Wiener design of adaptation algorithms with time-invariant gains

A design method is presented that extends least mean squared (LMS) adaptation of time-varying parameters by including general linear time-invariant filters that operate on the instantaneous gradient vector. The aim is to track time-varying parameters of linear regression models in situations where the regressors are stationary or have slowly time-varying properties. The adaptation law is optimized with respect to the steady-state parameter error covariance matrix for time-variations modeled as vector-ARIMA processes. The design method systematically uses prior information about time-varying parameters to provide filtering, prediction, or fixed lag smoothing estimates for arbitrary lags. The method is based on a transformation of the adaptation problem into a Wiener filter design problem. The filter works in open loop for slow parameter variations, whereas a time-varying closed loop has to be considered for fast variations. In the latter case, the filter design is performed iteratively. The general form of the solution at each iteration is obtained by a bilateral Diophantine polynomial matrix equation and a spectral factorization. For white gradient noise, the Diophantine equation has a closed-form solution. Further structural constraints result in very simple design equations. Under certain model assumptions, the Wiener designed adaptation laws reduce to LMS adaptation. Compared with Kalman estimators, the channel tracking performance becomes nearly the same in mobile radio applications, whereas the complexity is, in general, much lower     

Tracking of time-varying mobile radio channels. II. A case study

For pt.I see ibid., vol.49, p.2207-17 (2001). Low-complexity Wiener LMS (WLMS) adaptation algorithms, of use for channel estimation, have been derived in Lindbom et al. (2001). They are here evaluated on the fast fading radio channels encountered in IS-136 TDMA systems, with the aim of clarifying several issues: How much can channel estimation performance be improved with these tools, as compared to LMS adaptation? When can an improved tracking MSE be expected to result in a meaningful reduction of the bit error rate? Will optimal prediction of future channel estimates significantly improve the equalization? Can one single tracker with fixed gain be used for all encountered Doppler frequencies and SNRs, or must a more elaborate scheme be adopted? These questions are here investigated both analytically and by simulation. An exact analytical expression for the tracking MSE on two-tap FIR channels is presented and utilized. With this tool, the MSE performance and robustness of WLMS algorithms based on different statistical models can be investigated. A simulation study then compares the uncoded bit error rate of detectors, where channel trackers are used in decision directed mode in conjunction with Viterbi algorithms. A Viterbi detector combined with WLMS, based on second order autoregressive fading models possibly combined with integration, provides good performance and robustness at a reasonable complexity     

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.     

Mean-square performance of a family of affine projection algorithms

Affine projection algorithms are useful adaptive filters whose main purpose is to speed the convergence of LMS-type filters. Most analytical results on affine projection algorithms assume special regression models or Gaussian regression data. The available analysis also treat different affine projection filters separately. This paper provides a unified treatment of the mean-square error, tracking, and transient performances of a family of affine projection algorithms. The treatment relies on energy conservation arguments and does not restrict the regressors to specific models or to a Gaussian distribution. Simulation results illustrate the analysis and the derived performance expressions.     

Improved Soft Iterative Channel Estimation for Turbo Equalization of Time Varying Frequency Selective Channels

In this paper, we present computationally efficient iterative channel estimation algorithms for Turbo equalizer-based communication receiver. Least Mean Square (LMS) and Recursive least Square (RLS) algorithms have been widely used for updating of various filters used in communication systems. However, LMS algorithm, though very simple, suffers from a relatively slow and data dependent convergence behaviour; while RLS algorithm, with its fast convergence rate, finds little application in practical systems due to its computational complexity. Variants of LMS algorithm, Variable Step Size Normalized LMS (VSSNLMS) and Multiple Variable Step Size Normalized LMS algorithms, are employed through simulation for updating of channel estimates for turbo equalization in this paper. Results based on the combination of turbo equalizer with convolutional code as well as with turbo codes alongside with iterative channel estimation algorithms are presented. The simulation results for different normalized fade rates show how the proposed channel estimation based-algorithms outperformed the LMS algorithm and performed closely to the well known Recursive least square (RLS)-based channel estimation algorithm.     

发射分集OFDM系统中的一种维纳LMS信道估计和跟踪算法

针对发射分集的OFDM(orthogonal frequency divided multiplexing)系统,提出了一种维纳LMS信道估计和跟踪算法。该算法利用维纳预测技术提高了跟踪信道变化的能力。仿真结果表明,该算法实现了信道的可靠估计,与传统的LMS算法和LIYe提出的简化信道估计算法相比,性能得到较大的提高,这种性能的提高在信道变化较快的条件下更为明显。     

Direct blind MMSE channel equalization based on second-orderstatistics

A family of new MMSE blind channel equalization algorithms based on second-order statistics are proposed. Instead of estimating the channel impulse response, we directly estimate the cross-correlation function needed in Wiener-Hopf filters. We develop several different schemes to estimate the cross-correlation vector, with which different Wiener filters are derived according to minimum mean square error (MMSE). Unlike many known sub-space methods, these equalization algorithms do not rely on signal and noise subspace separation and are consequently more robust to channel order estimation errors. Their implementation requires no adjustment for either single- or multiple-user systems. They can effectively equalize single-input multiple-output (SIMO) systems and can reduce the multiple-input multiple-output (MIMO) systems into a memoryless signal mixing system for source separation. The implementations of these algorithms on SIMO system are given, and simulation examples are provided to demonstrate their superior performance over some existing algorithms     

