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Near-Field Rainbow: Wideband Beam Training for XL-MIMO

Mingyao Cui, Linglong Dai, Zhaocheng Wang, Shidong Zhou, Ning Ge

arXiv:2205.03543v1cs.ITeess.SP

TL;DR

Wideband XL-MIMO suffers severe array-gain loss because different frequencies focus at different locations. This paper controls that near-field beam split with time-delay beamforming, enabling simultaneous multi-location search and near-optimal beam training with low overhead.

  • Problem

    Wideband XL-MIMO experiences near-field beam split, which causes severe array-gain loss when beams at different frequencies focus at different locations.

  • Method

    The paper derives a near-field rainbow beamforming vector that controls frequency-dependent beam locations and generates multiple focused beams through one RF chain.

  • Results

    The proposed scheme achieves near-optimal near-field beam training with much-reduced training overhead, while covering multiple angles and distances across the bandwidth.

  • Takeaways & Limitations

    Near-field beam split can be exploited rather than only compensated, making it useful for fast wideband near-field beam training.

Abstract

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Wideband extremely large-scale multiple-input-multiple-output (XL-MIMO) is a promising technique to achieve Tbps data rates in future 6G systems through beamforming and spatial multiplexing. Due to the extensive bandwidth and the huge number of antennas for wideband XL-MIMO, a significant near-field beam split effect will be induced, where beams at different frequencies are focused on different locations. The near-field beam split effect results in a severe array gain loss, so existing works mainly focus on compensating for this loss by utilizing the time delay (TD) beamformer. By contrast, this paper demonstrates that although the near-field beam split effect degrades the array gain, it also provides a new possibility to realize fast near-field beam training. Specifically, we first reveal the mechanism of the near-field controllable beam split effect. This effect indicates that, by dedicatedly designing the delay parameters, a TD beamformer is able to control the degree of the near-field beam split effect, i.e., beams at different frequencies can flexibly occupy the desired location range. Due to the similarity with the dispersion of natural light caused by a prism, this effect is also termed as the near-field rainbow in this paper. Then, taking advantage of the near-field rainbow effect, a fast wideband beam training scheme is proposed. In our scheme, the close form of the beamforming vector is elaborately derived to enable beams at different frequencies to be focused on different desired locations. By this means, the optimal beamforming vector with the largest array gain can be rapidly searched out by generating multiple beams focused on multiple locations simultaneously through only one radio-frequency (RF) chain. Finally, simulation results demonstrate the proposed scheme is able to realize near-optimal nearfield beam training with a very low training overhead.

I. INTRODUCTION

Wideband XL-MIMO can support Tbps-rate 6G communications, but its large aperture and bandwidth create near-field beam split that causes array-gain loss. Existing approaches mainly compensate for this loss, while the paper identifies near-field beam split as an opportunity for faster near-field CSI acquisition and beam training.

  • Motivation: XL-MIMO combines many antennas and wide bandwidth to provide spatial multiplexing and beamforming gains for future Tbps-rate 6G communications.Its large aperture can compensate for severe mmWave or THz path loss while enabling tens of GHz-wide bandwidth.
  • Motivation: Near-field propagation becomes essential for high-frequency XL-MIMO because electromagnetic wavefronts must be modeled as spherical rather than planar.The near-field boundary grows with the square of array aperture and signal frequency; a 1-meter array at 30 GHz has a 200-meter near-field region.
  • Problem: In wideband XL-MIMO, frequency-independent phase shifters focus different frequencies at different locations, producing near-field beam split and severe array-gain loss.The resulting beams cannot all align with a user at one target location.
  • Prior Work: Time-delay beamformers can compensate for near-field beam split by partitioning the array into subarrays and correcting spherical-wave group delays.This can focus the full bandwidth at the desired spatial angle and distance.
  • Research Gap: Existing near-field CSI acquisition methods either mismatch spherical near-field propagation, neglect wideband beam split, or require potentially unacceptable pilot overhead to estimate angle and distance jointly.The literature had not studied accurate wideband XL-MIMO near-field CSI acquisition with acceptable pilot overhead.

