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3D User Localization for Planar Arrays in LoS Near- and Far-Fields via Summed Phase Differences

Sergey Isaev, Nikola Zlatanov

Abstract

This paper presents a phase-difference-based scheme for three-dimensional (3D) line-of-sight (LoS) user localization using a uniform planar array (UPA), applicable to both near-field and far-field regimes under the exact spherical-wave model. Unlike the previously studied two-dimensional (2D) uniform linear array (ULA) case, the 3D UPA case requires jointly exploiting the two array axes in order to recover the user's range, azimuth, and zenith angle. Adjacent-antenna phase-differences are first estimated from uplink pilots and then summed along the array axes to obtain unwrapped phase-differences between widely separated antenna elements. These summed phase-differences enable the construction of multiple three-equation systems whose solutions yield the user's range, azimuth, and zenith angle. We quantify the number of such equation systems, provide a representative closed-form estimator that uses only three phase-difference sums, and propose an all-data nonlinear least-squares estimator that exploits all available sums. Numerical results show that the least-squares estimator, when initialized by the closed-form estimate, achieves Cramér--Rao bound accuracy. Moreover, unlike state-of-the-art baseline schemes, whose performance depends on well-tuned hyperparameters, the proposed estimators are hyperparameter-free.

3D User Localization for Planar Arrays in LoS Near- and Far-Fields via Summed Phase Differences

Abstract

This paper presents a phase-difference-based scheme for three-dimensional (3D) line-of-sight (LoS) user localization using a uniform planar array (UPA), applicable to both near-field and far-field regimes under the exact spherical-wave model. Unlike the previously studied two-dimensional (2D) uniform linear array (ULA) case, the 3D UPA case requires jointly exploiting the two array axes in order to recover the user's range, azimuth, and zenith angle. Adjacent-antenna phase-differences are first estimated from uplink pilots and then summed along the array axes to obtain unwrapped phase-differences between widely separated antenna elements. These summed phase-differences enable the construction of multiple three-equation systems whose solutions yield the user's range, azimuth, and zenith angle. We quantify the number of such equation systems, provide a representative closed-form estimator that uses only three phase-difference sums, and propose an all-data nonlinear least-squares estimator that exploits all available sums. Numerical results show that the least-squares estimator, when initialized by the closed-form estimate, achieves Cramér--Rao bound accuracy. Moreover, unlike state-of-the-art baseline schemes, whose performance depends on well-tuned hyperparameters, the proposed estimators are hyperparameter-free.

Paper Structure

This paper contains 7 sections, 20 equations, 4 figures.

Table of Contents

  1. Introduction
  2. System Model and Problem Formulation
  3. System and Channel Model
  4. Problem Formulation
  5. Proposed Phase-Difference Localization Scheme
  6. Numerical Results
  7. Conclusion

Figures (4)

  • Figure 1: RMSE of the distance estimate $r$ (in meters) versus the number of uplink pilots $K$ for a $21 \times 21$ UPA ($N=10$). User locations: $r\in\{5,40\}$ m, $\phi=\pi/4$, $\theta=\pi/6$.
  • Figure 2: RMSE of the azimuth-angle estimate $\theta$ (in radians) versus the number of uplink pilots $K$ for a $21 \times 21$ UPA ($N=10$). User locations: $r\in\{5,40\}$ m, $\phi=\pi/4$, $\theta=\pi/6$.
  • Figure 3: RMSE of the distance estimate $r$ (in meters) versus the UPA size parameter $N$ for $K=50$ pilots. User locations: $r\in\{5,40\}$ m, $\phi=\pi/4$, $\theta=\pi/6$.
  • Figure 4: RMSE of the zenith-angle estimate $\phi$ (in radians) versus the UPA size parameter $N$ for $K=50$ pilots. User locations: $r\in\{5,40\}$ m, $\phi=\pi/4$, $\theta=\pi/6$.