Complex Elliptically Symmetric Research Articles

Robust and sparse M-estimation of DOA

A robust and sparse Direction of Arrival (DOA) estimator is derived for array data that follows a Complex Elliptically Symmetric (CES) distribution with zero-mean and finite second-order moments. The derivation allows to choose the loss function and four loss functions are discussed in detail: the Gauss loss which is the Maximum-Likelihood (ML) loss for the circularly symmetric complex Gaussian distribution, the ML-loss for the complex multivariate t-distribution (MVT) with ν degrees of freedom, as well as Huber and Tyler loss functions. For Gauss loss, the method reduces to Sparse Bayesian Learning (SBL). The root mean square DOA error of the derived estimators is discussed for Gaussian, MVT, and ϵ-contaminated data. The robust SBL estimators perform well for all cases and nearly identical with classical SBL for Gaussian array data.

Phase synchronization sensitivity for widely separated MIMO radar in CES disturbance

This paper investigates the sensitivity to phase synchronization errors through the Hybrid Cramér–Rao Lower Bounds (HCRLB) for joint estimation of target location and velocity in coherent Multiple-Input Multiple-Output (MIMO) radars with distributed antennas. In particular, we derived the HCRLB of range and velocity of a target embedded in Complex Elliptically Symmetric (CES) distributed clutter when the phase error on each receiver and transmitter is modeled as a Gaussian random variable. Closed-form expressions of the HCRLBs are derived and evaluated as a function of the phase error variance.

Censoring outliers in elliptically symmetric multivariate distribution

In this paper, we discuss the problem of censoring outliers in a dataset with a class of complex elliptically symmetric (CES) distribution, which has various engineering applications. To this end, the regularized cost function is formulated by the maximum likelihood (ML) setting. Since the ML estimation of the free outlier subset requires solving a combinatorial problem, an efficient and low complexity algorithm is proposed that guarantees the convergence to a stationary point. Two independent and dependent penalty terms to the selected subset are considered. The conventional generalized inner product (GIP) and the submodular minimization approaches are employed in the derivation of the proposed algorithm in the independent and dependent cases, respectively. The dependent case improves performance by slightly increasing complexity. As a special case for Gaussian assumption and using a non-regularized cost function, the proposed algorithm reduces to the fast minimum covariance determinant (FMCD) algorithm, which has impressive efficiency and received considerable attention in statistics and computer science. Hence, the proposed algorithm extends/generalizes the FMCD to be applicable to the non-Gaussian distributions. The simulation results highlight that the proposed algorithms exhibit satisfactory performance in CES environments.

CRLBs for Location and Velocity Estimation for MIMO Radars in CES-Distributed Clutter

In this article, we investigate the problem of jointly estimating target location and velocity for widely separated multiple-input multiple-output (MIMO) radar operating in correlated non-Gaussian clutter, modeled by a complex elliptically symmetric (CES) distribution. More specifically, we derive the Cramér–Rao lower bounds (CRLBs) when the target is modeled by the Swerling 0 model and the clutter is complex t-distributed. We thoroughly analyze the impact of the clutter correlation and spikiness to provide accurate performance estimation. Index terms—Cramér–Rao lower bounds (CRLBs), MIMO radar, location and velocity estimation, performance analysis, complex elliptically symmetric (CES) distributed, and complex t-distribution.

Refinement and derivation of statistical resolution limits for circular or rectilinear correlated sources in CES data models

This paper analyzes the deterministic (DCRB) and the stochastic (SCRB) Cramér-Rao bound on direction-of-arrival (DOA) estimation for two equi-powered correlated complex circular or rectilinear sources affected by complex circular white noise in different complex elliptically symmetric (CES) data models. Beginning by decomposing these CRBs, into factors depending on signal and noise parameters, and on geometric parameters of the array, some new interpretable closed-form expressions are derived in particular scenarios. These expressions provide useful insight into the behavior of these CRB’s dependence on the correlation factor. Approximate closed-form expressions of these CRBs for small DOA separation are also derived. These results lead to new formulas for statistical resolution limit (SRL) based on the Smith criterion at which an unbiased DOA estimation algorithm can resolve two closely-spaced circular or rectilinear sources. These SCRB-derived formulas are much less optimistic than those which have so far been deduced only from the DCRB under the assumption of known sources.

