Analysis of stability and performance of adaptation algorithms with time-invariant gains

Adaptation laws that track parameters of linear regression models are investigated. The considered class of algorithms apply linear time-invariant filtering on the instantaneous gradient vector and includes least mean squares (LMS) as its simplest member. The asymptotic stability and steady-state tracking performance for prediction and smoothing estimators is analyzed for parameter variations described by stochastic processes with time-invariant statistics. The analysis is based on a novel technique that decomposes the inherent feedback of adaptation algorithms into one time-invariant loop and one time-varying loop. The impact of the time-varying feedback on the tracking error covariance can be neglected under certain conditions, and the performance analysis then becomes straightforward. Performance analysis in the presence of a non-negligible time-varying feedback is performed for algorithms that use scalar measurements. Convergence in mean square error (MSE) and the MSE tracking performance is investigated, assuming independent consecutive regression vectors. Closed-form expressions for the tracking MSE are thereafter derived without this independence assumption for a subclass of algorithms applied to finite impulse response (FIR) models with white inputs. This class includes Wiener LMS adaptation.     

A hybrid array minimizing the effects of the random weight vectorerrors in the LMS array and the Applebaum array

A hybrid adaptive array that combines the least mean square-error (LMS) array and the Applebaum array is presented. The array minimizes the effect of the random errors in the weight vectors of the LMS and Applebaum arrays. These weight vectors containing random errors are scaled and combined to yield a novel weight vector. The mean square error (MSE) is used as a measure of performance to derive optimal weighting factors. An algorithm is devised to adjust the weighting factors automatically by an iterative procedure based on the complex LMS algorithm to achieve the optimum weighting factors. It is shown that the hybrid array performs better than the Applebaum array or the LMS array. In addition, it is less sensitive to the random weight vector errors     

基于连续块处理的改进LMS算法研究

在基于连续数据块处理的硬件系统中,若干块数据通过LMS迭代算法处理后得到多组权值矢量。若各组权矢量波动过大,系统性能将会急剧下降。针对此问题,本文提出了一种基于连续块处理的改进LMS算法,该方法首先利用LMS算法对一块数据进行迭代计算,得到此块数据对应的更新权矢量,将该权矢量乘以一个复系数使其向上一块数据对应计算所得的权矢量进行归一（幅度和相位上的归一）,然后将归一化处理后的权矢量作为下一块数据迭代计算的初值。经过上述处理后,权值收敛速度得到了提高,同时减小了数据块间更新权矢量的波动,进而减小系统其他相关参数的波动,提高硬件系统稳定性。最后仿真实验结果验证了本文算法的有效性和正确性。      

Convergence of the delayed normalized LMS algorithm with decreasingstep size

In several practical applications of the LMS algorithm, including certain VLSI implementations, the coefficient adaptation can be performed only after some fixed delay. The resulting algorithm is known as the delayed LMS (DLMS) algorithm in the literature. Previous published analyses of this algorithm are based on mean and moment convergence under the independence assumption between successive input vectors. These analyses are interesting and give valuable insights into the convergence properties but, from a practical viewpoint, they do not guarantee the correct performance of the particular realization with which the user must live. We consider a normalized version of this algorithm with a decreasing step size μ(n) and prove the almost sure convergence of the nonhomogeneous algorithm, assuming a mixing input condition and the satisfaction of a certain law of large numbers     

Optimum sampling vectors for Wiener filter noise reduction

Sampling is a very important and basic technique for signal processing. In the case that noise is added to a signal in the sampling process, we may use a reconstruction and noise reduction filter such as the Wiener filter. The Wiener filter provides a restored signal of which the mean square error is minimized. However, the mean square error by the Wiener filter depends on the sampling vectors. We may have the freedom to construct sampling vectors. We provide optimum sampling vectors under the condition that the Wiener filter is used for noise reduction for two cases wherein the noise is added before/after sampling. The sampling vectors provided in this paper may not be practical since they are very complicated. However, the minimum mean square error, which we provide theoretically, can be used for evaluating other sampling vectors. We provide all proofs of the theorems and lemmas. Furthermore, by experimental results, we show their advantages     

基于FPGA的自适应谱线增强系统设计

在此基于Altera公司的现场可编程门阵列（FPGA）芯片EP2C8F256C6,采用最小均方算法设计了自适应谱线增强（ALE）处理系统。以FPGA为处理核心,实现数据采样控制、数据延时控制、LMS核心算法和输出存储控制等。充分利用FPGA高速的数据处理能力和丰富的片内乘法器,设计了LMS算法的流水线结构,保证整个系统具有高的数据吞吐能力和处理速度。并且通过编写相应的VHDL程序在QuartusⅡ软件上进行仿真,仿真结果表明该设计可以快速、准确地实现自适应谱线增强。     

