A Statistical Characterization of Localization Performance in Wireless Networks

TL;DR

This paper derives an analytical distribution for the Cramer-Rao lower bound in wireless localization, considering random anchor and target positions, providing a comprehensive statistical characterization of localization performance.

Contribution

It introduces a novel analytical framework for the distribution of the CRLB in Time-of-Arrival localization, accounting for random anchor and target locations.

Findings

Derived a closed-form expression for the CRLB distribution conditioned on anchor count.

Established a connection between internodal angles and localization accuracy.

Provided a network-wide statistical characterization of localization error.

Abstract

Localization performance in wireless networks has been traditionally benchmarked using the Cramer-Rao lower bound (CRLB), given a fixed geometry of anchor nodes and a target. However, by endowing the target and anchor locations with distributions, this paper recasts this traditional, scalar benchmark as a random variable. The goal of this work is to derive an analytical expression for the distribution of this now random CRLB, in the context of Time-of-Arrival-based positioning. To derive this distribution, this work first analyzes how the CRLB is affected by the order statistics of the angles between consecutive participating anchors (i.e., internodal angles). This analysis reveals an intimate connection between the second largest internodal angle and the CRLB, which leads to an accurate approximation of the CRLB. Using this approximation, a closed-form expression for the distribution of the CRLB, conditioned on the number of participating anchors, is obtained. Next, this conditioning is eliminated to derive an analytical expression for the marginal CRLB distribution. Since this marginal distribution accounts for all target and anchor positions, across all numbers of participating anchors, it therefore statistically characterizes localization error throughout an entire wireless network. This paper concludes with a comprehensive analysis of this new network-wide-CRLB paradigm.