Weighted Multidimensional Scaling Localization Method With Bias Reduction Based on TOA

In this article, the source localization problem by utilizing time-of-arrival measurements is addressed. The weighted multidimensional scaling (MDS) is an efficient approach and superior in antinoise capability. However, the weighting matrix is lack of accuracy due to the ignoring of second-order noise terms. Meanwhile, according to the Gauss–Markov theorem, the least-squares solution is not optimum when the system matrix in solving contains noise. Attributing to these drawbacks, a considerable bias occurred in the presence of high noise level as well as poor coordinate geometry. The aim of this article is to enhance the performance of weighted MDS method. It starts through deriving a novel weighting matrix, which considers second-order noise terms for a more precise estimation. Second, the expectation bias of weighted MDS is analyzed and determined. To suppress this bias, a straightforward approach is proposed by subtracting the expectation bias from the weighted MDS result. Finally, two alternative methods are proposed by following the bias reduction scheme while differing in solving approach to balance time consumption and estimation performance. All of these proposed methods can reduce the bias to some degree and achieve the Cramér–Rao lower bound accuracy according to theoretical performance analysis and numerical simulation results.