Theoretical Guarantees for AOA-based Localization: Consistency and Asymptotic Efficiency

We study the problem of signal source localization using angle of arrival (AOA) measurements. We begin by presenting verifiable geometric conditions for sensor deployment that ensure the model's asymptotic localizability. Then we establish the consistency and asymptotic efficiency of the maximum likelihood (ML) estimator. However, obtaining the ML estimator is challenging due to its association with a non-convex optimization problem. To address this, we propose an asymptotically efficient two-step estimator that matches the ML estimator's asymptotic properties while achieving low computational complexity (linear in the number of measurements). The primary challenge lies in obtaining a consistent estimator in the first step. To achieve this, we construct a linear least squares problem through algebraic operations on the measurement nonlinear model to first obtain a biased closed-form solution. We then eliminate the bias using the data to yield an asymptotically unbiased and consistent estimator. In the second step, we perform a single Gauss-Newton iteration using the preliminary consistent estimator as the initial value, achieving the same asymptotic properties as the ML estimator. Finally, simulation results demonstrate the superior performance of the proposed two-step estimator for large sample sizes.