Estimating second-order Volterra system parameters from noisy measurements based on an LMS variant or an errors-in-variables method

This paper deals with the identification of a nonlinear SISO system modelled by a second-order Volterra series expansion when both the input and the output are disturbed by additive white Gaussian noises. Two methods are proposed. Firstly, we present an unbiased on-line approach based on the LMS. It includes a bias correction scheme which requires the variance of the input additive noise. Secondly, we suggest solving the identification problem as an errors-in-variables issue, by means of the so-called Frisch scheme. Although its computational cost is high, this approach has the advantage of estimating the Volterra kernels and the variances of both the additive noises and the input signal, even if the signal-to-noise ratios at the input and the output are low.     

二阶Volterra变数据块长LMS算法

二阶Volterra数据块LMS算法利用当前时刻及其以前时刻更多输入信号和误差信号的信息提高了算法的收敛速度,但由于其固定数据块长取值的不同导致了算法的收敛速度和稳态误差此消彼长。针对这个问题,本文提出一种二阶Volterra变数据块长LMS算法,通过时刻改变输入信号数据块长度提高算法性能。本算法首先采用两个并行的二阶Volterra滤波器,其输入信号数据块长差值始终保持一个单位;然后将其各自的输出误差信号同时输入到数据块长判决器,通过判决器得到下一时刻各个滤波器输入信号的数据块长度;最后以第1个二阶Volterra滤波器的输出作为整个滤波系统的输出,从而改善了算法性能。将本算法应用于非线性系统辨识,计算机仿真结果表明,高斯噪声背景下本算法的收敛速度和稳态性能都得到了明显的提高。      

一种抗冲击噪声的对数总体最小二乘自适应滤波算法

在未知系统输入信号和输出信号均含有噪声的环境中,传统的自适应滤波算法,如最小均方(LMS)算法,会产生有偏估计。总体最小二乘(TLS)算法能够同时最小化输入信号与输出信号的噪声干扰,是解决此类问题的重要方法。然而,在许多实际应用中,干扰噪声可能具有冲击特性,这使得传统基于2阶统计量的自适应滤波算法,包括总体最小二乘算法性能严重恶化,以至于不能正常工作。为了解决这个问题,该文在总体最小二乘法的基础上,利用对数函数对其改进,提出了一种能够抗冲击干扰的对数总体最小二乘(L-TLS)算法。最后,通过计算机仿真实验验证了该新算法的有效性。     

Low-rank estimation of higher order statistics

Low-rank estimators for higher order statistics are considered in this paper. The bias-variance tradeoff is analyzed for low-rank estimators of higher order statistics using a tensor product formulation for the moments and cumulants. In general, the low-rank estimators have a larger bias and smaller variance than the corresponding full-rank estimator, and the mean-squared error can be significantly smaller. This makes the low-rank estimators extremely useful for signal processing algorithms based on sample estimates of the higher order statistics. The low-rank estimators also offer considerable reductions in the computational complexity of such algorithms. The design of subspaces to optimize the tradeoffs between bias, variance, and computation is discussed, and a noisy input, noisy output system identification problem is used to illustrate the results     

Asymptotically optimal blind fractionally spaced channel estimationand performance analysis

When the received data are fractionally sampled, the magnitude and phase of most linear time-invariant FIR communications channels can be estimated from second-order output only statistics. We present a general cyclic correlation matching algorithm for known order FIR blind channel identification that has closed-form expressions for calculating the asymptotic variance of the channel estimates. We show that for a particular choice of weights, the weighted matching estimator yields (at least for large samples) the minimum variance channel estimator among all unbiased estimators based on second-order statistics. Furthermore, the matching approach, unlike existing methods, provides a useful estimate even when the channel is not uniquely identifiable from second-order statistics     

A two-stage algorithm for MIMO blind deconvolution of nonstationarycolored signals

