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Constrained adaptive filtering algorithms have been extensively studied in many applications. Most existing constrained adaptive filtering algorithms are developed under the mean square error (MSE) criterion, which is an ideal optimality criterion under Gaussian noises. This assumption however fails to model the behavior of non-Gaussian noises found in practice. Motivated by the robustness and simplicity of maximum correntropy criterion (MCC) for non-Gaussian impulsive noises, this paper proposes a new adaptive filtering algorithm called constrained maximum correntropy criterion (CMCC). Specifically, CMCC incorporates a linear constraint into a MCC filter to solve a constrained optimization problem explicitly. The proposed adaptive filtering algorithm is easy to implement, has low computational complexity, and can significantly outperform those MSE based constrained adaptive algorithms in heavy-tailed impulsive noises. Additionally, the mean square convergence behaviors are studied under energy conservation relation, and a sufficient condition to ensure the mean square convergence and the steady-state mean square deviation (MSD) of the CMCC algorithm are obtained. Simulation results confirm the theoretical predictions under both Gaussian and non-Gaussian noises, and demonstrate the excellent performance of the novel algorithm by comparing it with other conventional methods. [ABSTRACT FROM AUTHOR]

Nonlinear similarity measures defined in kernel space, such as correntropy, can extract higher order statistics of data and offer potentially significant performance improvement over their linear counterparts especially in non Gaussian signal processing and machine learning. In this paper, we propose a new similarity measure in kernel space, called the kernel risk-sensitive loss (KRSL), and provide some important properties. We apply the KRSL to adaptive filtering and investigate the robustness, and then develop the MKRSL algorithm and analyze the mean square convergence performance. Compared with correntropy, the KRSL can offer a more efficient performance surface, thereby enabling a gradient-based method to achieve faster convergence speed and higher accuracy while still maintaining the robustness to outliers. Theoretical analysis results and superior performance of the new algorithm are confirmed by simulation. [ABSTRACT FROM PUBLISHER]

As a robust nonlinear similarity measure in kernel space, correntropy has received increasing attention in domains of machine learning and signal processing. In particular, the maximum correntropy criterion (MCC) has recently been successfully applied in robust regression and filtering. The default kernel function in correntropy is the Gaussian kernel, which is, of course, not always the best choice. In this paper, we propose a generalized correntropy that adopts the generalized Gaussian density (GGD) function as the kernel, and present some important properties. We further propose the generalized maximum correntropy criterion (GMCC) and apply it to adaptive filtering. An adaptive algorithm, called the GMCC algorithm, is derived, and the stability problem and steady-state performance are studied. We show that the proposed algorithm is very stable and can achieve zero probability of divergence (POD). Simulation results confirm the theoretical expectations and demonstrate the desirable performance of the new algorithm. [ABSTRACT FROM AUTHOR]

Kernel adaptive filters (KAF) are a class of powerful nonlinear filters developed in Reproducing Kernel Hilbert Space (RKHS). The Gaussian kernel is usually the default kernel in KAF algorithms, but selecting the proper kernel size (bandwidth) is still an open important issue especially for learning with small sample sizes. In previous research, the kernel size was set manually or estimated in advance by Silverman׳s rule based on the sample distribution. This study aims to develop an online technique for optimizing the kernel size of the kernel least mean square (KLMS) algorithm. A sequential optimization strategy is proposed, and a new algorithm is developed, in which the filter weights and the kernel size are both sequentially updated by stochastic gradient algorithms that minimize the mean square error (MSE). Theoretical results on convergence are also presented. The excellent performance of the new algorithm is confirmed by simulations on static function estimation, short term chaotic time series prediction and real world Internet traffic prediction. [ABSTRACT FROM AUTHOR]

