A Two-Stage Method for Short-Wave Target Localization Using DOA and TDOA Measurements

The ionosphere can make the short-wave source localization problem further non-linear, leading to a complicated solution process. This paper proposed a coordinated positioning algorithm based on the ionospheric virtual height (IVH) model to jointly estimate the target position and ionosphere reflection height using direction of arrival (DOA) and time difference of arrival (TDOA) measurements. The method is divided into two stages, the first stage is DOA localization and the second stage is TDOA localization. For the difficulty of establishing the pseudo-linear equation of elevation angle in the first stage, this paper proposed solving quadratic equations to establish the pseudo-linear equation. Moreover, since the TDOA pseudo-linear equations require the target position, the TDOA pseudo-linear equations can be established based on the estimates of DOA stage, which can lead to cooperative localization. Based on the pseudo-linear equations, the short-wave source localization problem is modeled as an optimization problem with double quadratic equation constraints. This paper proposed to solve the optimization problem combining the differentiable exact penalty method. The proposed method has a larger convergence region and requires fewer iterations to converge than the Lagrange method. Theoretical analysis shows that the localization performance of the proposed method can reach the Cramér-Rao lower bound (CRLB) and has higher accuracy than single parameter localization. Finally, the validity of theoretical analysis and the differentiable exact penalty method is verified by simulations.