Sparse leaky‐LMS algorithm for system identification and its convergence analysis

In this paper, a novel adaptive filter for sparse systems is proposed. The proposed algorithm incorporates a log‐sum penalty into the cost function of the standard leaky least mean square (LMS) algorithm, which results in a shrinkage in the update equation. This shrinkage, in turn, enhances the performance of the adaptive filter, especially, when the majority of unknown system coefficients are zero. Convergence analysis of the proposed algorithm is presented, and a stability criterion for the algorithm is derived. This algorithm is given a name of zero‐attracting leaky‐LMS (ZA‐LLMS) algorithm. The performance of the proposed ZA‐LLMS algorithm is compared to those of the standard leaky‐LMS and ZA‐LMS algorithms in sparse system identification settings, and it shows superior performance compared to the aforementioned algorithms. Copyright © 2013 John Wiley & Sons, Ltd.     

A unifying framework for the analysis of proportionate NLMS algorithms

Despite being a de facto standard in sparse adaptive filtering, the two most important members of the class of proportionate normalised least mean square (PNLMS) algorithms are introduced empirically. Our aim is to provide a unifying framework for the derivation of PNLMS algorithms and their variants with an adaptive step‐size. These include algorithms with gradient adaptive learning rates and algorithms with adaptive regularisation parameters. Convergence analysis is provided for the proportionate least mean square (PLMS) algorithm in both the mean and mean square sense and bounds on its parameters are derived. An alternative, more insightful approach to the convergence analysis is also presented and is shown to provide an estimate of the optimal step‐size of the PLMS. Incorporating the so obtained step‐size into the PLMS gives the standard PNLMS together with a unified framework for introducing other adaptive learning rates. Simulations on benchmark sparse impulse responses support the approach. Copyright © 2014 John Wiley & Sons, Ltd.     

Steiglitz–McBride adaptive notch filter based on a variable‐step‐size LMS algorithm and its application to active noise control

This paper proposes a new Steiglitz–McBride (SM) adaptive notch filter (SM‐ANF) based on a robust variable‐step‐size least‐mean‐square algorithm and its application to active noise control (ANC). The proposed SM‐ANF not only has fast convergence but also has small misadjustment. The variable‐step‐size algorithm uses the sum of the squared cross correlation between the error signal and the delayed inputs corresponding to the adaptive weights. The cross correlation provides robustness to the broadband signal, which plays the role of noise. The proposed SM‐ANF is computationally simpler than the existing Newton/recursive least‐squares‐type ANF. The frequency response of the new SM‐ANF has a notch depth of about ?25 dB (for each of the three frequencies considered) and has spectral flatness within 5 dB (peak to peak). This robust notch filter algorithm is used as an observation noise canceller for the secondary path estimation of an ANC system based on the SM method. The ANC with proposed SM‐ANF provides not only faster convergence but also an 11‐dB improvement in noise attenuation over the SM‐based ANC without such a SM‐ANF. Copyright © 2015 John Wiley & Sons, Ltd.     

Compressive sensing‐based adaptive sparse predistorter design for power amplifier linearization

Greedy algorithms in the compressive sensing theory have been formed the essential method for pruning power amplifier (PA) behavioral models and digital predistorters (DPDs). However, the inherent batch mode of these algorithms limits their application in adaptive digital predistortion framework. In this paper, a powerful subspace pursuit greedy scheme combined with stochastic gradient descent adaptive algorithm is proposed to design a class of adaptive sparse DPDs. According to the given sparsity level, the proposed approach can obtain the sparse terms of DPDs and extract the corresponding coefficients adaptively. Performance improvement of the proposed method is validated by simulation results on the adaptive DPD excited by 15‐MHz 3‐carrier Long‐Term Evolution signals and 50‐MHz 16 amplitude/phase‐shift keying signals. Meanwhile, measurement results on a Doherty PA excited by 30‐MHz 3‐carrier Long‐Term Evolution signals are also performed to verify the advantage of the proposed approach. Simulation and experimental results show that proposed algorithm can efficiently construct the adaptive sparse DPD models with only a small number of parameters; both nonlinear distortions and memory effects in the PA can be almost completely removed. A comparison with the nonsparsity aware DPD techniques and batch mode compressive sensing pruning techniques has been demonstrated that the proposed method exhibit faster convergence, improving tracking capabilities and reduced computational complexity.     

