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A Unified Codebook Design for Curvature-Reconfigurable Apertures: Seamless Near to Far Field Coverage

Zhoujie You, Shu Sun, Ruifeng Gao, Jue Wang, Xianghao Yu

Abstract

Beam training for extremely large-scale arrays with curvature-reconfigurable apertures (CuRAs) faces the critical challenge of severe, geometry-dependent angle-range coupling. While most existing designs compartmentalize near field and far field scenarios, we propose a unified, distance-adaptive hierarchical codebook framework for 1-D and 2-D CuRAs that seamlessly bridges both propagation regimes. Under a spherical-wave model, we first characterize the beamforming-gain correlation in a polar angular domain, deriving an angle-dependent angular sampling rule to capture the varying curvature. To achieve full-range coverage, we introduce a direction-dependent effective Rayleigh distance (ERD) as a soft boundary to gate the range sampling. Crucially, by sampling uniformly in the reciprocal-range domain, the proposed codebook provides precise, dense focusing within the ERD and automatically degenerates into sparse, angle-only steering beyond it. This mechanism eliminates the need for hard mode-switching between near- and far-field operations. Simulation results demonstrate that our unified design consistently outperforms representative baselines in spectral efficiency and alignment accuracy, offering a comprehensive solution for full-range CuRA communications.

A Unified Codebook Design for Curvature-Reconfigurable Apertures: Seamless Near to Far Field Coverage

Abstract

Beam training for extremely large-scale arrays with curvature-reconfigurable apertures (CuRAs) faces the critical challenge of severe, geometry-dependent angle-range coupling. While most existing designs compartmentalize near field and far field scenarios, we propose a unified, distance-adaptive hierarchical codebook framework for 1-D and 2-D CuRAs that seamlessly bridges both propagation regimes. Under a spherical-wave model, we first characterize the beamforming-gain correlation in a polar angular domain, deriving an angle-dependent angular sampling rule to capture the varying curvature. To achieve full-range coverage, we introduce a direction-dependent effective Rayleigh distance (ERD) as a soft boundary to gate the range sampling. Crucially, by sampling uniformly in the reciprocal-range domain, the proposed codebook provides precise, dense focusing within the ERD and automatically degenerates into sparse, angle-only steering beyond it. This mechanism eliminates the need for hard mode-switching between near- and far-field operations. Simulation results demonstrate that our unified design consistently outperforms representative baselines in spectral efficiency and alignment accuracy, offering a comprehensive solution for full-range CuRA communications.

Paper Structure

This paper contains 16 sections, 3 theorems, 61 equations, 9 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. System Model of the 1-D CuRA
  3. Geometry of the 1-D CuRA
  4. Near field Downlink Channel Model of the 1-D CuRA
  5. Codebook Design for the 1-D CuRA
  6. Polar Coordinate Transformation
  7. Analysis of Sampling Method for 1-D CuRA
  8. Joint Angular Domain
  9. Distance Domain
  10. Generalization to the 2-D CuRA
  11. Geometry of the 2-D CuRA
  12. Codebook Design for the 2-D CuRA
  13. Joint angular-domain correlation
  14. Distance Domain
  15. Simulation Results
  16. ...and 1 more sections

Key Result

Lemma 1

For a fixed distance $r$, the angular-domain beamforming gain between two directions $(\rho_1,\varphi_1)$ and $(\rho_2,\varphi_2)$ can be approximated as where $\mathbf a(\rho,\varphi)$ denotes the far field steering vector obtained by applying a first-order Taylor expansion of the near field focusing vector $\mathbf b(r,\rho,\varphi)$ with respect to $1/r$. Moreover,$\Theta(\rho,\varphi) = \sqr

Figures (9)

  • Figure 1: System model of the 1-D CuRA.
  • Figure 2: Geometry of the 2-D CuRA.
  • Figure 3: Beamforming gain in angular domain $(\theta-\phi)$ vs. distance and bending angle $\beta$ for the 1-D CuRA.
  • Figure 4: Beamforming gain on the range--elevation plane $(r,\theta)$ for the 1-D CuRA with fixed azimuth $\phi=0$.
  • Figure 5: Beamforming gain over the angular domain $(\theta,\phi)$ for the 2-D CuRA with $M=8$, $N=64$, $\theta=\pi/2$.
  • ...and 4 more figures

Theorems & Definitions (3)

  • Lemma 1
  • Lemma 2
  • Corollary 1