In this letter, we propose a hyperparameter optimization method for adaptive filtering based on deep unrolling, termed the deep unrolling affine projection (DAP) algorithm. The core idea is to reformulate the iterative structure of the traditional affine projection (AP) algorithm as a multilayer neural network, where each layer corresponds to one iteration and the step size is treated as a trainable parameter. These parameters are optimized through end-to-end supervised learning to enhance convergence speed and steady-state performance. While maintaining the interpretability and computational structure of the original algorithm, DAP leverages modern deep learning techniques to automatically learn hyperparameters from training data. Simulation results for the system identification task demonstrate that DAP outperforms the conventional AP algorithm in both convergence rate and accuracy. Importantly, DAP introduces no additional computational burden since the test phase involves only forward propagation. This makes it an efficient and practical solution for real-time adaptive filtering in engineering applications.

Traditional single-kernel or fixed-center multi kernel collaborative correntropies fundamentally assume that errors primarily cluster around a central point (typically zero). However, in real-world complex noise environments—such as those generated by mixed interference sources with diverse mechanisms—errors may exhibit multi-modal or highly asymmetric statistical characteristics. In such cases, a single central point or multi-kernels fixed at the origin cannot effectively capture the true shape of the error distribution. To address these problems, this letter proposes a novel robust learning algorithm by introducing variable-center multi-kernel correntropy into an asymmetric correntropy framework, where the kernel centers can be positioned at arbitrary locations. Compared with the maximum asymmetric correntropy criterion (MACC) algorithm, the proposed approach offers a more generalized formulation that enhances its capability to handle more complex error distributions, thereby improving algorithm performance. Notably, existing literature has not yet provided theoretical analysis for such variable-center multi-kernel asymmetric correntropy robust algorithms. Therefore, the main contributions of this work include: conducting the first theoretical analysis of the proposed algorithm, and validating the effectiveness of the analytical methodology.

Building upon the mean$p$-power error (MPE) criterion, the normalized subband$p$-norm (NSPN) algorithm demonstrates superior robustness in$\alpha$-stable noise environments ($1< \alpha \leq 2$) through effective utilization of low-order moment hidden in robust loss functions. Nevertheless, its performance degrades significantly when processing noise input or additive noise characterized by$\alpha$-stable processes ($0< \alpha \leq 1$). To overcome these limitations, we propose a novel fractional-order NSPN (FoNSPN) algorithm that incorporates the fractional-order stochastic gradient descent (FoSGD) method into the MPE framework. Additionally, this paper also analyzes the convergence range of its step-size, the theoretical domain of values for the fractional-order$\beta$, and establishes the theoretical steady-state mean square deviation (MSD) model. Simulations conducted in diverse impulsive noise environments confirm the superiority of the proposed FoNSPN algorithm against existing state-of-the-art algorithms.

As one of the most advanced variants in the correntropy family, the multi-kernel correntropy criterion demonstrates superior accuracy in handling non-Gaussian noise, particularly with multimodal distributions. However, current approaches suffer from key limitations-namely, reliance on a single type of sensitive Gaussian kernel and the manual selection of free parameters. To address these issues and further boost robustness, this paper introduces the concept of multi-kernel mixture correntropy (MKMC), along with its key properties. MKMC employs a flexible kernel function composed of a mixture of two Student'st-Cauchy functions with adjustable (non-zero) means. Building on this criterion within multi-sensor networks, we propose a robust distributed extended Kalman filter-AMKMMC-RDEKF based on adaptive multi-kernel mixture maximum correntropy. To reduce communication overhead, a consensus averaging strategy is incorporated. Furthermore, an adaptive mechanism is introduced to mitigate the impact of manually tuned free parameters. At the same time, the computational complexity and convergence ability of the proposed algorithm are analyzed. The effectiveness of the proposed algorithm is validated through challenging scenarios involving power system and land vehicle state estimation.

The cubature Kalman filter (CKF), while theoretically rigorous for nonlinear estimation, often suffers performance degradation due to model-environment mismatches in practice. To address this limitation, we propose CKFNet-a hybrid architecture that synergistically integrates recurrent neural networks (RNN) with the CKF framework while preserving its cubature principles. Unlike conventional modeldriven approaches, CKFNet embeds RNN modules in the prediction phase to dynamically adapt to unmodeled uncertainties, effectively reducing cumulative error propagation through temporal noise correlation learning. Crucially, the architecture maintains CKF's analytical interpretability via constrained optimization of cubature point distributions. Numerical simulation experiments have confirmed that our proposed CKFNet exhibits superior accuracy and robustness compared to conventional model-based methods and existing KalmanNet algorithms.

