Efficiency of subspace-based estimators for elliptical symmetric distributions

Abstract

Subspace-based algorithms that exploit the orthogonality between a sample subspace and a parameterdependent 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 Mestimates 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.

Resolving power of MUSIC-like algorithms for circular or rectilinear correlated sources in CES data models

“…Existence, uniqueness of the solution of (10) and convergence in probability of the sequence Γ y,T to R y have been proved in [28]. These properties have been extended in [29] to the sequence Γ ỹ,T which converges in probability to R ỹ.…”

“…However, these ML estimates cannot be obtained for arbitrary distributed x t and arbitrary C-CES distributed n t in (2). To overcome this difficulty, we consider here an alternative model used in [31], [29] and [33], where the observations y t in (2) are CES distributed. In this case the distributions of x t and n t are not specified, but only their second-order statistics are imposed.…”

“…It is a distribution-free estimator within the family of C-CES/NC-CES distributions. However, it has been proved for real elliptically symmetric (RES) distributions in [36] and for C-CES distributions in [37], then extended to NC-CES distributions in [29], that after normalizing, the solutions Γ u y,T and Γ u ỹ,T of ( 10) also converges in probability to σ u R y and σ u R ỹ, respectively with σ u = 1.…”

“…where Q x , Q s , Q n and Q ñ are the quadrivariance 3 of x t , s t , n t and n t , respectively. For C-CES distributed noise Q n and Q ñ are given in [31] and [29], respectively by…”

Low complexity sparse beamspace DOA estimation via single measurement vectors for uniform circular array

“…We compare the RMSEs of the proposed method, other methods and the stochastic Cramér-Rao lower bound (CRLB) [30], in which the number of sensors is 13 and the radius of the UCA is 0.5λ. Figure 5 shows the RMSEs of different methods under different SNR conditions.…”