Sparsity-Aware Multi-Source TDOA Localization

The problem of source localization from time-difference-of-arrival (TDOA) measurements is in general a non-convex and complex problem due to its hyperbolic nature. This problem becomes even more complicated for the case of multi-source localization where TDOAs should be assigned to their respective sources. We simplify this problem to an $\ell_1$-norm minimization by introducing a novel TDOA fingerprinting and grid design model for a multi-source scenario. Moreover, we propose an innovative trick to enhance the performance of our proposed fingerprinting model in terms of the number of identifiable sources. An interesting by-product of this enhanced model is that under some conditions we can convert the given underdetermined problem to an overdetermined one that could be solved using classical least squares (LS). Finally, we also tackle the problem of off-grid source localization as a case of grid mismatch. Our extensive simulation results illustrate a good performance for the introduced TDOA fingerprinting paradigm as well as a significant detection gain for the enhanced model.

I. Introduction

Determining the position of multiple sources in a two-dimensional or three-dimensional (2-D or 3-D) space is a fundamental problem which has received an upsurge of attention recently [1]. Many different approaches have been proposed in literature to recover the source locations based on time-of-arrival (ToA), time-difference-of-arrival (TDOA) or received-signal-strength (RSS) measurements between the source nodes (SNs) and some fixed receivers or access points (APs). A traditional wisdom in RSS-based localization tries to extract distance information from the RSS measurements. However, this approach fails to provide accurate location estimates due to the complexity and unpredictability of the wireless channel. This has motivated another category of RSS-based positioning, the so-called location fingerprinting, which discretizes the physical 2-D or 3-D space into grid points (GPs) and creates a map representing the space by assigning to every GP a set of location-dependent RSS parameters, one for every AP. The location of the source is then estimated by comparing real-time measurements with the fingerprinting map at APs, for instance using K-nearest neighbors (KNN) [2] or Bayesian classification (BC) [3].