Probability of divergence for the least-mean fourth algorithm

In this paper, it is shown that the least-mean fourth (LMF) adaptive algorithm is not mean-square stable when the regressor input is not strictly bounded (as happens, for example, if the input has a Gaussian distribution). For input distributions with infinite support, even for the Gaussian distribution, the LMF always has a nonzero probability of divergence, no matter how small the step-size is chosen. This result is proven for a slight modification of the Gaussian distribution in a one-tap filter and corroborated with several simulations. In addition, an upper bound is given for the probability of divergence of LMF as a function of the filter length, input power, step-size, and noise variance, for the case of Gaussian regressors. The results reported in this paper provide tools for designers to better understand the behavior of the LMF algorithm and decide on the convenience or not of its use for a given application.     

一种改进的解相关变步长LMS算法

讨论了一类针对传统LMS算法进行改进的变步长自适应算法,分析其性能,对原有算法进行改进,并针对输入信号高度相关时算法收敛速度下降导致性能下降的问题,引入了解相关原理,用输入向量的正交分量来更新滤波器权系数,有效加快了算法的收敛速度,并保持了原算法的良好性能。     

归一化子带自适应滤波器步长控制

定步长子带自适应滤波器必须在快的收敛速度和低的稳态失调之间进行折中。根据自适应滤波器系数向量均方偏差与步长之间的函数关系,该文采用使自适应滤波器系数向量均方偏差在每次迭代更新时最速下降的方法,提出一种步长控制算法来解决上述问题。该算法可以兼得快的收敛速度和低的稳态失调。实验结果验证了该方法的有效性。     

一种全最小二乘算法及其在非线性滤波器中的应用

针对输入输出观测数据均含有噪声的滤波问题,提出了一种稳定的总体最小二乘自适应算法。该算法以系统的增广权向量的瑞利商(RQ)与对增广权向量的最后元素的约束的和作为总损失函数,利用梯度最陡下降原理导出权向量的自适应迭代算法,并将该算法应用于非线性Volterra滤波器。研究了算法的稳定性能,提出的算法不仅有良好的收敛性能,而且在权向量的自适应迭代时不需要标准化处理,使得算法的实施更为简单。仿真实验表明,无论在线性系统或非线性系统,本文算法的收敛性能,鲁棒抗噪性能和稳态收敛精度明显高于其它同类总体最小二乘算法。     

Split Wiener filtering with application in adaptive systems

This paper proposes a new structure for split transversal filtering and introduces the optimum split Wiener filter. The approach consists of combining the idea of split filtering with a linearly constrained optimization scheme. Furthermore, a continued split procedure, which leads to a multisplit filter structure, is considered. It is shown that the multisplit transform is not an input whitening transformation. Instead, it increases the diagonalization factor of the input signal correlation matrix without affecting its eigenvalue spread. A power normalized, time-varying step-size least mean square (LMS) algorithm, which exploits the nature of the transformed input correlation matrix, is proposed for updating the adaptive filter coefficients. The multisplit approach is extended to linear-phase adaptive filtering and linear prediction. The optimum symmetric and antisymmetric linear-phase Wiener filters are presented. Simulation results enable us to evaluate the performance of the multisplit LMS algorithm.     

LMS algorithm for programming an analogue memory cell

This article presents the optimal performance of a nonvolatile analogue memory cell fabricated in 1.2?µm CMOS process, which is programmed using a LMS (least mean square) algorithm to implement an adaptive FIR filter used to identify an unknown signal. The memory cell is programmed to store and update the weight in the filter as charge in the floating gate of a pMOS transistor (FGMOS). Programming is linear using a pulse density modulation scheme by means of tunnelling and hot injection electrons. The behavior of the memory is included and programming method is developed. The LMS algorithm performed very well, and does not require the signal to be piecewise stationary, and requires no manual operation other than selection of the step-size of the adaptive parameter.     

A New Robust Variable Step-Size NLMS Algorithm

A new framework for designing robust adaptive filters is introduced. It is based on the optimization of a certain cost function subject to a time-dependent constraint on the norm of the filter update. Particularly, we present a robust variable step-size NLMS algorithm which optimizes the square of the a posteriori error. We also show the link between the proposed algorithm and another one derived using a robust statistics approach. In addition, a theoretical model for predicting the transient and steady-state behavior and a proof of almost sure filter convergence are provided. The algorithm is then tested in different environments for system identification and acoustic echo cancelation applications.     