The normalized subband adaptive filtering algorithm provides fast convergence rate for colored input signals as compared to the normalized least mean square algorithm, but it suffers from a poor convergence issue in the α -stable noise. In light of this, the normalized subband p-norm (NSPN) algorithm, which is based on the least mean p-power error (MPE) criterion, is proposed in this study. This technique is not only robust against impulsive noise samples, but it also maintains a fast convergence rate when colored input signals are used. In addition to this, we develop both the steady-state and the transient models of the NSPN algorithm and provide some insights. Then, in order to solve the problem of making a choice regarding the order p in the NSPN algorithm, we design an autonomous system and come up with the NSPN algorithm with a variable p-norm (VP-NSPN). In addition, we offer the TFC-based VP-NSPN algorithm with a fast convergence rate and low steady-state misadjustment simultaneously by making use of the tap-weights feedback-based convex combination (TFC) scheme. In conclusion, simulation results on system identification and acoustic echo cancellation are undertaken in order to validate the superiority of the proposed algorithms and validate the usefulness of the theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper presents a new normalized least mean square (NLMS) algorithm for sparse system identification where the input signal is corrupted by white measurement noise. The proposed algorithm, which is called bias-compensated zero attracting NLMS (BC-ZA-NLMS) algorithm, introduces the bias-compensation vector to get rid of the bias resulting from noisy input and introduces an l 1 -norm penalty in the cost function of the NLMS algorithm to make full use of the special property of the sparse system. In addition, to address the time variant sparsity, the bias-compensated reweight ZA-NLMS (BC-RZA-NLMS)) algorithm is also proposed, where the l 1 -norm penalty in the cost function of BC-ZA-NLMS algorithm is replaced by a log-sum function. Owing to the zero attractors in update equation, the proposed algorithms are superior to the conventional NLMS and bias-compensated NLMS (BC-NLMS) algorithms in the application of identifying the sparse system. A transient analysis of the proposed algorithms is also derived, which is able to accurately predict the behaviors of proposed algorithms. In addition, a stability analysis is introduced. Monte Carlo (MC) simulations are conducted to demonstrate the advantage of the proposed algorithms and to validate the theoretical results. [ABSTRACT FROM AUTHOR]

Total least squares (TLS) method has been widely used in errors-in-variables (EIV) modeling tasks in which both input and output data are disturbed by noises, and adaptive filtering algorithm using TLS has shown significantly superior performance to classical least squares (LS) method in EIV system. The TLS essentially extends the minimum mean square error (MMSE) criterion to EIV model, which, however, may work poorly when noise is non-Gaussian (especially heavy-tailed distribution). Recently, an information theoretic learning (ITL) based minimum total error entropy (MTEE) adaptive filtering algorithm has been proposed, which extends the minimum error entropy (MEE) criterion to EIV model and shows desirable performance in non-Gaussian noise environments. However, due to complex mathematical expression, MTEE is computationally expensive and difficult to carry out the theoretical analysis. In this paper, we propose a new ITL-based criterion called maximum total correntropy (MTC) and develop a gradient-based MTC adaptive filtering algorithm. We analyze theoretically the local stability and steady-state performance of the proposed algorithm. Simulation results confirm the theoretical analysis and show the superior performance of MTC in heavy-tailed noises. Further, simulation comparisons between MTC and MTEE are presented. Compared with the MTEE, the MTC is mathematically more tractable and computationally much simpler while achieving similar or even better performance. [ABSTRACT FROM AUTHOR]

As a robust adaptation criterion, the maximum correntropy criterion (MCC) has gained increased attention due to its successful application in the adaptation, especially in nonlinear and non-Gaussian situations. In this paper, the second-order Volterra (SOV) filter based on MCC is derived, which is called MCC-SOV. It combines the advantages of the SOV filter and MCC. However, similar to the conventional adaptive algorithm, the proposed MCC-SOV filter has a tradeoff between convergence rate and steady-state error. In order to solve this problem, we present a combination of the MCC-SOV filter by combining Volterra kernels, which is called CK-MCC-SOV. In addition, a weight transfer method is applied to improve the performance of the proposed filter in terms of the convergence rate and the steady-state error. Finally, simulations are carried out to demonstrate the advantages of the proposed filters. [ABSTRACT FROM AUTHOR]

To improve the robustness of subband adaptive filter (SAF) against impulsive interferences, we propose two modified SAF algorithms with an individual scale function for each subband, which are derived by maximizing correntropy-based cost function and minimizing logarithm-based cost function, respectively, called MCC-SAF and LC-SAF. Whenever the impulsive interference happens, the subband scale functions can sharply drop the step size, which eliminate the influence of outliers on the tap-weight vector update. Therefore, the proposed algorithms are robust against impulsive interferences and exhibit the faster convergence rate and better tracking capability than the sign SAF (SSAF) algorithm. Besides, in impulse-free interference environments, the proposed algorithms achieve similar convergence performance as the normalized SAF (NSAF) algorithm. Simulation results have demonstrated the performance of our proposed algorithms. [ABSTRACT FROM AUTHOR]

Proposed is a novel variable step size normalized subband adaptive filter algorithm, which assigns an individual step size for each subband by minimizing the mean square of the noise-free a posterior subband error. Furthermore, a noniterative shrinkage method is used to recover the noise-free priori subband error from the noisy subband error signal. Simulation results using the colored input signals have demonstrated that the proposed algorithm not only has better tracking capability than the existing subband adaptive filter algorithms, but also exhibits lower steady-state error. [ABSTRACT FROM AUTHOR]

