Localization performance in wireless networks has been traditionallybenchmarked using the Cramer-Rao lower bound (CRLB), given a fixed geometry ofanchor nodes and a target. However, by endowing the target and anchor locationswith distributions, this paper recasts this traditional, scalar benchmark as arandom variable. The goal of this work is to derive an analytical expressionfor the distribution of this now random CRLB, in the context ofTime-of-Arrival-based positioning. To derive this distribution, this work first analyzes how the CRLB isaffected by the order statistics of the angles between consecutiveparticipating anchors (i.e., internodal angles). This analysis reveals anintimate 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 thenumber of participating anchors, is obtained. Next, this conditioning is eliminated to derive an analytical expression forthe marginal CRLB distribution. Since this marginal distribution accounts forall target and anchor positions, across all numbers of participating anchors,it therefore statistically characterizes localization error throughout anentire wireless network. This paper concludes with a comprehensive analysis ofthis new network-wide-CRLB paradigm.
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