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Hierarchical Beam Training and Codebook Design for Movable Antenna-Assisted Near-Field Systems

Meihui Liu, Qian Zhang, Xuejun Cheng, Yuhui Jiao, Yunxiao Li, Ju Liu

arXiv:2609.03776v1cs.IT

TL;DR

Near-field beam training requires joint angle-distance search, creating high overhead, while existing hierarchical methods lack multi-resolution codebooks tailored to MA-assisted systems. The paper develops hierarchical training and a joint angle-distance codebook using movable antenna positions, achieving focused beams and rates close to exhaustive search with substantially less training overhead.

  • Problem

    Near-field spherical-wave propagation requires joint angle-distance beam training, while existing hierarchical methods lack multi-resolution codebooks for MA-assisted systems.

  • Method

    The paper develops hierarchical beam training with a multi-resolution codebook that jointly designs BS precoding and movable antenna positions over the angle-distance domain.

  • Results

    The proposed MA codebook provides substantial gains over FPA and far-field codebooks, while hierarchical training approaches exhaustive-search achievable rates with substantially less overhead.

  • Takeaways & Limitations

    Movable antenna reconfiguration supplies additional spatial degrees of freedom for improved near-field beam focusing within a low-overhead hierarchical strategy.

Abstract

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As sixth-generation (6G) communication systems evolve toward higher frequency bands and larger array apertures, the near-field range expands rapidly, making near-field channel estimation increasingly important and challenging. Beam training has been recognized as an effective approach for channel state information (CSI) acquisition. However, because of the propagation characteristics of spherical waves, beam training needs to perform a joint search in the angle and distance domains, which results in unaffordable beam training overhead. By flexibly reconfiguring antenna positions, movable antenna (MA) technology can fully exploit the spatial variations of wireless channels and achieve more accurate beam focusing, thereby providing additional flexibility for efficient beam training design. Therefore, based on MA-assisted near-field systems, we develop a hierarchical beam training strategy that combines reduced training overhead with high beam gain and design a corresponding hierarchical codebook. This codebook forms focused beams over the joint angle-distance domain, maximizing beam gain within the target region while suppressing energy leakage into non-target regions. Simulation results confirm substantial performance gains of the method over the conventional fixed-position antenna (FPA) system and the far-field beam training method.

I. INTRODUCTION

MA-assisted near-field systems address the hardware and complexity limits of large fixed-position arrays, where spherical-wave propagation requires joint angle-distance beam training. The paper develops a multi-resolution hierarchical codebook and jointly optimizes precoding and antenna positions for focused near-field beams.

  • Large fixed-position arrays increase hardware expenditure, energy consumption, and signal-processing complexity, constraining further wireless-network performance improvements.
  • Movable antennas exploit location-dependent channel variations within a bounded region without requiring additional RF chains.
  • Near-field spherical-wave propagation makes conventional far-field transmission schemes and angular-only beam training unsuitable.
  • Existing hierarchical methods lack a multi-resolution codebook jointly covering angle and distance, especially for MA-assisted near-field systems.
  • The paper develops hierarchical beam training and a corresponding multi-resolution codebook for joint angle-distance coverage under MA-assisted near-field propagation.
  • The codebook optimization alternates BS precoding and antenna-position updates, while simulations show improved focusing over FPA and far-field codebooks with training performance close to exhaustive search.

B. Signal Model

The signal model represents transmission from the BS to a single-antenna user using a precoding vector and additive white Gaussian noise.

  • The received signal is modeled from the BS transmit symbol, near-field channel, and BS precoding vector.
  • The precoding vector w ∈ C^N×1 controls BS transmission, while z ∼ CN(0, σ^2) represents the user's AWGN.

A. Hierarchical Beam Training

Hierarchical beam training searches the user’s location progressively, using wide beams first and narrower beams in later levels to reduce training overhead.

  • The first level uses a small number of wide beams to obtain a coarse user-location estimate across the service region.
  • Subsequent levels use narrower beams to refine the search within the region selected at the preceding level.
  • With R regions searched across F levels, hierarchical training requires R×F beam tests before the target region is determined.

B. Hierarchical Codebook Design

The hierarchical codebook samples each level’s angle-distance coverage region and designs movable-antenna beams to match a desired focused response while respecting physical constraints.

  • Each codeword is optimized so the synthesized beam pattern closely approximates a desired ideal beam pattern at its assigned resolution.
  • At level l, the coverage region is specified by angular and distance intervals and sampled using K angular points and S distance points.
  • The desired amplitude gain assigns constant target gain C_g inside the target angle-distance region and zero gain elsewhere.
  • This target pattern promotes high gain within the target region while suppressing energy leakage into other areas.
  • The BS precoding vector and MA position vector are jointly designed using sampled complex responses with target amplitudes and auxiliary phase alignment.
  • The optimization constrains antenna positions to the aperture and enforces a minimum inter-antenna spacing D_0.

