Coexistence of Near-Field and Far-Field Sources: the Angular Resolution Limit

TL;DR

This paper derives a closed-form analytical expression for the Angular Resolution Limit in mixed near-field and far-field source scenarios, providing insights into the fundamental resolution limits in passive source localization.

Contribution

It introduces a novel analytical derivation of the ARL for combined near-field and far-field sources using Smith's equation and CRB, applicable under low noise conditions.

Findings

Closed-form expression for the CRB in mixed NF and FF sources

Analytic ARL derived as a 4th-order polynomial under small separation

Qualitative insights into ARL behavior in realistic scenarios

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 Cram\'er-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.