Bearings-Only Localization with NLOS Reflected AoAs

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1 June 2013
Xiufeng Song, Peter Willett, Shengli Zhou

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Bearings-only localization with light-of-sight (LOS) propagation is well understood. This paper concentrates on bearings-only localization with non-line-of-sight (NLOS) measurements, where target images arrive at a network of sensors each after a single specular reflection. The reflecting surface can be 1) flat or 2) circular (inner side of a circle), and is assumed known. In this paper, we derive the least squares (LS), Stansfield, and maximum likelihood (ML) estimators for both cases. As to the former, their estimation performances are similar to their counterparts in LOS localization: Stansfield is very close to ML, and both are usually significantly better than LS. As regards the second, since the target-sensor geometry has multiple possibilities, the ML solution is extremely intricate. However, if a concentric opaque circle (such as the earth) lies within the reflecting one, e.g. the earth within the ionospheric layer, the propagation path becomes unique; a grid search based ML is available for such a circumstance. ML is computationally intensive for a circularly reflecting surface; two suboptimal algorithms, LS and Stansfield, are developed based on small angle approximation. These algorithms perform differently from those for the flat case: ML significantly outperforms LS and Stansfield, especially for a large observation error; however, Stansfield is not necessarily better than LS.