A new two-stage algorithm is proposed for the deconvolution of multi-input multi-output (MIMO) systems with colored input signals. While many blind deconvolution algorithms in the literature utilize high order statistics of the output signal for white input signals, the additional information contained in colored input signals allows the design of second-order statistical algorithms. In fact, practical signal sources such as speech signals do have distinct, nonstationary, colored power spectral densities. We present a two-stage signal separation approach in which the first step utilizes a matrix pencil between output auto-correlation matrices at different delays, whereas the second stage adopts a subspace method to identify and deconvolve MIMO systems     

A nonparametric phase estimation method for SIMO systems based onsecond-order and higher order statistics

We present a nonparametric phase estimation algorithm for linear single-input multiple-output (SIMO) channels. Given an unknown stationary input signal with known statistics, our approach is to obtain the joint minimum mean square phase estimation based on the polyspectra and the cross-spectra of the SIMO channel outputs. By utilizing both higher order and second-order statistics of the channel outputs, our approach is shown to be more accurate and reliable than methods based on higher order statistics alone. It can be applied to SIMO channels with common zeros     

Adaptive instantaneous frequency estimation of multicomponent FMsignals using quadratic time-frequency distributions

An adaptive approach to the estimation of the instantaneous frequency (IF) of nonstationary mono- and multicomponent FM signals with additive Gaussian noise is presented. The IF estimation is based on the fact that quadratic time-frequency distributions (TFDs) have maxima around the IF law of the signal. It is shown that the bias and variance of the IF estimate are functions of the lag window length. If there is a bias-variance tradeoff, then the optimal window length for this tradeoff depends on the unknown IF law. Hence, an adaptive algorithm with a time-varying and data-driven window length is needed. The adaptive algorithm can utilize any quadratic TFD that satisfies the following three conditions: First, the IF estimation variance given by the chosen distribution should be a continuously decreasing function of the window length, whereas the bias should be continuously increasing so that the algorithm will converge at the optimal window length for the bias-variance tradeoff, second, the time-lag kernel filter of the chosen distribution should not perform narrowband filtering in the lag direction in order to not interfere with the adaptive window in that direction; third, the distribution should perform effective cross-terms reduction while keeping high resolution in order to be efficient for multicomponent signals. A quadratic distribution with high resolution, effective cross-terms reduction and no lag filtering is proposed. The algorithm estimates multiple IF laws by using a tracking algorithm for the signal components and utilizing the property that the proposed distribution enables nonparametric component amplitude estimation. An extension of the proposed TFD consisting of the use of time-only kernels for adaptive IF estimation is also proposed     

Asymptotic Bias and Variance of Conventional Bispectrum Estimates for 2-D Signals

In this paper, we derive the asymptotic bias and variance of conventional bispectrum estimates of 2-D signals. Two methods have been selected for the estimation: the first one – the indirect method – is the Fourier Transform of the weighted third order moment, while the second one – the direct method – is the expectation of the Fourier component product. Most of the developments are known for 1-D signals and the first contribution of this paper is the rigorous extension of the results to 2-D signals. The calculation of the bias of the direct method is a totally original contribution. Nevertheless, we did all calculations (bias and variance) for both method in order to be able to compare the results. The second contribution of this paper consists of the comparison of the theoretical bispectrum estimate bias and variance with the measured bias and variance for two 2-D signals. The first studied signal is the output of a non-minimal phase linear system driven by a non-symmetric noise. The second signal is the output of a non-linear system with Gaussian input data. In order to assess the results, we performed the comparison for both methods with different sets of parameters. We show that the maximum bias coefficient is the one of the 1-D case multiplied by the dimensionality of the signal for both methods. We also show that the estimate variance coefficient is the 1-D case coefficient with a power equal to the signal dimensionality.Received October 21, 2002; Revised December 2003; Accepted March 25, 2004; First Online Version published in December 2004     

Fast identification of autoregressive signals from noisy observations

The purpose of this brief is to derive, from the previously developed least-squares (LS) based method, a faster convergent approach to identification of noisy autoregressive (AR) stochastic signals. The feature of the new algorithm is that in its bias correction procedure, it makes use of more autocovariance samples to estimate the variance of the additive corrupting noise which determines the noise-induced bias in the LS estimates of the AR parameters. Since more accurate estimates of this corrupting noise variance can be attained at earlier stages of the iterative process, the proposed algorithm can achieve a faster rate of convergence. Simulation results are included that illustrate the good performances of the proposed algorithm.     

