A comprehensive analysis of the geometry of TDOA maps in localization problems

In this paper we consider the well-established problem of time differences of arrival (TDOA)-based source localization and propose a comprehensive analysis of its solution for arbitrary sensor measurement and placement. More specifically, we define the TDOA map from the physical space of source locations to the space of range measurements (TDOAs), in the specific case of three receivers in 2D space. We then study the identifiability of the model, giving a complete analytical characterization of the image of this map and its invertibility. This analysis has been conducted in a completely mathematical fashion, using many different tools which make it valid for every sensor configuration. These results are the first step toward the solution of more general problems involving, for example, a larger number of sensors, uncertainty in their placement, or lack of synchronization.