TDOA--based localization in two dimensions: the bifurcation curve

In this paper, we complete the study of the geometry of the TDOA map that encodes the noiseless model for the localization of a source from the range differences between three receivers in a plane, by computing the Cartesian equation of the bifurcation curve in terms of the positions of the receivers. From that equation, we can compute its real asymptotic lines. The present manuscript completes the analysis of [Inverse Problems, Vol. 30, Number 3, Pages 035004]. Our result is useful to check if a source belongs or is closed to the bifurcation curve, where the localization in a noisy scenario is ambiguous.Comment: 11 pages, 3 figures, to appear in Fundamenta Informatica

The algebro-geometric study of range maps

Localizing a radiant source is a widespread problem to many scientific and technological research areas. E.g. localization based on range measurements stays at the core of technologies like radar, sonar and wireless sensors networks. In this manuscript we study in depth the model for source localization based on range measurements obtained from the source signal, from the point of view of algebraic geometry. In the case of three receivers, we find unexpected connections between this problem and the geometry of Kummer's and Cayley's surfaces. Our work gives new insights also on the localization based on range differences.Comment: 38 pages, 18 figure

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

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.Comment: 51 pages (3 appendices of 12 pages), 12 figure

Tail universalities in rank distributions as an algebraic problem: the beta-like function

Although power laws of the Zipf type have been used by many workers to fit rank distributions in different fields like in economy, geophysics, genetics, soft-matter, networks etc., these fits usually fail at the tails. Some distributions have been proposed to solve the problem, but unfortunately they do not fit at the same time both ending tails. We show that many different data in rank laws, like in granular materials, codons, author impact in scientific journal, etc. are very well fitted by a beta-like function. Then we propose that such universality is due to the fact that a system made from many subsystems or choices, imply stretched exponential frequency-rank functions which qualitatively and quantitatively can be fitted with the proposed beta-like function distribution in the limit of many random variables. We prove this by transforming the problem into an algebraic one: finding the rank of successive products of a given set of numbers