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The presence of noise is a pervasive issue that significantly impacts the performance of adaptive filtering algorithms. To address this challenge, the bias compensation technique has recently emerged, involving the incorporation of an additional term into the update equation. Despite the apparent simplicity of the bias-compensated least-mean-squares algorithm, conducting a comprehensive theoretical analysis of its performance is a complex task. In this paper, we undertake an exact expectation analysis to demonstrate the asymptotic unbiasedness of the algorithm, even when disregarding the commonly assumed theory of independence between adaptive coefficients and input data. Furthermore, to improve the understanding of the algorithm's stability, we employ a stochastic model that assumes independence between the radial and angular distributions of the input vector. The resulting model is intricate, necessitating heuristic approximations to derive practical insights. Notably, our analysis reveals that the upper bound on the step size of the bias-compensated least-mean-squares algorithm consistently remains smaller than that of the least-mean-squares algorithm. These findings gain robust support from extensive simulations. [ABSTRACT FROM AUTHOR]

Nonlinear adaptive filters, such as Volterra and functional link adaptive filter (FLAF), face increased computational complexity with rising input dimensionality or expansion order. To enhance computational efficiency, this paper introduces a nonlinear adaptive algorithm based on Rectified Linear Unit (ReLU). Applying ReLU to a tonal signal generates components corresponding to the original frequency and even harmonics. Additionally, applying ReLU to the sum of two tonal signals yields a spectrum encompassing intermodulation distortion frequencies. These traits enable the algorithm to mitigate harmonic and intermodulation distortions while maintaining judicious computational complexity. The numerical simulation results for the identification of nonlinear systems validate the algorithm's ability to fit harmonic and intermodulation nonlinear distortions. In simulations of nonlinear echo cancellation with diverse test signals and various distortion models, our proposed algorithm consistently demonstrates competitive performance, notably excelling in music and speech scenarios. [ABSTRACT FROM AUTHOR]

The direction-of-arrival (DOA) estimation problem with a few noisy snapshots can be formulated as a problem of finding a joint sparse representation of multiple measurement vectors (MMV), and some algorithms based on compressive sensing (CS), such as the joint ℓ 0 approximation DOA (JLZA-DOA) and Multiple Snapshot Matching Pursuit Direction of Arrival (MSMPDOA) algorithms, have recently been proposed for solving this problem. Compared with the conventional DOA methods, the CS-based methods can achieve super-resolution by using only small number of snapshots, without the necessity of an accurate initialization, with small sensitivity to the correlation of the source signals. However, these CS-based algorithms usually do not work well in low signal-noise ratio (SNR) environment. In addition, the increased number of sensors in massive multiple-input-multiple-output (MIMO) systems lead to a huge matrix, and the matrix inversion operation in each iteration of the CS-based algorithm results in a relatively high computational cost. The purpose of this paper is to propose a novel adaptive filtering algorithm, i.e., the ℓ 2 , 0 -least mean square (ℓ 2 , 0 -LMS) algorithm, which can be viewed as a generalization of the ℓ 0 -LMS algorithm for single measurement vector (SMV) problem. Our proposed algorithm incorporates a mixed norm (ℓ 2 , 0 -norm) to treat the joint sparsity and inherits the robustness against noise and the low complexity of the ℓ 0 -LMS algorithm, and can thus work well for massive MIMO systems. Numerical experiments demonstrate that the proposed algorithm can achieve much better estimation performance with a lower computational cost than the existing ones. [ABSTRACT FROM AUTHOR]

A stochastic analysis of the probabilistic least-mean-square (Prob-LMS) algorithm would be a useful guideline for designing the adaptive filter. However, no analytical expressions for the stochastic analysis of the Prob-LMS algorithm have been reported in the literature. Hence, this paper analyzes the mean-deviation and mean-square-deviation (MSD) behavior of the Prob-LMS algorithm for the general case of an unknown Gauss-Markov channel. Analytical expressions are derived for the transient and steady-state MSD of the Prob-LMS algorithm. Monte Carlo simulations for fixed and time varying channels show excellent agreement between the simulated and theoretical MSD for a wide range of parameters such as SNR, filter length and input signal statistics. Monte Carlo MSD simulation results are presented for the Prob-LMS algorithm which compare favorably to several well-known VSS algorithms. [ABSTRACT FROM AUTHOR]