System modeling in distributed adaptive networks remains challenging under sensor aging-induced nonlinear data censoring and non-Gaussian interference. This brief proposes a robust hierarchical control framework that incorporates a Laguerre-based random Fourier filter to capture network nonlinearities. To address censored measurements and impulsive noise, a robust Laguerre RFF Diffusion Hierarchical (LRFF-DH) algorithm is developed. It leverages maximum likelihood estimation to correct bias and introduces a connectivity-adaptive robustness strategy based on node connectivity, enhancing edge-node resilience while reducing computational load on critical nodes. Simulations on a 20-node ad-hoc network verify the algorithm's effectiveness in interference suppression and mean squared error reduction under complex nonlinear conditions.

The correntropy contains all the even-order information of data and is well suited for dealing with non-Gaussian noise. The subband adaptive filtering algorithm combined with the maximum correntropy criterion (MCC-SAF) has been proposed to efficiently handle colored input signals and impulsive noise. Compared with other similar algorithms, the MCC-SAF algorithm exhibits faster convergence speed and lower steady-state error in non-Gaussian noise environments. However, to the best of our knowledge, the transient and steady-state statistical performance of this algorithm is still missing. In this letter, the mean-square performance of the MCC-SAF is analyzed under Gaussian, binary, and uniform noises using vectorization method, in which the mathematical expectation of the exponential term is computed by means of the moment generating function (MGF), and finally the theoretical prediction models for mean square deviation (MSD) are derived and verified by computer simulations.

Recently, the proposal of the least mean square (LMS) and recursive least squares (RLS) algorithm for graph signal processing (GSP) provides excellent solutions for processing signals defined on irregular structures such as sensor networks. The existing work has completed the steady state error analysis of the GSP LMS algorithm and GSP RLS algorithm in Gaussian noise scenarios, and a range of values for the step size of the GSP LMS algorithm has also been given. Meanwhile, the transient error analysis of the GSP LMS algorithm and GSP RLS algorithm is also important and challenging. Completing the above work will help to quantitatively analyze the performance of the graph signal adaptive estimation algorithm at transient moments, which is what this paper is working on. By using formula derivation and mathematical induction, the transient errors expressions of the GSP LMS and GSP RLS algorithm are given in this paper. Based on the Brazilian temperature dataset, the related simulation experiments are executed, which strongly demonstrate the correctness of our proposed theoretical analysis.

Currently, adaptive filtering algorithms have been widely applied in frequency estimation for power systems. However, research on diffusion tasks remains insufficient. Existing diffusion adaptive frequency estimation algorithms exhibit certain limitations in handling input noise and lack robustness against impulsive noise. Moreover, traditional adaptive filtering algorithms designed based on the strictly-linear (SL) model fail to effectively address frequency estimation challenges in unbalanced three-phase power systems. To address these issues, this letter proposes an improved diffusion augmented complex maximum total correntropy (DAMTCC) algorithm based on the widely linear (WL) model. The proposed algorithm not only significantly enhances the capability to handle input noise but also demonstrates superior robustness to impulsive noise. Furthermore, it successfully resolves the critical challenge of frequency estimation in unbalanced three-phase power systems, offering an efficient and reliable solution for diffusion power system frequency estimation. Finally, we analyze the stability of the algorithm and computer simulations verify the excellent performance of the algorithm.

