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

TL;DR

This work addresses the problem of locating a signal source from AOA measurements by establishing verifiable geometric conditions for asymptotic localizability and proving that the ML estimator is consistent and asymptotically efficient. To overcome non-convexity, the authors introduce a two-step estimator: a $ oot n$-consistent bias-eliminated LS (BELS) initial estimator followed by a single Gauss-Newton refinement, achieving the same asymptotic performance as ML with linear complexity in the number of measurements, $n$. The framework is extended from 2-D to 3-D, including a 3-D BE-based initialization for the first coordinates and a GN refinement, along with practical procedures to estimate noise-variance terms when unknown. Extensive simulations demonstrate that the proposed two-step estimator attains the Cramér–Rao lower bound in large-sample regimes across fixed, random, coplanar, and noncoplanar sensor configurations, while maintaining a scalable $O(n)$ computational cost, highlighting its suitability for large-scale AOA localization tasks.

Abstract

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.

Figures (7)

• Figure 1: Illustration of AOA measurements in the 2-D scenario. The red dot represents the signal source while the blue dots represent the sensors.
• Figure 2: 2-D: biases and RMSEs of the estimators for fixed sensors.
• Figure 3: 2-D: RMSE comparison of BELS and BELS+GN estimators between the true variance of the sine of the noise and the estimated one.
• Figure 4: 2-D: Biases and RMSEs of the estimators for random sensors.
• Figure 5: 3-D: biases and RMSEs of the estimators for fixed sensors.

Theorems & Definitions (28)

• Definition 1
• Definition 2
• Theorem 1
• Lemma 1
• Theorem 2
• Proposition 1
• Lemma 2
• Proposition 2
• Proposition 3
• Theorem 3