Elliptic localization of multiple objects without position and synchronization of the transmitter

Multi-object elliptic localization is a challenging problem, considering that the association between the indirect path measurements and the objects is not known, and the transmitter is un-coordinated without synchronization with the receivers. Realizing that the direct path measurements do not have the data association issue, we propose a two-stage estimation method for this problem. In the first stage, a minimum measurement solution (MMS) method is developed to obtain a coarse estimate of the transmitter position and range offset coming from the unsynchronized transmitter and receivers, and the best linear unbiased estimator (BLUE) follows to improve the coarse estimate. In the second stage, we first propose a data association method based on the MMS for elliptic localization with known transmitter position, and then propose a BLUE for estimating the object positions using the indirect path measurements and the transmitter position and range offset estimate. The proposed method does not require any prior information about the object positions. Moreover, it has closed-form solutions in both stages, and thus is very computationally efficient. Mean square error analysis is conducted to show that the performance achieves the Cramer-Rao lower bound under correct data association and uncorrelated noise between the direct and indirect paths. Simulation results validate the good performance of the proposed method.