Convergence analysis of adaptive filtering algorithms with singulardata covariance matrix

The paper provides a rigorous analysis of the behavior of adaptive filtering algorithms when the covariance matrix of the filter input is singular. The analysis is done in the context of adaptive plant identification. The considered algorithms are LMS, RLS, sign (SA), and signed regressor (SRA) algorithms. Both the signal and weight behavior of the algorithms are considered. The signal behavior is evaluated in terms of the moments of the excess output error of the filter. The weight behavior is evaluated in terms of the moments of the filter weight misalignment vector. It is found that the RLS and SRA diverge when the input covariance matrix is singular. The steady-state signal behavior of the LMS and SA can be made arbitrarily fine by using sufficiently small step sizes of the algorithms. Indeed, the long-term average of the mean square excess error of the LMS is proportional to the algorithm step size. The long-term average of the mean absolute excess error of the SA is proportional to the square root of the algorithm step size. On the other hand, the steady-state weight behavior of both the LMS and SA have biases that depend on the weight initialization. The analytical results of the paper are supported by simulations     

Stochastic Modeling of the Transform-Domain $varepsilon {rm LMS}$ Algorithm

This paper presents a statistical analysis of the transform-domain least-mean-square (TDLMS) algorithm, resulting in a more accurate model than those discussed in the current open literature. The motivation to analyze such an algorithm comes from the fact that the TDLMS presents a higher convergence speed for correlated input signals, as compared with other adaptive algorithms possessing a similar computational complexity. Such a fact makes it a highly competitive alternative to some applications. Approximate analytical models for the first and second moments of the filter weight vector are obtained. The TDLMS algorithm has an orthonormal transformation stage, accomplishing a decomposition of the input signal into distinct frequency bands, in which the interband samples are practically uncorrelated. On the other hand, the intraband samples are correlated; the larger the number of bands, the higher their correlation. The model is then derived taking into account such a correlation, requiring that a high-order hyperelliptic integral be computed. In addition to the proposed model, an approximate procedure for computing high-order hyperelliptic integrals is presented. A regularization parameter is also considered in the model expressions, permitting to assess its impact on the adaptive algorithm behavior. An upper bound for the step-size control parameter is also obtained. Through simulation results, the accuracy of the proposed model is assessed.     

主从结构自适应噪声对消器MSANC的最优步长函数选择

ＭＳＡＮＣ是一种新型优化的主从结构自适应声对消器,此系统具有两个自适应滤波器,主滤波器和从滤波器。从滤波器用于估计输入信噪比,主滤波器进行真正的自适应滤波。主滤波器的步长是输入信噪比的函数,对这个步长函数的选取,我们根据函数类型考虑了几种不情况,正指数型函数能取得最佳对消效果,是针对于此系统的最优选择。     

Mean-square performance analysis of the family of selective partial update and selective regressor affine projection algorithms

In this paper we present a general formalism for the establishment and mean-square performance analysis of the family of selective partial update affine projection (SPU-AP), selective regressor affine projection (SR-AP), and selective partial update subband adaptive filter (SPU-SAF) algorithms. This analysis is based on energy conservation arguments and does not need to assume a Gaussian or white distribution for the regressors. We demonstrate through simulations that the results are useful in predicting the performance of these adaptive filter algorithms.     

智能天线辅助的PN 码同步捕获新方法研究

为有效解决强干扰环境下长PN 码的同步捕获问题,研究了基于自适应滤波器的PN 码同步捕获方法,给出了基于自适应滤波器权矢量范数的同步捕获判决准则,与传统的基于均方误差的判决相比,这种方法可以大幅提高判决的正确概率。在此基础上,研究了智能天线权值与PN 码同步联合捕获算法,并对其性能进行了计算机仿真验证,结果表明这种空时联合的捕获算法可以有效实现低SINR 环境下的长PN 码捕获。     

Behavior of the partial correlation coefficients of a least squareslattice filter in the presence of a nonstationary chirp input

This paper studies the performance of the a posteriori recursive least squares lattice filter in the presence of a nonstationary chirp signal. The forward and backward partial correlation (PARCOR) coefficients for a Wiener-Hopf optimal filter are shown to be complex conjugates for the general case of a nonstationary input with constant power. Such an optimal filter is compared to a minimum mean square error based least squares lattice adaptive filter. Expressions are found for the behavior of the first stage of the adaptive filter based on the least squares algorithm. For the general nth stage, the PARCOR coefficients of the previous stages are assumed to have attained Wiener-Hopf optimal steady state. The PARCOR coefficients of such a least squares adaptive filter are compared with the optimal coefficients for such a nonstationary input. The optimal lattice fitter is seen to track a chirp input without any error, and the tracking lag in such an adaptive filter is due to the least squares update procedure. The expression for the least squares based PARCOR coefficients are found to contain two terms: a decaying convergence term due to the weighted estimation procedure and a tracking component that asymptotically approaches the optimal coefficient value. The rate of convergence is seen to depend inversely on the forgetting factor. The tracking lag of the filter is derived as a function of the rate of nonstationarity and the forgetting factor. It is shown that for a given chirp rate there is a threshold adaptation constant below which the total tracking error is negligible. For forgetting factors above this threshold, the error increases nonlinearly. Further, this threshold forgetting factor decreases with increasing chirp rate. Simulations are presented to validate the analysis     

