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This letter deals with similarity parameter selection for knowledge-aided covariance matrix estimation in adaptive radar signal processing. Starting from the observation that the maximum likelihood estimate of the interference covariance matrix under a similarity constraint admits a closed-form expression, which depends on the similarity parameter, an adaptive procedure is devised to get a parameter free estimator. The technique is based on the expected likelihood principle and requires the solution of an implicit equation, which can be efficiently pursued via the bisection method due a monotonicity property. The analysis of the estimator, conducted also in comparison with the counterpart based on the cross-validation method confirms its effectiveness in terms of both performance and computational complexity.

This paper considers the problem of censoring outliers from the secondary dataset in a radar scenario where the sample support is limited. To this end, the generalized regularized likelihood function (GRLF) criterion is used and the corresponding regularized maximum likelihood (RML) estimate of the outlier subset is derived. Since the exact RML estimate involves the solution of a combinatorial optimization problem, a reduced complexity but approximate RML (ARML) procedure is also designed. As to the selection of the regularization parameter, both the expected likelihood (EL) principle and the cross-validation (CV) technique are exploited. At the analysis stage, the performance of the RML and ARML procedure is evaluated based on simulated data in comparison with some previously proposed methods. The results highlight that the RML and ARML algorithm achieves, in general, a satisfactory performance level whereas the previously proposed techniques often experience some performance degradation when the volume of training data is dramatically limited.

Regularization, which consists in shrinkage of the sample covariance matrix to a target matrix, is a commonly used and effective technique in low sample support covariance matrix estimation. Usually, a target matrix is chosen and optimization of the shrinkage factor is carried out, based on some relevant metric. In this letter, we rather address the choice of the target matrix. More precisely, we aim at evaluating, from observation of the data matrix, whether a given target matrix is a good regularizer. Towards this end, the expected likelihood (EL) approach is investigated. At a first step, we re-interpret the regularized covariance matrix estimate as the minimum mean-square error estimate in a Bayesian model where the target matrix serves as a prior. The likelihood function of the data is then derived, and the EL principle is subsequently applied. Over-sampled and under-sampled scenarios are considered.

In \cite{Abramovich04}, it was demonstrated that the likelihood ratio (LR) for multivariate complex Gaussian distribution has the invariance property that can be exploited in many applications. Specifically, the probability density function (p.d.f.) of this LR for the (unknown) actual covariance matrix $\R_{0}$ does not depend on this matrix and is fully specified by the matrix dimension $M$ and the number of independent training samples $T$. Since this p.d.f. could therefore be pre-calculated for any a priori known $(M,T)$, one gets a possibility 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 the a priori unknown actual covariance matrix. This ``expected likelihood'' (EL) quality assessment allows for significant improvement of MUSIC DOA estimation performance in the so-called ``threshold area'' \cite{Abramovich04,Abramovich07d}, and for diagonal loading and TVAR model order selection in adaptive detectors \cite{Abramovich07,Abramovich07b}. Recently, a broad class of the so-called complex elliptically symmetric (CES) distributions has 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 of CES, 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 $\mSigma_{0}$. Furthermore, we derive fixed point regularized covariance matrix estimates using the generalized expected likelihood methodology. This first part is devoted to the conventional scenario ($T \geq M$) while Part 2 deals with the under-sampled scenario ($T \leq M$).

Expected likelihood (EL) technique for quality assessment of parameter estimates of signals embedded in Gaussian noise is extended in this paper over the case where useful signals are immersed in a mixture of compound Gaussian and Gaussian-distributed noises. The main problem here is that analytical expressions for distributions of such mixtures do not exist in most cases. Moreover, in some cases like $K$ -distributed noise only, where closed-form expressions for the data distribution are available, the traditional Cramér-Rao bound does not exist. This makes the EL technique even more important for parameter estimation performance assessment. In this paper, for the so-called conditional model, we introduce test statistics whose distribution for the true (actual) parameters does not depend on these parameters and specifics of texture distribution, which makes them applicable for EL applications. We illustrate the utility of this EL technique by studying and predicting the performance breakdown of some direction of arrival estimators in a mixture of $K$-distributed and Gaussian noise. Refereed/Peer-reviewed

In the first part of this series of 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 $\mSigma_{0}$ 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 $T \geq M$. Additionally, regularized scatter matrix estimates based on the EL methodology were derived. In this second part, we consider the under-sampled scenario ($T \leq M$) which deserves a 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 $\mSigma_{0}$ 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.

We consider the problem of covariance matrix estimation for strongly non-homogeneous data (clutter), modelled as spherically invariant complex random vectors. When normalized, such data are described by a complex angular Gaussian distribution. For a number of independent identically distributed (i.i.d.) samples with this distribution, we introduce the likelihood ratio and demonstrate that for the true covariance matrix, this LR has a p.d.f. which does not depend on the true matrix itself, but rather just its dimension and sample support. This "scenario-invariance" is used to select regularizing parameters that reduce the LR of the MLE estimate to a level statistically equal to the likelihood of the true covariance matrix. We apply this approach to select the loading factor in fixed point loaded covariance matrix estimates, and the order in the fixed point time-varying auto regressive (TVAR) covariance matrix estimate. Refereed/Peer-reviewed

We consider the problem of detection-estimation (resolution) of a number of sources impinging upon an antenna array, where the covariance matrix of each source has a limited signal eigen-subspace, spanned by a known linear subspace. In general, the direction of arrival (DoA) and the subspace rank of each source is not known a priori and must be estimated, along with detection of the total number of such sources. We introduce the application of a GLRT-based ("expected likelihood" (EL)) detection-estimation framework to this problem in conjunction with a novel "multi-rank MUSIC" technique for initial DoA estimation Refereed/Peer-reviewed

Refereed/Peer-reviewed