A general class of nonlinear normalized adaptive filteringalgorithms

The normalized least mean square (NLMS) algorithm is an important variant of the classical LMS algorithm for adaptive linear filtering. It possesses many advantages over the LMS algorithm, including having a faster convergence and providing for an automatic time-varying choice of the LMS stepsize parameter that affects the stability, steady-state mean square error (MSE), and convergence speed of the algorithm. An auxiliary fixed step-size that is often introduced in the NLMS algorithm has the advantage that its stability region (step-size range for algorithm stability) is independent of the signal statistics. In this paper, we generalize the NLMS algorithm by deriving a class of nonlinear normalized LMS-type (NLMS-type) algorithms that are applicable to a wide variety of nonlinear filter structures. We obtain a general nonlinear NLMS-type algorithm by choosing an optimal time-varying step-size that minimizes the next-step MSE at each iteration of the general nonlinear LMS-type algorithm. As in the linear case, we introduce a dimensionless auxiliary step-size whose stability range is independent of the signal statistics. The stability region could therefore be determined empirically for any given nonlinear filter type. We present computer simulations of these algorithms for two specific nonlinear filter structures: Volterra filters and the previously proposed class of Myriad filters. These simulations indicate that the NLMS-type algorithms, in general, converge faster than their LMS-type counterparts     

On the convergence behavior of the LMS and the normalized LMSalgorithms

It is shown that the normalized least mean square (NLMS) algorithm is a potentially faster converging algorithm compared to the LMS algorithm where the design of the adaptive filter is based on the usually quite limited knowledge of its input signal statistics. A very simple model for the input signal vectors that greatly simplifies analysis of the convergence behavior of the LMS and NLMS algorithms is proposed. Using this model, answers can be obtained to questions for which no answers are currently available using other (perhaps more realistic) models. Examples are given to illustrate that even quantitatively, the answers obtained can be good approximations. It is emphasized that the convergence of the NLMS algorithm can be speeded up significantly by employing a time-varying step size. The optimal step-size sequence can be specified a priori for the case of a white input signal with arbitrary distribution     

基于DFP的二阶Volterra模型及其对Rössler混沌序列的预测

为克服最小二乘法或归一化最小二乘法在二阶Volterra建模时参数选择不当引起的问题,在最小二乘法基础上,应用一种基于后验误差假设的可变收敛因子技术,构建了一种基于Davidon-Fletcher-Powell算法的二阶Volterra模型（DFPSOVF）.给出参数估计中自相关逆矩阵估计的递归更新公式,并对其正定性、有界性和τ（n）的作用进行了研究.将DFPSOVF模型应用于Rössler混沌序列的单步预测,仿真结果表明其能够保证算法的稳定性和收敛性,不存在最小二乘法和归一化最小二乘法的发散问题.     

Comments on `Convergence and performance analysis of the normalizedLMS algorithm with uncorrelated Gaussian data'

Noting that a fine analysis is presented for the convergence and misadjustment of the normalized least-mean-square (NLMS) algorithm in the paper by Tarrab and Feuer (see ibid., vol.3, no.4, p.468091, July 1988), the commenter claims that the results and comparisons with the LMS algorithm are not in a form that readily enables the reader to draw practical conclusions. He points out that plotting mean-square error on a linear, instead of logarithmic (dB), scale hides the important detail of the error as it converges to its minimum value, which is exactly the region where the practical engineer requires detailed knowledge to assess performance. Moreover, in the comparison of the NLMS and LMS algorithm convergence rate and misadjustment, the practitioner wants to know how fast the algorithm will converge when the misadjustment is constrained to a specified value     

回声消除基于NLMS改进算法的研究

为了提高VoIP的通信质量,减少回声干扰,对LMS算法、NLMS算法进行阐述,基于NLMS提出了一种运算量小并且提高收敛性能的改进的自适应滤波算法。通过在Matlab下的仿真研究和对误差曲线的分析,证明了该改进算法的收敛速度快,均方误差小。用改进的算法对语音回声信号进行消除,仿真得到消除回声后的信号效果明显,为IP电话中回声消除的自适应滤波问题提供了一个较好的算法。     

