Unified Near-Field and Far-Field TDOA Source Localization Without the Knowledge of Signal Propagation Speed

We address the unified near-field and far-field time-difference-of-arrival source localization problem, in which the signal propagation speed is unknown, and the prior knowledge that whether the source is in the near-field or far-field is unavailable. Using the unified near-field and far-field model obtained by expressing the source position in the modified polar representation (MPR), two different methods, the semidefinite relaxation (SDR) method that achieves noise resilience and the two-step closed-from solution method that accomplishes low complexity, are proposed to jointly estimate the MPR coordinates and the propagation speed. For the SDR method, we express the propagation speed as the sum of a pre-selected constant and the residual to form a non-convex constrained weighted least squares problem, which is then relaxed into a convex semidefinite program by applying SDR. The two-step method obtains a coarse estimate in Step 1 by solving the quadratic programming problem without considering the relations among the variables, and then refines the coarse estimate in Step 2 by estimating the correction to compensate the error resulting from ignoring the relations. Theoretical mean square error analysis confirms that the proposed methods can reach the Cramer-Rao Lower Bound performance, which is also validated by using both simulated and real data.