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Two-Stage Hierarchical Beam Training for Near-Field Communications

Chenyu Wu, Changsheng You, Yuanwei Liu, Li Chen, Shuo Shi

arXiv:2302.12511v1cs.ITeess.SPmath.IT

TL;DR

Near-field beam training must handle energy spread and search across both angle and distance, making exhaustive search costly. The paper uses a two-stage hierarchical method that first estimates coarse direction with a central sub-array, then refines direction and distance. It reports over 99% overhead reduction versus exhaustive search with comparable rate performance.

Abstract

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Extremely large-scale array (XL-array) has emerged as a promising technology to improve the spectrum efficiency and spatial resolution of future wireless systems. However, the huge number of antennas renders the users more likely to locate in the near-field (instead of the far-field) region of the XL-array with spherical wavefront propagation. This inevitably incurs prohibitively high beam training overhead since it requires a two-dimensional (2D) beam search over both the angular and distance domains. To address this issue, we propose in this paper an efficient two-stage hierarchical beam training method for near-field communications. Specifically, in the first stage, we employ the central sub-array of the XL-array to search for a coarse user direction in the angular domain with conventional far-field hierarchical codebook. Then, in the second stage, given the coarse user direction, we progressively search for the fine-grained user direction-and-distance in the polar domain with a dedicatedly designed codebook. Numerical results show that our proposed two-stage hierarchical beam training method can achieve over 99% training overhead reduction as compared to the 2D exhaustive search, yet achieving comparable rate performance.

I. INTRODUCTION

Near-field propagation makes beam training depend on both angle and distance, while energy spread can undermine far-field methods. The paper proposes a two-stage hierarchical design and reports substantial overhead reductions with comparable rate performance.

I. INTRODUCTION

  • Near-field channels use spherical wavefronts, and applying far-field training directly can spread beam energy across angles and degrade performance.The near-field channel depends on both spatial angle and effective distance.Source: §I · p. 1, 2) Conventional Far-field Beam Training · p. 2

3) Near-field Beam Training:

  • Prior near-field approaches include polar exhaustive search, two-phase estimation, learning-based prediction, wideband beam-split training, and rectangular-coordinate hierarchy.A further hierarchical scheme in is customized for digital precoding.Source: 3) Near-field Beam Training · p. 2

B. Our Contributions

  • The first stage uses a central sub-array with far-field hierarchical training for coarse direction; a dedicated polar-domain hierarchy then refines direction and effective distance.The sub-array mitigates the near-field energy-spread effect.Source: B. Our Contributions · p. 2
  • The proposed training overhead scales as O(log2(N)) with the number of XL-array antennas.The paper also discusses extensions to planar arrays and metasurfaces.Source: B. Our Contributions · p. 2
  • Over 99% and 95% beam training overhead reduction versus exhaustive-search and two-phase schemes, respectively, with comparable rate performance.The proposed method's reductions are reported against the two near-field benchmarks.Source: B. Our Contributions · p. 2

II. SYSTEM MODEL

The system model considers a narrow-band downlink with a base-station ULA and a single-antenna user. Its near-field channel uses spherical-wave propagation, and beam training seeks a codeword matching the optimal beamforming vector.

II. SYSTEM MODEL

  • The model considers a narrow-band downlink in which an N-antenna base-station ULA communicates with a single-antenna user.The antennas are placed along the y-axis with spacing d = λc/2.Source: §II · pp. 2–3
  • The near-field line-of-sight channel is modeled with spherical-wave propagation, with the steering vector parameterized by spatial angle and user distance.The user is assumed to lie in the Fresnel region, where amplitude variations across antennas are negligible.Source: §II · p. 3
  • The optimal base-station beamforming vector is the near-field steering vector, and training finds the codeword that matches it.The model also notes that at sufficiently large distances the near-field steering vector becomes equivalent to the conventional far-field steering vector.Source: §II · p. 3

III. BENCHMARK SCHEMES

The benchmark schemes search for the near-field beam across angle and distance, either exhaustively or in two sequential phases. The two-phase method reduces the search but retains overhead proportional to the number of antennas.

III. BENCHMARK SCHEMES

  • Near-field training must search both angular and distance domains, and exhaustive search incurs overhead equal to NS.The benchmark section presents exhaustive and two-phase methods as existing near-field schemes with overhead limitations.Source: A. Exhaustive-Search Based Near-field… · p. 3, §III · p. 3

A. Exhaustive-Search Based Near-field Beam Training

  • With N = 512 antennas and S = 6 sampled distances, exhaustive search requires 3072 training symbols.The polar-domain codebook uniformly samples angle and uses nonuniform distance sampling, with wider distance intervals at larger ranges.Source: A. Exhaustive-Search Based Near-field… · p. 3

B. Two-phase Near-field Beam Training

  • The two-phase method first sweeps far-field beams, then searches distance for each candidate angle selected from the dominant angular region.Its overhead is N + KS; at N = 512, S = 6, and K = 1, the cited example requires 518 symbols.Source: B. Two-phase Near-field Beam Training · p. 4

C. Conventional Far-field Hierarchical Beam training

Conventional far-field hierarchical beam training cannot directly resolve near-field users because full-array far-field beams spread energy across a wide angular region. The proposed method first uses a central sub-array for coarse angular search, then refines direction and effective distance with a polar-domain codebook.