The safety, reliability and rapidity of the transient stability assessment (TSA) method are important for improving the reliability and security of power system operation. TSA methods based on deep learning have good evaluation performance, but the increasing number of network layers leads to the model training time being continuously extended. Therefore, a TSA method based on broad learning system (BLS) is proposed to improve the efficiency of power system TSA. Meanwhile, in order to overcome the deficiency of BLS in feature extraction, a TSA model based on spatio-temporal broad learning system (STBLS) is established, which combines BLS with graph convolutional network (GCN) and temporal convolutional network (TCN). First, the grid topology map and electrical measurement data are used as model inputs, and the graph convolution module and the temporal convolution module are used to extract the spatial features and temporal features of the system, respectively. The extracted spatio-temporal features are then used as inputs to the BLS classification module, which is utilized for the rapid assessment of transient stability. Simulation results demonstrate the effectiveness of the proposed model and highlights its superiority in terms of training time.Note to Practitioners—This paper is motivated by the realiza-tion of a fast and accurate assessment of power system transient stability (TSA). The main goal is to shorten the training time of the model as much as possible while ensuring that the assessment model has a high accuracy. In this paper, the broad learning system (BLS) is introduced into the field of TSA, and a TSA model based on spatio-temporal broad learning system (STBLS) is proposed by combining the BLS with graph convolutional net-work (GCN) and temporal convolutional network (TCN), which greatly shortens the training time of the model. Experimental results show that this method has higher efficiency and practical application value. In future research, we will conduct more com-prehensive validation of such methods and further optimize the model to improve its stability and generalization ability.

The continuous development of new adaptive filters (AFs) based on novel cost functions (CFs) is driven by the demands of various application scenarios and noise environments. However, these algorithms typically demonstrate optimal performance only in specific conditions. In the event of the noise change, the performance of these AFs often declines, rendering simple parameter adjustments ineffective. Instead, a modification of the CF is necessary. To address this issue, the robust generalized adaptive AF (RGA-AF) with strong adaptability and flexibility is proposed in this paper. The flexibility of the RGA-AF’s CF allows for smooth adaptation to varying noise environments through parameter adjustments, ensuring optimal filtering performance in diverse scenarios. Moreover, we introduce several fundamental properties of negative RGA (NRGA) entropy, present the negative asymmetric RGA AF (NARGA-AF) and kernel recursive NRGA AF (KRNRGA-AF). These AFs address asymmetric noise distribution and nonlinear filtering issues, respectively. Simulations of linear system identification and time-series prediction for Chua’s circuit under different noise environments demonstrate the superiority of the proposed algorithms in comparison to existing techniques.

Non-Gaussian noise and the uncertainty of noise distribution are the common factors that reduce accuracy in dynamic state estimation of power systems (PS). In addition, the optimal value of the free coefficients in the unscented Kalman filter (UKF) based on information theoretic criteria is also an urgent problem. In this article, a robust adaptive UKF (AUKF) under generalized minimum mixture error entropy with fiducial points (GMMEEFs) over improve Snow Geese algorithm is proposed to overcome the above difficulties. The estimation process of the proposed algorithm is based on several key steps including augmented regression error model (AREM) construction, adaptive state estimation, and free coefficients optimization. Specifically, an AREM consisting of state prediction and measurement errors is established at the first step. Then, GMMEEF-AUKF is developed by solving the optimization problem based on GMMEEF, which uses a generalized Gaussian kernel combined with mixture correntropy to enhance the flexibility further and resolve the data problem with complex attributes and update the noise covariance matrix according to the AREM framework. Finally, the ISGA is designed to automatically calculate the optimal value of coefficients, such as the shape coefficients of the kernel in the GMMEEF criterion, the coefficients selection sigma points in unscented transform, and the update coefficient of the noise covariance matrices fit with the PS model. Simulation results on the IEEE 14, 30, and 57-bus test systems in complex scenarios have confirmed that the proposed algorithm outperforms the MEEF-UKF and UKF by an average efficiency of 26% and65%, respectively.

The censored regression (CR)-based normalized subband adaptive algorithm (CR-NSAF) model has been recently introduced for processing signals with censored data. However, the effectiveness of this algorithm declines when dealing with noisy input signals in impulsive noise environments. To resolve this challenge, we propose a robust bias-compensated CR-NSAF algorithm (RBC-CRNSAF). This algorithm alleviates the negative impacts of the CR system and improves robustness by employing a logarithmic cost function approach. It also minimizes estimation bias from input noise by incorporating new compensation terms into the weights update function. Additionally, we analyze the computational complexity, convergence characteristics, and stability conditions of the algorithm. Finally, computer simulations indicate that RBC-CRNSAF considerably outperforms other similar algorithms in impulsive noise environments, validating its enhanced performance.

