Analysis of the combined effects of finite samples and model errorson array processing performance

The principal sources of estimation error in sensor array signal processing applications are the finite sample effects of additive noise and imprecise models for the antenna array and spatial noise statistics. While the effects of these errors have been studied individually, their combined effect has not yet been rigorously analyzed. The authors undertake such an analysis for the class of so-called subspace fitting algorithms. In addition to deriving first-order asymptotic expressions for the estimation error, they show that an overall optimal weighting exists for a particular array and noise covariance error model. In a companion paper, the optimally weighted subspace fitting method is shown to be asymptotically equivalent with the more complicated maximum a posteriori estimator. Thus, for the model in question, no other method can yield more accurate estimates for large samples and small model errors. Numerical examples and computer simulations are included to illustrate the obtained results and to verify the asymptotic analysis for realistic scenarios     

Sensor array processing based on subspace fitting

Algorithms for estimating unknown signal parameters from the measured output of a sensor array are considered in connection with the subspace fitting problem. The methods considered are the deterministic maximum likelihood method (ML), ESPRIT, and a recently proposed multidimensional signal subspace method. These methods are formulated in a subspace-fitting-based framework, which provides insight into their algebraic and asymptotic relations. It is shown that by introducing a specific weighting matrix, the multidimensional signal subspace method can achieve the same asymptotic properties as the ML method. The asymptotic distribution of the estimation error is derived for a general subspace weighting, and the weighting that provides minimum variance estimates is identified. The resulting optimal technique is termed the weighted subspace fitting (WSF) method. Numerical examples indicate that the asymptotic variance of the WSF estimates coincides with the Cramer-Rao bound. The performance improvement compared to the other techniques is found to be most prominent for highly correlated signals     

A generalization of weighted subspace fitting to full-rank models

The idea of subspace fitting provides a popular framework for different applications of parameter estimation and system identification. Previously, some algorithms have been suggested based on similar ideas, for a sensor array processing problem where the underlying data model is not low rank. We show that two of these algorithms (DSPE and DISPARE) fail to give consistent estimates and introduce a general class of subspace fitting-like algorithms for consistent estimation of parameters from a possibly full-rank data model. The asymptotic performance is analyzed, and an optimally weighted algorithm is derived. The result gives a lower bound on the estimation performance for any estimator based on a low-rank approximation of the linear space spanned by the sample data. We show that in general, for full-rank data models, no subspace-based method can reach the Cramer-Rao lower bound (CRB)     

Performance analysis of the approximate dynamic programmingalgorithm for parameter estimation of superimposed signals

We consider the classical problem of fitting a model composed of multiple superimposed signals to noisy data using the criteria of maximum likelihood (ML) or subspace fitting, jointly termed generalized subspace fitting (GSF). We analyze a previously proposed approximate dynamic programming algorithm (ADP), which provides a computationally efficient solution to the associated multidimensional multimodal optimization problem. We quantify the error introduced by the approximations in ADP and deviations from the key local interaction signal model (LISMO) modeling assumption in two ways. First, we upper bound the difference between the exact minimum of the GSF criterion and its value at the ADP estimate and compare the ADP with GSF estimates obtained by exhaustive multidimensional search on a fine lattice. Second, motivated by the similar accuracy bounds, we use perturbation analysis to derive approximate expressions for the MSE of the ADP estimates. These various results provide, for the first time, an effective tool to predict the performance of the ADP algorithm for various signal models at nonasymptotic conditions of interest in practical applications. In particular, they demonstrate that for the classical problems of sinusoid retrieval and array processing, ADP performs comparably to exact (but expensive) maximum likelihood (ML) over a wide range of signal-to-noise ratios (SNRs) and is therefore an attractive algorithm     

A performance analysis of subspace-based methods in the presence ofmodel error. II. Multidimensional algorithms

For pt.I, see ibid., vol.40, no.7, p.1758-74 (1992). In pt.I the performance of the MUSIC algorithms for narrowband direction-of-arrival (DOA) estimation when the array manifold and noise covariance are not correctly modeled was investigated. This analysis is extended to multidimensional subspace-based algorithms including deterministic (or conditional) maximum likelihood, MD-MUSIC, weighted subspace fitting (WSF), MODE, and ESPRIT. A general expression for the variance of the DOA estimates that can be applied to any of the above algorithms and to any of a wide variety of scenarios is presented. Optimally weighted subspace fitting algorithms are presented for special cases involving random unstructured errors of the array manifold and noise covariance. It is shown that one-dimensional MUSIC outperforms all of the above multidimensional algorithms for random angle-independent array perturbations     

A subspace method for direction of arrival estimation ofuncorrelated emitter signals

