Robust M-estimate adaptive filtering

An M-estimate adaptive filter for robust adaptive filtering in impulse noise is proposed. Instead of using the conventional least-square cost function, a new cost function based on an M-estimator is used to suppress the effect of impulse noise on the filter weights. The resulting optimal weight vector is governed by an M-estimate normal equation. A recursive least M-estimate (RLM) adaptive algorithm and a robust threshold estimation method are derived for solving this equation. The mean convergence performance of the proposed algorithm is also analysed using the modified Huber (1981) function (a simple but good approximation to the Hampel's three-parts-redescending M-estimate function) and the contaminated Gaussian noise model. Simulation results show that the proposed RLM algorithm has better performance than other recursive least squares (RLS) like algorithms under either a contaminated Gaussian or alpha-stable noise environment. The initial convergence, steady-state error, robustness to system change and computational complexity are also found to be comparable to the conventional RLS algorithm under Gaussian noise alone     

A recursive least M-estimate algorithm for robust adaptive filtering in impulsive noise: fast algorithm and convergence performance analysis

This paper studies the problem of robust adaptive filtering in impulsive noise environment using a recursive least M-estimate algorithm (RLM). The RLM algorithm minimizes a robust M-estimator-based cost function instead of the conventional mean square error function (MSE). Previous work has showed that the RLM algorithm offers improved robustness to impulses over conventional recursive least squares (RLS) algorithm. In this paper, the mean and mean square convergence behaviors of the RLM algorithm under the contaminated Gaussian impulsive noise model is analyzed. A lattice structure-based fast RLM algorithm, called the Huber Prior Error Feedback-Least Squares Lattice (H-PEF-LSL) algorithm is derived. Part of the H-PEF-LSL algorithm was presented in ICASSP 2001. It has an order O(N) arithmetic complexity, where N is the length of the adaptive filter, and can be viewed as a fast implementation of the RLM algorithm based on the modified Huber M-estimate function and the conventional PEF-LSL adaptive filtering algorithm. Simulation results show that the transversal RLM and the H-PEF-LSL algorithms have better performance than the conventional RLS and other RLS-like robust adaptive algorithms tested when the desired and input signals are corrupted by impulsive noise. Furthermore, the theoretical and simulation results on the convergence behaviors agree very well with each other.     

Adaptive H∞ channel equalization for wirelesspersonal communications

In this paper, a new adaptive H_∞ filtering algorithm is developed to recursively update the tap-coefficient vector of a decision feedback equalizer (DFE) in order to adaptively equalize the time-variant dispersive fading channel of a high-rate indoor wireless personal communication system. Different from conventional L ₂ (such as the recursive least squares (RLS)) filtering algorithms which minimize the squared equalization error, the adaptive H _∞ filtering algorithm is a worst case optimization. It minimizes the effect of the worst disturbances (including input noise and modeling error) on the equalization error. Hence, the DFE with the adaptive H_∞ filtering algorithm is more robust to the disturbances than that with the RLS algorithm. Computer simulation demonstrates that better transmission performance can be achieved using the adaptive H_∞ algorithm when the received signal-to-noise ratio (SNR) is larger than 20 dB     

EM-based recursive estimation of channel parameters

Recursive (online) expectation-maximization (EM) algorithm along with stochastic approximation is employed in this paper to estimate unknown time-invariant/variant parameters. The impulse response of a linear system (channel) is modeled as an unknown deterministic vector/process and as a Gaussian vector/process with unknown stochastic characteristics. Using these models which are embedded in white or colored Gaussian noise, different types of recursive least squares (RLS), Kalman filtering and smoothing and combined RLS and Kalman-type algorithms are derived directly from the recursive EM algorithm. The estimation of unknown parameters also generates new recursive algorithms for situations, such as additive colored noise modeled by an autoregressive process. The recursive EM algorithm is shown as a powerful tool which unifies the derivations of many adaptive estimation methods     

QR-based TLS and mixed LS-TLS algorithms with applications to adaptive IIR filtering