The sensitivity of the broad learning system (BLS) based on the minimum-mean-square error (MMSE) criterion to non-Gaussian noise limits the application of this system. In order to solve this problem, this article proposes robust BLS (GC-BLS) based on generalized maximum correntropy criterion (GMCC). GMCC has a more general kernel function, which can effectively extract high-dimensional features of the data. In the non-Gaussian noise environment, GC-BLS shows excellent robustness. However, if the kernel width deviates from its optimal value, the performance of the system constructed based on the entropy criterion may fluctuate or even degrade. Since robust regression methods can improve robustness and are independent of kernel width, this motivates us to further construct MGC-BLS based on M-estimator functions. MGC-BLS can effectively suppress the performance degradation caused by kernel width deviation and has better robustness when selecting an appropriate M-estimator function. The two algorithms are optimized by fixed-point iteration, the update strategy in the iterative process is analyzed, and the sufficient conditions for the convergence of the algorithms are given. Compared with other BLS variants, GC-BLS and MGC-BLS can show better performance in the tests. Experiments on University of California Irvine (UCI) regression datasets and time-series datasets show the effectiveness of the two new algorithms.

Kernel adaptive filters (KAF) have emerged as a prominent method for nonlinear system identification (NSI). However, the KAF becomes computationally intensive as the input signal grows. In complex systems, traditional KAF with a unit time-delay structure may struggle with insufficient control capability. Moreover, KAFs adapted to second-order statistics can be susceptible to non-Gaussian noise. In this letter, we introduce the Laguerre kernel adaptive filter (LKAF) for NSI, using a block-oriented nonlinear model. The LKAF leverages the Laguerre series to approximate the linear block, benefiting from infinite impulse response (IIR) characteristics and a simple feedforward structure. To address non-Gaussian noise, the LKAF employs an arctangent (AT) criterion. This integration leads to the development of the Laguerre kernel arctangent least mean square (L-KATLMS) algorithm and its variations, which utilize random Fourier approximation. Simulation results demonstrate the superiority of our proposed algorithms for NSI.

The recursive general mixed-norm (RGMN) algorithm achieved excellent convergence performance in traditional unknown parameter identification application. However, it produces obvious estimation bias when dealing with the censored data collected from many practical scenarios. In this article, an RGMN algorithm for censored regression (CR-RGMN) is proposed for censored signal processing scenarios. The CR-RGMN algorithm utilizes a probit regression model to compensate for the sample bias caused by the sampling device. Then, a general form of the adaptive filter algorithm based on minimizing the convex mixture of$l_{a}$and$l_{b}$norms of the error is developed to identify the unknown parameter. The theoretical steady-state analysis in mean sense proves that the proposed CR-RGMN algorithm is unbiased, and the steady-state mean square deviation (MSD) is derived to predict the convergence behavior of the proposed algorithm. The computational complexities of the CR-RGMN and other algorithms are provided as well. Computer simulation results under robust parameter identification and channel equalization applications show that the proposed CR-RGMN algorithm achieves better performance in terms of convergence speed and steady-state MSD than the RGMN algorithm in censored data processing. And the theoretical analysis results are also verified.

The individual-weighting-factors sign subband adaptive filtering (IWF-SSAF) algorithm has acquired attention due to its quick convergence and brilliant robustness against impulsive interference. However, its statistics behavior has not been investigated under errors-in-variables (EIV) model, which system input and output are all corrupted by noise. Therefore, in this letter, the statistics analysis including transient and steady-state behaviors of IWF-SSAF is derived in mean and mean-square senses by virtue of some frequently-utilized assumptions and Price's theorem. Finally, numerical simulations under system identification scenario manifest that the simulated and theoretical results are in good agreement in different environments, which corroborate the accurateness of the derived theoretical models.

