Array Shape Observability From Time Differences of Arrival

In this paper, we study the observability conditions for the relative position of a set of passive sensors from the measured time differences of arrival (TDOA) of signals from a set of noncooperative sources lying in the same plane. The sources are broadband to measure the TDOA and far enough to consider each source is detected in the same line of sight by each sensor. We show that a necessary observability condition is to have at least three sensors and three sources. Then, the sufficient observability conditions are established in this minimal configuration. Finally, we propose an algorithm to compute both sensors' location and the direction of arrival (DOA) of the sources, from the noise-free TDOA.