Direct Position Determination of Multiple Sources Using Nonuniform Linear Arrays With Gain-Phase Errors in a Moving Platform

Passive localization of signal sources has been widely used in applications such as target monitoring, navigation, and electronic reconnaissance. Compared with conventional two-step methods, direct position determination (DPD) can achieve higher localization accuracy by directly establishing the relationship between the source positions and the received signals. For a linear array affected by gain-phase errors, however, the joint estimation of direction-of-arrival (DOA) parameters and array gain-phase errors does not admit a unique solution without prior calibration information. In contrast, DPD does not require explicit estimation of the DOA parameters. In this paper, we prove that, under appropriate conditions, the source positions and array gain-phase errors can be uniquely determined within the DPD framework using a single moving platform. Based on this theoretical result, a joint localization and calibration algorithm is proposed for a single moving platform equipped with a sparse linear array affected by gain-phase errors. The proposed method employs a virtual interpolated array to estimate the source positions in a coarray-based virtual domain, while the array gain-phase errors are estimated in the physical domain associated with the actual array. By alternately updating the source-position and array-error estimates, the proposed method progressively improves both the localization and array calibration results. Furthermore, the well-posedness conditions of the proposed algorithm are derived, and the uniqueness of the global solution to the corresponding objective function is analyzed. Simulation results demonstrate the effectiveness of the proposed method and show that its estimation performance approaches the Cramér–Rao bound (CRB).