Localization of a Moving Source by Frequency Measurements

This paper investigates the localization of a moving source in position and velocity, by observing the emitted frequency from the source that is subject to the Doppler shift effect at a number of stationary sensors. Previous attempts rely on exhaustive grid search or numerical polynomial optimization to obtain a solution. We shall propose a constrained optimization to formulate the localization problem, which enables the problem to be solved efficiently using the linear optimization method to reach a closed-form solution or the semi-definite relaxation technique to achieve a noise resilient estimate. The algorithms are developed for the single-time and multiple-time observations. The presence of errors in the source frequency and the sensor positions are considered. The non-cooperative scenario where the source frequency is completely not known is also addressed. Analysis validates the proposed closed-form solution in reaching the Cramer-Rao Lower Bound accuracy under Gaussian noise over the small error region. Simulations support the performance of the proposed solutions.