Coexistence of near-field and far-field sources: the angular resolution limit
Résumé
Passive source localization is a well known inverse problem in which we convert the observed measurements into information about the direction of arrivals. In this paper we focus on the optimal resolution of such problem. More precisely, we propose in this contribution to derive and analyze the Angular Resolution Limit (ARL) for the scenario of mixed Near-Field (NF) and Far-Field (FF) Sources. This scenario is relevant to some realistic situations. We base our analysis on the Smith's equation which involves the Cramér-Rao Bound (CRB). This equation provides the theoretical ARL which is independent of a specific estimator. Our methodology is the following: first, we derive a closed-form expression of the CRB for the considered problem. Using these expressions, we can rewrite the Smith's equation as a 4-th order polynomial by assuming a small separation of the sources. Finally, we derive in closed-form the analytic ARL under or not the assumption of low noise variance. The obtained expression is compact and can provide useful qualitative informations on the behavior of the ARL. 1. Introduction Very few works are related to the study of the realistic situation where there exists coexisting far-field (FF) and near-field (NF) sources [2] such as speaker localization using microphone arrays and guidance (homing) systems. At the contrary, we can find a plethora of contributions on the localization of far-field sources [1]. More recently, the problem of localization of near-field sources has been tackled in reference [4, 5] for instance. In the context of the problem of source localization, one can see three contributions: (1) propose new efficient algorithms/estimators [1], (2) study the estimation performance independently of a specific algorithm thanks to the lower bound on the Mean Square error (MSE) [3, 5] and (3) derive and study the theoretical resolution, i.e., the minimal angular distance to resolve/discriminate two closely spaced emitted signals in terms of their direction of arrivals. Our contribution belongs to the third point. More precisely, based on the Smith's equation [6, 8, 9] which involves the Cramér-Rao Bound (CRB) [3], we derive and analyze the Angular Resolution Limit (ARL) for the realistic scenario where we have two sources, one located in the far-field of the array and another in the near-field of the array.
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