CRLB Trace Analysis Under Random Sensor Deployments for Hybrid RSS/AOA Localization

Abstract

This article analyzes the Cramér–Rao lower bound (CRLB) trace of 2-D hybrid received signal strength (RSS)/angle of arrival (AOA) localization under line-of-sight (LOS) conditions and random sensor deployments. By exploiting the radial-tangential structure of the hybrid Fisher information matrix (FIM), we obtain a compact representation in terms of sensing precision parameters and distance-weighted geometric descriptors. This representation leads to an exact closed-form trace formula and a baseline-penalty decomposition. The baseline term describes distance-weighted information accumulation, whereas the penalty factor captures the coupling between RSS/AOA sensing imbalance and normalized angular anisotropy. We prove that this penalty is bounded whenever both modalities provide nonzero Fisher information, indicating that angular anisotropy alone cannot cause unbounded CRLB degradation. Under an isotropic annular deployment model, the hybrid CRLB trace is further treated as a random variable induced by sensor geometry. Its mean decays as $\mathcal {\textit {O}}\text {(}\textit {n}^{-{1}}\text {)}$ , while its standard deviation decays as $\mathcal {\textit {O}}\text {(}\textit {n}^{-{3}/{2}}\text {)}$ , showing increasing concentration as the number of sensors grows. A second-order finite-sample mean correction is then derived, which separates radial fluctuations from imbalance-dependent angular effects. A refined Gaussian approximation is also obtained for moderate network sizes. Numerical results verify the exact trace formula, bounded penalty representation, scaling laws, finite-sample refinement, and the advantage of hybrid RSS/AOA localization over RSS-only and AOA-only benchmarks. Estimator-level simulations further illustrate the relevance of the derived CRLB trace as a benchmark for practical weighted least-squares (WLSs) localization.