Abstract
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.
Highlights
Very few works are related to the study of the realistic situation where there exists coexisting farfield (FF) and near-field (NF) sources [2] such as speaker localization using microphone arrays and guidance systems
Analytical expression of the Cramer-Rao Bound The Cramer-Rao Bound (CRB) verifies the covariance inequality principle [3]. This bound is largely used in the signal processing community since it gives the best performance in term of Mean Square Error (MSE) at high Signal to Noise Ratio (SNR)
Analytic solutions of R(x): The resolution of a 4-th order polynomial is not straightforward but we propose a solution to this problem
Summary
Very few works are related to the study of the realistic situation where there exists coexisting farfield (FF) and near-field (NF) sources [2] such as speaker localization using microphone arrays and guidance (homing) systems. 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. Based on the Smith’s equation [6, 8, 9] which involves the Cramer-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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