Performance of LMS-based adaptive filters in tracking atime-varying plant

The article gives a general analysis of the LMS-based adaptive filters when used for tracking a class of time varying plants. The algorithms covered are the conventional LMS, the transform-domain normalized LMS, and the LMS/Newton algorithm. An important fact that we observe is that in comparing these algorithms, better initial convergence does not necessarily mean better tracking. A few special cases are presented that show the contrary     

Statistical properties of the LMS fourier analyzer in the presence of frequency mismatch

The statistical performances of the conventional adaptive Fourier analyzers, such as the least mean square (LMS), the recursive least square (RLS) algorithms, and so forth, may degenerate significantly, if the signal frequencies given to the analyzers are different from the true signal frequencies. This difference is referred to as frequency mismatch (FM). We analyze extensively the performance of the conventional LMS Fourier analyzer in the presence of FM. Difference equations governing the dynamics and closed-form steady-state expression for the estimation mean square error (MSE) of the algorithm are derived in detail. It is revealed that the discrete Fourier coefficient (DFC) estimation problem in the LMS eventually reduces to a DFC tracking one due to the FM, and an additional term derived from DFC tracking appears in the closed-form MSE expression, which essentially deteriorates the performance of the algorithm. How to derive the optimum step size parameters that minimize or mitigate the influence of the FM is also presented, which can be used to perform robust design of step size parameters for the LMS algorithm in the presence of FM. Extensive simulations are conducted to reveal the validity of the analytical results.     

Superresolution restoration of an image sequence: adaptivefiltering approach

This paper presents a new method based on adaptive filtering theory for superresolution restoration of continuous image sequences. The proposed methodology suggests least squares (LS) estimators which adapt in time, based on adaptive filters, least mean squares (LMS) or recursive least squares (RLS). The adaptation enables the treatment of linear space and time-variant blurring and arbitrary motion, both of them assumed known. The proposed new approach is shown to be of relatively low computational requirements. Simulations demonstrating the superresolution restoration algorithms are presented.     

Multiprocessor Implementation of Adaptive Digital Filters

Adaptive filters, employing the transversal filter structure and the least mean square (LMS) adaptation algorithm, or its variations, have found wide application in data transmission equalization, echo cancellation, prediction, spectral estimation, on-line system identification, and antenna arrays. Recently, in response to requirements of fast start-up, or fast tracking of temporal variations, fast recursive least squares (FRLS) adaptation algorithms for both transversal and lattice filter structures have been proposed. These algorithms offer faster convergence than is possible with the LMS/ transversal adaptive filters, at the price of a five-to-tenfold increase in the number of multiplications, divisions, and additions. Here we discuss architectures and implementations of the LMS/transversal, fast-converging FRLS filter, and lattice filter algorithms which minimize the required hardware speed. We show how each of these algorithms can be partitioned so as to be realizable with an architecture based on multiple parallel processors.     

Tracking in Wireless Sensor Networks Using Particle Filtering: Physical Layer Considerations

In this paper, a new framework for target tracking in a wireless sensor network using particle filters is proposed. Under this framework, the imperfect nature of the wireless communication channels between sensors and the fusion center along with some physical layer design parameters of the network are incorporated in the tracking algorithm based on particle filters. We call this approach ldquochannel-aware particle filtering.rdquo Channel-aware particle filtering schemes are derived for different wireless channel models and receiver architectures. Furthermore, we derive the posterior Cramer-Rao lower bounds (PCRLBs) for our proposed channel-aware particle filters. Simulation results are presented to demonstrate that the tracking performance of the channel-aware particle filters can reach their theoretical performance bounds even with relatively small number of sensors and they have superior performance compared to channel-unaware particle filters.     

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     

Inverse halftoning using binary permutation filters

The problem of reconstructing a continuous-tone image given its ordered dithered halftone or its error-diffused halftone image is considered. We develop a modular class of nonlinear filters that can reconstruct the continuous-tone information preserving image details and edges that provide important visual cues. The proposed nonlinear reconstruction algorithms, denoted as binary permutation filters, are based on the space and rank orderings of the halftone samples provided by the multiset permutation of the "on" pixels in a halftone observation window. For a given window size, we obtain a wide range of filters by varying the amount of space-rank ordering information utilized in the estimate. For image reconstructions from ordered dithered halftones, we develop periodically space-varying filters that can account for the periodical nature of the underlying screening process. A class of suboptimal but simpler space-invariant reconstruction filters are also proposed and tested. Constrained LMS type algorithms are employed for the design of reconstruction filters that minimize the reconstruction mean squared error. We present simulations showing that binary permutation filters are modular, robust to image source characteristics, and that they produce high visual quality image reconstruction.     

