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Two-Stage Weighted Projection for Reliable Low-Complexity Cooperative and Non-Cooperative Localization

Harish K. Dureppagari, R. Michael Buehrer, Harpreet S. Dhillon

TL;DR

The paper tackles reliable, low-complexity localization in TDOA settings under challenging GDOP and SNR conditions and limited anchor visibility. It introduces the two-stage weighted projection method (TS-WPM), which iteratively refines both the reference anchor range and the user location, using diagonal covariance-based weights to maintain efficiency. The authors extend TS-WPM to cooperative localization via TW-TOA, derive the error covariance and MSE, and establish asymptotic optimality in high-SNR regimes, while comparing against WNLS and IPPM within a 3GPP-compliant analytical framework that models multipath and NLOS bias. A novel NLOS bias model links propagation conditions and SNR to bias and variance, enabling realistic simulations and performance evaluation. The results show TS-WPM achieving near-CRLB performance, outperforming state-of-the-art methods in high GDOP and multipath environments, with tangible gains in cooperative scenarios and favorable computational efficiency for real-time deployment in 5G/6G networks.

Abstract

In this paper, we propose a two-stage weighted projection method (TS-WPM) for time-difference-of-arrival (TDOA)-based localization, providing provable improvements in positioning accuracy, particularly under high geometric dilution of precision (GDOP) and low signal-to-noise ratio (SNR) conditions. TS-WPM employs a two-stage iterative refinement approach that dynamically updates both range and position estimates, effectively mitigating residual errors while maintaining computational efficiency. Additionally, we extend TS-WPM to support cooperative localization by leveraging two-way time-of-arrival (TW-TOA) measurements, which enhances positioning accuracy in scenarios with limited anchor availability. To analyze TS-WPM, we derive its error covariance matrix and mean squared error (MSE), establishing conditions for its optimality and robustness. To facilitate rigorous evaluation, we develop a 3rd Generation Partnership Project (3GPP)-compliant analytical framework, incorporating 5G New Radio (NR) physical layer aspects as well as large-scale and small-scale fading. As part of this, we derive a generalized Cramér-Rao lower bound (CRLB) for multipath propagation and introduce a novel non-line-of-sight (NLOS) bias model that accounts for propagation conditions and SNR variations. Our evaluations demonstrate that TS-WPM achieves near-CRLB performance and consistently outperforms state-of-the-art weighted nonlinear least squares (WNLS) in high GDOP and low SNR scenarios. Moreover, cooperative localization with TS-WPM significantly enhances accuracy, especially when an insufficient number of anchors (such as 2) are visible. Finally, we analyze the computational complexity of TS-WPM, showing its balanced trade-off between accuracy and efficiency, making it a scalable solution for real-time localization in next-generation networks.

Two-Stage Weighted Projection for Reliable Low-Complexity Cooperative and Non-Cooperative Localization

TL;DR

The paper tackles reliable, low-complexity localization in TDOA settings under challenging GDOP and SNR conditions and limited anchor visibility. It introduces the two-stage weighted projection method (TS-WPM), which iteratively refines both the reference anchor range and the user location, using diagonal covariance-based weights to maintain efficiency. The authors extend TS-WPM to cooperative localization via TW-TOA, derive the error covariance and MSE, and establish asymptotic optimality in high-SNR regimes, while comparing against WNLS and IPPM within a 3GPP-compliant analytical framework that models multipath and NLOS bias. A novel NLOS bias model links propagation conditions and SNR to bias and variance, enabling realistic simulations and performance evaluation. The results show TS-WPM achieving near-CRLB performance, outperforming state-of-the-art methods in high GDOP and multipath environments, with tangible gains in cooperative scenarios and favorable computational efficiency for real-time deployment in 5G/6G networks.

Abstract

In this paper, we propose a two-stage weighted projection method (TS-WPM) for time-difference-of-arrival (TDOA)-based localization, providing provable improvements in positioning accuracy, particularly under high geometric dilution of precision (GDOP) and low signal-to-noise ratio (SNR) conditions. TS-WPM employs a two-stage iterative refinement approach that dynamically updates both range and position estimates, effectively mitigating residual errors while maintaining computational efficiency. Additionally, we extend TS-WPM to support cooperative localization by leveraging two-way time-of-arrival (TW-TOA) measurements, which enhances positioning accuracy in scenarios with limited anchor availability. To analyze TS-WPM, we derive its error covariance matrix and mean squared error (MSE), establishing conditions for its optimality and robustness. To facilitate rigorous evaluation, we develop a 3rd Generation Partnership Project (3GPP)-compliant analytical framework, incorporating 5G New Radio (NR) physical layer aspects as well as large-scale and small-scale fading. As part of this, we derive a generalized Cramér-Rao lower bound (CRLB) for multipath propagation and introduce a novel non-line-of-sight (NLOS) bias model that accounts for propagation conditions and SNR variations. Our evaluations demonstrate that TS-WPM achieves near-CRLB performance and consistently outperforms state-of-the-art weighted nonlinear least squares (WNLS) in high GDOP and low SNR scenarios. Moreover, cooperative localization with TS-WPM significantly enhances accuracy, especially when an insufficient number of anchors (such as 2) are visible. Finally, we analyze the computational complexity of TS-WPM, showing its balanced trade-off between accuracy and efficiency, making it a scalable solution for real-time localization in next-generation networks.

Paper Structure

This paper contains 22 sections, 6 theorems, 78 equations, 8 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contributions
  4. Novel Two-Stage Weighted Projection Method
  5. Novel TS-WPM: TDOA Localization
  6. Novel TS-WPM: Cooperative Localization
  7. Optimality and Robustness of TS-WPM: Analytical Study
  8. Key Preliminaries
  9. Optimality and Robustness of TS-WPM
  10. Alternative Intuition: Robustness of TS-WPM
  11. Analytical Evaluation Framework
  12. CRLB Analysis: CIR and AWGN
  13. NLOS Bias Modeling
  14. Results and Discussion
  15. Positioning Performance
  16. ...and 7 more sections

Key Result

Lemma 1

The error covariance of TS-WPM, denoted by ${\rm cov}(\hat{\boldsymbol{\theta}}_{\mathrm{tswpm}}-\boldsymbol{\theta})$, and its MSE, denoted by ${\rm MSE}_{\mathrm{tswpm}}$, can be obtained as: where $\mathbf{C}_2$, $\mathbf{H}_2$, and $\mathbf{F}_{2,[x,y]}$ are as given in eq:cov_tdoa_scen2, eq:jacobian_pseudo_toa, and eq:efim_tdoa_scen2, respectively, and $\lambda_i\left(\mathbf{F}_{2,[x,y]}\r

Figures (8)

  • Figure 1: TOA error comparison: Comparing TOA error observed in AWGN and multipath.
  • Figure 2: Anchor - UE drop: Good Geometry.
  • Figure 3: CDF of position error evaluated at 3.5 GHz with 5 MHz bandwidth in AWGN and multipath: Good geometry and high SNR regime
  • Figure 4: Anchor - UE drop: Non-ideal Geometry.
  • Figure 5: CDF of position error evaluated at 3.5 GHz with 5 MHz bandwidth in AWGN and multipath: Non-ideal geometry
  • ...and 3 more figures

Theorems & Definitions (6)

  • Lemma 1: Error Covariance and MSE of TS-WPM
  • Theorem 1: Optimality of TS-WPM
  • Lemma 2: Error Covariance and MSE of WNLS
  • Proposition 1: Robustness of TS-WPM
  • Theorem 2: CRLB of TOA for Multipath
  • Corollary 2.1: CRLB of TOA for AWGN