C. Conventional Far-field Hierarchical Beam training

  • Conventional far-field hierarchical training uses wide beams followed by narrower beams, but is not directly applicable to near-field users because of the energy-spread effect.For N = 512, its stated overhead is 2 log2(N) = 18 training beams.Source: C. Conventional Far-field Hierarchical… · p. 4

A. Key Idea

  • The full-array near-field energy-spread effect creates strong gains across a wide angular region, making the optimal beam direction difficult to distinguish by received power.The effect arises because the user is in the far field of individual sub-arrays but the near field of the full array, giving sub-arrays different pointing directions.Source: A. Key Idea · p. 4
  • A central sub-array produces a sharp beam-gain peak at the true user direction while approximating the full-array normalized beam gain.The example uses an XL-array with N = 512 antennas and a central sub-array with Nc = 128 antennas.Source: A. Key Idea · p. 4, Figure 2 · p. 4
  • The proposed method uses a central sub-array for coarse angular search, followed by a dedicated lower-level codebook to refine direction and effective distance in the angular and distance domains.The hierarchical codebook has Lt = log2(N) layers split between upper- and lower-level codebooks.Source: A. Key Idea · pp. 4–5

B. Stage 1: Coarse User Direction Estimation

Stage 1 estimates a coarse user direction with an angular-domain hierarchical codebook. Its codewords use central sub-arrays and binary-tree search, with two tested beams per layer.

B. Stage 1: Coarse User Direction Estimation

  • Stage 1 estimates the coarse user direction with an upper-level, one-dimensional hierarchical codebook in the angular domain.The method follows the conventional far-field hierarchical beam-training approach for this initial search.Source: 1) Upper-level 1D Hierarchical… · p. 5

1) Upper-level 1D Hierarchical Codebook Design:

  • Each upper-level layer divides the angular domain equally, covers the full distance domain, and nests each beam's coverage within the union of two beams at the next layer.At layer ℓ, the design activates Nℓ = 2ℓ central antennas to form far-field beams.Source: 1) Upper-level 1D Hierarchical… · p. 5

2) Hierarchical Beam Training of Stage 1:

  • Binary-tree search tests two beams per layer, estimates the direction from the best codeword, and incurs total Stage 1 overhead of 2L beams.The activated sub-array is chosen from the array center because a non-central sub-array may not point to the desired direction and can increase estimation error.Source: 2) Hierarchical Beam Training of Stage 1 · p. 5

C. Stage 2: Fine-grained Direction-and-distance Estimation

After the coarse direction estimate, Stage 2 searches for fine-grained direction and distance using a lower-level two-dimensional hierarchical codebook. The design supports progressive binary-tree search over angular and distance coverage.

C. Stage 2: Fine-grained Direction-and-distance Estimation

  • Exhaustively searching all remaining direction-and-distance pairs requires training overhead equal to the product of candidate directions and sampled distances.The lower-level hierarchical codebook is designed to reduce this Stage 2 search overhead.Source: C. Stage 2: Fine-grained… · p. 6

1) Lower-Level 2D Hierarchical Codebook Design:

  • Each lower-level codeword covers an angular region and a distance region, with each layer jointly covering the full angular and distance domains and nesting coverage across successive layers.The codebook covers angular domain [−1, 1] and distance domain [rmin, ∞].Source: 1) Lower-Level 2D Hierarchical… · p. 6
  • Lower-level codewords correspond to near-field direction-and-distance pairs, while sampled distances for each direction are progressively divided in a binary-tree manner.The design activates central antennas at each layer and samples directions across the angular domain.Source: 1) Lower-Level 2D Hierarchical… · p. 6, §7 · p. 6

2) Hierarchical Beam Training of Stage 2:

Stage 2 progressively tests direction-and-distance codewords, using four candidates in each intermediate layer and a separate final-layer search. The worked example ends with a selected beam steering near the user's stated location.

