Cramér-Rao Bound Analysis of Spatial Effects in 2D TOA Localization With Directive Antennas

Although Time-of-Arrival (TOA) localization is widely studied in the literature, most analyses assume omnidirectional antennas, which oversimplifies system performance and leaves the impact of antenna directivity underexplored. This paper introduces a new, more realistic and comprehensive model that accounts for directive antennas, offering a significant improvement over conventional TOA localization theory. Expressions are derived for the Fisher Information Matrix (FIM) terms corresponding to the location coordinates of a passive target, explicitly incorporating arbitrary antenna radiation patterns. Our analysis reveals that the FIM singularity conditions depend on the derivatives of the anchor antenna radiation pattern, which introduces new geometrical insights into observability. While our analysis is applicable for any TOA setup, we focus on scenarios that extend conventional TOA theory and have not been previously reported. TOA studies often conclude that 2D localization is only possible with at least two anchors, and that collinear anchor-target layouts leads to insufficient observability. We prove that a single anchor with a directive antenna can suffice for localization using multiple range measurements, and that the FIM can be inverted even in the collinear anchor-target geometry. In both of these cases, observability emerges from the directional noise introduced by non-isotropic antenna radiation patterns. The results are validated via Maximum Likelihood Estimator (MLE) simulations for both single anchor and collinear anchor cases. In regions where the FIM is near-singular, the MLE bias is significant and the Cramér-Rao Bound (CRB) is not a lower bound due to insufficient information. Conversely, in non-singular regions, the MLE is unbiased and achieves the CRB.