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Closed-form solution for TDOA-based joint source and sensor localization in two-dimensional space

In this paper, we propose a closed-form solution for time-difference-of-arrival (TDOA) based joint source and sensor localization in two-dimensional space (2D). This closed-form solution is a combination of two closed-form solutions for time-of-arrival information recovery and time-of-arrival (TOA)-based joint source and sensor localization in 2D. In our previous works, we derived closed-form solutions for TOA-based joint source and sensor localization and near-closed-form solutions for TOA information recovery in three-dimensional space (3D). Since the localization in 2D is simpler than that in 3D, closed-form solutions for both problems in 2D are derived in this paper. The root-mean-square errors (RMSEs) achieved by the proposed closed-form solution are compared with the Cramér-Rao lower bound (CRLB) in synthetic experiments. The results show that the proposed solution works well in both low-noise and noisy cases and with both small and large numbers of sources and sensors.