2) Hierarchical Beam Training of Stage 2:

  • Each intermediate Stage 2 layer tests four codewords, combining two steering directions with two steering distances, then carries the best beam forward.The process repeats in the next layer to progressively resolve the user's fine-grained direction and distance.Source: 2) Hierarchical Beam Training of Stage 2 · pp. 6–7
  • In the last layer, the method selects two directions and tests codewords whose steering distances fall within the interval determined by the preceding layer's best beam.The remaining final-layer codewords are tested to determine the best near-field beam.Source: 2) Hierarchical Beam Training of Stage 2 · p. 7
  • In the N = 512 example with user location [θ, r] = [0, 40 m], the selected beam steers to [θ, r] = [−0.0019, 35 m].The example uses Lt = 9, L = 7, and S∆ = 70; the final selected codeword yields the maximum received signal power.Source: 2) Hierarchical Beam Training of Stage 2 · p. 7

V. DISCUSSIONS AND EXTENSIONS

This section discusses key parameters of the proposed beam-training design and extensions to planar arrays and metasurfaces.

  • The discussion covers beam-training parameters and extensions to systems with planar arrays and metasurfaces.Source: §V · p. 7

A. Beam Training Overhead

The proposed method has logarithmic training-overhead scaling, and its Stage 1/Stage 2 boundary balances training cost against near-field accuracy.

A. Beam Training Overhead

  • Training overhead scales as O(log2(N)), significantly lower than the benchmark near-field beam-training schemes.Stage 1 generally uses two training beams per layer, followed by four per Stage 2 layer.Source: A. Beam Training Overhead · p. 7
  • In the stated random-location example, the proposed design uses 24 training beams.The overhead is calculated as 2 × 7 + 4 + 6.Source: A. Beam Training Overhead · p. 7

B. Proper Value of L

  • For users in the Fresnel region, L can be set to Lt −2; starting Stage 2 earlier adds overhead with little spatial-resolution gain.The codeword beam gains for sampled distances of 9 m and 68 m are reported to be similar for N = 512 and L = 7.Source: B. Proper Value of L · p. 7
  • For users farther than Rc, the activated Stage 1 antenna count should keep the user within that aperture’s Fresnel boundary.The corresponding Stage 1 layer count is L = ⌈log2(NL)⌉.Source: B. Proper Value of L · pp. 7–8

C. Extensions to UPA and IRS:

The proposed beam-training method extends to UPAs by separating horizontal and vertical training, and to IRS-aided systems by reducing cascaded-channel training to near-field training for the IRS.

1) UPA:

  • For a UPA, joint 3D training can be split into sequential horizontal and vertical beam training for two perpendicular ULAs.The UPA is considered in the y-z plane, with the user in the x-y plane.Source: 1) UPA · p. 8

2) IRS/RIS:

  • The IRS extension addresses a setting where the IRS-user channel is near-field and the BS–IRS link is assumed to be far-field.Source: 2) IRS/RIS · p. 8
  • For IRS-aided training, aligning the BS beam with the BS–IRS channel and counteracting its near-field steering vector yields a pseudo phase-shift vector that serves as the transmit beam.The resulting cascaded-channel training has the form of near-field beam training for the IRS.Source: 2) IRS/RIS · p. 9

VI. NUMERICAL RESULTS

The numerical evaluation defines beam-identification success rate and achievable rate, and compares the proposed method with near-field and far-field training benchmarks.

  • The evaluation uses N = 512 antennas and f = 100 Ghz, with λc = 0.003 m, S∆ = 68.27, S = 6, and rmin = 6.14 m.Source: §VI · p. 9
  • Success rate measures whether the identified polar-domain codeword matches the optimal codeword, while achievable rate is reported in bps/Hz.Numerical results are averaged over 5000 random channel realizations.Source: §VI · p. 9
  • The benchmark set includes perfect-CSI beamforming, exhaustive-search near-field training, two-phase near-field training with K = 1, and conventional far-field hierarchical training.Source: §VI · p. 9

1) Overhead Comparison:

The proposed method reduces training overhead while retaining rate performance, with reported success and rate behavior across distance, effective SNR, and antenna count. It first finds a coarse direction, then resolves fine-grained direction and distance.

1) Overhead Comparison:

3) Impact of the Effective SNR:

2) Impact of BS-user Distance:

  • Success rate remains above 80% across user distances, while the conventional far-field scheme performs poorly at short distances below 50 m.The proposed method is described as not sensitive to BS-user distance and applicable to both far-field and near-field scenarios.Source: 2) Impact of BS-user Distance · p. 10
  • At short distances, the proposed method approaches exhaustive-search rate performance and stays close to the two-phase scheme with 24 versus 518 training symbols.At large distances, its rate performance degenerates to that of the far-field hierarchical scheme.Source: 2) Impact of BS-user Distance · p. 10

4) Impact of the Number of BS Antennas:

VII. CONCLUSION

  • The two-stage method finds a coarse user direction first, then resolves fine-grained direction and distance using stage-specific hierarchical codebooks.Source: §VII · p. 11
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