Regularized Covariance Estimation for Polarization Radar Detection in Compound Gaussian Sea Clutter

This paper investigates regularized estimation of Kronecker-structured covariance matrices (CM) for polarization radar in sea clutter scenarios where the data are assumed to follow the complex, elliptically symmetric (CES) distributions with a Kronecker-structured CM. To obtain a well-conditioned estimate of the CM, we add penalty terms of Kullback-Leibler divergence to the negative log-likelihood function of the associated complex angular Gaussian (CAG) distribution. This is shown to be equivalent to regularizing Tyler's fixed-point equations by shrinkage. A sufficient condition that the solution exists is discussed. An iterative algorithm is applied to solve the resulting fixed-point iterations and its convergence is proved. In order to solve the critical problem of tuning the shrinkage factors, we then introduce two methods by exploiting oracle approximating shrinkage (OAS) and cross-validation (CV). The proposed estimator, referred to as the robust shrinkage Kronecker estimator (RSKE), is shown to achieve better performance compared with several existing methods when the training samples are limited. Simulations are conducted for validating the RSKE and demonstrating its high performance by using the IPIX 1998 real sea data.

Joint Estimation of Location and Scatter in Complex Elliptically Symmetric Distributions

The joint estimation of the location vector and the shape matrix of a set of independent and identically Complex Elliptically Symmetric (CES) distributed observations is investigated from both the theoretical and computational viewpoints. This joint estimation problem is framed in the original context of semiparametric models allowing us to handle the (generally unknown) density generator as an infinite-dimensional nuisance parameter. In the first part of the paper, a computationally efficient and memory saving implementation of the robust and semiparmaetric efficient R-estimator for shape matrices is derived. Building upon this result, in the second part, a joint estimator, relying on the Tyler’s M-estimator of location and on the R-estimator of shape matrix, is proposed and its Mean Squared Error (MSE) performance compared with the Semiparametric Cramér-Rao Bound (SCRB).

On the performance of robust plug-in detectors using M-estimators

This paper provides an original asymptotic analysis of robust adaptive detectors performance in the context of non-Gaussian observations. We focus on a single-steering case in homogeneous environment and analyze the properties of different adaptive detectors such as Adaptive (Normalized) Matched Filter (AMF/ANMF), Kelly’s GLRT, and Rao test when an estimator of the covariance matrix is plugged-in. When the noise distribution turns to be non-Gaussian, the detectors relying on the Sample Covariance Matrix (SCM) can perform poorly and an interesting alternative is the use of M-estimators. In this context, we show that, from Complex Elliptically Symmetric (CES) samples, the distribution of a detection statistic built with M-estimators can be accurately approximated by the one of the same statistic built from the SCM of an equivalent Gaussian setting. The loss due to this approximation is theoretically derived and shown to be negligible in most cases. This explicit equivalent statistic is especially interesting since it allows to tune robust plug-in detectors with well established results from the Gaussian detection framework. Furthermore, this approach provides new insights on the robust estimation tools behavior. Finally, some simulations illustrate the interest of the proposed results.

Maximum-Likelihood Direction Finding Under Elliptical Noise Using the EM Algorithm

Unlike subspace-based solutions of direction-of-arrival (DOA) estimation under non-Gaussian noise, where the only optional difference with the Gaussian case is the scatter/covariance matrix estimation method, maximum-likelihood (ML)-based DOA solutions need a different treatment under the non-Gaussianity assumption. In this letter, we derive a particular ML-based DOA solution, called the expectation–maximization (EM) estimator, under the wide class of complex elliptically symmetric (CES) distributions.

Direction of Arrival Estimation in Elliptical Models via Sparse Penalized Likelihood Approach.

