In this paper, we analyze the performance limit of sensor localization from a novel perspective. We consider distance-based single-hop sensor localization with noisy distance measurements by time of arrival (TOA). Differently from the existing studies, the anchors are assumed to be randomly deployed, with the result that the trace of the associated Cramer-Rao Lower Bound (CRLB) matrix becomes a random variable. We adopt this random variable as a scalar metric for the performance limit and then focus on its statistical attributes. By the Central Limit Theorems for U-statistics, we show that as the number of anchors goes to infinity, this scalar metric converges to a random variable which is an affine transformation of a chi-square random variable of degree 2. In addition, we provide the quantitative relationship among the mean, the standard deviation, the number of anchors, parameters of communication channels and the distribution of the anchors. Extensive simulations are carried out to confirm the theoretical results. On the one hand, our study reveals some fundamental features of sensor localization; on the other hand, the conclusions we draw can in turn guide us in the design of wireless sensor networks.
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