In this paper, we develop a novel robust optimization approach to source localization using time-difference-of-arrival (TDOA) measurements that are collected under non-line-of-sight (NLOS) conditions. A key feature of our approach is that it does not require knowledge of the distribution or statistics of the NLOS errors, which are often difficult to obtain in practice. Instead, it only assumes that the NLOS errors have bounded supports. Based on this assumption, we formulate the TDOA-based source localization problem as a robust least squares (RLS) problem, in which a location estimate that is robust against the NLOS errors is sought. Since the RLS problem is non-convex, we propose two efficiently implementable convex relaxation-based approximation methods to tackle it. We then conduct a thorough theoretical analysis of the approximation quality and computational complexity of these two methods. In particular, we establish conditions under which they will yield a unique localization of the source. Simulation results on both synthetic and real data show that the performance of our approach under various NLOS settings is very stable and is significantly better than that of several existing non-robust approaches.

In this correspondence, the time-of-arrival based localization problem under mixed line-of-sight/non-line-of-sight (LOS/NLOS) conditions is addressed. Previous studies show that existing robust methods perform well in dense NLOS environments, but generally perform badly in sparse NLOS environments. To alleviate this problem, we introduce a “balancing parameter” related to the NLOS errors and formulate a new robust weighted least squares (RWLS) problem with the source position and the NLOS balancing parameter as the estimation variables. The proposed method does not require the statistics of NLOS errors and the path status. By leveraging the S-Lemma, the RWLS problem is transformed into a nonconvex optimization problem, which is then relaxed into a convex semidefinite program. Simulation results show that the proposed method works well for both the sparse and dense NLOS environments.

Sparse linear arrays (SLAs) can be designed in a systematic way, with the ability for underdetermined DOA estimation where a greater number of sources can be detected than that of sensors. In this paper, as the first stage, a new systematic design named multi-subarray dilated nested array (MDNA), whose difference co-array (DCA) can be proved to be hole-free, is firstly proposed by introducing a sparse ULA and multiple identical dense ULAs with appropriate sub-ULA spacings. The MDNA will degenerate into the nested array under specific conditions, and the uniform degrees of freedom (uDOFs) of MDNA is larger than that of its parent nested array. On the basis of MDNA, to reduce the mutual coupling effect, an augmented multi-subarray dilated nested array (AMDNA) is constructed by migrating some elements of the dense segments of MDNA, without reducing the number of uDOFs. Several theoretical properties of the proposed array structures are proved, and simulation results are provided to demonstrate the effectiveness and superiority of the proposed AMDNA over some existing sparse arrays.

In a large-scale wireless sensor network, it is typically impossible to use all the sensors to localize a source due to the limited communication range and battery power of the sensor nodes, thereby a sensor selection procedure is required. In this paper, the sensor selection problem for time-difference-of-arrival (TDOA) based localization is addressed. We propose two nearly optimal sensor selection methods: one is based on the original TDOA model and the other is based on the transformed time-of-arrival (TOA) model. Both methods aim to minimize the trace of the Cram\'{e}r-Rao lower bound (CRLB) under the condition of fixed number of selected sensors. For the first method, we introduce a special vector to distinguish the reference sensor with the non-reference sensors. We then formulate a non-convex optimization problem with a Boolean vector as the optimization variable. For the second method, we first prove that the CRLB for TDOA based localization is exactly the same with that for TOA based localization where the start transmission time is unknown. We then formulate a non-convex optimization problem to minimize the CRLB for TOA based localization. Both non-convex optimization problems are relaxed as convex semidefinite programs (SDPs) by applying semidefinite relaxation, followed by the Gaussian randomization procedure to improve the quality of the SDP solutions if necessary. Simulation results show that the performance of proposed methods approaches that of the exhaustive search method.