Consistent and Asymptotically Efficient Localization from Bearing-only Measurements

2025-07-10

AI Summary

This paper addresses the problem of source localization using only angle-of-arrival (AOA) measurements, tackling the computational intractability and lack of asymptotic optimality inherent in conventional maximum likelihood estimation (MLE) due to its non-convex objective. We propose a two-step algebraic-iterative estimator: first, a consistent initial estimate of the noise variance is obtained via the reciprocal of the largest eigenvalue of a specially constructed matrix, yielding a high-accuracy initial localization; second, a single Gauss–Newton iteration suffices to achieve asymptotic consistency and efficiency identical to MLE. The method achieves low computational complexity while preserving theoretical optimality. In large-sample regimes, it significantly outperforms existing AOA-based localization algorithms. Simulation results confirm its high accuracy, strong robustness to noise and model mismatch, and asymptotic efficiency.

Technology Category

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Application Category

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Abstract

We study the problem of signal source localization using bearing-only measurements. Initially, we present easily verifiable geometric conditions for sensor deployment to ensure the asymptotic identifiability of the model and demonstrate 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 a two-step estimator that shares the same asymptotic properties as the ML estimator while offering low computational complexity, linear in the number of measurements. The primary challenge lies in obtaining a preliminary 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. The key to this process is obtaining a consistent estimator of the variance of the sine of the noise by taking the reciprocal of the maximum eigenvalue of a specially constructed matrix from the data. 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.

Problem

• Localizing signal source using bearing-only measurements efficiently
• Overcoming non-convex optimization in maximum likelihood estimation
• Developing low-complexity estimator with asymptotic ML properties

Innovation

• Two-step estimator with low computational complexity
• Linear least-squares for preliminary consistent estimator
• Gauss-Newton iteration for asymptotic ML properties