Elliptic Localization With Unknown Transmitter Positions and Imperfect Synchronizations

In this article, we tackle the challenging elliptic localization problem, in which the transmitters emit signals that are reflected by the object and then received by the receivers. For practical consideration, we consider a scenario where the transmitter positions are unknown, the transmitter-receiver synchronization is absent, and the receiver positions are imperfectly known but have random position errors. The idea of joint estimation of the object and transmitter positions, and the time offsets of the transmitters is proposed. To achieve this, we begin by modifying the time delay (TD) measurement models in both direct and indirect paths. Subsequently, we develop a constrained weighted least-squares (CWLS) problem based on these modified models, with the correlations of errors taken into consideration. The nonconvex nature of the CWLS problem poses a significant challenge. To tackle this issue, we employ the semidefinite relaxation (SDR) technique, which transforms the CWLS problem into a more manageable semidefinite programming (SDP) problem. Furthermore, we conduct the mean square error (mse) analysis, revealing that our CWLS solution closely approaches the Cramer-Rao lower bound (CRLB), especially in scenarios with low noise levels. Through simulation results, we validate the mse analysis and confirm the superior performance of our proposed method as compared to existing approaches.