IV. BS PRECODING AND ANTENNA POSITION OPTIMIZATION

The coupled BS precoding and antenna-position variables are optimized using block coordinate descent, with alternating updates across the two variable blocks.

  • IV. BS PRECODING AND ANTENNA POSITION OPTIMIZATION: BCD alternately updates the BS precoding vector w and antenna position vector t.The iteration index is denoted by ℓ.
  • IV. BS PRECODING AND ANTENNA POSITION OPTIMIZATION: The alternating strategy addresses the coupled variables in the beam-pattern-matching objective.
  • IV. BS PRECODING AND ANTENNA POSITION OPTIMIZATION: Each update is formulated at iteration ℓ + 1 before proceeding to the next block.

A. BS Precoding Optimization

For fixed antenna positions, the BS precoding problem is expressed over sampled steering responses and target responses, then solved under a transmit-power constraint using KKT conditions.

  • A. BS Precoding Optimization: The precoding subproblem is represented in compact matrix form and incorporated into a Lagrangian with multiplier μ ≥ 0 for the power constraint.
  • A. BS Precoding Optimization: The steering matrix A contains responses at all sampled points, while p contains their target responses and Q = K × S is the sample count.
  • A. BS Precoding Optimization: KKT conditions separate the solution into cases according to whether the unconstrained precoder satisfies the power limit.
  • A. BS Precoding Optimization: When the unconstrained solution meets the power limit, μ = 0 and the optimal precoder is the unconstrained least-squares solution w★ = (AᴴA)^−1Aᴴp.
  • A. BS Precoding Optimization: When the unconstrained precoder exceeds the power limit, the optimal solution satisfies ∥w★∥2 = Pmax and μ is found by one-dimensional line search.
  • A. BS Precoding Optimization: Eigenvalue decomposition reduces computational complexity by avoiding repeated matrix inversions when determining μ.

B. Antenna Position Optimization

The antenna-position subproblem minimizes beam-response mismatch under aperture and spacing constraints, using projected gradient descent for its highly nonconvex objective.

  • B. Antenna Position Optimization: With w updated, antenna positions t are optimized to minimize mismatch between synthesized and desired beam responses.
  • B. Antenna Position Optimization: The position optimization is constrained by the physical aperture L and minimum inter-element spacing D0, represented by t ∈ CFR ∩ CAC.
  • B. Antenna Position Optimization: Nonlinear phase terms involving antenna positions make f(t) highly nonconvex, motivating the use of projected gradient descent.
  • B. Antenna Position Optimization: The objective is written through residuals e_k,i(t), whose position dependence arises through the near-field steering vector.
  • B. Antenna Position Optimization: The full antenna-position gradient is assembled from per-antenna partial derivatives and used to update positions along the negative gradient direction.
  • B. Antenna Position Optimization: Armijo backtracking selects the step size, after which the intermediate positions are projected onto the feasible set through a convex quadratic program.

V. SIMULATION RESULTS

Simulations show that movable-antenna codebooks improve near-field beam-training performance over FPA and far-field codebooks while approaching exhaustive-search performance with lower overhead.

  • Both MA and FPA objective curves decrease monotonically and converge within a few iterations.The MA codebook converges to a significantly lower objective value than the FPA codebook.
  • The MA codebook achieves the best achievable rate under three-level hierarchical beam training, while the far-field codebook performs worst.The far-field codebook suffers significant phase mismatch from its plane-wave assumption.
  • Subsequent hierarchical stages further improve performance within the region selected at the preceding stage.The multi-resolution codebook supports accurate user localization.
  • Increasing the BS antenna count from 128 to 256 improves achievable rate at the same training overhead.The improvement is attributed to increased array gain and finer near-field focusing capability.
  • The MA codebook achieves higher achievable rate than the FPA codebook for the same number of antennas.
  • The MA codebook closely approaches the exhaustive-search upper bound while requiring substantially less training overhead across SNR levels.

VI. CONCLUSION

The paper develops low-overhead hierarchical beam training and a multi-resolution codebook for accurate joint angle-distance focusing in MA-enabled near-field transmission. Simulations show substantial gains over FPA and far-field codebooks, while achievable rate closely approaches exhaustive search with substantially less overhead.

  • The proposed scheme combines low-overhead hierarchical beam training with a multi-resolution codebook for joint angle-distance beam focusing.
  • The MA codebook provides substantial performance gains over conventional FPA and far-field codebooks.The gains arise from additional spatial degrees of freedom enabled by flexible antenna-position reconfiguration.
  • The hierarchical strategy closely approaches exhaustive-search achievable rate while requiring substantially less training overhead.This balances system performance and implementation complexity.
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