Abstract This paper studies the stochastic behavior of the LMS algorithm in a system identification framework for a cyclostationary colored input without assuming a Gaussian distribution for the input. The input cyclostationary signal is modeled by a colored random process with periodically time-varying power. The generation of the colored non-Gaussian random process is parametrized in novel manner by passing a Gaussian random process through a coloring filter followed by a zero memory nonlinearity. The unknown system parameters are fixed in most of the cases studied here. Mathematical models are derived for the behavior of the mean and mean-square-deviation (MSD) and the excess mean-square error (EMSE) of the adaptive weights as a function of the input cyclostationarity. The models display the dependence of the algorithm upon the input nonlinearity and coloration. Three nonlinearities that are studied in detail with Monte Carlo simulations provide strong support for the theory. [ABSTRACT FROM AUTHOR]

This paper analyzes the statistical behavior of the bias-compensated sign subband adaptive filtering algorithm with individual-weighting-factors (BC-IWF-SSAF) under errors-in-variables (EIV) model. The theoretical expressions for transient and steady-state behaviors of BC-IWF-SSAF in the mean and mean-square senses are derived through utilizing some reasonable assumptions and Price's theorem. Furthermore, the stability ranges of algorithm are provided. To accomplish performance enhancement, the variable step-size version (VSS) of BC-IWF-SSAF, called VSS-BC-IWF-SSAF algorithm, is developed based on the transient model via minimizing the mean-square deviation (MSD) at every iteration to tackle the contradictory requirements of quick convergence rate and high filtering accuracy caused by fixed step-size (FSS). Finally, Monte-Carlo simulations under system identification and acoustic echo cancellation applications certificate the effectiveness of statistical analysis and variable step-size strategy. [ABSTRACT FROM AUTHOR]

The classical proportionate adaptive filtering (PAF) algorithms achieve a fast initial convergence for sparse impulse response. But the small coefficients receive very little gain so that the time needed to reach steady-state misalignment is increased. In addition, the PAF algorithms converge much slower than the original adaptive filtering (OAF) algorithms when the impulse response is dispersive. In order to address these problems, this paper proposes a family of gain-combined PAF (GC-PAF) algorithms. The gain-combined matrix of the proposed GC-PAF algorithms is implemented by using a sigmoidal activation function to adaptively combine the proportionate matrix and identity matrix, which can retain the advantages of both the PAF algorithms in the context of sparse impulse response and the OAF algorithms in the context of dispersive impulse response. Meanwhile, to be also applicable to the family of sign algorithms against impulsive noise, a general framework for the update of the sigmoidal activation function is obtained by using the gradient descent method to minimize the L 1 -norm of the system output error. Simulations in the contexts of three different sparsity impulse responses have shown that the proposed GC-PAF algorithms perform much better than the OAF, PAF and improved PAF (IPAF) algorithms. [ABSTRACT FROM AUTHOR]

In recent years, the distributed estimation problem based on the diffusion strategy has received more and more attention, where the node filters cooperate with other in-network node filters to estimate the unknown parameter vector. In this paper, we propose a Diffusion Bias Compensated Recursive Maximum Correntropy Criterion (DBCRMCC) algorithm based on the idea of distributed estimation for adaptive filtering in network containing input noise and non-Gaussian output noise. The Recursive Maximum Correntropy Criterion (RMCC) is an adaptive filtering algorithm with Maximum Correntropy Criterion (MCC) as the cost function which is robust to large outliers. Considering that the estimates directly obtained by the RMCC algorithm are biased when the input is disturbed by noise, this paper proposes the DBCRMCC algorithm under some reasonable assumptions by using the principle of unbiased estimation and combining it with a diffusion strategy. Through bias-compensation for biased estimates and cooperation among node filters, the asymptotic unbiased estimates of the unknown parameters can be obtained. The simulation results show that the proposed DBCRMCC algorithm has acceptable convergence speed and estimation accuracy in the environment where the input signal is noisy and the output noise is non-Gaussian. [ABSTRACT FROM AUTHOR]