In this paper, an -penalized maximum likelihood (ML) approach is developed for estimating the directions of arrival (DOAs) of source signals from the complex elliptically symmetric (CES) array outputs. This approach employs the -norm penalty to exploit the sparsity of the gridded directions, and the CES distribution setting has a merit of robustness to the uncertainty of the distribution of array output. To solve the constructed non-convex penalized ML optimization for spatially either uniform or non-uniform sensor noise, two majorization-minimization (MM) algorithms based on different majorizing functions are developed. The computational complexities of the above two algorithms are analyzed. A modified Bayesian information criterion (BIC) is provided for selecting an appropriate penalty parameter. The effectiveness and superiority of the proposed methods in producing high DOA estimation accuracy are shown in numerical experiments.

Semiparametric CRB and Slepian-Bangs Formulas for Complex Elliptically Symmetric Distributions

The main aim of this paper is to extend the semiparametric inference methodology, recently investigated for Real Elliptically Symmetric (RES) distributions, to Complex Elliptically Symmetric (CES) distributions. The generalization to the complex field is of fundamental importance in all practical applications that exploit the complex representation of the acquired data. Moreover, the CES distributions has been widely recognized as a valuable and general model to statistically describe the non-Gaussian behaviour of datasets originated from a wide variety of physical measurement processes. The paper is divided in two parts. In the first part, a closed form expression of the constrained Semiparametric Cram\'{e}r-Rao Bound (CSCRB) for the joint estimation of complex mean vector and complex scatter matrix of a set of CES-distributed random vectors is obtained by exploiting the so-called \textit{Wirtinger} or $\mathbb{C}\mathbb{R}$-\textit{calculus}. The second part deals with the derivation of the semiparametric version of the Slepian-Bangs formula in the context of the CES model. Specifically, the proposed Semiparametric Slepian-Bangs (SSB) formula provides us with a useful and ready-to-use expression of the Semiparametric Fisher Information Matrix (SFIM) for the estimation of a parameter vector parametrizing the complex mean and the complex scatter matrix of a CES-distributed vector in the presence of unknown, nuisance, density generator. Furthermore, we show how to exploit the derived SSB formula to obtain the semiparametric counterpart of the Stochastic CRB for Direction of Arrival (DOA) estimation under a random signal model assumption. Simulation results are also provided to clarify the theoretical findings and to demonstrate their usefulness in common array processing applications.

Detection of Unknown Signals Under Complex Elliptically Symmetric Distributions

Detection of unknown signals is considered under complex elliptically symmetric (CES) distributions, a wide family that includes several well-known distributions as special cases. We study the detection problem for unknown deterministic signals and random CES signals, both corrupted by CES noise. For detection of unknown deterministic signals, we form the generalized likelihood ratio test and reduce it to a sufficient test statistic, which is either the norm of the received vector or a function of it. Performance analysis is provided under both known and estimated noise scatter matrix. For detection of random CES signals, we form the likelihood ratio test, provide performance analysis, and establish conditions under which the sufficient test statistic is given by the norm of the received vector. Interestingly, this norm test statistic is proven to be the uniformly most powerful test for the detection of all CES distributed signals.

Covariance Matrix Estimation Under Degeneracy for Complex Elliptically Symmetric Distributions

In several adaptive algorithms, the data covariance matrix must be frequently estimated, particularly in modern wireless systems where, due to high mobility, signal statistics change in time. When the underlying distribution is degenerate, likelihood-based estimators of the covariance matrix and/or scatter matrix may not exist for non-Gaussian signals. A well-known class of non-Gaussian distributions is the family of complex elliptically symmetric (CES) distributions, which includes several well-known distributions as special cases. In this paper, we give some results on the degeneracy of CES distributions and derive the maximum likelihood (ML) estimator and $M$ -estimators of the scatter matrix for degenerate CES vectors. Estimation with insufficient data is addressed by a regularization approach. We also formulate the Cramer–Rao lower bound (CRLB) on the elements of the scatter matrix for degenerate CES vectors. Our results coincide with those of the nondegenerate CES vectors whenever the scatter matrix is full rank. Numerical results are presented to justify the efficiency of the proposed method.

