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Hierarchical Codebook Design and Low-Overhead Beam Training for Near-Field Communications With Uniform Circular Arrays

Gen Luo, Hang Yuan, Xiaozheng Gao, Minwei Shi, Chong Han, Kai Yang

arXiv:2609.04836v1eess.SP

TL;DR

Near-field XL-MIMO beam training must resolve coupled angle–distance channels efficiently, especially for UCA systems whose resolution-aware design remains underdeveloped. The paper derives UCA minimum resolvable distance, builds a hierarchical DRBF–FP codebook, and applies HDA-BAR for two-stage training. For the considered configuration, HDA-BAR uses 384 probing slots and reduces overhead by 99.66% versus exhaustive near-field search.

  • Problem

    Near-field UCA systems require codebooks and beam training that exploit joint angle–distance resolution while reducing exhaustive-search overhead.

    Source: §I · p. 1, A. Prior Works · p. 2

  • Method

    The paper derives minimum resolvable distance from a spherical-wave model, designs hierarchical DRBF and FP codebooks, and trains them with two-stage HDA-BAR.

    Source: §II · p. 3, B. BAR-Based Fine-Grained Beam Search · p. 8

  • Results

    384 probing slots achieve a 99.66% overhead reduction relative to exhaustive scanning of 114368 FP codewords while outperforming near-field TPBT and far-field exhaustive search.

    Source: §VI · p. 12

  • Takeaways & Limitations

    The proposed hierarchy maintains near-exhaustive beamforming performance with substantially reduced training overhead for the considered UCA configuration.

    Source: §VI · p. 12, §VII · p. 12

Abstract

from arXiv · show

Extremely large-scale multiple-input multiple-output (XL-MIMO) enables near-field location-specific beam focusing for sixth-generation (6G) communications. Uniform circular arrays (UCAs), with rotational symmetry and uniform azimuth coverage, have emerged as a key enabling architecture for near-field XL-MIMO systems. In this paper, we propose a resolution-aware hierarchical codebook for near-field UCA systems, along with an efficient two-stage beam training scheme to significantly reduce the training overhead. Specifically, we characterize the minimum resolvable distance of UCA systems in the near-field region based on a geometric spherical-wave propagation model, revealing their spatial resolution capability in the joint angle--distance domain. Guided by this result, we design a UCA-specific hierarchical codebook, where a power-efficient distance-robust beamforming (DRBF) codebook provides coarse azimuth localization and a full-precision (FP) codebook sampled according to the minimum resolvable distance enables refined angle--distance beam search. The regularized modal compensation suppresses weak-mode amplification and provides a controllable tradeoff between absolute amplitude gain under unit-norm transmission and distance robustness. Based on this codebook, we develop a hierarchical decoupled-architecture Bayesian regression (HDA-BAR) scheme for fast and accurate near-field beam training. For the considered array configuration, the resulting HDA-BAR training procedure requires $384$ probing slots, corresponding to an approximately \(99.66\%\) overhead reduction relative to the conventional near-field exhaustive-search benchmark.

I. INTRODUCTION

Near-field XL-MIMO requires spherical-wave modeling because angular codebooks are inadequate, motivating UCA-specific resolution-aware codebooks and low-overhead beam training. The paper develops a hierarchical design that combines coarse distance-robust localization with refined angle–distance search.

Motivation

  • Spherical-wave propagation makes near-field beamforming jointly dependent on angle and distance, enabling location-specific focusing and distance-domain resolution.Source: §I · p. 1

Prior limitations

  • ULA-based near-field codebooks use nonuniform distance sampling, while UCA-specific resolution characterization remains insufficiently established.Source: A. Prior Works · pp. 1–2

UCA motivation

  • UCAs offer rotational symmetry and more uniform near-field responses across azimuth directions than ULAs.Source: A. Prior Works · pp. 1–2

Proposed design

  • The proposed hierarchical codebook uses a regularized distance-robust beamforming layer for coarse sectors and a full-precision layer sampled by minimum resolvable distance.Source: B. Our Contributions · p. 2

Proposed training

  • HDA-BAR first localizes azimuth with DRBF beams, then performs Bayesian fine search over a pruned full-precision codebook.Source: B. Our Contributions · p. 2

III. NEAR-FIELD SPATIAL RESOLUTION ANALYSIS OF UCA

The resolution analysis models separability between nearby near-field users through their channel matrix and effective degrees of freedom. It derives a minimum resolvable distance that varies with range, relative azimuth, and array aperture.