In all presented analyses of the normalized subband adaptive filter (NSAF) algorithm, there is a common assumption that the length of the adaptive filter is equal to that of the unknown system. In many practices, however, the adaptive filter usually works in an under-modeling situation. Namely, the length of the adaptive filter is less than that of the unknown system. Therefore, for this case, the existing analysis results for the NSAF algorithm are not applicable. In this paper, we analyze the performance of the deficient length NSAF algorithm based on some reasonable assumptions and approximations. More precisely, the expressions that characterize the transient-state and steady-state mean-square behavior of the algorithm are presented. Simulation results in various scenarios support our theoretical expressions. In addition, based on our analyses, a variable step size NSAF algorithm suitable for the under-modeling case is developed, which improves the performance. [ABSTRACT FROM AUTHOR]

This paper presents an architecture namely the Mirror MFx (MMFx) for adapting adaptive filtering algorithms for Active Noise Control (ANC) with on-line secondary path modelling. The proposed architecture is used in conjunction with the LMS algorithm, resulting in the MMFxLMS algorithm. A time domain analysis of the algorithm is presented, showing that the algorithm converges regardless of secondary path modeling errors. Simulations of the algorithm in different conditions but with the same parameters result in 100% convergence. The algorithm is especially suited to deal with large and sudden changes in the secondary path when the ANC system is in operation. Comparisons with competing algorithms are made, showing that they do not reach the same performance. [ABSTRACT FROM AUTHOR]

This paper proposes new transform-domain (TD) adaptive algorithms for long order adaptive filters. The scheme is based on decomposing the long adaptive filter into smaller subfilters. Each subfilter uses a different error signal in its update equation thus providing reduced coupling between subfilters and consequent improvement in convergence speed. We consider in this paper least mean-square adaptation of each subfilter along with power normalization, and derive an analytical model for the mean-square performance of the algorithm. Analysis assumes correlated Gaussian input data and will show that gains in convergence speed come at the expense of high misadjustment. A hybrid of the proposed algorithm and the TDLMS is described that offers significant performance improvements since it can be tuned to achieve very fast convergence while allowing low misadjustment. Performance advantages of the hybrid algorithm and theoretical results are verified by simulation examples. [Copyright &y& Elsevier]

System identification is often encountered in applications such as echo cancellation, active noise control, and channel equalization. If the unknown system is sparse, using sparsity-induced methods can improve the convergence performance. Recently, the l 2 , 0 -norm constraint was used to derive a block-sparse LMS (BS-LMS) algorithm to accelerate convergence for identifying multi-clustering sparse systems. In some cases, output of the unknown system is contaminated by impulsive noise, and BS-LMS performs poorly or even diverges when identifying such systems. To address this problem, this paper constructs a loss function by combining the absolute error and the l 2 , 0 -norm of adaptive filter weight vector, and then uses the subgradient descent method to develop a block-sparse sign algorithm (BS-SA). Its mean and mean-square performance is also analyzed based on the Gaussian-Bernoulli impulsive noise model under some frequently used assumptions. Finally, simulations are performed to test the robustness of BS-SA against impulsive noise and to evaluate the accuracy of theoretical expressions derived for statistical performance. [ABSTRACT FROM AUTHOR]

This paper proposes a novel diffusion subband adaptive filtering algorithm for distributed networks. To achieve a fast convergence rate and small steady-state errors, a variable step size and a new combination method is developed. For the adaptation step, the upper bound of the mean-square deviation (MSD) of the algorithm is derived and the step size is adaptive by minimizing it in order to attain the fastest convergence rate on every iteration. Furthermore, for a combination step realized by a convex combination of the neighbor-node estimates, the proposed algorithm uses the MSD, which contains information on the reliability of the estimates, to determine combination coefficients. Simulation results show that the proposed algorithm outperforms the existing algorithms in terms of the convergence rate and the steady-state errors. [ABSTRACT FROM AUTHOR]

The Normalized Least Mean Square (NLMS) algorithm with a filtered-x structure (FxNLMS) is a widely used adaptive algorithm for Active Noise Control (ANC) due to its simplicity and ease of implementation. One of the major drawbacks is its slow convergence. A modified filtered-x structure (MFxNLMS) can be used to moderately improve the speed of convergence, but it does not offer a huge improvement. A greater increase in the speed of convergence can be obtained by using the MFxNLMS algorithm with orthogonal correction factors (M-OCF), but the usage of orthogonal correction factors also increases the computational complexity and limits the usage of the M-OCF in massive real-time applications. However, Graphics Processing Units (GPUs) are well known for their potential for highly parallel data processing. Therefore, GPUs seem to be a suitable platform to ameliorate this computational drawback. In this paper, we propose to derive the M-OCF algorithm to a partitioned block-based version in the frequency domain (FPM-OCF) for multichannel ANC systems in order to better exploit the parallel capabilities of the GPUs. The results show improvements in the convergence rate of the FPM-OCF algorithm in comparison to other NLMS-type algorithms and the usefulness of GPU devices for developing versatile, scalable, and low-cost multichannel ANC systems. [ABSTRACT FROM AUTHOR]

