ToA-based Localization of Far-Away Targets: Equi-DOP Surfaces, Asymptotic Bounds, and Dimension Adaptation

This paper studies the Dilution of Precision (DOP) in the Time-of-arrival (ToA)-based localization of targets outside the anchors convex hull. In the far-away target regime, we derive a closed-form expression of the DOP that reveals a linear asymptotic scaling law. We characterize the asymptotic DOP bounds, equi-DOP surfaces/contours in 3D/2D localization scenarios, which quantifies the reliability of location estimates on a trajectory. Motivated by vehicular applications, we propose a range-aided dimension adaptation scheme. Here the localization dimension is adapted in real-time using a single range measurement such that the maximum or root-mean-square DOP does not exceed a threshold. Since high-accuracy localization of far-away targets is infeasible due to linear DOP scaling with distance, this scheme prioritizes high-performance tracking of nearby targets while monitoring far-away targets with range-only measurements.