Extension of LMS stability condition over a wide set of signals

A sufficient condition for least mean squares (LMS) algorithm stability with a small set of assumptions is derived in this paper. The derivation is not, contrary to the majority of currently known conditions, based on the independence assumption or other statistic properties of the input signals. Moreover, it does not make use of the small‐step‐size assumption, neither does it assume the input signals are stationary. Instead, it uses a theory of discrete systems and properties of a discrete state‐space matrix. Therefore, the result can be applied to a wide set of signals, including deterministic and nonstationary signals. The location of all eigenvalues of the matrix responsible for the LMS algorithm stability has been calculated. Simulation experiments, where the step size reaches a couple of hundreds without loss of stability, are shown to support the theory. On the other hand, simulation where the calculations based on the small‐step‐size theory provide a too large estimation of the upper bound for the step size, while the new condition gives a proper solution, is also presented. Therefore, the new condition may be used in cases where fast adaptation is necessary and when the independence theory or the small‐step‐size assumptions do not hold. Copyright © 2014 John Wiley & Sons, Ltd.     

基于自适应滤波算法的谐波仿真分析

为改善自适应滤波算法的滤波效果,减小稳态误差,提高跟踪响应速度,并验证改进的变步长LMS滤波算法的有效性和优越性,利用电磁暂态仿真软件PSCAD/EMTDC,构建电网谐波仿真计算模型,并与传统的基于瞬时无功功率理论的p-q算法以及定步长LMS算法进行仿真比较,根据不同系统条件下的滤波仿真波形,进行快速傅里叶分析,验证了此改进变步长LMS算法在计算量增加不多的前提下,可以同时获得较好的跟踪速度和较小的稳态误差,证实了该算法的有效性。     

A simple variable step size LMS adaptive algorithm

A new LMS based variable step size adaptive algorithm is presented. The step size is incremented or decremented by a small positive value, whenever the instantaneous error is positive or negative, respectively. The algorithm is simple, robust and efficient. It is characterized by fast convergence and low steady state mean squared error. The performance of the algorithm is analysed for a stationary zero‐mean white‐Gaussian input. MC simulations are provided to demonstrate its improved performance over the conventional LMS (Proc. IEEE 1976; 64 :1151–1162) and some other variable step size adaptive algorithms (IEEE Trans. Signal Process. 1992; 40 :1633–1642; IEEE Trans. Signal Process. 1997; 45 :631–639) within a range of statistical environments. For a non‐stationary input, the proposed algorithm behaves similar to these algorithms. A modified version of the algorithm is presented to perform in the presence of abrupt changes. Copyright © 2004 John Wiley & Sons, Ltd.     

基于自适应变步长最小均方的谐波检测算法的研究

从数字信号处理中的自适应噪声对消原理出发,介绍了一种改进的变步长最小均方(LMS)算法,该算法根据基波电流和谐波电流的不相关性,利用误差信号和参考输入的互相关估计来控制迭代步长,使得步长的更新不受谐波电流的影响。该算法原理简单,运算量小,易于实现,仿真结果表明了该谐波电流检测算法的有效性。     