Recently, the sign subband adaptive filter (SSAF) algorithm has obtained great attention, due to its robustness against impulsive noises and decorrelating property for correlated input signals. However, the performance of the algorithm in the steady-state is not analyzed. In this paper, we study the steady-state mean-square-deviation (MSD) behavior of the SSAF algorithm by using energy conservation relation, Price׳s theorem and some reasonable assumptions. Simulation results in different system identification scenarios (including the input signals, tap lengths, impulsive noises, number of subbands, and step sizes) are provided to support our theoretical analysis. [ABSTRACT FROM AUTHOR]

Recently, the kernel adaptive filtering algorithm with the generalized maximum correntropy criterion (GMCC) has gained attention due to its excellent nonlinear system modeling ability and robustness to non-Gaussian noise. However, since the default center of the GMCC is located at zero, it hampers its performance in the non-zero mean noise environment. To overcome the problem, this paper proposes a kernel adaptive filtering algorithm based on the GMCC with variable center (GMCC-VC) called generalized kernel maximum correntropy criterion with variable center (GKMCC-VC). In addition, the online vector quantization (VQ) method is introduced to the GKMCC-VC algorithm for suppressing the increasing network size. The probability of divergence, the mean-square steady-state behavior, and the convergence condition of the GKMCC-VC algorithm are also obtained under some assumptions. The potential relationship between the GMCC-VC and the generalized minimum error entropy (GMEE) criteria is discussed in the paper. Finally, the simulation results verify the superiority of the proposed algorithm in modeling nonlinear systems and the correctness of the theoretical model. • This paper proposes the GKMCC-VC algorithm for non-zero mean and non-Gaussian noise. • The steady-state performance of the proposed algorithm is analyzed. • Simulation results show the effectiveness and feasibility of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Partial discharge (PD) detection plays a vital role in on-line condition monitoring of electrical apparatus in the power systems. However, the noise of PD measurements significantly degrades the performance of detection algorithms. In this paper, we focus on developing an adaptive filtering technique for the PD denoising problem. Heretofore, there are just a few literature reviews addressing the PD denoising based on such a method. The proposed recursive continuous S-shaped (RCSS) algorithm integrates the advantages of recursive strategy and continuous S-shaped function into adaptive noise cancellation (ANC) system, yielding enhanced filtering performance. The proposed algorithm can tackle PD noises in both Gaussian and impulsive scenarios, which is easy to implement in practical applications. The convergence behavior is also analyzed. Extensive simulation and experimental results confirm that the proposed algorithm can address polluted PD pulses, even in excessively noisy conditions, resulting in smaller mean square error (MSE) as compared to other state-of-the-art algorithms. [ABSTRACT FROM AUTHOR]

An adaptive estimation algorithm based on the Lorentzian norm is proposed for echo cancellation in vehicle hands-free communication systems and video teleconferencing systems, namely the affine-projection Lorentzian (APL) algorithm. By minimizing the Lorentzian norm of the a posteriori error vector with a suitable constraint on the weight vector and providing a dynamic Lorentzian-norm-controlling parameter, the proposed APL algorithm achieves robustness against impulsive disturbances and speeds up convergence for colored input signals. The computational complexity of the APL algorithm is analyzed and a fast recursive filtering method is employed to reduce its complexity. The stability analysis, based on energy-conservation arguments, shows that the APL algorithm converges. Furthermore, its tracking behavior is also investigated and a step size optimizing the tracking performance is derived. Simulation results agree well with the theoretical analysis. Simulation results for channel estimation and in-car echo cancellation scenarios demonstrate that the APL algorithm achieves better performance compared to the maximum-correntropy-criterion, affine-projection-generalized-maximum-correntropy, affine-projection-sign, affine-projection-like M-estimate, and Lorentzian adaptive filtering algorithms in various impulsive interference environments. [ABSTRACT FROM AUTHOR]

• A variant of the quantized minimum error entropy (QMEE) algorithm is proposed. • The mean square convergence performance analysis for QMEE algorithm is given. • The theoretical values of the steady-state EMSE for QMEE algorithm are validated by simulations. • QMEE algorithm is applied in electroencephalo (EEG) de-noising task. Adaptive filtering algorithms have been widely used in many areas, among which the minimum error entropy (MEE) algorithm is a superior choice, due to its excellent performance in the non-Gaussian noise situations. However, the computational complexity of the MEE algorithm is expensive, which leads to the computational bottlenecks, especially for large-scale datasets. In order to address the problem, we propose an adaptive filtering algorithm based on the quantized minimum error entropy (QMEE) criterion with an online quantization method, named QMEE algorithm. Moreover, we analyze the transient behavior characteristic and derive an approximate analytical expression for the steady-state excess mean square error (EMSE) based on the Taylor expansion. The extensive simulation results in linear modeling and electroencephalogram (EEG) denoising task demonstrate that the proposed method can outperform other robust adaptive filtering algorithms. [ABSTRACT FROM AUTHOR]