To acquire an improvement of the performance of the subband adaptive filter with impulsive interference, the normalized subband adaptive filter (NSAF) algorithm-based maximum correntropy criterion (MCC), called MCC-NSAF, has been developed. However, it is hard for the MCC criterion to take into account both the fast convergence rate and low steady-state error with a fixed kernel bandwidth. To handle this dilemma, in this paper, the normalized subband adaptive filter algorithm-based maximum mixture correntropy criterion (MMC) algorithm, called MMC-NSAF, is proposed. The MMC-NSAF algorithm integrates two correntropies with complementary control sizes of Gaussian kernel and can not only converge quickly but also has a small misalignment. Yet the MMC-NSAF still has some limitations because of its fixed step size. Therefore, a convexly combined MMC-NSAF algorithm, namely MMC-CNSAF, is proposed. The MMC-CNSAF algorithm utilizes two Gaussian kernels of different size and can combine overall improvement between convergence rate and steady-state error simultaneously. Eventually, the convergence and the computational complexity are analyzed. To achieve the test and verification of the robustness of proposed algorithms, simulations are carried out to discuss the effect of the parameters algorithm under colored input signals in impulsive interference environments for system identification. Meanwhile, the performances of the proposed algorithms under the influence of the acoustic echo cancellation in practical application are studied.

Currently, the generalized maximum correntropy criterion (GMCC) is extensively used in adaptive filtering arithmetic to handle generalized Gaussian noise. However, when the input signal is also subject to noise interference, the performance of the GMCC algorithm is degraded. In addition, the algorithms developed based on GMCC standards are facing high steady-state error problems. To address these issues, the generalized maximum Blake–Zisserman total correntropy (GMBZTC) algorithm based on the generalized maximum Blake–Zisserman (GMBZ) robust loss function is proposed in this article. More importantly, this article gives a detailed performance evaluation of the GMBZTC algorithm under generalized Gaussian noise conditions, obtains the conditions that guarantee the stability of the algorithm, and calculates the steady-state mean square deviation (S-MSD) of the algorithm. Finally, the excellence of the GMBZTC algorithm compared with other algorithms and the correctness of the theoretical analysis are demonstrated by simulation.

The adaptive filtering algorithm based on the maximum correntropy criterion (MCC) is very effective in suppressing non-Gaussian noises and therefore attracts widespread attentions. At present, some works have been done for study the convergence and steady-state performance analysis of the MCC algorithm, but its transient performance analysis is still an open problem. To provide a comprehensive theoretical foundation for the MCC algorithm, we propose a method for transient performance analysis based on moment generating function (MGF). Since this method can efficiently calculate the expected value of the exponential term in the iterative update equation, it can avoid the discrepancies caused by introducing some approximation methods such as Taylor expansions in the analysis process. To date, there is no precedent for using this method to analyze the transient performance of the MCC algorithm. In addition, the steady-state performance and stability conditions of the MCC algorithm are discussed based on this method. Finally, the proposed analytical method is applied to the system identification problem, and the results show that the theoretical analysis results are agree well with the Monte Carlo simulation results.

The random Fourier filter-based filtered-x least mean square (RF-FxLMS) algorithm has been proposed for the nonlinear active noise control (NANC) system to reduce the computational burden of the kernel filter. However, the RF-FxLMS algorithm markedly fluctuates when dealing with impulsive noise. In addition, the computing cost for the RF-FxLMS algorithm is still pricey in practice. In this work, a random Fourier filter based filtered-x generalized hyperbolic secant function (RF-FxGHSF) algorithm is presented to deal with impulsive noise. In virtue of the bilinear scheme, a cascaded random Fourier filter model is designed for concise computations, and the cascaded RF-FxGHSF (CRF-FxGHSF) algorithm is derived. Moreover, the steady-state convergence conditions are analyzed. The calculation complexity of the proposed algorithms is compared, and experiments emphatically analyze the principle for the presented model. Numerical simulations with α-stable noise and real noise carried out in different nonlinear path scenarios verify the convergence ability of proposed algorithms.

An adaptive least mean M-estimate algorithm for censored regression (CR-LMM) is presented for the robust parameter estimation of the censored regression system. To correct the bias produced by censored observation, the estimated error derived from the probit regression model is employed to construct an M-estimate cost function. It can expel the adverse impact of the impulsive noise and is solved by the unconstrained optimization method. Furthermore, the robust variable step-size (VSS) strategy, which is also predicted on the robust cost function, is also utilized to improve the convergence performance of the proposed CR-LMM algorithm, i.e., convergence speed and steady-state mean square deviation. The condition which guarantees the CR-LMM algorithm stability is obtained by analyzing the convergence in the mean and mean-square sense, and the theoretical steady-state result is also derived. Computer simulations in system identification scenarios are carried out to demonstrate that the proposed algorithms are superior to the existing algorithms in the impulsive environment with different background noise and the theoretical results are verified.

