A comprehensive analysis of the geometry of TDOA maps in localisation problems
This foundational analysis addresses source localization for applications like sensor networks, though it is incremental as it builds on well-established TDOA problems.
The paper tackles the problem of TDOA-based source localization by analyzing the geometry of TDOA maps for three receivers in 2D space, providing a complete analytical characterization of the map's image and invertibility.
In this manuscript we consider the well-established problem of TDOA-based source localization and propose a comprehensive analysis of its solutions for arbitrary sensor measurements and placements. 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 towards the solution of more general problems involving, for example, a larger number of sensors, uncertainty in their placement, or lack of synchronization.