A novel eigenstructure-based method for direction estimation is presented. The method assumes that the emitter signals are uncorrelated. Ideas from subspace and covariance matching methods are combined to yield a noniterative estimation algorithm when a uniform linear array is employed. The large sample performance of the estimator is analyzed. It is shown that the asymptotic variance of the direction estimates coincides with the relevant Cramer-Rao lower bound (CRB). A compact expression for the CRB is derived for the ease when it is known that the signals are uncorrelated, and it is lower than the CRB that is usually used in the array processing literature (assuming no particular structure for the signal covariance matrix). The difference between the two CRBs can be large in difficult scenarios. This implies that in such scenarios, the proposed methods has significantly better performance than existing subspace methods such as, for example, WSF, MUSIC, and ESPRIT. Numerical examples are provided to illustrate the obtained results     

A new subspace identification algorithm for high-resolution DOA estimation

In this paper, we propose a new direction of arrival (DOA) estimator for sensor-array processing. The estimator is based on a linear algebraic connection between the standard subspace model of the array correlation matrix and a special signal-plus-interference model, which we develop in this paper. The estimator we propose is a signal subspace scaled MUSIC algorithm, which we call SSMUSIC. It is not a subspace weighted MUSIC, because the scaling depends on the eigenstructure of the estimated signal subspace. SSMUSIC has the advantage of simultaneously estimating the DOA and the power of each source. We employ a second-order perturbation analysis of the estimator and derive stochastic representations for its bias and squared-error. We compare the new DOA estimator with the MUSIC estimator, based on these representations. Numerical results demonstrate the superior performance of SSMUSIC relative to MUSIC and the validity of the perturbation results.     

用投影算子改善信号子空间方向估计算法的稳健性

基于信号子空间的超分辨方法一般对接收器阵列误差非常敏感。本文提出的投影变换法利用目标方向的初始估计和阵列流形的先验知识，可以显著减小信号子空间方法的误差灵敏度。文中给出了数值仿真和实际阵列数据测试的结果。     

Application of signal subspace approach to high-resolution narrowband array processing

The paper first briefly reviews some subspace techniques for high-resolution array processing. It is shown that existing high-resolution techniques like the MUSIC algorithm are based on visual inspection of the spatial spectrum. It is not a scientifically valid means of assessing resolution of a spectrum estimator. The paper then proposes a technique based on a combination of optimal processing and signal subspace extraction for high-resolution array processing. Numerical results show that the proposed technique not only achieves superresolution of the spectrum, but also provides power estimates of the arrivals.     

In-variance of subspace based estimators

Subspace-based estimates, i.e., estimates obtained by exploiting the orthogonality between a sample subspace and a parameter-dependent subspace, have proved useful in many applications, including array processing and system identification. The purpose of this paper is to complement the already available theoretical results generally obtained in specific contexts. We discuss the generalization of the optimal weighted subspace fitting approach introduced by Viberg (1989) in the DOA estimation context; we exhibit some invariance properties of optimally weighted estimate, and we show the equivalence between subspace fitting and subspace matching     

Efficient Subspace-Based Estimator for Localization of Multiple Incoherently Distributed Sources

In this paper, a new subspace-based algorithm for parametric estimation of angular parameters of multiple incoherently distributed sources is proposed. This approach consists of using the subspace principle without any eigendecomposition of the covariance matrix, so that it does not require the knowledge of the effective dimension of the pseudosignal subspace, and therefore the main difficulty of the existing subspace estimators can be avoided. The proposed idea relies on the use of the property of the inverse of the covariance matrix to exploit approximately the orthogonality property between column vectors of the noise-free covariance matrix and the sample pseudonoise subspace. The resulting estimator can be considered as a generalization of the Pisarenko's extended version of Capon's estimator from the case of point sources to the case of incoherently distributed sources. Theoretical expressions are derived for the variance and the bias of the proposed estimator due to finite sample effect. Compared with other known methods with comparable complexity, the proposed algorithm exhibits a better estimation performance, especially for close source separation, for large angular spread and for low signal-to-noise ratio.     

Array interpolation and DOA MSE reduction

Interpolation or mapping of data from a given real array to data from a virtual array of more suitable geometry is well known in array signal processing. This operation allows arrays of any geometry to be used with fast direction-of-arrival (DOA) estimators designed for linear arrays. In an earlier companion paper , a first-order condition for zero DOA bias under such mapping was derived and was also used to construct a design algorithm for the mapping matrix that minimized the DOA estimate bias. This bias-minimizing theory is now extended to minimize not only bias, but also to consider finite sample effects due to noise and reduce the DOA mean-square error (MSE). An analytical first-order expression for mapped DOA MSE is derived, and a design algorithm for the transformation matrix that minimizes this MSE is proposed. Generally, DOA MSE is not reduced by minimizing the size of the mapping errors but instead by rotating these errors and the associated noise subspace into optimal directions relative to a certain gradient of the DOA estimator criterion function. The analytical MSE expression and the design algorithm are supported by simulations that show not only conspicuous MSE improvements in relevant scenarios, but also a more robust preprocessing for low signal-to-noise ratios (SNRs) as compared with the pure bias-minimizing design developed in the previous paper.     