A bias‐compensated normalised least mean square (BCNLMS) algorithm with variable step size (denoted as BCVNLMS) is developed in this work to improve the accuracy and convergence speed for the three‐phase power system (TPPS) frequency estimation in circumstance of the noise and harmonic interference. Taking into account the problem of bias estimation caused by the mixed noises in the acquisition signal of the TPPS, the NLMS algorithm with an unbiased criterion (called BCNLMS) was employed to compensate the bias and improve the estimate accuracy. Furthermore, an adaptive variable step scheme is also incorporated into the BCNLMS algorithm to improve the convergence speed and stability during the process of adaptively frequency estimation. The experimental results show that the proposed BCVNLMS algorithm outperforms other well‐known methods for frequency estimation of the TPPS in several scenarios such as severe noise pollution, harmonic interference and frequency mutation. [ABSTRACT FROM AUTHOR]

• The kernel adaptive filter with generalized tanh function (KAF-GT) is proposed, which shows favourable performance in combating impulsive noise. • The convergence analysis of the proposed KAF-GT algorithm is conducted using energy conservation relation. • By using the pre-tuned dictionary strategy to model the unknown nonlinear system, we perform the EMSE analysis and arrive at a closed-form solution for predicting the steady-state behavior. • To further improve the performance of KAF-GT, an optimization scheme for scale factor is designed. Recently, online learning algorithms in machine learning have been imposed much attention. As a typical family, kernel adaptive filtering algorithms receive particular interest due to their sequential learning-based features. However, the kernel least mean square (KLMS) algorithm is not suitable for nonlinear tasks corrupted by non-Gaussian noise, especially impulsive noise. This is because the derivation of the KLMS algorithm is on the basis of the mean square error (MSE) criterion which only captures information of second-order statistics. In this paper, motivated by tanh function, we develop its generalized variant by introducing a scale factor for better representation capability; then we incorporate kernel adaptive filter with the generalized tanh function to propose a robust sequential learning algorithm. Based on establishing the energy conservation relation, we derive a sufficient condition for ensuring the algorithm convergence. In addition, to perform the steady-state excess mean square error (EMSE) analysis, we use the pre-tuned dictionary strategy to model the unknown nonlinear system in form of a finite-order combination; by Taylor expansion, we arrive at a closed-form solution for predicting the steady-state behavior. To further improve the algorithm performance, we design an optimization scheme for scale factor. Simulations for nonlinear time series prediction show that the designed schemes yield better performance than some state-of-art algorithms. The steady-state EMSE analysis is validated to provide accurate prediction results. [ABSTRACT FROM AUTHOR]

The adaptive variable step-size technique is an effective way to improve the convergence rate of an adaptive filtering algorithm. However, for most existing variable step-size sign algorithm against impulsive noise, either the performance is not good enough, or the calculation is complex. In this paper, a correntropy inspired variable step-size method for sign algorithm is proposed. The new variable step-size method is computationally simple and robust to impulsive noises. Theoretical analysis and simulation results demonstrate that the proposed algorithm can achieve desirable performance with low computational complexity in presence of impulsive noises. [ABSTRACT FROM AUTHOR]

For high background noise environment, an improved affine projection subband adaptive filter (IAPSAF) is proposed, which is derived by minimizing the sum of the squared Euclidean norms of the differences between the updated tap-weight vector and past tap-weight vectors. Due to the use of the past tap-weight vectors, the IAPSAF can achieve lower steady-state misalignment relative to the improved multiband-structured subband adaptive filter (IMSAF). Furthermore, a combined step size IAPSAF (CSS-IAPSAF) is developed to improve the performance of the IAPSAF. Simulation results in an acoustic echo cancellation (AEC) context show that the IAPSAF achieves lower steady-state misalignment than the existing algorithms, such as normalized subband adaptive filter (NSAF), IMSAF, and improved NSAF (INSAF). The improvement of the CSS-IAPSAF is also demonstrated in terms of convergence rate and steady-state misalignment. [ABSTRACT FROM AUTHOR]