An analytical derivation for second-order blind separation of two signals

In this paper, we propose analytical formulas that involve second-order statistics for separating two signals. The method utilizes source decorrelation and correlation function diversity. In particular, the proposed SOBAS (second-order blind analytical separation) algorithm differs from the ASOBI (analytical second-order blind identification) algorithm in that it does not require prior knowledge or estimation of the noise variance. Computer simulations demonstrate the effectiveness of the proposed method.     

Adaptive Input Design for Identification of Output Error Model with Constrained Output

Optimal input design for system identification is an area of intensive modern research. This paper considers the identification of output error (OE) model, for the case of constrained output variance. The constraint plays a very important role in the process industry, in the reduction of degradation of product quality. In this paper, it is shown, in the form of a theorem, that the optimal input signal, with constrained output, is achieved by a minimum variance controller together with a stochastic reference. The key problem is that the optimal input depends on the system parameters to be identified. In order to overcome this problem, a two-stage adaptive procedure is proposed: obtaining an initial model using PRBS as input signal; application of adaptive minimum variance controller together with the stochastic variable reference, in order to generate input signals for system identification. Theoretical results are illustrated by simulations.     

Blind identification and equalization based on second-orderstatistics: a time domain approach

A new blind channel identification and equalization method is proposed that exploits the cyclostationarity of oversampled communication signals to achieve identification and equalization of possibly nonminimum phase (multipath) channels without using training signals. Unlike most adaptive blind equalization methods for which the convergence properties are often problematic, the channel estimation algorithm proposed here is asymptotically ex-set. Moreover, since it is based on second-order statistics, the new approach may achieve equalization with fewer symbols than most techniques based only on higher-order statistics. Simulations have demonstrated promising performance of the proposed algorithm for the blind equalization of a three-ray multipath channel     

光电跟踪伺服系统的自适应差分进化算法辨识

为了获得准确的光电跟踪伺服系统的模型,采用自适应差分进化算法对光电跟踪伺服系统进行辨识研究,该算法根据辨识误差平方和自动调整变异、交叉因子。在输入为正弦离散数字信号下辨识系统的离散模型。为了验证算法的有效性,在频域内与扫频法辨识的一、二阶模型和系统实际输出比较研究。实验结果表明:在相同正弦离散信号条件下,辨识输出与系统实际输出基本一致,与扫频法的RMSE相比减小了20.33%,二阶模型在高频段偏离系统实际输出稍大些,一阶系统输出与系统实际输出基本一致。研究结果表明,自适应差分进化算法计算量小,方法简便,辨识准确,在光电跟踪伺服控制领域具有一定的工程应用价值。     

A least-squares approach to blind channel identification

Conventional blind channel identification algorithms are based on channel outputs and knowledge of the probabilistic model of channel input. In some practical applications, however, the input statistical model may not be known, or there may not be sufficient data to obtain accurate enough estimates of certain statistics. In this paper, we consider the system input to be an unknown deterministic signal and study the problem of blind identification of multichannel FIR systems without requiring the knowledge of the input statistical model. A new blind identification algorithm based solely on the system outputs is proposed. Necessary and sufficient identifiability conditions in terms of the multichannel systems and the deterministic input signal are also presented     

Blind identification of digital communication channels withcorrelated noise

Blind identification consists of estimating the impulse response of a linear, time-invariant channel used for transmission of digital data by observing the channel output without knowledge of the transmitted symbol sequence. We show how a previously proposed algorithm (Moulines et al., 1995) based on second-order statistics of the received signal can be modified to account for correlated noise     

