Frame-Theoretic AOA Network Augmentation with the Minimum Number of Additional Sensors and Optimal Bearings

In line-of-sight sensing and surveillance systems, angle-of-arrival (AOA) localization accuracy depends strongly on sensor geometry. The spatially distributed sensors considered here may be electromagnetic, acoustic, optical, or other directional sensing devices that provide bearing measurements with known nominal accuracy. Existing frame-theoretic sensor-augmentation studies usually optimize the added-sensor bearings for a fixed number of sensors, whereas practical network design often requires determining the minimum number of sensors needed to satisfy a prescribed accuracy constraint. Poorly balanced bearing geometry can make the Fisher information matrix (FIM) ill-conditioned and increase localization uncertainty. This paper addresses this problem using the A-optimal Cramér-Rao lower bound trace as the accuracy measure, because it represents the sum of the lower bounds on the coordinate estimation-error variances. The existing Fisher information matrix is described by its total information, strong information direction, and eigenvalue gap. Each additional sensor is represented as a weighted vector in a double-angle plane, which converts the bearing-design problem into a planar vector-balancing problem. This representation separates the effects of total information and directional imbalance on localization performance. A polygon condition is used to determine the minimum achievable imbalance and A-optimal cost for fixed sensor weights. For equal-weight sensors, a direct rule is derived for finding the minimum required number of additional sensors, together with explicit bearing constructions for weak-direction compensation and tight-frame completion. The analysis is also extended to unequal sensor weights. Numerical optimization and Monte Carlo localization experiments confirm the analytical results and show that the proposed method improves information balance, FIM conditioning, empirical localization accuracy, and tail-error performance compared with the reference configurations. Random-network and target-position mismatch experiments further evaluate the design beyond the nominal network geometry.