Unified Subspace Fitting Framework and Its Performance Analysis for Direct Position Determination in the Presence of Multipath Propagation

The cost functions and their performances of direct position determination (DPD) methods in the presence of multipath propagation are investigated. We first establish a general DPD (GDPD) model in the presence of multi-path propagation and point out that the existing cost functions cannot get the emitter positions correctly because of the singularity of the manifold matrix in a multipath propagation scenario. Eight cost functions are developed for the GDPD model and formulated in a unified subspace fitting (USF)-based framework, which provides insight into their algebraic and asymptotic relations. Moreover, we derive the closed-form expressions of the asymptotic distributions of the estimation errors, which are optimized by those cost functions. Besides, the optimal cost function for achieving an optimal asymptotic performance is derived based on the optimization theory. Finally, the numerical simulations and Cramér–Rao lower bound are provided to verify the analytical results and show that: 1) the cost functions which work well in the single-path DPD model cannot find the emitters correctly in a multipath scenario; 2) the signal subspace fitting cost functions and noise subspace fitting cost functions, which are proposed in this paper, find the emitters accurately in the multipath propagation scenarios; 3) the optimal-weighted-subspace-fitting cost function holds the best asymptotic performance under the USF framework; and 4) the asymptotic performance of a multiple dimension cost function is better than a 1D cost function.