In this brief, a robust and sparse recursive adaptive filtering algorithm, called convex regularized recursive maximum correntropy (CR-RMC), is derived by adding a general convex regularization penalty term to the maximum correntropy criterion (MCC). An approximate expression for automatically selecting the regularization parameter is also introduced. Simulation results show that the CR-RMC can significantly outperform the original recursive maximum correntropy (RMC) algorithm especially when the underlying system is very sparse. Compared with the convex regularized recursive least squares (CR-RLS) algorithm, the new algorithm also shows strong robustness against impulsive noise. The CR-RMC also performs much better than other LMS-type sparse adaptive filtering algorithms based on MCC. [ABSTRACT FROM AUTHOR]

To overcome the performance degradation of adaptive filtering algorithms in the presence of impulsive noise, a novel normalized sign algorithm (NSA) based on a convex combination strategy, called NSA-NSA, is proposed in this paper. The proposed algorithm is capable of solving the conflicting requirement of fast convergence rate and low steady-state error for an individual NSA filter. To further improve the robustness to impulsive noises, a mixing parameter updating formula based on a sign cost function is derived. Moreover, a tracking weight transfer scheme of coefficients from a fast NSA filter to a slow NSA filter is proposed to speed up the convergence rate. The convergence behavior and performance of the new algorithm are verified by theoretical analysis and simulation studies. [ABSTRACT FROM AUTHOR]

Proposed is a new subband adaptive filter (SAF) algorithm by minimizing Huber’s cost function that is robust to impulsive noises. Generally, this algorithm works in the mode of the normalized SAF (NSAF) algorithm, while it behaves like the sign SAF (SSAF) algorithm only when the impulsive noises appear. To further improve the robustness of this algorithm against impulsive noises, the subband cutoff parameters are updated in a recursive way. Moreover, the proposed algorithm can be interpreted as a variable step size NSAF algorithm, thus it exhibits faster convergence rate and lower steady-state error than the NSAF. Simulation results, using different colored input signals in both impulsive and free-impulsive noise environments, show that the proposed algorithm works better than some existing algorithms. [ABSTRACT FROM AUTHOR]

To improve the performance and reduce the computational complexity, a novel sign adaptation scheme for convex combination of adaptive filters is proposed. By using a sign adaptation scheme, the proposed scheme slightly reduces the computational complexity of the basic combination of mixing parameter, and improves the robustness of mixing parameter. Moreover, to obtain fast convergence property during the period of convergence transition in different environments, the instantaneous transfer scheme is applied to the proposed algorithm. The instantaneous transfer scheme has lower computational complexity than that of the basic combination of least mean square (basic-CLMS) algorithms. Simulation results in the context of system identification under time-invariant and time-variant systems demonstrate that the proposed algorithm based on the instantaneous transfer scheme has lower computational complexity and faster convergence rate than that of the basic-CLMS algorithm during period of convergence transitions. [ABSTRACT FROM AUTHOR]

Ø The KRSL based adaptive filtering algorithm with sinusoidal signal is derived; Ø A recursive MKRSL based harmonic estimation method is developed; Ø A RMKRSL with adaptive gain factor is proposed to enhance the tracking ability. The development of effective harmonics estimation method plays a crucial role for safe operation of power system. Recently several existing parametric estimation approaches, however, are mainly developed and discussed under the assumption of Gaussian noise which is always unsuitable in real power systems. This work is mainly concerned with the robust parametric estimation methods. The kernel risk sensitive loss (KRSL) shows an advantage in that its performance surface is flat around the optimal solution while sharp far away from the optimal solution and it may exhibit outstanding performance in terms of faster convergence rate and higher estimation accuracy under non-Gaussian noise cases. Therefore, a minimum KRSL (MKRSL) algorithm with sinusoidal signal is designed, and we further develop a novel recursive MKRSL (RMKRSL) algorithm for harmonics estimation in non-Gaussian noise cases. In addition, an adaptive gain factor is introduced into the gain matrix of RMKRSL algorithm to enhance its tracking capability. Some simulations by using the signal generated from the given models and IEEE-1159-PQE databases are conducted to illustrate the performance of the proposed methods under different conditions. [ABSTRACT FROM AUTHOR]

Part I of this paper reviewed the development of the linear active noise control (ANC) technique in the past decade. However, ANC systems might have to deal with some nonlinear components and the performance of linear ANC techniques may degrade in this scenario. To overcome this limitation, nonlinear ANC (NLANC) algorithms were developed. In Part II, we review the development of NLANC algorithms during the last decade. The contributions of heuristic ANC algorithms are outlined. Moreover, we emphasize recent advances of NLANC algorithms, such as spline ANC algorithms, kernel adaptive filters, and nonlinear distributed ANC algorithms. Then, we present recent applications of ANC technique including linear and nonlinear perspectives. Future research challenges regarding ANC techniques are also discussed. [ABSTRACT FROM AUTHOR]