A Closed-Form Solution for 3D Source Localization Using Angles and Doppler Shifted Frequencies

Moving source localization in 3-D space using frequencies of arrival (FOA) have recently made significant progress with the development of closed-form solutions. However, the closed-form solution algorithms for source localization using FOA only introduces a large number of auxiliary variables and results in a significant jump in the minimum number of receivers. In order to overcome the inherent defect of source localization using FOA measurements only, we address locating the moving source by jointly using AOA and FOA measurements in this article. A closed-form solution is designed for AOA-FOA based source localization, where the AOA and FOA measurement noises, the carry frequency error, as well as the receiver location uncertainties are considered, and the two-stage weighted least squares (TSWLS) technique is applied to overcome the nonlinear relation between the measurements and the source location parameters. The proposed solution can give the source position and velocity estimate with much fewer receivers and nuisance parameters, and lower computational complexity, in comparison with the state-of-the-art FOA localization algorithm. The proposed solution is shown analytically and numerically to achieve the Cramer-Rao lower bound (CRLB) performance under small Gaussian noise conditions and outperform the FOA-only localization algorithm in terms of both CRLB and root mean square error under the same noise conditions.