Single-step localization using multiple moving arrays in the presence of observer location errors

Direct position determination (DPD) is a promising technique that offers superior performance compared with conventional two-step localization methods. Existing DPD methods presume that the observer locations are known exactly, whereas in practical environments, a small error in the observer locations will lead to an erroneous localization. This study considers the localization of a stationary transmitter by separated moving arrays from passive measurements taken at different points along the trajectory. The precise locations and velocities of the observers are not available, but their errors are assumed to be Gaussian distributed. Using this probability distribution, we propose maximum likelihood-based DPD approaches in the presence of observer location errors for both unknown and known signals. The proposed DPDs rely on alternating iteration schemes, which reduce the multidimensional nonlinear optimization problem to optimizations of dimensions that are much smaller than the number of unknowns. As opposed to the conventional two-step methods that extract measurement parameters and then estimate the positions from them, the proposed DPDs achieve the localization in a single step by exploiting the information of angles, time delays, and Doppler frequency shifts, but without computing them. Additionally, we derive the Cramér-Rao bound (CRB) formula for this DPD problem in the presence of observer location errors. The simulation results prove that the performance of our methods attains the associated CRB. Moreover, they are more robust than the conventional two-step approaches with respect to observer location errors. We demonstrate our methods for the scenario of multiple moving arrays, but these methods can easily be extended to DPD problems accounting for observer location errors in different scenarios.