New Semidefinite Programming Joint Localization and Synchronization Using Sequential One-Way TOAs and Doppler Shifts

In a time-division (TD) broadcast localization and synchronization (TDBLAS) system, moving user nodes (UNs) with the clock offset and the clock drift usually use the sequential one-way time-of-arrival (TOA) and Doppler shift measurements from the anchor nodes (ANs) to resolve the joint localization and synchronization (JLAS) problem in the presence of the AN position error. The objective function of the maximum likelihood (ML) method to solve the above JLAS problem in a TDBLAS system is high-dimensional nonlinear and nonconvex. Thus, the existing iterative method for solving the above ML estimation problem usually encounters issues like local minima or nonconvergence when an accurate initial guess is missing. In this article, we propose a new semidefinite programming (SDP) method to address this issue, which can guarantee the global optimal solution without requiring initialization. We introduce the optimization variable and formulate a nonconvex-constrained weighted least-squares (CWLS) minimization problem. Subsequently, we propose a novel semidefinite relaxation (SDR) approach to relax the complex and nonconvex CWLS problem into a convex and tractable SDP problem. Through theoretical estimation error analysis, we demonstrate that the CWLS solution can achieve the Cramér-Rao lower bound (CRLB) under small Gaussian noise. Simulation results in a 2-D scenario show that the proposed SDP method reaches the CRLB under small Gaussian noise. Compared to the conventional iterative method, the proposed SDP method exhibits greater robustness, achieving the global optima without requiring initialization under small Gaussian noise.