higher convergence rate. The BCMEE utilizes a kernel function. In the kernel function, the slicing window size takes certain numbers of past errors during adaptation processing in BCMEE’s updating, whereas the other classical algorithm relies only on the current error signal and takes advantage of bias-compensated vector to compensate the bias of filter caused by the input noise. Simulation show the excellent performance of the proposed algorithms.

Bias-compensated normalized maximum correntropy criterion algorithm for system identification with noisy input

Abstract This paper proposes a bias-compensated normalized maximum correntropy criterion (BCNMCC) algorithm charactered by its low steady-state misalignment for system identification with noisy input in an impulsive output noise environment. The normalized maximum correntropy criterion (NMCC) is derived from a correntropy based cost function, which is rather robust with respect to impulsive noises. To deal with the noisy input, we introduce a bias-compensated vector to the NMCC algorithm, and then an unbiasedness criterion and some reasonable assumptions are used to compute the bias-compensated vector. Taking advantage of the bias-compensated vector, the bias caused by the input noise can be effectively suppressed. System identification simulation results demonstrate that the proposed BCNMCC algorithm can outperform other related algorithms with noisy input especially in an impulsive output noise environment.

A Variable Step-Size Shrinkage Set-Membership Affine Projection Algorithm for Noisy Input

To solve the conflicting requirement of fast convergence and low steady-state misalignment, a variable step-size shrinkage set-membership affine projection algorithm is proposed, which is efficient for the correlated input signal and noisy input environments. The new variable step size is derived by minimizing the square of noise-free a posteriori error, and the shrinkage method is employed to estimate the second-order statistics of the noise-free a priori error vector. Moreover, the stability analysis of the algorithm is conducted. Simulations demonstrate the effectiveness of the proposed algorithm for various noisy input environments.

Statistical Convergence Analysis for Optimal Control of DFT-Domain Adaptive Echo Canceler

The frequency-domain adaptive filter (FDAF) is widely used in echo cancellation systems due to its low complexity and fast convergence rate. However, the FDAF algorithm with a fixed step size exhibits a tradeoff among the convergence rate, steady-state misalignment, tracking ability, and robustness to near-end speech interferences. Several variable step-size FDAF algorithms were presented to address this problem. However, the state-of-the-art variable step-size FDAF algorithms did not handle this problem comprehensively. This paper presents a new robust variable step-size control approach to the FDAF algorithm. Based on a statistical analysis of the FDAF algorithm, an optimal step size for each frequency bin is derived by minimizing the mean-square deviation (MSD) between the true weight vector and estimated weight vector at each frame. Calculation of the step size requires the system distance and the observation noise power spectral density (PSD). The system distance is estimated using the deterministic recursive equations of MSD and the noise PSD is computed using the magnitude squared coherence function between the far-end signal and error signal. Moreover, a close link between the proposed FDAF and the frequency-domain Kalman filter is revealed. Specifically, the work presented here can be understood as a means to adaptively monitor and control the underlying acoustic state space of the Kalman filter, including means for fast readaptation of the adaptive filter after abrupt echo path changes. Simulation results demonstrate that the proposed algorithm can achieve fast convergence and low steady-state misalignment. Furthermore, the algorithm is robust to the double-talk interferences, but it does not require an explicit double-talk detector.