Convex Relaxation Methods for Unified Near-Field and Far-Field TDOA-Based Localization

This paper develops two convex relaxation solutions for the unified localization of a signal source using time difference of arrival measurements, regardless of whether the source is in the near field for coordinate positioning or in the far field for the direction of arrival estimation. The previous study on unified estimation only derived an iterative solution, which is sensitive to initialization. Albeit a coarse initialization was supplied to start the iteration, it may not be sufficient to ensure convergence to the global solution especially when the source is close to the sensors. The proposed solutions come from two novel formulations for optimization, one using the weighted least squares with the modified polar representation of the source position as variable and the other applying fractional programming with the Cartesian coordinate representation instead. Both the optimization problems are solved by first performing semidefinite relaxation and then tightening the relaxed problem by including a set of second-order cone constraints. The two formulations are created from different approaches. Nevertheless, we are able to prove that both the formulations reduce to solving exactly the same mixed semidefinite/second-order cone program and thus establish their equivalence. Furthermore, the proposed solution method is extended to the more practical scenario when sensor position errors are present. The results from both the simulated and real experiments show that the proposed method achieves almost the same performance of the iterative maximum likelihood estimator under ideal initialization.