Closed-Form and Near Closed-Form Solutions for TDOA-Based Joint Source and Sensor Localization

In this paper, we derive closed-form and near closed-form solutions for joint source and sensor localization from time-difference-of-arrival (TDOA) measurements. In our previous works, we derived closed-form and near closed-form solutions for joint source and sensor localization from time-of-arrival (TOA) measurements. On the basis of these results, the main idea in this paper is to recover the TOA information only from the given TDOA measurements. We show that the TOA information can be recovered by using the low-rank property of the difference of square TOA-distance matrix in a closed-form or a near closed-form based on the linear method of solving polynomial equations. Since the low-rank property is reliable even in noisy cases, the TOA recovery works well under both small and large amounts of noise. The root-mean-squared errors achieved by our proposed algorithms are compared with the Cramer-Rao lower bound in synthetic experiments. The results show that the proposed methods work well for both small and large amounts of noise and for small and large numbers of sources and sensors.