Statistical performance analysis of direct position determination method based on doppler shifts in presence of model errors

The direct position determination (DPD) method based on Doppler shifts for narrowband emitter is first presented by Amar and Weiss. Compared to the conventional differential Doppler localization method, this DPD method has higher localization accuracy. In this paper, the statistical performance of this DPD method in presence of model errors (i.e., the observer position and velocity uncertainty) is analyzed mathematically. First, the DPD method for narrowband emitter using Doppler shifts is introduced. Then, applying the matrix eigen-perturbation theory that relates the eigenvalue perturbations to the additive noise on the Hermitian matrix, the first-order asymptotic expression of the direct localization errors is derived. As a consequence, the theoretical variance of position estimation can be approximately determined. Subsequently, two kinds of exact formulas for calculating the localization success probability are deduced using the analytical results obtained in the previous section of this paper. Furthermore, the Cramér-Rao bound (CRB) expression for the DPD method in absence of model errors is presented, which takes on more explicit form than the formulation given by Amar and Weiss. The obtained CRB can be viewed as a reasonable benchmark to assess the performance degradation due to model mismatch. Simulation results support the validation of the theoretical analysis in this paper.