The least squares (LS), total least squares (TLS), and mixed LS-TLS approaches are compared as to their properties and performance on several classical filtering problems. Mixed LS-TLS is introduced as a QR-decomposition-based algorithm for unbiased, equation error adaptive infinite impulse response (IIR) filtering. The algorithm is based on casting adaptive IIR filtering into a mixed LS-TLS framework. This formulation is shown to be equivalent to the minimization of the mean-square equation error subject to a unit norm constraint on the denominator parameter vector. An efficient implementation of the mixed LS-TLS solution is achieved through the use of back substitution and inverse iteration. Unbiasedness of the system parameter estimates is established for the mixed LS-TLS solution in the case of uncorrelated output noise, and the algorithm is shown to converge to this solution. LS, TLS, and mixed LS-TLS performance is then compared for the problems of echo cancellation, noise reduction, and frequency equalization.     

自适应滤波算法在微弱振动测量中的应用

将递归最小二乘自适应滤波算法应用于激光多普勒测振技术中,搭建了相应的微弱振动测量装置。模拟仿真与实验中,通过与设计的切比雪夫低通滤波算法对比,结果表明:该递归最小二乘自适应滤波算法能够有效抑制随机高斯白噪声,还原出原始信号;能够对简谐振动信号实现有效滤波,并且可以还原出淹没在噪声中的低频20 Hz信号;文中算法可以去除语音噪声,使声音更加纯净,增强语音信号,以此验证了该算法在外差振动测量中的可行性。该算法简单易用、收敛性强、速度快,尤其对于随机噪声的去除比普通的低通滤波器更加有效。     

A fast recursive total least squares algorithm for adaptive IIR filtering

This work develops a new fast recursive total least squares (N-RTLS) algorithm to recursively compute the total least squares (TLS) solution for adaptive infinite-impulse-response (IIR) filtering. The new algorithm is based on the minimization of the constraint Rayleigh quotient in which the first entry of the parameter vector is fixed to the negative one. The highly computational efficiency of the proposed algorithm depends on the efficient computation of the gain vector and the adaptation of the Rayleigh quotient. Using the shift structure of the input data vectors, a fast algorithm for computing the gain vector is established, which is referred to as the fast gain vector (FGV) algorithm. The computational load of the FGV algorithm is smaller than that of the fast Kalman algorithm. Moreover, the new algorithm is numerically stable since it does not use the well-known matrix inversion lemma. The computational complexity of the new algorithm per iteration is also O(L). The global convergence of the new algorithm is studied. The performances of the relevant algorithms are compared via simulations.     

Adaptive Volterra filtering with complete lattice orthogonalization

The article presents a new recursive least squares (RLS) adaptive nonlinear filter, based on the Volterra series expansion. The main approach is to transform the nonlinear filtering problem into an equivalent multichannel, but linear, filtering problem. Then, the multichannel input signal is completely orthogonalized using sequential processing multichannel lattice stages. With the complete orthogonalization of the input signal, only scalar operations are required, instability problems due to matrix inversion are avoided and good numerical properties are achieved. The avoidance of matrix inversion and vector operations reduce the complexity considerably, making the filter simple, highly modular and suitable for VLSI implementations. Several experiments demonstrating the fast convergence properties of the filter are also included     

New forms of Levinson and Schur algorithms

The Levinson and Schur solutions to the adaptive filtering and parameter estimation problem of recursive least squares processing are described. Unnormalized versions of a newly developed Schur RLS adaptive filter are presented. A systolic array of the Schur RL adaptive filter is devised and its performance is illustrated with a typical example. The classical Levinson and Schur algorithms drop out as special cases of the more general Levinson and Schur RLS adaptive filtering algorithms. The recently introduced split Levinson and Schur algorithms, which are obtained by exploiting the symmetry in the Toeplitz-structured extended normal equations, are reviewed     

Modified LMS algorithm for unbiased impulse response estimation innonstationary noise

In the presence of input interference, the Wiener solution for impulse response estimation is biased. It is proved that bias removal can be achieved by proper scaling of the optimal filter coefficients and a modified least mean squares (LMS) algorithm is then developed for accurate system identification in noise. Simulation results show that the proposed method outperforms two total least squares (TLS) based adaptive algorithms under nonstationary interference conditions     

An algorithm for efficient, unbiased, equation-error infiniteimpulse response adaptive filtering