Resolution metric

  • The analysis quantifies two-user spatial separability through the channel matrix and effective degrees of freedom in the joint angle–distance domain.Source: §III · p. 4

Geometric model

  • The UCA geometry is parameterized using a reference user and a nearby user whose radial and angular displacements are Δr = d cos φr and Δϕ = d sin φr.Source: §III · p. 4

Resolution criterion

  • The gain matrix and its eigenvalues connect channel energy distribution to the EDoF criterion for resolving the two users.Source: §III · p. 4

Resolution limit

  • The minimum resolvable distance dms(φr, r) is the smallest positive separation satisfying the prescribed EDoF threshold condition.Source: §III · p. 4

Resolution behavior

  • The derived resolution generally improves with larger array radius, worsens with greater user range, and varies with relative azimuth.Source: §III · p. 5

IV. PROPOSED HIERARCHICAL CODEBOOK DESIGN

The proposed hierarchical codebook uses a distance-robust first layer for coarse angular localization and a resolution-aware full-precision second layer for joint angle–distance refinement. Regularized modal compensation improves absolute gain under unit-norm transmission while preserving distance robustness.

Hierarchical design

  • The two-layer codebook uses DRBF beams for coarse angular localization and resolution-aware FP beams for refined joint angle–distance search.The design exploits UCA phase-mode structure to reduce redundant beam search.Source: §IV · p. 5

Layer 1: DRBF codebook

  • DRBF codewords combine phase-mode truncation with regularized modal compensation to improve worst-case absolute amplitude gain while retaining distance robustness under unit-norm transmission.Power efficiency is defined through coherent array-power gain under a fixed unit-norm transmit constraint.Source: A. Layer-1 Power-Efficient DRBF Codebook · p. 5

Layer 2: FP codebook

  • The FP codebook is formed by sampling angular and radial domains according to the derived minimum resolvable distance.The phase-mode representation uses circular-mode indices m, retains modes |m| ≤ M, and assumes 2M < N for alias-free orthogonality.Source: A. Layer-1 Power-Efficient DRBF Codebook · p. 5

Layer 1: DRBF codebook

  • Phase-mode truncation suppresses dominant distance-dependent phase variation because the retained-mode phase is only a fraction (M/(k0Rt))^2 of the conventional far-field phase error.This establishes the distance-robustness mechanism used by the DRBF design.Source: A. Layer-1 Power-Efficient DRBF Codebook · p. 5
  • Regularized modal compensation suppresses weak-mode amplification while preserving truncation-induced distance robustness.The regularization parameter controls the resulting modal weighting and gain behavior.Source: A. Layer-1 Power-Efficient DRBF Codebook · p. 6

Codebook construction

  • The DRBF design searches candidate modal orders and regularization values using exact spherical-wave absolute gain, then generates all sectors by UCA rotation.Only one reference sector requires offline optimization because of rotational symmetry.Source: A. Layer-1 Power-Efficient DRBF Codebook · p. 6

B. Layer-2: Full-Precision Codebook

The full-precision codebook samples the joint angular–radial domain using local resolution expressions and validates coverage with exact spherical-wave gain. Its Cartesian-product construction provides explicit local guarantees while retaining tunable resolution and offline complexity.

