Cramér–Rao Lower Bound Analysis for Elliptic Localization With Random Sensor Placements

Elliptic localization (EL) based on time-sum-of-arrival (TSOA) measurements has become popular due to its widespread applications in wireless sensor networks (WSNs) and distributed radar systems. While the performance limit of EL characterized by the Cramér–Rao lower bound (CRLB) has been thoroughly studied in literature when the sensor [transmitter and receiver (Rx)] positions are modeled as fixed deterministic quantities, the bound in the random network scenario has not been studied. This article introduces a methodology to investigate the TSOA-based localization performance, for the scenario where the sensors are randomly placed having their positions modeled by random parameters with their probability density functions specified. A tractable expression of the metric that approximates the CRLB and its distribution is analytically derived to characterize the fundamental limits of TSOA-based localization, which can be applied to both conventional WSNs and the special case in which the Rxs form a uniform linear array. Simulation results validate the theoretical development and demonstrate how the performance of EL is affected by the randomness of the sensor positions and different network parameters.