Search

• 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]

• This paper derives interpretable unified closed-form expressions for the threshold ASNR along the Cox and the Sharman and Durrani criteria associated with the conventional MUSIC and NC MUSIC algorithms in the context of arbitrary circular or rectilinear distributed correlated sources and circular CES distributed noise. • Using these expressions, we investigate the impact of the non-Gaussianity of the noise, as well as of the phase and magnitude of the correlation of the sources. • In particular, we prove for the first time that the phase of the correlation, which is the non-circularity phase separation for rectilinear sources, may have a strong impact on the resolution and a zero phase may lead to overly optimistic resolution. • We quantify the resolution benefit provided when the SCM is replaced by M - estimates of the covariance matrix for CES observations. The concept of threshold array signal-to-noise ratio (ASNR) which is defined as the minimal SNR at which specific high-resolution algorithms are able to resolve two closely spaced far-field sources, allows to quantify and to compare sensors array performance in localizing remote targets. This paper generalizes and extends the expressions of the threshold ASNR given in the literature for the conventional and non-circular (NC) MUSIC direction-of-arrival (DOA) estimation algorithms in the context of uncorrelated stochastic circular or rectilinear Gaussian sources and circular complex Gaussian (C-CG) noise, in a more general stochastic framework. We assume that the sources are correlated with an arbitrary distribution, which is inherent in a context of multipath or smart jammers, and that the noise is circular complex elliptically symmetric (C-CES) distributed, which can model impulsive noise with heavy-tailed distributions. The C-CES and NC-CES distributed observations are also considered to quantify the gain in resolution provided when the sample covariance matrix (SCM) of the observations is replaced by M -estimates of this matrix. Asymptotic approaches and perturbation analysis have been performed to derive closed-form expressions of the mean null spectra of the two considered MUSIC algorithms for both observation models, which allow us to derive, for the first time, general unified explicit analytical expressions of the threshold ASNR along the Cox and the Sharman and Durrani criteria. These expressions allow us to quantify the impact of the non-Gaussianity of noise and observations, as well as of the phase and magnitude of the correlation on the resolution threshold, and to quantify the benefit provided when the SCM is replaced by M -estimates of this covariance matrix for CES observations. Finally, numerical illustrations are included to support our theoretical analysis. [ABSTRACT FROM AUTHOR]

• This paper has shown that all the NC subspace-based algorithms built from the SCM designed for uncorrelated rectilinear sources embedded in spatially white CCG noise can be also applied for correlated rectilinear sources in the contexts of SCM estimate with C-CES noise and M-estimate with NC-CES observations. • A perturbation analysis has been performed to derive closed-form expressions for the asymptotic covariance matrices of DOA estimates for three NC MUSIC-like algorithms in two CES data models. • Interpretable closed-form expressions of the asymptotic variance of the estimated DOA of two equi-power correlated sources has been derived for the first time. • A number of properties that highlight how the asymptotic variances of NC MUSIClike DOA estimation algorithms depend on key parameters such as SNR, DOA and phase separations, correlation factor and C-CES noise parameters were derived. • Analytical robustness results were illustrated proving that the use of robust Mestimators enhances the robustness of the subspace-based DOA estimation algorithms against heavy-tailed C-CES observations model deviations, with negligibleloss in performance for C-CG distributed observations. This paper focuses on a theoretical performance analysis of subspace-based algorithms for the localization of spatially correlated rectilinear sources embedded in circular complex elliptically symmetric (C-CES) distributed noise model and also when the observations are non-circular CES (NC-CES) distributed with dependent scatter matrices on the direction of arrival (DOA) parameters. A perturbation analysis has been performed to derive closed-form expressions for the asymptotic covariance matrices of DOA estimates for non-circular subspace-based algorithms in two CES data models. Robustness of subspace-based algorithms is theoretical evaluated using robust covariance matrix estimators (instead of the sample covariance matrix (SCM)). We prove, for the first time, interpretable closed-form expressions of the asymptotic variance of the estimated DOA of two equi-power correlated sources, which allows us to derive a number of properties describing the DOA variance's dependence on signals parameters and non-Gaussian distribution of the noise. Different robustness properties are theoretically analyzed. In particular, we prove in the framework of NC-CES distributed observations, that Tyler's M -estimator enhances the performance for heavy-tailed distributions w.r.t. the SCM, with negligible loss in performance for circular Gaussian distributed observations. Finally, some Monte Carlo illustrations are given for quantifying this robustness and specifying the domain of validity of our theoretical asymptotic results. [ABSTRACT FROM AUTHOR]

• 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]