State estimation of a moving frequency source from observations at multiple receivers

The task of position and velocity estimation of a moving transmitter (with either a known or unknown frequency) is a common problem arising in many different application domains. Based on the Doppler effect, this work presents a direct solution using only the frequency measured by a multitude of receivers with a known state. A natural rewriting of the problem as a system of polynomial equations allows for the use of homotopy continuation to find the global solution without any a priori information about the frequency source. We show that the data from six or seven receivers is sufficient in case of known or unknown frequency, respectively. After a brief development of the mathematics, two simple examples are provided: (1) position and velocity estimation of a vocalizing dolphin emitting an acoustic signal and (2) initial orbit determination of a satellite emitting an electromagnetic signal.