一种改进的自适应谐波电流检测方法

针对现有的基于双曲正切函数变步长LMS算法的谐波电流检测仍存在稳态误差和收敛速度不能同时满足要求的问题,分析了一种在基于双曲正切函数变步长LMS算法的基础上改进的变步长算法,利用误差的时间均值估计建立步长与误差之间的新型双曲正切函数关系以控制步长的更新,降低稳态误差,提高算法的检测精度。并且同时对权值采用两次迭代更新,将两次迭代的结果作为新的权值,以加快权值的更新速度,提高算法的收敛速度。该算法具有较高的检测精度的同时还有较快的响应速度。Matlab/Simulink的仿真结果证明了该算法用于谐波电流检测具有很好的效果。     

一种改进的变步长LMS自适应算法

为了得到一种抗干扰性能强、收敛速度快、稳态误差小的变步长LMS算法,本文分析了传统LMS算法、变步长LMS算法及其改进算法,并对现有变步长LMS算法进行分类归纳,在列出几种具有代表性算法的基础上,提出了一种改进的变步长LMS算法.新方法引入修正系数,通过当前和过去估计误差以及误差的时域均值来调节自适应算法的步长,具有收敛速度快,稳态误差小的优点,并且提高了LMS算法的抗干扰性能.文中最后给出了仿真结果,仿真结果与理论分析一致.     

Single channel informed signal separation using artificial‐stereophonic mixtures and exemplar‐guided matrix factor deconvolution

In this paper, a method is proposed to tackle the problem of single channel audio separation. The proposed method leverages on the exemplar source is used to emulate the targeted speech signal. A multicomponent nonnegative matrix factor 2D deconvolution (NMF2D) is proposed to model the temporal and spectral changes and the number of spectral basis of the audio signals. The paper proposes an artificial auxiliary channel to imitate a pair of stereo mixture signals, which is termed as “artificial‐stereophonic mixtures.” The artificial‐stereophonic mixtures and the exemplar source are jointly used to guide the factorization process of the NMF2D. The factorization is adapted under a hybrid framework that combines the generalized expectation–maximization algorithm with multiplicative update adaptation. The proposed algorithm leads to fast and stable convergence and ensures the nonnegativity constraints of the solution are satisfied. Adaptive sparsity has also been introduced on each sparse parameter in the multicomponent NMF2D model when the exemplar deviates from the target signal. Experimental results have shown the competence of the proposed algorithms in comparison with other algorithms.     

A joint echo cancellation algorithm for quick suppression of howls in hearing aids

When the echo path of a hearing aid suddenly changes, howls easily occur. To quickly suppress the howls, a joint echo cancellation (JEC ) algorithm, which combines the variable step normalized least mean square (VNLMS ) algorithm with the notch filter algorithm, is proposed. According to whether the hearing aid howls or not, different strategies are used. First, when there are no howls, the echo signal is estimated using VNLMS and the step factor is computed according to three types of filter states, which are defined based on the normalized distance between the short‐term average and the long‐term average of the filter coefficients. Then, different step factors are used for different states. Second, when there are howls, the update of VNLMS is frozen to stabilize the howl frequency. To improve the detection accuracy, a howling detection algorithm based on the zoom‐fast Fourier transformation (ZoomFFT ) is proposed. The ZoomFFT algorithm can analyze the spectrum of a narrowband signal in a specified high sampling frequency. Then, the notch filters based on the estimated howl frequencies are dynamically generated to restrain the howls. Finally, when the howls are suppressed, VNLMS is reactivated. Compared to other echo cancellation algorithms, the proposed algorithm can quickly suppress the howls, and JEC has the best comprehensive performance. Furthermore, the quality of the processed speech is high, and the operation time is short. Thus, the proposed algorithm is suitable for low‐power‐consumption and small‐volume products such as hearing aids. © 2017 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.     