Recently, maximum correntropy criterion (MCC) has been widely and successfully used in robust signal processing and machine learning, in which the correntropy is maximized instead of minimizing the popular mean square error (MSE) to improve the robustness with respect to outliers or impulsive noises. A lot of efforts have been devoted to derive different adaptive algorithms under MCC, but to date, little insight has been gained as to how the MCC solution will be influenced by outliers. In this paper, we investigate this problem and our focus is mainly on the parameter estimation of a simple linear errors-in-variables (EIVs) model with scalar variables. Under some conditions, we derive an upper bound on the absolute value of the estimation error and show that the MCC solution can get very close to the true value of the unknown parameter even with arbitrarily large outliers in both the input and output variables. Illustrative examples are provided to verify and clarify the theory.

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.

This study proposes a block‐sparse non‐uniform norm constraint normalised subband adaptive filter (BS‐NNCNSAF) for the block‐sparse system identification problem, which is obtained by minimising a novel cost function involving the non‐uniform mixed l 2, p norm like a constraint. It can achieve better performance compared with the existing algorithms in the block‐sparse system identification. To further enhance the performance of the algorithm, the shrinkage BS‐NNCNSAF (SH‐BS‐NNCNSAF) algorithm is proposed. The proposed SH‐BS‐NNCNSAF algorithm is derived by taking the priori and the posteriori subband errors to achieve the time‐varying subband step sizes. Finally, simulations have been carried out to verify the performance of proposed algorithms. The simulation results verify that the proposed algorithms improve the performance of the filter, in terms of system identification in sparse systems.

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.

Anomaly detection has attracted much attention in recent years since it plays a crucial role in many domains. Various anomaly detection approaches have been proposed, among which one-class support vector machine (OCSVM) is a popular one. In practice, data used for anomaly detection can be distributively collected via wireless sensor networks. Besides, as the data usually arrive at the nodes sequentially, online detection method that can process streaming data is preferred. In this paper, we formulate a distributed online OCSVM for anomaly detection over networks and get a decentralized cost function. To get the decentralized implementation without transmitting the original data, we use a random approximate function to replace the kernel function. Furthermore, to find an appropriate approximate dimension, we add a sparse constraint into the decentralized cost function to get another one. Then we minimize these two cost functions by stochastic gradient descent and derive two distributed algorithms. Some theoretical analysis and experiments are performed to show the effectiveness of the proposed algorithms. Experimental results on both synthetic and real datasets reveal that both of the proposed algorithms achieve low misdetection rates and high true positive rates. Compared with other state-of-the-art anomaly detection methods, the proposed distributed algorithms not only show good anomaly detection performance, but also require relatively short running time and low CPU memory consumption.

This work develops a robust diffusion recursive least squares algorithm to mitigate the performance degradation often experienced in networks of agents in the presence of impulsive noise. This algorithm minimizes an exponentially weighted least-squares cost function subject to a time-dependent constraint on the squared norm of the intermediate estimate update at each node. With the help of side information, the constraint is recursively updated in a diffusion strategy. Moreover, a control strategy for resetting the constraint is also proposed to retain good tracking capability when the estimated parameters suddenly change. Simulations show the superiority of the proposed algorithm over previously reported techniques in various impulsive noise scenarios.

In order to improve the performances of recently presented improved normalised subband adaptive filter (INSAF) and proportionate INSAF algorithms for highly noisy system, this study proposes their set‐membership versions by exploiting the theory of set‐membership filtering. Apart from obtaining smaller steady‐state error, the proposed algorithms significantly reduce the overall computational complexity. In addition, to further improve the steady‐state performance for the algorithms, their smooth variants are developed by using the smoothed absolute subband output errors to update the step sizes. Simulation results in the context of acoustic echo cancellation have demonstrated the superiority of the proposed algorithms.

In this letter, a novel diffusion Volterra (DV) algorithm is proposed for distributed in-network system identification in the presence of α-stable noise. The proposed algorithm is based on the logarithmic least mean pth-power criterion, which makes it robust against impulsive interferences, at the price of increased complexity. To overcome this shortcoming, we further develop the diffusion interpolated Volterra algorithm, which provides computational savings and good performance in comparison with the DV algorithm. Simulations results show that the proposed adaptive algorithms achieve better performance than the state-of-the-art approaches for distributed nonlinear system identification in impulsive noise.

