Minimal Solvers for Unsynchronized TDOA Sensor Network Calibration using Far Field Approximation

Calibration of network nodes using only time differences of arrival (TDOA) measurements opens up for interesting applications in wireless ad-hoc sensor networks, e.g. finding the positions of cell phones by only ambient sounds or radio. We present two novel approaches for the problem of self-calibration of network nodes using only TDOA when both receivers and transmitters are unsynchronized. We consider the previously unsolved minimum problem of far field localization in three dimensions, which is to locate four receivers by the signals of nine unknown transmitters, for which we assume that they originate from far away. The first approach, the Ellipsoid TDOA method, is a geometric representation based on the fact that the time differences between four receivers characterize an ellipsoid. We calculate by linear least-squares regression the ellipsoid from the observed measurements of nine or more transmitters, by which the constellation of receivers is characterized. In the second approach we propose using linear algebra techniques on the matrix of unsynchronized TDOA measurements, enabling us to solve a set of linear equations for a parametrization of the unknowns. This approach is extended to more than four receivers and nine transmitters in a straightforward manner. In extensive experiments we evaluate and compare both approaches and analyze specific failure modes of the algorithms. Here, we demonstrate that the algorithms are robust to moderate Gaussian measurement noise and that the far field assumption is reasonable if the distance between transmitters and receivers is at least four times the distance between the receivers. In an indoor experiment using sound we reconstruct the microphone positions up to a mean error of 5 cm.