B. Our Contributions

The paper turns near-field beam split from an array-gain problem into a controllable mechanism for fast wideband near-field beam training. By designing time delays, it generates multiple frequency-dependent beams across controllable angular and distance ranges, reducing training overhead while retaining strong performance.

  • Controllable near-field beam split: Time-delay circuits can flexibly control both the angular and distance coverage of beams across different frequencies, enabling controllable near-field beam split.The paper terms this controllable effect the near-field rainbow, by analogy with prism-induced dispersion.
  • Controllable near-field beam split: The near-field rainbow enables multiple beams to focus on multiple locations simultaneously using only one RF chain.This capability provides a new use for near-field beam split beyond compensating its array-gain loss.
  • Fast beam training scheme: The proposed training scheme generates multiple beams over angular ranges within a distance range, then searches different distance ranges across time slots by adjusting delays.Spatial angle is searched in a frequency-division manner, while distance is searched in a time-division manner.
  • Fast beam training scheme: Unlike exhaustive training, the scheme searches multiple locations in each time slot, significantly reducing beam-training overhead.Exhaustive near-field training searches only one location per time slot.
  • Simulation findings: Simulations show satisfactory average-rate performance at significantly reduced training overhead, outperforming existing far-field schemes in the near-field range.The scheme also works in the far-field range because it automatically decays to far-field beam training there.
  • Simulation findings: The proposed near-field beam-training scheme automatically decays to far-field beam training in far-field scenarios.This supports operation in both near-field and far-field ranges.

B. Near-Field Beam Split

In wideband XL-MIMO, frequency-independent phase-shifter beamforming focuses different frequencies at different locations, producing near-field beam split and severe array-gain loss. The effect becomes more pronounced as frequency moves away from the carrier.

  • Frequency-dependent wideband array responses mismatch frequency-independent phase-shifter beamfocusing vectors.This mismatch causes beams at different frequencies to focus on different locations.
  • The beam at frequency f_m focuses on a location determined by maximizing the array-gain function G.The optimum satisfies the phase and distance-related arguments of G simultaneously reaching zero.
  • Because the desired user location differs from the frequency-specific focus, wideband XL-MIMO suffers near-field beam split and array-gain loss.The misalignment is termed the near-field beam split effect.
  • The effect should be addressed particularly when the system bandwidth is very large.
  • 50% of subcarriers suffer more than 50% array gain loss with a 30 GHz carrier, 1 GHz bandwidth, and 256-element ULA.

C. Time-Delay Based Beamformer

A time-delay beamformer generates frequency-dependent beamforming vectors whose beams can remain focused on one desired location across the entire bandwidth. This mitigates near-field beam split when line-of-sight location information is available.

  • Time-delay beamformers are used to generate frequency-dependent beams that match frequency-dependent wideband channels.
  • The n-th antenna’s adjustable delay is represented through a frequency response [w_m]_n = 1/√N_t e^-j2πf_mτ(n)′.The delay parameter τ(n)′ is converted into an adjustable distance r(n)′ = cτ(n)′.
  • Setting r(n)′ = ndθ′ − n^2d^2α′ makes the delay-based beamforming vector share the form of the near-field array response.θ′ and α′ are adjustable delay parameters.
  • The proposed discussion uses a time-delay beamformer directly, while delay-phase precoding and other low-power architectures remain future work.
  • The beam at frequency f_m focuses on θ_m = θ′ and r_m = r′ = (1−θ′^2)/(2α′), independently of f_m.
  • Choosing θ′ = θ_0 and α′ = (1−θ_0^2)/(2r_0) focuses beams across the entire bandwidth at (r_0, θ_0).Under this setting, the near-field beam split effect can be mitigated.

III. MECHANISM OF NEAR-FIELD CONTROLLABLE BEAM SPLIT

The paper shows that time-delay parameters can control near-field beam split in both angular and distance dimensions. This controllable beam split enables simultaneous multi-location beam generation for fast near-field training with one RF chain.