The minimum error entropy (MEE) is an important and highly effective optimization criterion in information theoretic learning (ITL). For regression problems, MEE aims at minimizing the entropy of the prediction error such that the estimated model preserves the information of the data generating system as much as possible. In many real world applications, the MEE estimator can outperform significantly the well-known minimum mean square error (MMSE) estimator and show strong robustness to noises especially when data are contaminated by non-Gaussian (multimodal, heavy tailed, discrete valued, and so on) noises. In this brief, we present some theoretical results on the robustness of MEE. For a one-parameter linear errors-in-variables (EIV) model and under some conditions, we derive a region that contains the MEE solution, which suggests that the MEE estimate can be very close to the true value of the unknown parameter even in presence of arbitrarily large outliers in both input and output variables. Theoretical prediction is verified by an illustrative example.

A bias-compensated normalized subband adaptive filter (BC-NSAF) algorithm is proposed for system identification. In the proposed algorithm, a bias-compensation vector is derived to eliminate the bias caused by the noisy input signals. To estimate the input noise variance, a new estimation method is proposed, which does not require the input-output variance ratio in advance. Simulation results show that the proposed algorithm obtains better convergence performance than the existing algorithms in the presence of noisy input signals.

The maximum correntropy criterion (MCC) has received increasing attention in signal processing and machine learning due to its robustness against outliers (or impulsive noises). Some gradient based adaptive filtering algorithms under MCC have been developed and available for practical use. The fixed-point algorithms under MCC are, however, seldom studied. In particular, too little attention has been paid to the convergence issue of the fixed-point MCC algorithms. In this letter, we will study this problem and give a sufficient condition to guarantee the convergence of a fixed-point MCC algorithm.

An individual-activation-factor memory proportionate affine projection algorithm (IAF-MPAPA) is proposed for sparse system identification in acoustic echo cancellation (AEC) scenarios. By utilizing an individual activation factor for each adaptive filter coefficient instead of a global activation factor, as in the standard proportionate affine projection algorithm (PAPA), the adaptation energy over the coefficients of the proposed IAF-MPAPA can achieve a better distribution, which leads to an improvement of the convergence performance. Moreover, benefiting from the memory characteristics of the proportionate coefficients, its computational complexity is less than the PAPA and improved PAPA (IPAPA). In the context of AEC and stereophonic AEC (SAEC) for highly sparse impulse responses, simulation results indicate that the proposed IAF-MPAPA outperforms the PAPA, IPAPA, and memory IPAPA (MIPAPA) in terms of the convergence rate and tracking capability when the unknown impulse response suddenly changes.

A novel complex-valued non-linear equaliser-based pipelined decision feedback recurrent neural network (CPDFRNN) is proposed in this study for non-linear channel equalisation in wireless communication systems. The CPDFRNN with low computational complexity, a modular structure comprising a number of modules that are interconnected in a chained form, is an extension of the recently proposed real-valued pipelined decision feedback recurrent neural equalisers. Each module is implemented by a small-scale complex-valued decision feedback recurrent neural network (CDFRNN). Moreover, a decision feedback part in each module can overcome the unstable characteristic of the complex-valued recurrent neural network (CRNN). To suit the modularity of the CPDFRNN, an adaptive amplitude complex-valued real-time recurrent learning (CRTRL) algorithm is presented. Simulations demonstrate that the CPDFRNN equaliser using the amplitude CRTRL algorithm with less computational complexity not only eliminates the adverse effects of the nesting architecture, but also provides a superior performance over the CRNN and CDFRNN equalisers for non-linear channels in wireless communication systems.

This correspondence presents an extended pipelined second-order Volterra (EPSOV) filter for active control of nonlinear noise processes. The corresponding nonlinear filtered-x algorithms using the filter bank implementation are also suggested. Compared to the standard SOV filter using the filtered-x least mean square (SOVFXLMS), those modules of the EPSOV filter can be performed simultaneously in a pipelined parallelism fashion, and this would lead to a significant improvement in its total computational efficiency. Results obtained from computer simulations for nonlinear noise processes demonstrate that the proposed method outperforms the SOV.

