### A closed-form solution for TDOA source localization with calibration emitter in the presence of partial synchronization clock biases and sensor location uncertainties

• Received:2022-09-27 Revised:2023-01-13 Online:2023-01-18 Published:2023-01-18

Abstract: It is known that the synchronization clock biases and sensor location uncertainties can significantly degrade the target localization accuracy based on time difference of arrival (TDOA) measurements. In order to effectively restrain the influence of the two model errors, this paper proposes a new TDOA target localization method using calibration emitter for the partially synchronous model. Aiming at the highly nonlinear characteristic of the TDOA measurement model, a closed-form positioning method based on two stages (called stage-A and stage-B) is proposed, which can achieve the decoupling estimation of the unknowns and avoid iterative operation. Specifically, stage-A only uses the TDOA measurements from the calibration emitter and exploits the idea of linear elimination. The closed-form solutions for the sensor position and synchronization clock bias are obtained sequentially by utilizing the linear Bayesian estimator and weighted least squares (WLS) estimator, respectively. In stage-B, the TDOA measurements from the target are combined with the estimation results obtained in stage-A, and a set of new pseudo-linear observation equations is derived. Afterwards, the closed-form solution for the target location is produced by using the WLS estimator and Lagrange multiplier method. Finally, a more accurate joint closed-form solution for sensor position and synchronization clock bias is given based on the linear Bayesian estimator. Additionally, the relevant Cramér-Rao bound (CRB) is derived, and the asymptotic optimality of the proposed estimator is proved by mathematical analysis. Simulation results corroborate the validity of the theoretical development and illustrate the superiority of the new method.