Matched, mismatched, and robust scatter matrix estimation and hypothesis testing in complex t-distributed data

Scatter matrix estimation and hypothesis testing are fundamental inference problems in a wide variety of signal processing applications. In this paper, we investigate and compare the matched, mismatched, and robust approaches to solve these problems in the context of the complex elliptically symmetric (CES) distributions. The matched approach is when the estimation and detection algorithms are tailored on the correct data distribution, whereas the mismatched approach refers to the case when the scatter matrix estimator and the decision rule are derived under a model assumption that is not correct. The robust approach aims at providing good estimation and detection performance, even if suboptimal, over a large set of possible data models, irrespective of the actual data distribution. Specifically, due to its central importance in both the statistical and engineering applications, we assume for the input data a complex t-distribution. We analyze scatter matrix estimators derived under the three different approaches and compare their mean square error (MSE) with the constrained Cramer-Rao bound (CCRB) and the constrained misspecified Cramer-Rao bound (CMCRB). In addition, the detection performance and false alarm rate (FAR) of the various detection algorithms are compared with that of the clairvoyant optimum detector.

The Misspecified Cramer-Rao Bound and Its Application to Scatter Matrix Estimation in Complex Elliptically Symmetric Distributions

This paper focuses on the application of recent results on lower bounds under misspecified models to the estimation of the scatter matrix of complex elliptically symmetric (CES) distributed random vectors. Buildings upon the original works of Q. H. Vuong [Cramer-Rao Bounds for Misspecified Models, Div. of the Humanities and Social Sci., California Inst. of Technol., Pasadena, CA, USA, Working Paper 652, Oct. 1986] and Richmond–Horowitz [“Parameter Bounds on Estimation Accuracy Under Model Misspecification,” IEEE Trans. Signal Process., vol. 63, no. 9, pp. 2263–2278, May 2015], a lower bound, named misspecified Cramer–Rao bound (MCRB), for the error covariance matrix of any unbiased (in a proper sense) estimator of a deterministic parameter vector under misspecified models, is introduced. Then, we show how to apply these results to the problem of estimating the scatter matrix of CES distributed data under data mismodeling. In particular, the performance of the maximum likelihood (ML) estimator are compared, under mismatched conditions, with the MCRB and with the classical CRB in some relevant study cases.

Maximum likelihood covariance matrix estimation for complex elliptically symmetric distributions under mismatched conditions

This paper deals with the maximum likelihood (ML) estimation of scatter matrix of complex elliptically symmetric (CES) distributed data when the hypothesized and the true model belong to the CES family but are different, then under mismatched model condition. Firstly, we derive the Huber limit, or sandwich matrix expression, for a generic CES model. Then, we compare the performance of mismatched and matched ML estimators to the Huber limit and to the Cramér–Rao lower bound (CRLB) in some relevant study cases.

Regularized Covariance Matrix Estimation in Complex Elliptically Symmetric Distributions Using the Expected Likelihood Approach—Part 2: The Under-Sampled Case

In the first part of these two papers, we extended the expected likelihood approach originally developed in the Gaussian case, to the broader class of complex elliptically symmetric (CES) distributions and complex angular central Gaussian (ACG) distributions. More precisely, we demonstrated that the probability density function (p.d.f.) of the likelihood ratio (LR) for the (unknown) actual scatter matrix Σ0 does not depend on the latter: it only depends on the density generator for the CES distribution and is distribution-free in the case of ACG distributed data, i.e., it only depends on the matrix dimension M and the number of independent training samples T, assuming that T ≥ M. Additionally, regularized scatter matrix estimates based on the EL methodology were derived. In this second part, we consider the under-sampled scenario (T ≤ M) which deserves specific treatment since conventional maximum likelihood estimates do not exist. Indeed, inference about the scatter matrix can only be made in the T-dimensional subspace spanned by the columns of the data matrix. We extend the results derived under the Gaussian assumption to the CES and ACG class of distributions. Invariance properties of the under-sampled likelihood ratio evaluated at Σ0 are presented. Remarkably enough, in the ACG case, the p.d.f. of this LR can be written in a rather simple form as a product of beta distributed random variables. The regularized schemes derived in the first part, based on the EL principle, are extended to the under-sampled scenario and assessed through numerical simulations.