  • Time-delay circuits can control both the angular and distance coverage ranges of beams in near-field XL-MIMO.The resulting property is termed near-field controllable beam split.
  • For a time-delay beamformer, adjustable parameters θ′ and α′ determine the beam focus and its frequency-dependent split.The beam focus is characterized through the array-gain function G.
  • If p ≠ 0 or q ≠ 0, θ_m or α_m depends on frequency, so beams at different frequencies split toward different locations.
  • When θ′ lies within [−1, 1], p can be zero and the far-field beam split effect is eliminated.
  • When θ′ is outside [−1, 1], p cannot be zero, making θ_m a function of frequency and inducing beam split again.For θ′ = 1.5 at the center frequency, p = −1 gives θ_c = −0.5.
  • Unlike phase-shifter beamforming, whose split degree is fixed, time-delay beamforming controls split degree by adjusting θ′.This adjustment indirectly controls the integer p and the beam-split pattern.
  • Adjusting beam coverage through controllable split can support fast CSI acquisition and near-field beam training with one RF chain.

B. Near-Field Controllable Beam Split

The near-field controllable beam split mechanism lets a TD beamformer deliberately control frequency-dependent beam locations across angle and distance. This controllability, termed the near-field rainbow, supports flexible multi-location beam patterns.

  • Feature 1: When p ≠ 0 or q ≠ 0, beam locations depend on frequency, so different frequencies focus on different locations despite using a TD beamformer.The resulting split can be induced rather than merely compensated.
  • Feature 2: An abnormal distance parameter α′ < 0 forces q > 0, introducing frequency-dependent focusing distances, whereas actual α′ > 0 permits αm = α′ for all beams.Thus, α′ controls splitting along the distance dimension.
  • Feature 3: Near-field controllable beam split jointly adjusts beam angle θm and distance rm, whereas the far-field special case controls only the angle.The near-field setting therefore supports more flexible beam-location patterns.
  • Controllable patterns: Designing θ′ and α′ appropriately allows beams across the bandwidth to occupy multiple angles and distances simultaneously.The angle-only and distance-only patterns are illustrated in Fig. 5(a) and Fig. 5(b), respectively.
  • Near-field rainbow: The TD beamformer can flexibly adjust the angular and distance ranges covered across the bandwidth, motivating the name “near-field rainbow.”The paper compares this frequency-dependent spreading to prism-induced dispersion and links designed rainbow patterns to efficient beam management.

IV. PROPOSED BEAM TRAINING SCHEME

Exhaustive near-field beam training searches a codebook spanning sampled angles and distances, but its overhead grows with the full codebook size. The section establishes the training procedure and its practical overhead problem.

  • Near-field beam training: Near-field beam training selects a beamfocusing vector from a codebook whose codewords cover desired angles and distances.Each codeword corresponds to a unique location, and the procedure estimates the user’s location from measured received powers.
  • Codebook construction: The exhaustive scheme samples U angles and S distance rings, then searches the entire codebook to obtain the optimal beamfocusing vector.When S = 1, the near-field codebook reduces to the far-field codebook.
  • Training overhead: T1 = US time slots are required for exhaustive near-field beam training.The scheme measures one codebook location per time slot before selecting the location with the largest received power.
  • Beam selection: After estimating the strongest location, the base station generates a near-field beam aligned with that location for serving the user.The resulting beamfocusing gain is described as near-optimal.
  • Motivation: The exhaustive search has unacceptable practical overhead because the near-field codebook is much larger than the far-field codebook, with US ≫ U.The section therefore motivates a low-pilot-overhead alternative for XL-MIMO.

B. Proposed Near-Field Beam Training Scheme

The proposed near-field rainbow scheme uses frequency-dependent angle spreading to search angles in parallel and time division to search distance rings. It thereby reduces the search burden while retaining beam-gain-based selection.

