Exact solutions for source localization problem with minimal squared distance error

Abstract

There were many researches in source localization problem such as relative localization with GPS(Global Positioning System) and target tracking with wireless sensor network. When there is no noise or a little noise, there have been studies about an analytic solution. However, when the noise is not negligible, only the existence of local l(2) minimizing solution and the existence and uniqueness of l(1) minimization are known in particular conditions. This paper demonstrates the exact location of the source, which is the solution of l(1) minimization for squared distance errors with three measurements. It also shows that the number of sources is less than 3, and the nonunique cases with two or three solutions are classified in detail and presented along with some examples. We considered four critical points and their related singular points in the measurement circles. A few numerical implementations for the exact locations of the source are provided and compared with the approximated level set using many measurement grid points. (c) 2022 Published by Elsevier Inc.