Complex ambiguity functions using nonstationary higher ordercumulant estimates

The complex ambiguity function based on second-order statistics (CAF-SOS) has been used to simultaneously estimate the frequency-delay of arrival (FDOA) and time-delay of arrival (TDOA) between two signal measurements; its performance, however, is sensitive to the correlation between two additive noise sources. When the noise sources are assumed to be Gaussian, we develop a new complex ambiguity function based on higher order statistics (CAF-HOS) that reduces the unknown noise-correlation effect. The new CAF-HOS algorithm utilizes nonstationary higher order cross cumulant estimates and their Fourier transform. In fact, we suggest a nonstationary estimate of fourth-order cross-cumulants and obtain the analytical expressions for its mean value and variance. We compare the analytical expressions with results obtained by Monte Carlo runs. Also, we compare the performance of the new complex ambiguity function based on fourth-order statistics (CAF-FOS) against the CAF-SOS algorithm using different Gaussian noise sources, different signals of interest, different signal-to-noise ratios, and different lengths of data     

MAXIMUM LIKELIHOOD SOURCE SEPARATION FOR FINITE IMPULSE RESPONSE MULTIPLE INPUT—MULTIPLE OUTPUT CHANNELS IN THE PRESENCE OF ADDITIVE NOISE

Blind identification-blind equalization for finite Impulse Response(FIR)Multiple Input-Multiple Output(MIMO)channels can be reformulated as the problem of blind sources separation.It has been shown that blind identification via decorrelating sub-channels method could recover the input sources.The Blind Identification via Decorrelating Sub-channels(BIDS)algorithm first constructs a set of decorrelators,which decorrelate the output signals of subchannels,and then estimates the channel matrix using the transfer functions of the decorrelators and finally recovers the input signal using the estimated channel matrix.In this paper,a new qpproximation of the input source for FIR-MIMO channels based on the maximum likelihood source separation method is proposed.The proposed method outperforms BIDS in the presence of additive white Garssian noise.     

MIMO-AR System Identification and Blind Source Separation for GMM-Distributed Sources

The problem of blind source separation (BSS) and system identification for multiple-input multiple-output (MIMO) auto-regressive (AR) mixtures is addressed in this paper. Two new time-domain algorithms for system identification and BSS are proposed based on the Gaussian mixture model (GMM) for sources distribution. Both algorithms are based on the generalized expectation-maximization (GEM) method for joint estimation of the MIMO-AR model parameters and the GMM parameters of the sources. The first algorithm is derived under the assumption of unstructured input signal statistics, while the second algorithm incorporates the prior knowledge about the structure of the input signal statistics due to the statistically independent source assumption. These methods are tested via simulations using synthetic and audio signals. The system identification performances are tested by comparison between the state transition matrix estimation using the proposed algorithms and the well-known multidimensional Yule-Walker solution followed by an instantaneous BSS method. The results show that the proposed algorithms outperform the Yule-Walker based approach. The BSS performances were compared to other convolutive BSS methods. The results show that the proposed algorithms achieve higher signal-to-interference ratio (SIR) compared to the other tested methods.     

Robust Reduced-Rank Adaptive Algorithm Based on Parallel Subgradient Projection and Krylov Subspace

In this paper, we propose a novel reduced-rank adaptive filtering algorithm exploiting the Krylov subspace associated with estimates of certain statistics of input and output signals. We point out that, when the estimated statistics are erroneous (e.g., due to sudden changes of environments), the existing Krylov-subspace-based reduced-rank methods compute the point that minimizes a “wrong” mean-square error (MSE) in the subspace. The proposed algorithm exploits the set-theoretic adaptive filtering framework for tracking efficiently the optimal point in the sense of minimizing the “true” MSE in the subspace. Therefore, compared with the existing methods, the proposed algorithm is more suited to adaptive filtering applications. A convergence analysis of the algorithm is performed by extending the adaptive projected subgradient method (APSM). Numerical examples demonstrate that the proposed algorithm enjoys better tracking performance than the existing methods for system identification problems.