基于LMS算法的自适应滤波及在回声消除中的应用

介绍NLMS和NVLMS两种算法控制步长的思想。在此基础上，提出了新的变步长算法，同时使用误差积累和误差控制步长变化，并构建了基于自适应滤波算法的回声消除系统，将三种算法分别在此系统中应用，仿真验证了提出的算法具有更快的收敛速度和更小的稳态误差。并且在发生系统跳变时也能快速收敛。     

Variable step-size Griffiths’ algorithm for improved performance of feedforward/feedback active noise control

A new robust computationally efficient variable step-size LMS algorithm is proposed and it is applied for secondary path (SP) identification of feedforward and feedback active noise control (ANC) systems. The proposed variable step-size Griffiths’ LMS (VGLMS) algorithm not only uses a step-size, but also the gradient itself, based on the cross-correlation between input and the desired signal. This makes the algorithm robust to both stationary and non-stationary observation noise and the additional computational load involved for this is marginal. Further, in terms of convergence speed and error, it is better than those by the Normalized LMS (NLMS) and the Zhang’s method (Zhang in EURASIP J. Adv. Signal Process. 2008(529480):1–9, 2008). The convergence rate of the feedforward and feedback ANC systems with the VGLMS algorithm for SP identification is faster (by a factor of 2 and 3, respectively) compared with that using NLMS algorithm. For feedforward ANC, its convergence rate is faster (3 times) compared with Akhtar’s algorithm (Akhtar in IEEE Trans Audio Speech Lang Process 14(2), 2006). Also, for higher main path lengths compared with SP, the proposed algorithm is computationally efficient compared with Akhtar’s algorithm.     

外辐射源雷达抗直达波干扰技术研究

外辐射源雷达系统中,直达波干扰严重影响了雷达对目标的探测性能.文中针对直达波干扰问题,通过对LMS、NLMS、改进的NLMS算法的收敛速度、时变系统跟踪能力、失调量等的分析,将改进的归一化LMS(NLMS)自适应滤波算法应用于直达波干扰抑制,取得了较好的处理效果,其对消得益可达40 dB;分析了滤波器阶数、参数选择对对消性能和信噪比损失的影响,给出了典型参数值.最后,真实数据的处理结果验证了该方法的有效性.     

Performance of NLMS-based Parallel Interference Cancellation (PIC) For Up-Link CDMA Systems

Parallel interference cancellation (PIC) is a promising detection technique to suppress multiple access interference (MAI) for up-link direct-sequence CDMA (DS-CDMA) systems. Adaptive PICs are attractive due to its low implementation complexities and good performance. In this paper, we propose a general framework for analyzing the bit error rate (BER) and convergence performance of the normalized LMS (NLMS) based PIC. To further improve the convergence speed of the PIC system, a novel switch-mode noise-constrained NLMS (SNC-NLMS) algorithm for the adaptive multistage PIC (AMPIC) is also proposed. Simulation results show that the analytical results are reasonably accurate and the SNC-NLMS AMPIC outperforms the NLMS-based one, if the algorithm parameters are properly chosen.     

New Variable Step-Sizes Minimizing Mean-Square Deviation for the LMS-Type Algorithms

The least-mean-square-type (LMS-type) algorithms are known as simple and effective adaptation algorithms. However, the LMS-type algorithms have a trade-off between the convergence rate and steady-state performance. In this paper, we investigate a new variable step-size approach to achieve fast convergence rate and low steady-state misadjustment. By approximating the optimal step-size that minimizes the mean-square deviation, we derive variable step-sizes for both the time-domain normalized LMS (NLMS) algorithm and the transform-domain LMS (TDLMS) algorithm. The proposed variable step-sizes are simple quotient forms of the filtered versions of the quadratic error and very effective for the NLMS and TDLMS algorithms. The computer simulations are demonstrated in the framework of adaptive system modeling. Superior performance is obtained compared to the existing popular variable step-size approaches of the NLMS and TDLMS algorithms.     