Resolution-aware sampling

  • The FP codebook selects angular and radial sampling intervals according to the minimum resolvable distance.The sampling is derived from the local near-field gain model and its angular–radial displacement relations.Source: B. Layer-2: Full-Precision Codebook · p. 6
  • The local gain model maps angular and radial displacements to a normalized manifold correlation governed by the product of three zeroth-order Bessel terms.The displacement relations are Δr = d cos φr and Δϕ = d sin φr/r.Source: B. Layer-2: Full-Precision Codebook · p. 6
  • For τ = 0.75, the amplitude-gain criterion becomes |J0(dβ1)J0(dβ2)J0(dβ3)| = 0.5 in the local regime.This links the EDoF threshold to the amplitude-gain criterion used to determine FP sampling intervals.Source: B. Layer-2: Full-Precision Codebook · p. 6

Codebook construction

  • The resulting FP codebook is the Cartesian product of L1 angular and L2 radial grid points, with size Nc = L1L2.The angular grid is made distance-independent using the adopted approximation for Δϕτ.Source: B. Layer-2: Full-Precision Codebook · pp. 6–7

Coverage validation

  • The exact criterion evaluates angle–distance coupling throughout every cell and requires g2D_min ≥ 0.5 for τ = 0.75.Construction complexity is O(NL1L2), and resolution is controlled by τ.Source: B. Layer-2: Full-Precision Codebook · p. 7

V. PROPOSED NEAR-FIELD HDA-BAR BEAM TRAINING

The proposed HDA-BAR beam training scheme uses two sequential stages: coarse azimuth estimation with DRBF beams followed by BAR-based high-accuracy search over a reduced FP codebook.

Two-stage training

  • Stage 1 rapidly estimates azimuth using the coarse DRBF codebook, while Stage 2 uses BAR over the reduced FP codebook to identify the optimal near-field beam.Source: §V · p. 7

A. Energy-Correlation-Based Fractional Angle Refinement

The fractional-angle refinement exploits UCA rotational symmetry by matching local energy responses against an offline distance-averaged template. Centering and normalization remove common offset and scale, enabling pilot-free online refinement.

Sector localization

  • Stage 1 sweeps L uniformly rotated DRBF codewords across all azimuth sectors and uses the maximum-energy codeword for sector-level localization.The sweep uses an orthogonal training pilot with power Ptr.Source: A. Energy-Correlation-Based Fractional… · p. 7

Template matching

  • UCA rotational symmetry makes local energy responses circular shifts of a common reference template.Only neighboring codewords around the selected sector are used for local refinement.Source: A. Energy-Correlation-Based Fractional… · p. 7
  • The reference template averages exact spherical-wave responses over offline distance samples and compares them across a fractional-offset grid.This accounts for distance variation without requiring separate online templates.Source: A. Energy-Correlation-Based Fractional… · p. 7

Fractional refinement

  • Centering and normalization remove common energy offset and overall scale before fractional-offset estimation.The template set is generated offline, so refinement requires no additional pilot and costs O((2Q + 1)|E|) online operations.Source: A. Energy-Correlation-Based Fractional… · pp. 7–8

B. BAR-Based Fine-Grained Beam Search

The BAR-based fine-grained search prunes the FP codebook using a refined azimuth estimate, then adaptively probes correlated candidates with a DAC-kernel Bayesian regression model.

Codebook reduction

  • The reduced codebook retains candidates near the refined azimuth estimate and targets the maximum received signal power.The pruning interval preserves one Layer-1 angular interval on either side of the estimate.Source: B. BAR-Based Fine-Grained Beam Search · p. 8

Bayesian regression

  • The reduced-codebook beam-energy vector is modeled with Gaussian-process regression using a covariance matrix built from candidate-codeword correlations.The energy perturbation is approximated as Gaussian for tractable modeling in high-SNR large-scale antenna systems.Source: B. BAR-Based Fine-Grained Beam Search · pp. 8–9

DAC kernel

  • The DAC kernel models the X-shaped angular–range correlation and local smoothness of near-field UCA beam energies.Its hyperparameters control the correlation-pattern opening angle and widths.Source: B. BAR-Based Fine-Grained Beam Search · p. 8