APF中一种改进的变步长LMS自适应谐波检测算法

在有源电力滤波器(Active Power Filter, APF)的低信噪比(Signal Noise Ratio, SNR)环境下,为了提高变步长最小均方(Least Mean Square, LMS)自适应算法对谐波电流检测的跟踪速度及精度,提出改进的变步长LMS算法。该算法在MVSS-LMS算法的基础上,增加历史误差的遗忘加权和估计并控制步长更新,动态控制步长更新范围,采用滑动窗遗忘加权减小了计算复杂度。同时,对改进算法性能进行稳定性分析。实验结果表明,该算法不仅具有较快的动态响应速度,而且在APF的低信噪比情况下,稳态误差有所减小,具有较高的抗干扰能力,谐波电流检测效果较好。     

Convergence properties of bias‐eliminating algorithms for errors‐in‐variables identification

This paper considers the problem of dynamic errors‐in‐variables identification. Convergence properties of the previously proposed bias‐eliminating algorithms are investigated. An error dynamic equation for the bias‐eliminating parameter estimates is derived. It is shown that the convergence of the bias‐eliminating algorithms is basically determined by the eigenvalue of largest magnitude of a system matrix in the estimation error dynamic equation. When this system matrix has all its eigenvalues well inside the unit circle, the bias‐eliminating algorithms can converge fast. In order to avoid possible divergence of the iteration‐type bias‐eliminating algorithms in the case of high noise, the bias‐eliminating problem is re‐formulated as a minimization problem associated with a concentrated loss function. A variable projection algorithm is proposed to efficiently solve the resulting minimization problem. A numerical simulation study is conducted to demonstrate the theoretical analysis. Copyright © 2005 John Wiley & Sons, Ltd.     

Development of numerical linear algebra algorithms in dynamic fixed‐point format: a case study of Lanczos tridiagonalization

Proper range and precision analysis play an important role in the development of fixed‐point algorithms for embedded system applications. Numerical linear algebra algorithms used to find singular value decomposition of symmetric matrices are suitable for signal and image‐processing applications. These algorithms have not been attempted much in fixed‐point arithmetic. The reason is wide dynamic range of data and vulnerability of the algorithms to round‐off errors. For any real‐time application, the range of the input matrix may change frequently. This poses difficulty for constant and variable fixed‐point formats to decide on integer wordlengths during float‐to‐fixed conversion process because these formats involve determination of integer wordlengths before the compilation of the program. Thus, these formats may not guarantee to avoid overflow for all ranges of input matrices. To circumvent this problem, a novel dynamic fixed‐point format has been proposed to compute integer wordlengths adaptively during runtime. Lanczos algorithm with partial orthogonalization, which is a tridiagonalization step in computation of singular value decomposition of symmetric matrices, has been taken up as a case study. The fixed‐point Lanczos algorithm is tested for matrices with different dimensions and condition numbers along with image covariance matrix. The accuracy of fixed‐point Lanczos algorithm in three different formats has been compared on the basis of signal‐to‐quantization‐noise‐ratio, number of accurate fractional bits, orthogonality and factorization errors. Results show that dynamic fixed‐point format either outperforms or performs on par with constant and variable formats. Determination of fractional wordlengths requires minimization of hardware cost subject to accuracy constraint. In this context, we propose an analytical framework for deriving mean‐square‐error or quantization noise power among Lanczos vectors, which can serve as an accuracy constraint for wordlength optimization. Error is found to propagate through different arithmetic operations and finally accumulate in the last Lanczos vector. It is observed that variable and dynamic fixed‐point formats produce vectors with lesser round‐off error than constant format. All the three fixed‐point formats of Lanczos algorithm have been synthesized on Virtex 7 field‐programmable gate array using Vivado high‐level synthesis design tool. A comparative study of resource usage and power consumption is carried out. Copyright © 2015 John Wiley & Sons, Ltd.     