Convergence behavior of affine projection algorithms

A class of equivalent algorithms that accelerate the convergence of the normalized LMS (NLMS) algorithm, especially for colored inputs, has previously been discovered independently. The affine projection algorithm (APA) is the earliest and most popular algorithm in this class that inherits its name. The usual APA algorithms update weight estimates on the basis of multiple, unit delayed, input signal vectors. We analyze the convergence behavior of the generalized APA class of algorithms (allowing for arbitrary delay between input vectors) using a simple model for the input signal vectors. Conditions for convergence of the APA class are derived. It is shown that the convergence rate is exponential and that it improves as the number of input signal vectors used for adaptation is increased. However, the rate of improvement in performance (time-to-steady-state) diminishes as the number of input signal vectors increases. For a given convergence rate, APA algorithms are shown to exhibit less misadjustment (steady-state error) than NLMS. Simulation results are provided to corroborate the analytical results     

Normalized data nonlinearities for LMS adaptation

Properly designed nonlinearly-modified LMS algorithms, in which various quantities in the stochastic gradient estimate are operated upon by memoryless nonlinearities, have been shown to perform better than the LMS algorithm in system identification-type problems. The authors investigate one such algorithm given by W_k+l=W_k+μ(d_k-W_k^t X_k)X_kf(X_k) in which the function f(X_k) is a scalar function of the sum of the squares of the N elements of the input data vector X_k. This form of algorithm generalizes the so-called normalized LMS (NLMS) algorithm. They evaluate the expected behavior of this nonlinear algorithm for both independent input vectors and correlated Gaussian input vectors assuming the system identification model. By comparing the nonlinear algorithm's behavior with that of the LMS algorithm, they then provide a method of optimizing the form of the nonlinearity for the given input statistics. In the independent input case, they show that the optimum nonlinearity is a single-parameter version of the NLMS algorithm with an additional constant in the denominator and show that this algorithm achieves a lower excess mean-square error (MSE) than the LMS algorithm with an equivalent convergence rate. Additionally, they examine the optimum step size sequence for the optimum nonlinear algorithm and show that the resulting algorithm performs better and is less complex to implement than the optimum step size algorithm derived for another form of the NLMS algorithm. Simulations verify the theory and the predicted performance improvements of the optimum normalized data nonlinearity algorithm     

Convergence Behavior of NLMS Algorithm for Gaussian Inputs: Solutions Using Generalized Abelian Integral Functions and Step Size Selection

This paper studies the mean and mean square convergence behaviors of the normalized least mean square (NLMS) algorithm with Gaussian inputs and additive white Gaussian noise. Using the Price’s theorem and the framework proposed by Bershad in IEEE Transactions on Acoustics, Speech, and Signal Processing (1986, 1987), new expressions for the excess mean square error, stability bound and decoupled difference equations describing the mean and mean square convergence behaviors of the NLMS algorithm using the generalized Abelian integral functions are derived. These new expressions which closely resemble those of the LMS algorithm allow us to interpret the convergence performance of the NLMS algorithm in Gaussian environment. The theoretical analysis is in good agreement with the computer simulation results and it also gives new insight into step size selection.     

一种新的变步长NLMS盲自适应多用户检测算法

首先推导得到最小输出能力准则的无约束形式,得到易于NLMS算法实现形式;在该NLMS算法实现的基础上,通过重复归一化迭代运算推导,得到重复跌代的等效形式;基于该形式提出一种新的变步长NLMS算法,算法复杂度仍然为O(N),算法具有很好的健壮性,并且具有更好的检测综合性能。仿真表明,与文献［5］所提变参数的变步长NLMS算法相比,本文算法收敛速度更快且输出信干比更好。在同步CDMA系统下,其综合性能达到KALMAN,RLS相当水平。      

