Altitude constrained source localization using TDOA, FDOA and differential Doppler rate

Abstract

In this paper, we focus on locating known-altitude sources using multiple moving sensors. Besides the time difference of arrival (TDOA) and frequency difference of arrival (FDOA) measurements, the differential Doppler rate (i.e., the time derivative of FDOA) measurements are incorporated, together with the altitude constraint, to further enhance the localization performance. Two sets of computationally efficient estimators are developed, one of which applies to the case where the sensor positions are noise-free and the other one takes the sensor position errors into consideration. In the proposed algorithms, source localization is formulated into a nonlinear-constrained weighted least squares problem, which is solved in an iterative manner. During each iteration, the highly nonlinear constraints are linearized using estimates obtained in the previous iteration such that the source position could be updated explicitly. Since the proposed algorithms update the source position in closed form during each iteration, they are computationally much more efficient compared with the maximum likelihood (ML)-based algorithm. The Cramer-Ráo lower bounds (CRLBs) for the problems in consideration are also derived, along with brief comparisons, to reveal the optimal performance. Numerical simulations corroborate the effectiveness of the proposed algorithms under moderate noise levels and validate their performance improvement over existing algorithms.