Development and convergence analysis of new least‐mean‐square‐based algorithm equipped with exponential‐decay step size and disturbance compensation for active noise control

In real‐world active noise control (ANC) applications, disturbance can be picked up by error sensors and significantly degrade the steady‐state ANC performance. This study proposes two techniques in combination with a least‐mean‐square (LMS) based ANC algorithm, named normalized filtered‐x LMS/commutation error (NFxLMS/CE) algorithm, to deal with the disturbance that is independent of a reference signal. A new stochastic method to analyze convergence properties of the NFxLMS/CE algorithm under influence of the disturbance is first established. Given that the reference signal is persistently exciting of sufficient order, exponential convergence of the algorithm is derived with a step‐size condition. An exponential‐decay step size (EDSS) is then proposed to obtain a new ANC algorithm referred to as EDSS‐NFxLMS/CE algorithm. In addition, a disturbance‐compensation (DC) technique is developed for the EDSS‐NFxLMS/CE algorithm to obtain an EDSS‐NFxLMS/CE_DC algorithm such that the influence of the disturbance can be reduced. It is shown that the EDSS‐NFxLMS/CE_DC algorithm is exponentially convergent. Moreover, computer simulations show that the EDSS‐NFxLMS/CE_DC algorithm can achieve a better ANC performance in terms of convergence rate and level of noise reduction as compared with that using the EDSS‐NFxLMS/CE algorithm without DC and that using NFxLMS/CE_DC algorithm of constant step sizes. These results support the effectiveness of the proposed techniques and EDSS‐NFxLMS/CE_DC algorithm. Copyright © 2012 John Wiley & Sons, Ltd.     

一种新的变步长LMS自适应算法及其应用

传统的固定步长的LMS算法难于同时获取较快的收敛速度与较小的稳态误差,基于这一矛盾,本文在分析基本LMS算法的基础上,提出了一种新的变步长LMS算法,该算法具有较快的收敛速度和较小的稳态误差。理论分析和计算机仿真结果表明该算法在收敛速度与稳态误差的性能上均优于基本LMS算法。尤其是在低估噪比的情况下,其性能的优越性更为突出。与传统的LMS算法相比较,新算法更适合于应用到回波消除等实时性要求高的场合。     

On simplex‐based piecewise‐linear approximations of non‐linear mappings

Simplex‐based piecewise‐linear (PWL) approximations of non‐linear mappings are needed when the robust PWL analysis is used to directly solve non‐linear equations. This paper proposes a straightforward technique for transforming the well‐known approximations into another form. This new form is computationally more efficient, since it preserves the sparse structure of the original Jacobian matrix. Furthermore, this new form of PWL approximation explicitly relates the simplex‐based PWL analysis to the conventional formulation of the Katzenelson algorithm. The proposed transform technique is also extended to treat groupwise‐separable mappings and, finally, non‐separable but sparse mappings that arise in real‐life simulation of large electronic circuits. In this paper, all these (transformed) simplex‐based PWL approximations are compared in terms of their generality and efficiency. The computational efficiency of the PWL approximation that utilizes sparsity is validated with realistic simulations. Copyright © 2005 John Wiley & Sons, Ltd.     

一种新的变步长LMS算法

自适应滤波器在信号检测、信号恢复、数字通信等许多领域中被广泛应用,自适应算法一直是学术界一个重要研究课题,提出了一种新的变步长LMS算法.算法根据自适应滤波收敛程度的加深,逐渐减小步长.试验结果表明应用该算法设计的自适应滤波器与当今通用的变步长LMS算法相比具有运算简单,收敛速度更快等优点.     

Bias‐eliminating least‐squares identification of errors‐in‐variables models with mutually correlated noises

This paper proposes a bias‐eliminating least‐squares (BELS) approach for identifying linear dynamic errors‐in‐variables (EIV) models whose input and output are corrupted by additive white noise. The method is based on an iterative procedure involving, at each step, the estimation of both the system parameters and the noise variances. The proposed identification algorithm differs from previous BELS algorithms in two aspects. First, the input and output noises are allowed to be mutually correlated, and second, the estimation of the noise covariances is obtained by exploiting the statistical properties of the equation error of the EIV model. Copyright © 2012 John Wiley & Sons, Ltd.