Adaptive search

  • BAR iteratively selects probes by balancing posterior mean exploitation and uncertainty-driven exploration.Posterior statistics are updated after each observation, and the next candidate maximizes the acquisition function.Source: B. BAR-Based Fine-Grained Beam Search · pp. 8–9

Two-stage search

  • Stage 2 initializes with the outermost radial codeword nearest the refined azimuth, then explores both angle and range.This initialization compensates for Stage 1 providing an angular prior without a reliable range estimate.Source: B. BAR-Based Fine-Grained Beam Search · pp. 8–9

C. Complexity and Overhead Analysis

The HDA-BAR procedure reduces online training overhead by combining DRBF sweeping with reduced-codebook Bayesian search, while precomputing codebook and kernel quantities offline.

Online complexity

  • Incremental Cholesky updates give Stage 2 search complexity O(Tmax^3), with acquisition maximization contributing O(NredTmax).Only required marginal posterior variances are evaluated during the search.Source: C. Complexity and Overhead Analysis · p. 9

Complexity components

  • The online complexity includes O(L + (2Q + 1)|E|) for Stage 1 and linear candidate pruning in Nred.Codebooks, templates, and DAC-kernel entries are precomputed offline.Source: C. Complexity and Overhead Analysis · p. 9

Algorithm structure

  • Stage 1 sweeps the DRBF codebook and refines azimuth, while Stage 2 prunes the FP codebook and performs BAR probing.Algorithm 2 updates posterior statistics and selects subsequent probes adaptively.Source: C. Complexity and Overhead Analysis · p. 9

VI. SIMULATION RESULTS

The simulations verify the UCA resolution analysis, codebook coverage, DRBF robustness, and HDA-BAR performance. The proposed training achieves near-exhaustive beamforming performance with substantially lower overhead.

Simulation setup

  • M⋆ = 96 and ζrel = 39.81 produce the selected Layer-1 operating point using the exact spherical-wave manifold.This setting retains 193 modes and achieves a worst absolute amplitude gain of −13.71 dB over the assigned sector.Source: §VI · p. 10

Resolution verification

  • The UCA achieves a smaller minimum resolvable distance than the ULA under matched element count and physical aperture.The simulation trends also show that minimum resolvable distance increases with user distance and decreases with aperture.Source: §VI · p. 10

DRBF evaluation

  • Regularized DRBF improves worst-case absolute amplitude gain over strict inverse-modal beams while retaining limited distance variation.The strict inverse amplifies weak retained Bessel modes, whereas regularization suppresses that amplification; the far-field beam remains affected by range-dependent defocusing.Source: §VI · p. 10
  • 2.26° minimum DRBF half-amplitude width is 10.43× the 0.217° far-field width, with 1.51 dB excess worst-case loss and 0.51 dB range ripple.The wider DRBF beam maintains sector containment over the evaluated service region while trading angular sharpness for distance robustness.Source: §VI · pp. 10–11

FP codebook coverage

  • g2D_min = 0.781 (−2.15 dB) exceeds the prescribed 0.5 (−6.02 dB) threshold, verifying continuous Layer-2 FP coverage without angle–distance gaps.The audit uses exact-spherical unit-norm codewords and evaluates the complete region despite displaying only selected cells and radial samples.Source: §VI · p. 11

Achievable-rate evaluation

  • The proposed two-stage search approaches near-field exhaustive-search performance as reference SNR increases and consistently outperforms near-field TPBT and far-field exhaustive search.At lower SNR, noise-induced Stage-1 refinement errors propagate through candidate pruning, but the gap narrows rapidly with increasing SNR.Source: §VI · p. 11

