RSS-Based Localization: Ensuring Consistency and Asymptotic Efficiency

TL;DR

This work addresses RSS-based source localization by establishing verifiable geometric conditions that guarantee asymptotic localizability and by proposing a two-step estimator that matches the maximum likelihood estimator's asymptotic efficiency with linear computational complexity. The first step yields a $\sqrt{n}$-consistent estimator via linear LS formulations under known and unknown noise variance, while the second step applies a single Gauss-Newton iteration to reach ML performance. The key contributions include (i) proving asymptotic localizability under sensor-geometry constraints, (ii) deriving a $\sqrt{n}$-consistent LS-based initializer, and (iii) showing that one GN update suffices to achieve ML-like efficiency with $\mathcal{O}(n)$ complexity. Simulations across 2-D and 3-D scenarios validate the theory, demonstrating that LS+GN attains the Cramér–Rao-like bound (RCRLB) and scales linearly with the number of measurements, making the method suitable for massive RSS datasets.

Abstract

We study the problem of signal source localization using received signal strength 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, computing the ML estimator is challenging due to its reliance on solving a non-convex optimization problem. To overcome this, we propose a two-step estimator that retains the same asymptotic properties as the ML estimator while offering low computational complexity, linear in the number of measurements. The main challenge lies in obtaining a consistent estimator in the first step. To address this, we construct two linear least-squares estimation problems by applying algebraic transformations to the nonlinear measurement model, leading to closed-form solutions. In the second step, we perform a single Gauss-Newton iteration using the consistent estimator from the first step as the initialization, achieving the same asymptotic efficiency as the ML estimator. Finally, simulation results validate the theoretical property and practical effectiveness of the proposed two-step estimator.

Figures (7)

• Figure 1: Biases of all the estimators with varying numbers of measurements for 2-D RSS-based localization.
• Figure 2: RMSEs for RSS-based localization of 2-D fixed sensors.
• Figure 3: RMSE comparison of the LS and LS+GN estimators with their unknown-noise-variance counterparts for RSS-based localization of 2-D fixed sensors.
• Figure 4: Biases and RMSEs for RSS-based localization of 2-D random deployed sensors.
• Figure 5: Biases of the estimators with varying numbers of measurements for RSS-based localization of 3-D fixed sensors.

Theorems & Definitions (20)

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