Quasi-Closed-Form Algorithms for Spherical Angle-of-Arrival Source Localization

The ground curvature cannot be neglected when locating the source that generates long-range propagation signals, so the traditional angle-of-arrival (AOA) source localization evolves into spherical AOA source localization. For this problem, a quasi-closed-form algorithm, the spherical weighted pseudolinear estimator (SWPLE), is proposed in this paper. The analysis of the SWPLE reveals that the algorithm has bias problems. To mitigate the bias of the SWPLE, we propose an algorithm based on biased term estimation, the bias compensated spherical weighted pseudolinear estimator (BCSWPLE). The instrumental variable (IV) method is further introduced to solve the bias problem of the SWPLE, and the spherical weighted instrumental variables estimator (SWIVE) is proposed. The theoretical analysis shows that the SWIVE is asymptotically unbiased and achieves the Cramér-Rao lower bound (CRLB) when both the small Gaussian noise assumption and the first-order approximation are satisfied and the number of measurements is sufficiently large. The proposed algorithms can obtain quasi-closed-form solutions to the source location. Therefore, they can avoid convergence problems that are common in the maximum likelihood estimator (MLE) and achieve more robust performance. Moreover, the proposed algorithms have lower computational complexities than traditional algorithms. When there are sufficient measurements, two bias mitigation algorithms can significantly suppress the bias, among which the SWIVE can achieve similar or even better bias performance compared with the MLEs. Extensive simulations prove the proposals of the study.