一种改进的NLMS自适应滤波器的FPGA实现

针对当前广泛应用的自适应滤波器,提出了一种改进的变步长NLMS自适应算法,在不增加计算复杂度的条件下获得了更好的收敛速度。在硬件实现过程中,利用FPGA并行处理的特点,采用自上而下的设计方法和流水线设计技术,获得了较好的滤波效果和较快的处理速度,完全满足自适应信号处理领域中实时性的要求。     

采用归一化最小均方误差准则的LM-BP算法

传统神经网络通常以最小均方误差(LMS)或最小二乘（RLS）为收敛准则,而在自适应均衡等一些应用中,使用归一化最小均方误差（NLMS）准则可以使神经网络性能更加优越。本文在NLMS准则基础上,提出了一种以Levenberg-Marquardt（LM）训练的神经网络收敛算法。通过将神经网络的误差函数归一化,然后采用LM算法作为训练算法,实现了神经网络的快速收敛。理论分析和实验仿真表明,与采用最速下降法的NLMS准则和采用LM算法的LMS准则相比,本文算法收敛速度快,归一化均方误差更小,应用于神经网络水印系统中实现了水印信息的盲提取,能更好的抵抗噪声、低通滤波和重量化等攻击,性能平均提高了4%。      

一种变步长自适应算法在GPS空时抗干扰中的应用

GPS空时二维联合抗干扰处理算法能将强干扰以及多径干扰的强度抑制到接近噪底,并且不会对原信号产生严重的损失或者扭曲。研究发现它的干扰抑制性能优于纯空域滤波。但是此算法比纯空域算法的复杂度高,因为需要处理的采样数据协方差矩阵的维数很大,很难做到实时处理。为了解决这一问题,目前工程中普遍采用的是最小均方误差算法(LMS)。通过分析功率倒置(PI)算法,给出了一种变步长的最小均方误差算法(NLMS)。此算法的干扰抑制性能优于最小均方误差算法,能达到多级维纳滤波算法(MSWF)的抑制效果,满足工程需要,并且运算量低,运算时间短,具有更好的可行性和实用性。并将此算法与LMS和MSWF算法进行了仿真对比,验证了其有效性。     

一种约束稳定性最小均方噪声对消算法

研究了最小均方误差(LMS)算法、归一化的最小均方(NLMS)算法及变步长NLMS算法在自适应噪声干扰抵消器中的应用,针对目前这些算法在噪声对消器应用中的缺点,将约束稳定性最小均方(CS-LMS)算法应用到噪声处理中,并进一步结合变步长的思想提出来一种新的变步长CS-LMS算法。通过MATLAB进行仿真分析,结果证实提出的算法与其他算法相比,能很好地滤除掉噪声从而得到期望信号,明显的降低了稳态误差,并拥有好的收敛速度。     

一种改进的NLMS算法在声回波抵消中的应用

收敛速度和残余均方误差是衡量最小均方算法性能的重要指标。在声回波抵消算法中,为了寻求收敛速度快和计算量小的自适应算法,在归一化最小均方误差算法基础上,把当前时刻以前的误差引入归一化收敛因子中得到一种新算法,可以减小信号样本波动对权重带来的影响。该算法比传统的归一化最小均方算法收敛性能更好,稳态失调也比其小。计算机仿真结果表明,新算法在自适应回波抵消中的综合性能要优于传统的归一化最小均方误差算法。     

基于NLMS的CDMA盲自适应多用户检测算法研究

多用户检测是抑制DS-CDMA系统多址干扰最有效的技术之一。由于所需的先验知识仪有期望用户的地址码,盲多用户检测技术的研究尤受重视。最小输出能量(MOE)准则被广泛用于盲线性多用户检测。目前已提出的该类检测器多采用LMS或RLS算法。本文则研究基于NLMS算法的盲自适应检测技术,并进一步提出盲自适应变步长NLMS检测器和参数可变的盲自适应变步长NLMS检测器。它们具备很好的收敛速度和跟踪能力,以及较高的输出信干比,同时计算复杂度仅为O(3N)或O(4N),非常适合硬件实现。