Regularized Covariance Matrix Estimation in Complex Elliptically Symmetric Distributions Using the Expected Likelihood Approach— Part 1: The Over-Sampled Case

In Abramovich [“Bounds on Maximum Likelihood Ratio-Part I: Application to Antenna Array Detection-Estimation With Perfect Wavefront Coherence,” IEEE Trans. Signal Process., vol. 52, pp. 1524-1536, June 2004], it was demonstrated, for multivariate complex Gaussian distribution, that the probability density function (p.d.f.) of the likelihood ratio (LR) for the (unknown) actual covariance matrix R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> does not depend on this matrix and is fully specified by the matrix dimension M and the number of independent training samples T. This invariance property hence enables one to compare the LR of any derived covariance matrix estimate against this p.d.f., and eventually get an estimate that is statistically “as likely” as R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> . This “expected likelihood” quality assessment allowed significant improvement of MUSIC DOA estimation performance in the so-called “threshold area,” and for diagonal loading and TVAR model order selection in adaptive detectors. Recently, the so-called complex elliptically symmetric (CES) distributions have been introduced for description of highly in-homogeneous clutter returns. The aim of this series of two papers is to extend the EL approach to this class of CES distributions as well as to a particularly important derivative, namely the complex angular central distribution (ACG). For both cases, we demonstrate a similar invariance property for the LR associated with the true scatter matrix Σ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> . Furthermore, we derive fixed point regularized covariance matrix estimates using the generalized expected likelihood methodology. This first part is devoted to the conventional scenario (T ≥ M) while Part II deals with the undersampled scenario (T ≤ M).

Complex Elliptically Symmetric Distributions: Survey, New Results and Applications

Complex elliptically symmetric (CES) distributions have been widely used in various engineering applications for which non-Gaussian models are needed. In this overview, circular CES distributions are surveyed, some new results are derived and their applications e.g., in radar and array signal processing are discussed and illustrated with theoretical examples, simulations and analysis of real radar data. The maximum likelihood (ML) estimator of the scatter matrix parameter is derived and general conditions for its existence and uniqueness, and for convergence of the iterative fixed point algorithm are established. Specific ML-estimators for several CES distributions that are widely used in the signal processing literature are discussed in depth, including the complex t -distribution, K-distribution, the generalized Gaussian distribution and the closely related angular central Gaussian distribution. A generalization of ML-estimators, the M-estimators of the scatter matrix, are also discussed and asymptotic analysis is provided. Applications of CES distributions and the adaptive signal processors based on ML- and M-estimators of the scatter matrix are illustrated in radar detection problems and in array signal processing applications for Direction-of-Arrival (DOA) estimation and beamforming. Furthermore, experimental validation of the usefulness of CES distributions for modelling real radar data is given.

Complex Elliptically Symmetric Random Variables—Generation, Characterization, and Circularity Tests

Complex elliptically symmetric (CES) distributions constitute a flexible and broad class of distributions for many engineering applications and include the widely used complex Gaussian, complex -, complex generalized Gaussian and symmetric -stable distributions for example. Their careful statistical characterization is needed. Moreover, the mechanisms for generating random variables from these distributions are not well defined in literature. In this paper we provide such treatment in order to provide a better insight on their statistical properties and simplifying proofs and derivations. For example, insightful expressions for complex kurtosis coefficients of CES distributions are derived providing an interpretation on what complex kurtosis really measures. Also derived are asymptotic distributions of circularity measures, the sample circularity quotient and coefficient, assuming i.i.d. samples from an unspecified CES distribution. Also new Wald's type detectors of circularity are proposed that are valid within CES distributions with finite fourth-order moments. These results are accompanied with examples that are of interest in developing signal processing algorithms for complex-valued signals.