An algorithm for efficiently adjusting the coefficients of equation-error infinite impulse response (IIR) adaptive filters is described. Unlike the recursive least squares (RLS) algorithm, the proposed algorithm yields unbiased filter coefficients. Simulations involving the identification of unknown pole-zero systems demonstrate the algorithm's improved performance over the equation-error RLS algorithm     

A Stabilized Multichannel Fast RLS Algorithm for Adaptive Transmultiplexer Receivers

The transmultiplexer (TMUX) system has been studied for its application to multicarrier communications. The channel impairments including noise, interference, and distortion draw the need for adaptive reconstruction at the TMUX receiver. Among possible adaptive methods, the recursive least squares (RLS) algorithm is appealing for its good convergence rate and steady state performance. However, higher computational complexity due to the matrix operation is the drawback of utilizing RLS. A fast RLS algorithm used for adaptive signal reconstruction in the TMUX system is developed in this paper. By using the polyphase decomposition method, the adaptive receiver in the TMUX system can be formulated as a multichannel filtering problem, and the fast algorithm is obtained through the block Toeplitz matrix structure of received signals. In addition to the reduction of complexity, simulation results show that the adaptive TMUX receiver has a convergence rate close to that of the standard RLS algorithm and the performance approaches the minimum mean square error solution.     

A fast recursive least squares adaptive second order Volterrafilter and its performance analysis

A fast, recursive least squares (RLS) adaptive nonlinear filter modeled using a second-order Volterra series expansion is presented. The structure uses the ideas of fast RLS multichannel filters, and has a computational complexity of O(N³) multiplications, where N-1 represents the memory span in number of samples of the nonlinear system model. A theoretical performance analysis of its steady-state behaviour in both stationary and nonstationary environments is presented. The analysis shows that, when the input is zero mean and Gaussian distributed, and the adaptive filter is operating in a stationary environment, the steady-state excess mean-squared error due to the coefficient noise vector is independent of the statistics of the input signal. The results of several simulation experiments show that the filter performs well in a variety of situations. The steady-state behaviour predicted by the analysis is in very good agreement with the experimental results     

LMS与RLS算法仿真消噪对比研究

讨论了RLS（递归最小二乘）和LMS（最小均方）自适应滤波算法及原理,对两种算法进行了系统全面的分析,对比研究了各自的优势及不足,提出了两种算法在语音消噪仿真中的算法实现,对实际语音信号进行了仿真消噪,研究表明选用算法对语音消噪是明显有效的,RLS自适应消噪算法及LMS自适应噪声抵消算法具有很强的实际应用价值。     

一种快速递归全局最小二乘算法

针对FIR系统输入和输出信号均被噪声干扰的情况,提出一种快速递归全局最小二乘(XS-RTLS)算法用于迭代计算全局最小二乘解,算法沿着输入数据的符号方向并采用著名的快速增益矢量,搜索约束瑞利商(c-RQ)的最小值得到系统参数估计。算法关于方向更新矢量的内积运算可通过加减运算实现,有效降低了计算复杂度;另外XS-RTLS算法没有进行相关矩阵求逆递归运算,因而具有长期稳定性,算法的全局收敛性通过Laslle不变性原理得到证明。最后通过仿真比较了XS-RTLS算法和递归最小二乘(RLS)算法在非时变系统和时变系统中的性能,验证了XS-RTLS算法的长期稳定性。     

A new method for low-rank transform domain adaptive filtering

This paper introduces a least squares, matrix-based framework for adaptive filtering that includes normalized least mean squares (NLMS), affine projection (AP) and recursive least squares (RLS) as special cases. We then introduce a method for extracting a low-rank underdetermined solution from an overdetermined or a high-rank underdetermined least squares problem using a part of a unitary transformation. We show how to create optimal, low-rank transformations within this framework. For obtaining computationally competitive versions of our approach, we use the discrete Fourier transform (DFT). We convert the complex-valued DFT-based solution into a real solution. The most significant bottleneck in the optimal version of the algorithm lies in having to calculate the full-length transform domain error vector. We overcome this difficulty by using a statistical approach involving the transform of the signal rather than that of the error to estimate the best low-rank transform at each iteration. We also employ an innovative mixed domain approach, in which we jointly solve time and frequency domain equations. This allows us to achieve very good performance using a transform order that is lower than the length of the filter. Thus, we are able to achieve very fast convergence at low complexity. Using the acoustic echo cancellation problem, we show that our algorithm performs better than NLMS and AP and competes well with FTF-RLS for low SNR conditions. The algorithm lies in between affine projection and FTF-RLS, both in terms of its complexity and its performance     