LMS和归一化LMS算法收敛门限与步长的确定

从LMS算法失调量的准确表达式出发,根据输入信号特征值分布重新研究了LMS,归一化LMS(Normalized LMS,NLMS)算法收敛的必要条件,推导出LMS和NLMS 算法收敛的步长门限,并分析了输入信号特征值分布、滤波器阶数对算法收敛步长门限的影响,推导出满足性能失调下步长的自适应计算公式,减小了应用 LMS,NLMS算法时步长选取的盲目性,与已有的算法相比,具有计算简单、实用、自适应性能强,同时可获得满意失调量的特点,计算机模拟结果表明该方法的正确性。     

Exact expectation analysis of the LMS adaptive filter

In almost all analyses of the least mean square (LMS) adaptive filter, it is assumed that the filter coefficients are statistically independent of the input data currently in filter memory, an assumption that is incorrect for shift-input data. We present a method for deriving a set of linear update equations that can be used to predict the exact statistical behavior of a finite-impulse-response (FIR) LMS adaptive filter operating upon finite-time correlated input data. Using our method, we can derive exact bounds upon the LMS step size to guarantee mean and mean-square convergence. Our equation-deriving procedure is recursive and algorithmic, and we describe a program written in the MAPLE symbolic-manipulation software package that automates the derivation for arbitrarily-long adaptive filters operating on input data with stationary statistics. Using our analysis, we present a search algorithm that determines the exact step size mean-square stability bound for a given filter length and input correlation statistics. Extensive computer simulations indicate that the exact analysis is more accurate than previous analyses in predicting adaptation behavior. Our results also indicate that the exact step size bound is much more stringent than the bound predicted by the independence assumption analysis for correlated input data     

一种新的变步长自适应噪声消除算法

本文针对电力线噪声的特点,提出了一种新的变步长自适应噪声消除算法.在自适应算法的步长与梯度之间建立了新的关系,弥补了基于误差的变步长算法在自适应噪声消除方面的不足,克服了标准LMS算法的收敛性对输入信号的敏感性,并能根据梯度调整步长大小从而实现算法的快速收敛.通过理论分析设计了新的变步长自适应噪声消除算法,并进行了仿真和实测数据验证,证明了算法相对于其他算法的优势.     

Sign-sign LMS convergence with independent stochastic inputs

The sign-sign adaptive least-mean-square (LMS) identifier filter is a computationally efficient variant of the LMS identifier filter. It involves the introduction of signum functions in the traditional LMS update term. Consideration is given to global convergence of parameter estimates offered by this algorithm, to a ball with radius proportional to the algorithm step size for white input sequences, specially from Gaussian and uniform distributions     

An improved statistical analysis of the least mean fourth (LMF) adaptive algorithm

The paper presents an improved statistical analysis of the least mean fourth (LMF) adaptive algorithm behavior for a stationary Gaussian input. The analysis improves previous results in that higher order moments of the weight error vector are not neglected and that it is not restricted to a specific noise distribution. The analysis is based on the independence theory and assumes reasonably slow learning and a large number of adaptive filter coefficients. A new analytical model is derived, which is able to predict the algorithm behavior accurately, both during transient and in steady-state, for small step sizes and long impulse responses. The new model is valid for any zero-mean symmetric noise density function and for any signal-to-noise ratio (SNR). Computer simulations illustrate the accuracy of the new model in predicting the algorithm behavior in several different situations.     

Nonlinear effects in LMS adaptive equalizers

An adaptive transversal equalizer based on the least-mean-square (LMS) algorithm, operating in an environment with a temporally correlated interference, can exhibit better steady-state mean-square-error (MSE) performance than the corresponding Wiener filter. This phenomenon is a result of the nonlinear nature of the LMS algorithm and is obscured by traditional analysis approaches that utilize the independence assumption (current filter weight vector assumed to be statistically independent of the current data vector). To analyze this equalizer problem, we use a transfer function approach to develop approximate analytical expressions of the LMS MSE for sinusoidal and autoregressive interference processes. We demonstrate that the degree to which LMS may outperform the corresponding Wiener filter is dependent on system parameters such as signal-to-noise ratio (SNR), signal-to-interference ratio (SIR), equalizer length, and the step-size parameter     

Partial-Update L∞ -Norm Based Algorithms

The computational complexity of an adaptive filtering algorithm increases with increasing the filter tap length and therefore, the use of such a filter can become prohibitive for certain applications, especially for real-time implementation. In this paper, we develop low-complexity adaptive filtering algorithms by incorporating the concept of partial updating of the filter coefficients into the technique of finding the gradient vector in the hyperplane based on the L_infin-norm criterion. Two specific partial update algorithms based on the sequential and M-Max coefficient updating are proposed. The statistical analyses of the two algorithms are carried out, and evolution equations for the mean and mean-square of the filter coefficient misalignment as well as the stability bounds on the step size are obtained. It is shown that the proposed partial update algorithm employing the M-Max coefficient updating can achieve a convergence rate that is closest to that of the full update algorithm. Finally, simulations are carried out to validate the theoretical results and study the convergence rate of the proposed algorithms