To combat the linear and non-linear distortions for time-invariant and time-variant channels, a novel adaptive joint process equaliser based on a pipelined decision feedback recurrent neural network (JPDFRNN) is proposed in this paper. The JPDFRNN consists of a number of simple small-scale decision feedback recurrent neural network (DFRNN) modules and a linear combiner. The cascaded DFRNN provides pre-processing for the linear combiner. Moreover, each DFRNN can provide a local interpolation for M sample points; the final linear combiner presents a global interpolation with good localisation properties. Furthermore, since those modules of non-linear subsection can be performed simultaneously in a pipelined parallelism fashion, this would result in a significant improvement in the total computational efficiency. Simulation results show that the performance of the JPDFRNN using the modified real-time recurrent learning (RTRL) algorithm is superior to that of the DFRNN and RNN for the non-linear time-invariant and time-variant channels.

To reduce the computational complexity of the bilinear recurrent neural network (BLRNN), a novel low-complexity nonlinear adaptive filter with a pipelined bilinear recurrent neural network (PBLRNN) is presented in this paper. The PBLRNN, inheriting the modular architectures of the pipelined RNN proposed by Haykin and Li, comprises a number of BLRNN modules that are cascaded in a chained form. Each module is implemented by a small-scale BLRNN with internal dynamics. Since those modules of the PBLRNN can be performed simultaneously in a pipelined parallelism fashion, it would result in a significant improvement of computational efficiency. Moreover, due to nesting module, the performance of the PBLRNN can be further improved. To suit for the modular architectures, a modified adaptive amplitude real-time recurrent learning algorithm is derived on the gradient descent approach. Extensive simulations are carried out to evaluate the performance of the PBLRNN on nonlinear system identification, nonlinear channel equalization, and chaotic time series prediction. Experimental results show that the PBLRNN provides considerably better performance compared to the single BLRNN and RNN models.

In this letter, a novel pipelined decision feedback RNN equalizer (PDFRNE) with low computational complexity is proposed. Since each module is a DFRNN with the decision feedback structure so that it can eliminate the past error remaining in the network. Moreover, the performance can be further improved. At the same time, it can overcome the unstableness due to its nature of the infinite impulse response (IIR) structure.

A novel nonlinear adaptive filter with pipelined Chebyshev functional link artificial recurrent neural network (PCFLARNN) is presented in this paper, which uses a modification real-time recurrent learning algorithm. The PCFLARNN consists of a number of simple small-scale Chebyshev functional link artificial recurrent neural network (CFLARNN) modules. Compared to the standard recurrent neural network (RNN), those modules of PCFLARNN can simultaneously be performed in a pipelined parallelism fashion, and this would lead to a significant improvement in its total computational efficiency. Furthermore, contrasted with the architecture of a pipelined RNN (PRNN), each module of PCFLARNN is a CFLARNN whose nonlinearity is introduced by enhancing the input pattern with Chebyshev functional expansion, whereas the RNN of each module in PRNN utilizing linear input and first-order recurrent term only fails to utilize the high-order terms of inputs. Therefore, the performance of PCFLARNN can further be improved at the cost of a slightly increased computational complexity. In addition, due to the introduced nonlinear functional expansion of each module in PRNN, the number of input signals can be reduced. Computer simulations have demonstrated that the proposed filter performs better than PRNN and RNN for nonlinear colored signal prediction, nonstationary speech signal prediction, and chaotic time series prediction.

This paper proposes a novel computational efficient adaptive nonlinear equalizer based on combination of finite impulse response (FIR) filter and functional link artificial neural network (CFFLANN) to compensate linear and nonlinear distortions in nonlinear communication channel. This convex nonlinear combination results in improving the speed while retaining the lower steady-state error. In addition, since the CFFLANN needs not the hidden layers, which exist in conventional neural-network-based equalizers, it exhibits a simpler structure than the traditional neural networks (NNs) and can require less computational burden during the training mode. Moreover, appropriate adaptation algorithm for the proposed equalizer is derived by the modified least mean square (MLMS). Results obtained from the simulations clearly show that the proposed equalizer using the MLMS algorithm can availably eliminate various intensity linear and nonlinear distortions, and be provided with better anti-jamming performance. Furthermore, comparisons of the mean squared error (MSE), the bit error rate (BER), and the effect of eigenvalue ratio (EVR) of input correlation matrix are presented.