Noise-constrained least mean squares algorithm

We consider the design of an adaptive algorithm for finite impulse response channel estimation, which incorporates partial knowledge of the channel, specifically, the additive noise variance. Although the noise variance is not required for the offline Wiener solution, there are potential benefits (and limitations) for the learning behavior of an adaptive solution. In our approach, a Robbins-Monro algorithm is used to minimize the conventional mean square error criterion subject to a noise variance constraint and a penalty term necessary to guarantee uniqueness of the combined weight/multiplier solution. The resulting noise-constrained LMS (NCLMS) algorithm is a type of variable step-size LMS algorithm where the step-size rule arises naturally from the constraints. A convergence and performance analysis is carried out, and extensive simulations are conducted that compare NCLMS with several adaptive algorithms. This work also provides an appropriate framework for the derivation and analysis of other adaptive algorithms that incorporate partial knowledge of the channel     

Optimum Transmitter/Receiver Design for a Narrowband Overlay in Noncoordinated Subscriber Lines

Analytical Performance of the LMS Algorithm on the Estimation of Wide Sense Stationary Channels The performance of the least mean square (LMS) algorithm on the estimation of time-varying channels is analytically evaluated, using the estimation error-correlation matrix, the mean-square weight error (MSWE), and the mean-square estimation error (MSE) as parameters. Expressions for those parameters are obtained from a set of hypotheses usually adopted in the communication systems context. The channel is modeled as a wide sense stationary (WSS) discrete-time stochastic field with known autocorrelation. The expressions for the steady-state MSWE and MSE are specialized for the class of WSS channel models, and an original analysis of the optimum LMS step-size parameter for usual channel models is addressed. For the sake of comparison with other works, the analytical step-size optimization for random-walk models is also considered. Several estimates of MSWE curves obtained by computer simulation are compared with analytical results for validation purposes. A very good agreement between simulated and analytical results for both the MSWE expressions and the optimum value of the LMS step-size parameter is shown.     

ULV and generalized ULV subspace tracking adaptive algorithms

Traditional adaptive filters assume that the effective rank of the input signal is the same as the input covariance matrix or the filter length N. Therefore, if the input signal lives in a subspace of dimension less than N, these filters fail to perform satisfactorily. In this paper, we present two new algorithms for adapting only in the dominant signal subspace. The first of these is a low-rank recursive-least-squares (RLS) algorithm that uses a ULV decomposition (Stewart 1992) to track and adapt in the signal subspace. The second adaptive algorithm is a subspace tracking least-mean-squares (LMS) algorithm that uses a generalized ULV (GULV) decomposition, developed in this paper, to track and adapt in subspaces corresponding to several well-conditioned singular value clusters. The algorithm also has an improved convergence speed compared with that of the LMS algorithm. Bounds on the quality of subspaces isolated using the GULV decomposition are derived, and the performance of the adaptive algorithms are analyzed     

Tracking of time-varying channels using two-step LMS-type adaptive algorithm

This paper presents a modified version of the two-step least-mean-square (LMS)-type adaptive algorithm motivated by the work of Gazor. We describe the nonstationary adaptation characteristics of this modified two-step LMS (MG-LMS) algorithm for the system identification problem. It ensures stable behavior during convergence as well as improved tracking performance in the smoothly time-varying environments. The estimated weight increment vector is used for the prediction of weight vector for the next iteration. The proposed modification includes the use of a control parameter to scale the estimated weight increment vector in addition to a smoothing parameter used in the two-step LMS (G-LMS) algorithm, which controls the initial oscillatory behavior of the algorithm. The analysis focuses on the effects of these parameters on the lag-misadjustment in the tracking process. The mathematical analysis for a nonstationary case, where the plant coefficients are assumed to follow a first-order Markov process, shows that the MG-LMS algorithm contributes less lag-misadjustment than the conventional LMS and G-LMS algorithms. Further, the stability criterion imposes upper bound on the value of the control parameter. These derived analytical results are verified and demonstrated with simulation examples, which clearly show that the lag-misadjustment reduces with increasing values of the smoothing and control parameters under permissible limits.     

基于自适应Volterra滤波器的非线性系统辨识

为了识别非线性系统的参数,本文提出了一种基于自适应Volterra滤波器的非线性系统辨识方法。给出了自适应Volterra滤波器的LMS和RLS算法。数值仿真结果表明,该方法对于非线性定常和时变系统都有效。     

Analytical performance of the LMS algorithm on the estimation of wide sense stationary channels

The performance of the least mean square (LMS) algorithm on the estimation of time-varying channels is analytically evaluated, using the estimation error correlation matrix, the mean-square weight error (MSWE) and the mean-square estimation error (MSE) as parameters. Expressions for those parameters are obtained from a set of hypotheses usually adopted in the communication systems context. The channel is modeled as a wide sense stationary (WSS) discrete time stochastic field with known autocorrelation. The expressions for the steady-state MSWE and MSE are particularized for the class of WSS channel models, and an original analysis of the optimum LMS step-size parameter for usual channel models is addressed. For the sake of comparison with other works, the analytical step-size optimization for random-walk models is also considered. Several estimates of MSWE curves obtained by computer simulation are compared with analytical results for validation purposes. A very good agreement between simulated and analytical results for both the MSWE expressions and the optimum value of the LMS step-size parameter is shown.