On the estimation of bearing using a single hydrophone

It is well known that a single moving hydrophone can estimate the bearing of a pure tone source if the source frequency is known a priori. It is also well known that with two or more hydrophones, a moving array can provide an improved estimate of bearing by jointly estimation the source frequency and bearing with a Kalman filter. This is due to the introduction of the phase information contained in the Doppler. This leads to the question of whether by jointly estimating the bearing and source frequency with a Kalman filter, a single hydrophone will suffice. By using a form of the observability matrix adapted to the nonlinear case, it is shown that in general, it is not possible. The reason is shown to be related to the fact that the bearing information in the Doppler is not sufficient and the introduction of a spatial phase, by introducing at least one more hydrophone is necessary.