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• The asymptotic distribution of the projector associated with the SCM is given for arbitrary distributions of the useful signal and noise with finite fourth-order moments. • The asymptotic distribution of the DOA estimate given by the MUSIC algorithm is proved for both nonparametric non-Gaussian noise and parametric CES noise models. Theoretical evaluation of the loss of performance is given for heavy-tailed distributions of the noise which is significant for weak SNR and closely spaced sources. • For CES distributed observations, the asymptotic distribution of the projector associated with any M -estimate of the covariance matrix is given. This allows us to prove that this loss of performance can be alleviated. • These theoretical results are confirmed by some simulations performed with either a complex circular Student t -distribution or a complex generalized Gaussian distribution of the noise or the observations model. This paper addresses the theoretical analysis of the robustness of subspace-based algorithms with respect to non-Gaussian noise distributions using perturbation expansions. Its purpose is twofold. It aims, first, to derive the asymptotic distribution of the estimated projector matrix obtained from the sample covariance matrix (SCM) for arbitrary distributions of the useful signal and the noise. It proves that this distribution depends only of the second-order statistics of the useful signal, but also on the second and fourth-order statistics of the noise. Second, it derives the asymptotic distribution of the estimated projector matrix obtained from any M -estimate of the covariance matrix for both real (RES) and complex elliptical symmetric (CES) distributed observations. Applied to the MUSIC algorithm for direction-of-arrival (DOA) estimation, these theoretical results allow us to theoretically evaluate the performance loss of this algorithm for heavy-tailed noise distributions when it is based on the SCM, which is significant for weak signal-to-noise ratio (SNR) or closely spaced sources. These results also make it possible to prove that this performance loss can be alleviated by replacing the SCM by an M -estimate of the covariance for CES distributed observations, which has been observed until now only by numerical experiments. [ABSTRACT FROM AUTHOR]

• An extension of the matrix Slepian-Bangs formula to elliptical symmetric distributions under the assumption that the arbitrary density generator depends on unknown parameters is given. • This new matrix Slepian-Bangs formula involves a common structure to the existing one with a simple corrective coefficient. • Closed-form expressions of this coefficient are given for Student's t and generalized Gaussian distributions and are each compared according to different knowledge of the density generator. • This allows us to conclude that for an arbitrary parameterization, the Cramér-Rao bound may be very sensitive to the knowledge of the density generator for super-Gaussian distributions contrary to sub-Gaussian distributions. This paper mainly deals with an extension of the matrix Slepian-Bangs (SB) formula to elliptical symmetric (ES) distributions under the assumption that the arbitrary density generator depends on unknown parameters, aiming to rigorously quantify and understand the impact of this assumption on ES distributed parametric estimation models. This matrix SB formula is derived in a unified way within the framework of real (RES) and circular (C-CES) or noncircular (NC-CES) complex elliptically symmetric distributions, and then compared to the matrix SB formula obtained with fully known or completely unknown density generators. This new matrix SB formula involves a common structure to the existing one with a simple corrective coefficient. Closed-form expressions of this coefficient are given for Student's t and generalized Gaussian distributions and are each compared according to different knowledge of the density generator. This allows us to conclude that for an arbitrary parameterization, the Cramér-Rao bound (CRB) may be very sensitive to the knowledge of the density generator for super-Gaussian distributions contrary to sub-Gaussian distributions. Finally, we prove that for the parametrization with an unknown scale factor, the CRB for the estimation of the other parameters of the scatter matrix does not depend on the type of knowledge of the density generator. This latter result remains true for the specific noisy linear mixture data model where the parameter of interest is characterized by the range space of the mixing matrix. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the problem of testing impropriety (i.e., second-order noncircularity) of a sequence of complex-valued random variables (RVs) based on the generalized likelihood ratio test (GLRT) for Gaussian distributions is considered. Asymptotic (w.r.t. the data length) distributions of the GLR are given under the hypothesis that RVs are proper or improper, and under the true, not necessarily Gaussian distribution of the RVs. The considered RVs are independent but not necessarily identically distributed: assumption which has never been considered until now. This enables us to deal with the practical important situations of noncircular RVs disturbed by residual frequency offsets and additive circular noise. The receiver operating characteristic (ROC) of this test is derived as byproduct, an issue previously overlooked. Finally illustrative examples are presented in order to strengthen the obtained theoretical results. [Copyright &y& Elsevier]

Abstract: In this correspondence we mainly consider the asymptotic distribution of the estimator of circularity coefficients of scalar and multidimensional complex random variables. A particular attention is paid to rectilinear RV. After deriving new properties of the circularity coefficients, the maximum likelihood estimate of the circularity coefficients in the Gaussian case and asymptotic distribution of this estimate for arbitrary distributions are given. Finally, an illustrative example is presented in order to strengthen the obtained theoretical results. [Copyright &y& Elsevier]

• Asymptotic (in the number of measurements) distributions of estimates of the orthogonal projector associated with different M -estimates of the covariance matrix in the context of RES, C-CES, and NC-CES distributed observations whose covariance is low rank structured are given in the same framework. • The asymptotically minimum variance (AMV) subspace-based estimator of the parameter of interest characterized by the column subspace of the mixing matrix for general linear mixtures models, associated with the M -estimates of the covariance matrix is derived. • A common closed-form expression of the AMV bound which can be used as a benchmark against which any subspace-based algorithms are tested is derived. • It is proved that the AMV bound attains the stochastic CRB in the case of ML M -estimate of the covariance matrix for RES, C-CES, and NC-CES distributed observations • We specify the conditions for which the AMV bound based on Tyler's M -estimate attains this stochastic CRB for complex Student t and complex generalized Gaussian distributions. • It is proved that the stochastic CRB is equal to the semiparametric CRB recently introduced for this model. Subspace-based algorithms that exploit the orthogonality between a sample subspace and a parameter-dependent subspace have proved very useful in many applications in signal processing. The purpose of this paper is to complement theoretical results already available on the asymptotic (in the number of measurements) performance of subspace-based estimators derived in the Gaussian context to real elliptical symmetric (RES), circular complex elliptical symmetric (C-CES) and non-circular CES (NC-CES) distributed observations in the same framework. First, the asymptotic distribution of M -estimates of the orthogonal projection matrix is derived from those of the M -estimates of the covariance matrix. This allows us to characterize the asymptotically minimum variance (AMV) estimator based on estimates of orthogonal projectors associated with different M -estimates of the covariance matrix. A closed-form expression is then given for the AMV bound on the parameter of interest characterized by the column subspace of the mixing matrix of general linear mixture models. We also specify the conditions under which the AMV bound based on Tyler's M -estimate attains the stochastic Cramér-Rao bound (CRB) for the complex Student t and complex generalized Gaussian distributions. Finally, we prove that the AMV bound attains the stochastic CRB in the case of maximum likelihood (ML) M -estimate of the covariance matrix for RES, C-CES and NC-CES distributed observations, which is equal to the semiparametric CRB (SCRB) recently introduced. [ABSTRACT FROM AUTHOR]