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.

I. Introduction

Source localization has become an indispensable part of our modern life, including, but not limited to, daily commute, express delivery services, and even national security [1], [2], [3]. Particularly for military purpose, target position is critical intelligence information, which plays an important role in triumphing combat. Compared with the localization system based on assistance from source, an independent localization method [4] is more simple in structure and common adopted in sectors ranging from industry to military. Nevertheless, its accuracy suffers a lot from complex environment, which significantly reduces the received signal strength (RSS), such as poor weather from heavy wind to thunderstorms, intricate topography, and some kind of jamming technique. Therefore, how to strengthen the noise resilience has drawn considerable attention and has been researched in much literature.