Efficient subspace-based algorithm for adaptive bearing estimation and tracking

In some practical applications of array processing, the directions of the incident signals should be estimated adaptively, and/or the time-varying directions should be tracked promptly. In this paper, an adaptive bearing estimation and tracking (ABEST) algorithm is investigated for estimating and tracking the uncorrelated and correlated narrow-band signals impinging on a uniform linear array (ULA) based on the subspace-based method without eigendecomposition (SUMWE), where a linear operator is obtained from the array data to form a basis for the space by exploiting the array geometry and its shift invariance property. Specifically, the space is estimated using the least-mean-square (LMS) or normalized LMS (NLMS) algorithm, and the directions are updated using the approximate Newton method. The transient analyses of the LMS and NLMS algorithms are studied, where the "weight" (i.e., the linear operator) is in the form of a matrix and there is a correlation between the "additive noise" and "input data" that involve the instantaneous correlations of the received array data in the updating equation, and the step-size stability conditions are derived explicitly. In addition, the analytical expressions for the mean-square error (MSE) and mean-square deviation (MSD) learning curves of the LMS algorithm are clarified. The effectiveness of the ABEST algorithm is verified, and the theoretical analyses are corroborated through numerical examples. Simulation results show that the ABEST algorithm is computationally simple and has good adaptation and tracking abilities.     

慢衰落信道下的自适应MIMO检测算法

针对现有V-BLAST检测算法复杂度高的问题,本文提出了一种适用于慢衰落信道环境下的自适应MIMO检测算法。该算法首先利用基于最小均方误差准则的排序干扰抵消算法获得信号检测顺序和初始滤波器系数;利用信道之间的相关性,应用最小均方误差（LMS）算法到判决反馈均衡（DFE）结构中,完成滤波器系数的更新,从而避免了大量的矩阵求逆操作。仿真结果表明,与传统的MMSE-OSIC算法相比,本文算法在检测性能上略有下降,但具有更低的计算复杂度和更高的处理效率。      

基于自适应步长LMS算法的判决反馈均衡器研究

本文研究了一种基于自适应步长LMS算法的自适应判决反馈均衡器算法，此算法通过对前向滤波器、后向滤波器和锁相环参数分别进行实时的、自适应步长的调整，在均衡时变多径的同时高精度地锁定接收信号相位，因此适用于相干通信系统。仿真结果表明在相同多径条件下，该方法比基于固定步长LMS算法的判决反馈均衡器输出误码率低1-7倍。     

The Zak transform and sampling theorems for wavelet subspaces

The Zak transform is used for generalizing a sampling theorem of G. Waiter (see IEEE Trans. Informat. Theory, vol. 38, p. 881-884, 1992) for wavelet subspaces. Cardinal series based on signal samples f(a+n), n∈Z with a possibly unequal to 0 (Waiter's case) are considered. The condition number of the sampling operator and worst-case aliasing errors are expressed in terms of Zak transforms of scaling function and wavelet. This shows that the stability of the resulting interpolation formula depends critically on a     

Systolic array implementation of block LMS algorithm

The authors derive the systolic array implementation of the block LMS algorithm, consisting of N processing elements, where N is the filter order. The resulting array attains an order-independent sampling rate. Computer simulation results show that the block LMS algorithm is faster than the delayed LMS algorithm, which has previously been implemented on systolic arrays     

Wiener filtering and LMS algorithm of periodic nonuniformly sampledstochastic sequence

A study of Wiener filtering and an LMS algorithm of a periodic nonuniformly sampled stationary stochastic sequence is made. It is demonstrated that the Wiener-Hopf equation of such signals should be constructed and solved on each of the M staggered ordinals, a concept of periodic time-variant weights. Also shown is that in the LMS algorithm, M weight vectors should be adjusted respectively according to the staggered ordinals. Some conclusions and simulation results are given     

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Delta算子的SVD-LMS算法

由于传统的最小均方算法在性能上存在着一些缺陷,比如收敛速度慢等,这都会影响到整个系统的性能,因此研究了一种基于Delta算子描述的改进自适应滤波算法。利用矩阵的SVD(奇异值)分解技术推导出了Delta LMS算法的迭代步骤,这种算法能够降低设计的滤波器阶数,有效地减少计算量,并提高了收敛速度,改善了对误差的跟踪性能。运用Matlab对Delta SVD LMS算法进行仿真并与Delta LMS算法进行了比较,结果表明该算法的有效性。     

基于自适应分数阶傅里叶变换的线性调频信号检测及参数估计

该文提出了一种基于最小均方算法的自适应计算分数阶傅里叶变换的方法并将该方法应用到多分量chirp信号的检测与估计之中。该方法通过对连续型分数阶傅里叶反变换进行离散化采样,得到适合数值计算的离散形式,进而通过适当的选择输入向量和目标函数构造自适应滤波器,经过最小均方算法进行训练后所得的滤波器权系数即为分数阶傅里叶变换的结果。仿真实验表明,该方法可以用来计算分数阶傅里叶变换及对chirp信号进行检测和参数估计,且计算延时相对较小。     