RLS Algorithm with Variable Fogetting Factor for Decision Feedback Equalizer over Time-Variant Fading Channels

In a high-rate indoor wireless personal communication system, the delay spread due to multipath propagation results in intersymbol interference (ISI) which can significantly increase the transmission bit error rate (BER). Decision feedback equalizer (DFE) is an efficient approach to combating the ISI. Recursive least squares (RLS) algorithm with a constant forgetting factor is often used to update the tap-coefficient vector of the DFE for ISI-free transmission. However, using a constant forgetting factor may not yield the optimal performance in a nonstationary environment. In this paper, an adaptive algorithm is developed to obtain a time-varying forgetting factor. The forgetting factor is used with the RLS algorithm in a DFE for calculating the tap-coefficient vector in order to minimize the squared equalization error due to input noise and due to channel dynamics. The algorithm is derived based on the argument that, for optimal filtering, the equalization errors should be uncorrelated. The adaptive forgetting factor can be obtained based on on-line equalization error measurements. Computer simulation results demonstrate that better transmission performance can be achieved by using the RLS algorithm with the adaptive forgetting factor than that with a constant forgetting factor previously proposed for optimal steady-state performance or a variable forgetting factor for a near deterministic system.     

基于RLS的二次加权相关时延估计算法

针对测向定位中时延估计的问题,提出了一种基于递推最小二乘（Recursive Least Squares,RLS）算法的二次加权相关时延估计方法。该方法在二次相关算法基础上,一方面引入RLS算法,在二次相关前进行自适应滤波,提高系统抗噪能力,且具有较快的收敛速度;另一方面借鉴广义互相关的思路,引入加权函数,并且采用二次加权方式,提高时延估计的性能。仿真结果表明,在低信噪比环境下,基于RLS的二次加权相关时延估计法使谱峰更加尖锐,抑制了噪声的影响,提高了估计的精度。     

Optimal tracking of time-varying channels: a frequency domainapproach for known and new algorithms

In this paper, we developed a systematic frequency domain approach to analyze adaptive tracking algorithms for fast time-varying channels. The analysis is performed with the help of two new concepts, a tracking filter and a tracking error filter, which are used to calculate the mean square identification error (MSIE). First, we analyze existing algorithms, the least mean squares (LMS) algorithm, the exponential windowed recursive least squares (EW-RLS) algorithm and the rectangular windowed recursive least squares (RW-RLS) algorithm. The equivalence of the three algorithms is demonstrated by employing the frequency domain method. A unified expression for the MSIE of all three algorithms is derived. Secondly, we use the frequency domain analysis method to develop an optimal windowed recursive least squares (OW-RLS) algorithm. We derive the expression for the MSIE of an arbitrary windowed RLS algorithm and optimize the window shape to minimize the MSIE. Compared with an exponential window having an optimized forgetting factor, an optimal window results in a significant improvement in the h MSIE. Thirdly, we propose two types of robust windows, the average robust window and the minimax robust window. The RLS algorithms designed with these windows have near-optimal performance, but do not require detailed statistics of the channel     

Oversampled sigma-delta LMS adaptive FIR filters

Two filter designs for adaptive least mean squares (LMS) filtering with sigma-delta modulated input signals are described. One implementation is multibit multiplier-free and operates entirely at the oversampling frequency of the sigma-delta signals, in the other design only the FIR filter operates at the oversampled frequency while the adaptive filtering algorithm is performed at the Nyquist rate. To circumvent any aliasing problems that may be caused by the downsampling process in the architecture and ensure convergence of the adaptive FIR filter. It is necessary to attenuate the high-frequency sigma-delta quantisation noise that is present. To perform this task a multiplier-free, multistage IIR filter structure is used that requires considerably fewer computations than an equivalent FIR filter. The two adaptive LMS filter designs are analysed and their performance is compared with a conventional PCM system in terms of achievable minimum MSE and adaptation speed