Several techniques have been proposed in the literature to accelerate the convergence of adaptive algorithms for the identification of sparse impulse responses (i.e., with energy concentrated in a few coefficients). Among these techniques, the improved µ-law proportionate normalized least mean squares (IMPNLMS) algorithm is one of the most effective. This paper presents an accurate transient analysis of this algorithm and derives an estimate of its steady-state MSE, without requiring the assumption of white Gaussian input signals. [ABSTRACT FROM AUTHOR]

This paper presents a stochastic model of the least-mean-square for bilinear forms (LMS-BF) algorithm in which the bilinear term is defined with respect to the temporal and spatial impulse responses of a multiple-input/single-output (MISO) spatiotemporal system. Specifically, taking into account uncorrelated and correlated Gaussian input data, an analytical stochastic model is derived describing the behavior of the temporal and spatial (and, consequently, spatiotemporal) adaptive filters for both transient and steady-state phases. Based on the proposed model, some interesting insights about convergence and steady-state characteristics of the algorithm are discussed, thereby providing useful design guidelines. Moreover, an analytical relationship is established between the LMS and the LMS-BF algorithms, which makes it possible to obtain appropriate performance comparisons confirming the improved convergence characteristics achieved by the latter. Through simulation results, the accuracy of the proposed model as well as some features of the algorithm are verified under different operating scenarios. [ABSTRACT FROM AUTHOR]

In this paper, we proposed a novel online kernel-adaptive learning algorithm under the reproducing-kernel-Hilbert space (RKHS), which is called kernel q-Rényi (KqR) algorithm. The KqR algorithm is derived via constructing a q-Rényi kernel function as a cost function into kernel adaptive filtering algorithm (KAF). The vector-quantization (VQ) method is wildly used in the KAF algorithms, which is called quantized KAF. The quantized kernel q-Rényi (QKqR) algorithm is also proposed to significantly reduce computational cost via quantizing input space to curb the size of neural networks growth. The mean-square-convergence analysis of the QKqR algorithm is established, and the theoretical-value of the excess-mean-square-error (EMSE) are presented. The performance of the KqR and QKqR algorithms are investigated and demonstrated via tracking the Mackey-Glass (MG) time-series-prediction and the nonlinear channel equalization (NCE) under the non-Gaussian background interferences. [ABSTRACT FROM AUTHOR]

Adaptive filter has been successfully considered for the system identifications. Under some circumstances, due to the unknown system's intrinsic characteristics, nonnegativity is an desired constraint imposed on the parameters to be estimated, which will affect the performance of devised adaptive filtering. This paper studies a nonnegative least mean mixed-norm algorithm (NNLMMN) with fourth-order moment error and square-order error, which are jointly considered and devised to design a new cost function for dealing with the system identification problems with a nonnegativity constraint in mixed Gaussian noise environment. We also provide the theoretical behavior prediction of the mean weight, the transient analysis and steady-state excess mean square error (EMSE) of the NNLMMN algorithm. Experiment results show that the superior performance of the NNLMMN algorithm and verify the effectiveness of the presented theory analysis. [ABSTRACT FROM AUTHOR]

In real life, many engineering problems are modeled in the complex domain (CD). This paper proposes two complex-valued proportionate adaptive filtering (CPAF) algorithms in the CD for identifying the complex sparse systems in impulsive noises. The proposed CPAF algorithms are derived by maximizing the reuse of Versoria cost subjected to the weighted squared Euclidean norm of the filter tap-weight vector difference with the proportionate matrices as the weight of the weighted Euclidean norm. To address the tradeoff problem between the fast convergence rate and low steady-state misadjustment of the CPAF algorithms, two combined-step-size (CSS) CPAF algorithms are also proposed in the CD, which are obtained by introducing a time-varying step-size bound (TVSSB) into the constraint condition of the cost function of the CPAF algorithms. The TVSSB is achieved adaptively by using a modified sigmoidal function (MSF) to combine a large step-size and a small one. In order to be robust against impulsive noises, the MSF is adapted with the aid of a stochastic gradient ascent algorithm which is obtained by maximizing the Versoria cost function with respect to the real part of the a priori prediction error. Simulations on complex sparse system identification under impulsive noises in the CD demonstrate that the performance of the proposed CPAF algorithms is superior to that of the complex-valued variants of the affine projection Versoria, affine projection sign (APS), proportionate APS (PAPS), and improved PAPS (IPAPS) algorithms, also validate the effectiveness of the proposed CSS-CPAF algorithms. [ABSTRACT FROM AUTHOR]