基于自适应子空间追踪的层析SAR成像算法

在对分布式SAR进行数据降采样下会信号的三维处理增加不少难题。其中在解决频域距离弯曲校正时,由于方位向的降采样使数据不再满足奈奎斯特定理,导致在多普勒域计算距离偏移量时会出现数据的混叠。针对该问题,提出了基于LMS估计的距离弯曲校正算法,该方法根据最小均方估计思想估计权值系数完成方位向的插值,有效解决了该条件下的距离弯曲问题。针对高层成像中稀疏阵列导致基线数量有限且不均导致成像分辨率差的问题,提出了基于压缩感知的自适应子空间追踪方法来提高高度维成像的分辨性能,相比于正交匹配追踪算法,它能实现对迭代得到候选解的同步检验,避免了错误结果积累的问题,有效提高了成像的质量。     

Efficient Adaptive SVD Algorithm for MIMO Applications

Many multiple-input multiple-output (MIMO) applications require the computation of some or all of the nonzero singular values and the corresponding left and right singular vectors of a time-varying channel response matrix. An adaptive algorithm is derived to achieve this goal, based on a first-order perturbation, which updates a full or partial singular value decomposition (SVD) using input and noisy output vectors. The updates can be computed recursively, resulting in a highly efficient algorithm that has lower complexity than the earlier least-mean-square (LMS)-based algorithm and achieves better performance at low signal-to-noise ratio (SNR). The performance is demonstrated using measured MIMO channel data obtained in an urban microcellular environment.     

FPGA Based Implementation and Comparison of Beamformers for CDMA2000

For the integration of smart antennas into third generation code division multiple access (CDMA) base stations, it still remains as a challenging task to implement smart antenna algorithms on programmable processors. In this paper, we study implementations of some CDMA compatible beamforming algorithms, namely least mean square (LMS), constant modulus (CM), and space code correlator (SCC) algorithms, using Xilinx??s Virtex family FPGAs. This study exhibits feasibility of implementing even simple, practical, and computationally small algorithms based on today??s most powerful FPGA technologies. 16 and 32 bits floating point implementations of the algorithms are investigated using both Virtex II and Virtex IV FPGAs. CDMA2000 reverse link baseband signal format is used in the signal modeling. Randomly changing fading and Direction-of-arrivals (DOAs) of multipaths are considered as a channel condition. The implementation results in terms of beamforming accuracy, FPGA resource utilization, weight vector computation time, and DOA estimation error are presented. Beamformer weight vectors using LMS and CM can be computed within less than 20 ??s on Virtex II FPGA and 10 ??s on Virtex IV FPGA, and using SCC it can be achieved within less than 22 ??s on Virtex IV FPGA. These results show that FPGAs provide approximately 500 times faster speed in implementations than our previous work with DSPs.     

一种基于自适应阵列天线的波束赋形算法

自适应阵列天线中的数字波束赋形(DBF)技术是智能天线数字信号处理部分的核心.提出了一种可用于自适应阵列波束赋形的SMI-LMS算法--由SMI(采样协方差矩阵求逆)算法决定LMS(最小均方)算法的初始权向量.该算法充分结合了SMI算法收敛速度快和LMS算法稳态误差小的优点,能在较强干扰环境下,确保权向量的快速收敛和跟踪速度.与传统的LMS算法相比,SMI-LMS算法具有良好的收敛性能、较快的跟踪速度和较小的输出误差,并可以有效改善自适应方向图的副瓣性能.仿真结果验证了该结论.     

Tracking of time-varying mobile radio channels .1. The Wiener LMSalgorithm

Adaptation algorithms with constant gains are designed for tracking smoothly time-varying parameters of linear regression models, in particular channel models occurring in mobile radio communications. In a companion paper, an application to channel tracking in the IS-136 TDMA system is discussed. The proposed algorithms are based on two key concepts. First, the design is transformed into a Wiener filtering problem. Second, the parameters are modeled as correlated ARIMA processes with known dynamics. This leads to a new framework for systematic and optimal design of simple adaptation laws based on prior information. The algorithms can be realized as Wiener filters, called learning filters, or as "LMS/Newton" updates complemented by filters that provide predictions or smoothing estimates. The simplest algorithm, named the Wiener LMS, is presented. All parameters are here assumed governed by the same dynamics and the covariance matrix of the regressors is assumed known. The computational complexity is of the same order of magnitude as that of LMS for regressors which are either white or have autoregressive statistics. The tracking performance is, however, substantially improved