Localization by Doppler Derivatives and Doppler-Shifted Frequencies

When a narrow-band source travels over a receiver, the observed signal will experience a Doppler frequency shift that can be exploited to determine its location. The Doppler shift is not constant even if the motion is linear with constant velocity and it has a rate of change. This work explores the Doppler derivatives (DDs) in addition to the Doppler-shifted frequencies (DSFs) encountered by a number of static receivers to locate the source in position and velocity, for the cases where the emitting frequency is known and not known. The algorithms from the literature are sub-optimum and require a one-dimensional grid search if the emitting frequency is not known. We first analyze the benefit of incorporating DD for DSF localization in terms of complexity lessening, performance improvement, and blind geometry reduction. The paper continues to develop localization algorithms for the known and unknown emitted frequency cases, in which one is the closed-form solution that is computationally attractive and the other is the semidefinite programming (SDP) solution that is noise resilient. The closed-form estimator is shown by theory to be optimum under small Gaussian noise. Both algorithms do not require grid search and their computational complexities are analyzed. Simulations support the promising performance of the proposed algorithms.