Conditional and Unconditional Deterministic Bounds on the MSE of the Non-Uniform Linear Co-centered Orthogonal Loop and Dipole Array
Résumé
The co-centered orthogonal loop and dipole (COLD) array exhibits some interesting properties, which makes it ubiquitous in the context of polarized source localization. In the literature, one can find a plethora of estimation schemes adapted to the COLD array. Nevertheless, their ultimate performance in terms the so-called threshold region of mean square error (MSE), have not been fully investigated. In order to fill this lack, we focus, in this fast communication, on conditional and unconditional bounds that are tighter than the well known Cramér-Rao Bound (CRB). More precisely, we give some closed form expressions of the McAulay-Hofstetter, the Hammersley-Chapman-Robbins, the McAulay-Seidman bounds and the recent Todros-Tabrikian bound, for both the conditional and unconditional observation model. Finally, numerical examples are provided to corroborate the theoretical analysis and to reveal a number of insightful properties. II. INDEX TERMS Deterministic lower bounds, co-centered orthogonal loop and dipole array, mean square error, performance analysis, passive source localization, SNR threshold. III. INTRODUCTION Nowadays, the recent source localization systems need to operate in increasingly more crowded signal environments [1]. In this context, taking into account both the polarization diversity and the spatial diversity became ubiquitous in antenna array systems and their processing as wireless communication, radar, sonar systems etc. [1]-[3]. Among different types of polarization sensitive arrays, the co-centered orthogonal loop and dipole array is commonly used since it exhibits numerous interesting properties [4]-[6] (e.g., the constant norm of the polarization vector, the insensibility of the polarization vector w.r.t. the source localization in the plan of the antenna etc.) In the literature, one can find a plethora of estimation schemes adapted and/or designed particularly for the COLD array [5]. Nevertheless, their ultimate performance in terms of the mean square error (MSE), especially in the non-asymptotic region (meaning for low signal-to-noise ratio (SNR) or low observations), have not been fully investigated. We can cite [7], [8], in which the authors derived closed form expression of the approximated Cramér-Rao bound (CRB) for a sufficient large number of sensors in the context of a COLD linear and uniform array. Whereas in [9], [10] the authors derived, respectively, expressions of the CRB for a known single source and the resolution limit for two known sources, both for known polarization state parameters. Nevertheless, to the best of our knowledge, no results concerning the breakdown prediction for the COLD linea array (possible non-uniform) can be found in the literature. To fill this lack, we focus, in this fast communication, on lower bounds that are tighter than the CRB. More precisely, we give some closed form expressions of the McAulay-Hofstetter (MCB), the Hammersley-Chapman-Robbins (HCRB), the McAulay-Seidman (MSB) bounds and a recently proposed Todros-Tabrikian bound (TTB), for both the commonly assumed conditional (i.e., when the signals are assumed to be deterministic) and unconditional (i.e., when the signals are assumed to be driven by a Gaussian random process) observation models with unknown direction of arrival (DOA) and unknown polarization state parameters. Such bounds are known to be efficient to delimit and predict the optimal operating zone of estimators [11], [12] which is given by the threshold or breakdown point, i.e., when the estimator's MSE increases dramatically. Such deterministic lower bounds can be derived using one of the unifications given in [11], [13]-[15]. In this paper, we adopt the Todros and Tabrikian unification in which they propose a novel class of performance lower bounds by applying a proper integral transform [14]. Using an adequate choice of the kernel of the integral transform of the likelihood-ratio function, one obtain some well known lower bounds as the MCB, HCRB, MSB and TTB. Tao BAO is also with the
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