  • B. Proposed Near-Field Beam Training Scheme: A TD beamformer generates beams occupying the entire potential angular range across frequencies, allowing multiple physical locations to be measured in one time slot through one RF chain.Distance rings are varied across time slots.
  • B. Proposed Near-Field Beam Training Scheme: The scheme uses abnormal angle parameters and actual distance parameters to search the optimal angle by frequency division and the distance ring by time division.This replaces exhaustive joint searching with separated angle and distance searches.
  • B. Proposed Near-Field Beam Training Scheme: The abnormal delay parameter θ′ is designed so frequency-dependent angles θm = θ′ + (2pfc)/fm cover the potential angular range.The central-frequency beam is aligned with a predefined angle θc, and the resulting angle varies monotonically with frequency under the stated design.
  • B. Proposed Near-Field Beam Training Scheme: The distance search samples S distance rings over α ∈ [0, αmax] in different time slots while retaining the frequency-dependent angular pattern.Received signals are used to estimate beamfocusing gain and identify the best time slot and subcarrier.
  • B. Proposed Near-Field Beam Training Scheme: The estimated angle is recovered from the selected subcarrier frequency, after which the base station generates the beamfocusing vector for data transmission.The gain estimate does not require the specific value of ∥βm∥2, provided its relationship with frequency is available.
  • B. Proposed Near-Field Beam Training Scheme: The method’s training overhead is determined only by searching the optimal distance ring because angle search occurs through frequency division.The paper presents this as a significant reduction relative to exhaustive training.

C. Comparison on the Beam Training Overhead

The proposed scheme reduces beam-training overhead from exhaustive joint angle-distance search to distance-ring search alone. In the example configuration, this yields a large reduction in required time slots.

  • C. Comparison on the Beam Training Overhead: T1 = US time slots are required for exhaustive near-field beam training, whereas the proposed near-field rainbow scheme requires T2 = S.The proposed scheme removes the multiplicative angle-search factor from the time-slot overhead.
  • C. Comparison on the Beam Training Overhead: With N = 256, fc = 60 GHz, and ρmin = 2 m, U = 256 and S = 10, giving T2 = 10 versus T1 = 2560.The example directly quantifies the time-slot difference between the proposed and exhaustive schemes.

A. The Demonstration of Near-Field Controllable Beam Split

The TD beamformer produces a near-field rainbow by focusing different frequencies on multiple angles and distances. The proposed training scheme uses this effect to achieve near-optimal average rates with low overhead across near- and far-field conditions.

  • Different-frequency beams focus on multiple angles within the distance ring and cover the angular range [−0.2, 0.2].
  • Different-frequency beams focus on multiple distances at a fixed angle, covering the entire distance range.
  • The proposed scheme considers both angle and distance information while exploiting the near-field rainbow to avoid exhaustive angular search.
  • 96% of the average rate benchmark is achieved with only 8 training overhead.
  • The proposed scheme outperforms existing far-field and near-field schemes across SNR and achieves near-optimal average-rate performance.
  • Its performance remains robust across distances from the near-field to the far-field and across all considered angles.

VI. CONCLUSIONS

The paper reveals that TD circuits can control near-field beam splitting and uses this capability for fast beam training. Simulations verify broad spatial coverage, near-optimal rates with reduced overhead, and robustness to distance and angle.

  • TD circuits can flexibly control the degree of near-field beam split.
  • The proposed scheme searches the optimal angle in a frequency-division manner and the optimal distance in a time-division manner.
  • TD-generated beams over the entire bandwidth cover multiple angles and distances.
  • The scheme achieves a near-optimal average rate with much-reduced training overhead and remains robust to distance and angle.
  • The near-field beam split effect also provides a possibility for fast near-field CSI acquisition.
  • Future work will extend the near-field rainbow mechanism to delay-phase precoding architectures.

APPENDIX A. THE PERIODICITY OF G(x, y)

Appendix A proves that G(x, y) is periodic by showing that the relevant phase factor equals one for integer multiples of 2π.

  • G(x, y) is periodic because its period shift creates an integer-multiple-of-2π phase factor.
  • The identity e^j2πn(qn−p) = 1 establishes the periodicity result.
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