Beam training performance

  • DAC-based BAR converges faster and achieves higher rate than SE-based BAR and sequential exhaustive search over the displayed Stage-2 overhead range.The comparison uses the candidate set produced by the M = 96 Layer-1 estimator, with the performance gap narrowing as overhead increases.Source: §VI · pp. 11–12
  • 384 probing slots reduce overhead by 99.66% relative to scanning 114368 FP codewords while HDA-BAR approaches near-field exhaustive-search rate.The procedure uses 256 Stage-1 probes and at most 128 Stage-2 probes, and outperforms near-field TPBT and far-field exhaustive search.Source: §VI · p. 12

VII. CONCLUSION

The paper establishes UCA near-field resolution advantages and uses them to design a resolution-aware hierarchical codebook with low-overhead HDA-BAR training. For the considered configuration, the method preserves near-exhaustive performance while requiring only 384 probing slots.

Conclusion

  • Under matched antenna count and physical aperture, UCAs achieve smaller minimum resolvable distance than ULAs, indicating superior near-field spatial resolution and multiplexing potential.The conclusion attributes this advantage to exploiting UCA geometric properties in the joint angular and distance domains.Source: §VII · p. 12
  • The proposed hierarchical codebook combines Layer-1 regularized DRBF with Layer-2 FP sampling based on minimum resolvable distance.The DRBF design retains distance robustness while improving absolute amplitude gain under unit-norm transmission.Source: §VII · p. 12
  • HDA-BAR requires 384 probing slots, reducing training overhead by approximately 99.66% relative to near-field exhaustive search.This result is reported for the considered array configuration.Source: §VII · p. 12

APPENDIX A

The appendix simplifies the UCA gain-matrix derivation using Jacobi–Anger expansions, discrete phase-mode orthogonality, and small-argument Bessel approximations under the adopted system assumptions.

Gain-matrix simplification

  • Jacobi–Anger expansion transforms the UCA gain expressions into sums involving Bessel functions and phase-mode indices.The expansion is applied to obtain further expressions for the gain-matrix entries.Source: Appendix A · pp. 12–13

Mode orthogonality

  • UCA phase-mode orthogonality makes the summation equal N only when the combined mode index is an integer multiple of N; otherwise it is zero.For sufficiently large UCAs and dominant low-order modes, nonzero aliasing terms with q ≠ 0 are negligible.Source: Appendix A · p. 13
  • The dominant contribution follows from q = 0, giving m′ = −2m − 2ν and J−2m−2ν(x) = J2m+2ν(x).This relation produces the mode-index reduction used in the subsequent approximation.Source: Appendix A · p. 13

Small-argument approximation

  • The product of Bessel terms is approximated using the small-argument regime because δdβ1, δdβ2, and δdβ3 are sufficiently small.Retaining only the zeroth-order term, m = 0 and ν = 0, yields the approximate expression in (12).Source: Appendix A · p. 13

APPENDIX B

The appendix analyzes UCA phase-mode responses through stationary-phase approximations and compares finite-range behavior with the dense-UCA far-field limit, while controlling weak modes through regularization.

Stationary-phase analysis

  • The dense-UCA analysis projects the exact spherical-wave manifold onto the m-th phase-mode basis vector before applying a stationary-phase approximation.The comparison is developed around θ = 0 using a local expansion of the propagation distance.Source: Appendix B · p. 13
  • The dominant stationary regions near θ = 0 and θ = π acquire the same range-dependent phase increment relative to the far-field response.This allows the finite-to-far-field modal ratio to be formed after removing a mode-independent phase.Source: Appendix B · p. 13

Modal approximation

  • The modal approximation includes stationary-phase amplitude variation, higher-order path terms, and finite-array modal aliasing through εm.Equation (74) applies to retained modes away from zeros of Jm(k0Rt).Source: Appendix B · p. 13

Modal equalization

  • Strict inverse-modal equalization produces the normalized on-axis response, whose squared amplitude is analyzed using the expansion cos z = 1 − z^2/2 + O(z^4).The resulting substitutions complete the proof of Lemma 2, including the factor (M/(k0Rt))^2.Source: Appendix B · pp. 13–14

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