阵列存在未知有界扰动的鲁棒加权子空间拟合

由于阵列响应难于精确标定,因此,阵列信号处理算法的性能不仅受测量噪声的影响同时还受阵列响应扰动的影响。信号回波方向估计DOA受该系统扰动的影响很大。以往的研究均将模型的扰动假设为随机误差,且往往假设扰动是相关矩阵已知的高斯分布。然而,实际系统误差很难满足这一假设。模型误差的更合理的假设是未知有界(UBB)的假设,这里,研究模型扰动是未知有界时,信号的DOA估计问题,提出了鲁棒的信号子空间拟合原理,并用二次锥规划来求解。数值仿真表明该算法是有效的。     

Analysis of subspace fitting and ML techniques for parameterestimation from sensor array data

It is shown that the multidimensional signal subspace method, termed weighted subspace fitting (WSF), is asymptotically efficient. This results in a novel, compact matrix expression for the Cramer-Rao bound (CRB) on the estimation error variance. The asymptotic analysis of the maximum likelihood (ML) and WSF methods is extended to deterministic emitter signals. The asymptotic properties of the estimates for this case are shown to be identical to the Gaussian emitter signal case, i.e. independent of the actual signal waveforms. Conclusions concerning the modeling aspect of the sensor array problem are drawn     

Deconvolution estimation of motor unit conduction velocity distribution

A conduction velocity distribution (CVD) estimator that incorporates volume conductor modeling of the muscle voluntary response is introduced in this paper. The CVD estimates are obtained from two correlation functions, an autocorrelation and a cross, computed from myoelectric signal recorded at the skin surface. The performance of the proposed estimator is evaluated for simulated and experimental data. The study includes assessment of the estimator bias and standard deviation, as well as its sensitivity to errors in the model parameters. Simulations show its good performance in terms of estimator bias. A filtering technique also helps reduce its variance. However, the inaccuracy introduced in the estimation of model parameters considerably deteriorates the estimator performance.     

Derived PDF of maximum likelihood signal estimator which employs anestimated noise covariance

A probability density function (PDF) for the maximum likelihood (ML) signal vector estimator is derived when the estimator relies on a noise sample covariance matrix (SCM) for evaluation. By using a complex Wishart probabilistic model for the distribution of the SCM, it is shown that the PDF of the adaptive ML (AML) signal estimator (alias the SCM based minimum variance distortionless response (MVDR) beamformer output and, more generally, the SCM based linearly constrained minimum variance (LCMV) beamformer output) is, in general, the confluent hypergeometric function of a complex matrix argument known as Kummer's function. The AML signal estimator remains unbiased but only asymptotically efficient; moreover, the AML signal estimator converges in distribution to the ML signal estimator (known noise covariance). When the sample size of the estimated noise covariance matrix is fixed, it is demonstrated that there exists a dynamic tradeoff between signal-to-noise ratio (SNR) and noise adaptivity as the dimensionality of the array data (number of adaptive degrees of freedom) is varied, suggesting the existence of an optimal array data dimension that will yield the best performance     

一种新的基于 EEF 准则的空间声源个数估计算法

针对指数嵌入族（ Exponentially Embedded Families ,EEF）准则在快拍数小于阵元数情况下无法估计声源个数的问题,本文提出一种新的空间声源个数估计算法。首先通过球麦克风阵列采集空间声场高阶信息,建立球阵列信号模型,将声源个数估计扩展到三维空间。继而将观测信号空间分解为信号子空间和噪声子空间,利用最小均方差（ Minimum Mean-Squared Error ,MMSE）方法估计观测信号空间及噪声子空间的协方差矩阵,确保矩阵估计的一致性和准确性。在此基础上改进似然比函数,同时引入新的自由度计算,使得算法在快拍数小于阵元数的情况下能有效估计声源个数。仿真结果表明,在进行空间声源个数估计时,相对于EEF准则,新的算法不仅适用于快拍数小于阵元数情况,同时提高了估计准确率。     

Empirical-Bayes Estimation of Mean Life for a Censored Sample, Constant Hazard Rate Model

The estimation of mean life based on a censored sample from a population with a constant failure rate is considered in an empirical Bayes situation. Bayes risks of empirical Bayes estimates for various numbers of past samples are compared with the Bayes risk of the sample mean, and the minimum Bayes risk. A Monte Carlo simulation shows that the Bayes risk of the empirical Bayes estimator is very close to the minimum Bayes risk for as few as 5 past estimates of the mean life. Morever, it is better than the uniformly minimum variance unbiased estimator (the sample mean) in terms of the Bayes risk. Effects of misclassification of the prior are also considered.     

A robust adaptive array based on signal subspace approach

In this note, we propose a signal subspace approach that improves the performance of a beam steered adaptive array in the presence of steering errors due to look-direction error (LDE) and/or random steering error (RSE). In the method, the degrees of freedom (DOF) are reduced so as not to cancel the desired signal while preserving the optimal characteristic of the array, and thus the weights of the array are determined by a linear combination of the eigenvectors of the signal subspace. The proposed method works as far as the eigen decomposition of the input covariance matrix into signal and noise subspaces is possible. The proposed method improves noticeably the array performance of the beam steered array in the presence of steering errors and provides the optimum array performance in the absence of steering errors