Feasibility Study and Optimum Sensor Path Planning for Localization by Doppler Derivatives

We often rely on the emitted harmonic component for the localization of a non-cooperative source, through the Doppler shifted frequencies (DSFs) resulting from the relative motion with the sensors. Frequency mismatch in the local oscillators of the sensors can cause different offsets in the DSFs, making them unusable for localization. This paper explores the use of Doppler Derivative (DD), which is the rate of change of DSF and is free from frequency synchronization offset, for the localization of a static source by a few sensors in motion. We show that localization by DD is feasible, except for some blind scenarios where the conditions are established. We then continue to derive the optimum sensor placement for an optimum geometry that achieve the best localization performance by the D-Optimality criterion. The optimum geometries are extended to the optimum sensor path planning so that the optimality can be maintained as the sensors move. The cases of known and unknown center frequency, and 2-D and 3-D scenarios are considered. Simulations by the Maximum Likelihood Estimator validate well the theoretical developments.