Node localization in unsynchronized time of arrival sensor networks

We present a method for solving the previously unstudied problem of localizing a set of receivers and directions from transmitters placed far from the receivers, measuring unsynchronized time of arrival data. The same problem is present in node localization of microphone and antenna arrays. The solution algorithm using 5 receivers and 9 transmitters is extended to the overdetermined case in a straightforward manner. Degenerate cases are shown to be when the measurement matrix has rank 4 or less or the directions from the transmitters to the receivers lie on an intersection between the unit sphere and another quadric surface. In simulated experiments we explore how sensitive the solution is with respect to different degrees of far field approximations of the transmitters and with respect to noise in the data. Using real data we get a reconstruction of the receivers with a relative error of 14%.