Abstract: Normalized forms of adaptive algorithms are usually sought in order to obtain convergence properties independent of the input signal power. Such is the case of the well-known Normalized LMS (NLMS) algorithm. The Least-Mean Fourth (LMF) adaptive algorithm has been shown to outperform LMS in different situations. However, the LMF stability is dependent on both the signal power and on the adaptive weights initialization. This paper studies the behavior of two normalized forms of the LMF algorithm for Gaussian inputs. Contrary to what could be expected, the mean-square stability of both normalized algorithms is shown to be dependent upon the input signal power. Thus, the usefulness of the NLMF algorithm is open to question. [Copyright &y& Elsevier]

Abstract: In this paper, a new FIR adaptive filtering algorithm is proposed. The approach uses a variable step-size and the instantaneous value of the autocorrelation matrix in the coefficient update equation that leads to an improved performance. Convergence analysis of the algorithm has been presented. Simulation results show that the algorithm performs better than the Transform Domain LMS with Variable Step-Size (TDVSS) in stationary Additive White Gaussian Noise (AWGN) and Additive Correlated Gaussian Noise (ACGN) environments in a system identification setting. It is shown that the algorithm has a performance better than RLS and very similar to RRLS algorithm with a considerable reduction in computational complexity. Additionally, the performance of the proposed algorithm is shown to be superior to that of the Stabilized Fast Transversal Recursive Least Squares (SFTRLS) algorithm under the same conditions. [Copyright &y& Elsevier]

Abstract: The implementation of adaptive filters with fixed-point arithmetic requires computation quality evaluation. The accuracy may be determined by computing the global quantization noise power at the system output. In this paper, a new model for evaluating analytically the global noise power in LMS-based algorithms is presented. Thus, the model is developed for LMS and NLMS algorithms. The accuracy of our model is analyzed by simulations. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce a new approach to the method of non-parametric adaptive spectral analysis by using the Amplitude and Phase Estimation (APES) method, and taking into account the small sample errors of the sample covariance matrix. This approach is referred to as Adaptive Tuning Amplitude and Phase Estimation method (ATAPES). The main advantage of the ATAPES algorithm is its elimination of biased estimation exists with APES method, which is a biased peak location and corresponding problem of the biased amplitude estimation. The ATAPES method provides more accurate peak location and amplitude estimation with higher resolution than APES method. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a new time-varying step-size for the affine projection (AP) algorithm based on the minimization of the mean-square error (MSE) at each time instant. The step-size is dependent on accessible quantities and, therefore, does not need approximation. In addition, we show how the new step-size control equations can be incorporated into the fast AP (FAP) algorithm leading to a reduced complexity implementation of the proposed algorithm. The algorithm improved performance characteristics are verified by simulation experiments. [Copyright &y& Elsevier]

Recently, the kernel based constrained adaptive filtering algorithm has attracted a lot of attentions because of its robustness and superiority over traditional methods. As two classic kernel based algorithms, both constrained maximum correntropy criterion (CMCC) and constrained minimum error entropy (CMEE) have shown their superiorities in the case of non-Gaussian noise. However, both algorithms use only one kernel as the kernel function. To further improve the performance of the kernel based adaptive filtering algorithms, we first define the mixture kernel risk-sensitive loss (MKRSL) and study its properties. Then, we apply it to the constrained adaptive filtering and propose a novel constrained minimum MKRSL (CMM-KRSL) algorithm in this paper. Furthermore, we present the performance analysis of the CMM-KRSL algorithm, and provide the stability condition and the theoretical mean square deviation (MSD). Finally, we validate the accuracy of performance analysis and the superiorities of CMM-KRSL by simulations. [ABSTRACT FROM AUTHOR]