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Near-Field Wideband Beamforming for Extremely Large Antenna Arrays

Mingyao Cui, Linglong Dai

arXiv:2109.10054v4cs.ITeess.SP

TL;DR

Near-field beam split in wideband ELAA–THz systems makes different frequencies focus at distinct locations, causing substantial beamforming-gain loss. The paper approximates the channel with piecewise-far-field sub-arrays and applies PDF beamforming using joint phase-shifter and time-delayer control. It also introduces effective Rayleigh distance as a gain-loss-based measure of the practical near-field range.

  • Problem

    Near-field beam split causes different frequencies to focus at distinct locations, while existing mitigation methods target far-field beam split and rely on planar-phase properties unavailable in the near field.

  • Method

    The paper partitions the ELAA into small sub-arrays, approximates each local channel as far-field, and uses PDF with joint phase-shifter and time-delayer control.

  • Results

    The PDF method mitigates near-field beam split, and effective Rayleigh distance more accurately quantifies the practical near-field range than classical Rayleigh distance.

  • Takeaways & Limitations

    Beamforming-gain evaluation provides a practical basis for distinguishing far-field and near-field regions in ELAA communications.

  • Takeaways & Limitations

    The sub-array analysis assumes the user lies in the far-field region of each sub-array over the specified activity range.

Abstract

from arXiv · show

The natural integration of extremely large antenna arrays (ELAAs) and terahertz (THz) communications can potentially achieve Tbps data rates in 6G networks. However, due to the extremely large array aperture and wide bandwidth, a new phenomenon called "near-field beam split" emerges. This phenomenon causes beams at different frequencies to focus on distinct physical locations, leading to a significant gain loss of beamforming. To address this challenging problem, we first harness a piecewise-far-field channel model to approximate the complicated near-field wideband channel. In this model, the entire large array is partitioned into several small sub-arrays. While the wireless channel's phase discrepancy across the entire array is modeled as near-field spherical, the phase discrepancy within each sub-array is approximated as far-field planar. Built on this approximation, a phase-delay focusing (PDF) method employing delay phase precoding (DPP) architecture is proposed. Our PDF method could compensate for the intra-array far-field phase discrepancy and the inter-array near-field phase discrepancy via the joint control of phase shifters and time delayers, respectively. Theoretical and numerical results are provided to demonstrate the efficiency of the proposed PDF method in mitigating the near-field beam split effect.Finally, we define and derive a novel metric termed the "effective Rayleigh distance" by the evaluation of beamforming gain loss. Compared to classical Rayleigh distance, the effective Rayleigh distance is more accurate in determining the near-field range for practical communications.

I. INTRODUCTION

ELAA–THz integration promises Tbps-rate 6G communications but introduces near-field and wideband beam-split challenges. The paper proposes piecewise-far-field modeling, PDF beamforming, and effective Rayleigh distance analysis to address them.

  • Motivation: ELAA apertures and THz bandwidth support ultra-high-speed, potentially Tbps-rate 6G communications.THz antennas also facilitate ELAA deployment.
  • Motivation: ELAA scaling changes channel modeling from far-field planar waves to near-field spherical wavefronts across the array.Classical Rayleigh distance grows with the square of array aperture normalized by wavelength.
  • Problem: Wideband frequency dependence causes beams at different frequencies to focus at distinct locations, producing near-field beam split and beamforming-gain loss.Existing approaches mainly target far-field beam split and rely on planar-phase properties unavailable in the near field.
  • Contributions: The paper partitions the ELAA into small sub-arrays, decomposing phase discrepancy into inter-array near-field and intra-array far-field components.The model is implemented through a DPP architecture combining phase shifters and time delayers.
  • Contributions: The PDF method mitigates near-field beam split, while effective Rayleigh distance uses beamforming-gain loss to characterize practical near-field range more accurately than classical Rayleigh distance.The paper reports theoretical analysis and numerical validation.

B. Discussion on Near-Field Beam Split

Near-field wideband beamforming combines spherical-wave focusing with frequency-dependent phase mismatch. As a result, different frequencies focus at different physical locations and can substantially lose gain at the intended receiver location.

  • Far- and near-field models: Far-field approximation makes channel phase linear in antenna index and produces direction-dependent beamforming.The approximation becomes inaccurate for very large antenna indices.
  • Wideband beam split: For a 512-antenna ULA at 100 GHz, the Rayleigh distance is about 400 m, requiring the spherical-wave channel model over that near-field range.The near-field region expands substantially as antenna count increases.
  • Far- and near-field models: Near-field beams focus according to both receiver distance and direction, motivating the term beamfocusing.At the center frequency, energy is focused on the receiver location (r, θ).
  • Wideband beam split: Wideband phase mismatch prevents constructive addition at the intended location when fm ≠ fc, reducing normalized beamforming gain and splitting beam energy.The effect differs between far-field directional splitting and near-field location-dependent splitting.
  • Wideband beam split: A 512-antenna, 100-GHz array with B = 5 GHz has over 50% of subcarriers losing at least 60% beamforming gain.The example illustrates the practical severity of near-field beam split.

III. PROPOSED METHODS

The proposed method approximates the near-field channel by partitioning the ELAA into sub-arrays whose local channels are treated as far-field. This piecewise model enables PDF beamforming based on piecewise-linear phase structure.

  • III. Proposed Methods: The piecewise-far-field model provides a manageable, piecewise-linear approximation to the nonlinear near-field channel.The PDF method is built on this approximation to mitigate near-field beam split.
  • A. Piecewise-Far-Field Channel Model: With 256 antennas at 100 GHz, K = 4 sub-arrays produce a piecewise-far-field phase profile that closely approximates the true near-field channel.The figure compares far-field, near-field, and piecewise-far-field channel models and their phase versus antenna index.
  • A. Piecewise-Far-Field Channel Model: Partitioning the ELAA reduces each sub-array’s near-field range, allowing a receiver to be near-field relative to the entire array but far-field relative to each sub-array.This is the key modeling assumption behind the decomposition.
  • A. Piecewise-Far-Field Channel Model: The array is divided into K sub-arrays with P adjacent antennas each, satisfying N = KP.Sub-channel quantities are defined between each sub-array and each user.
  • A. Piecewise-Far-Field Channel Model: Within each sub-array, the approximated channel phase is linear in local antenna index and therefore has far-field character.Different sub-arrays can still have different distances and directions to the user.

B. Proposed Phase-Delay Focusing Method

The PDF method uses a piecewise-far-field model to separate inter-array near-field and intra-array far-field phase discrepancies. It jointly controls TTDs and PSs in a DPP architecture to focus broadband energy at the receiver location.

  • The piecewise-far-field model decomposes phase discrepancy into inter-array near-field and intra-array far-field components.The model treats each small sub-array as far-field while retaining the entire array’s near-field behavior.
  • PDF compensates inter-array phase using TTDs and intra-array phase using PS-based sub-arrays.TTDs provide frequency-dependent phase control, while PSs generate frequency-independent planar-wave compensation.
  • The proposed design focuses beam energy across the entire bandwidth on the receiver location (r, θ).This joint PS and TTD control is termed phase-delay focusing.
  • The kth TTD uses an adjustable distance parameter, while the kth PS-based sub-array uses an adjustable phase parameter.The resulting sub-array beamforming vector combines frequency-independent PS phase with frequency-dependent TTD phase.
  • The optimization selects distance parameters {r′_k} to maximize beamforming gain at user location (r, θ) across the entire bandwidth.The derived frequency-independent solution is optimal for all subcarriers under the stated formulation.

C. Analysis of Beamforming Gain Performance

The analysis expresses PDF’s average beamforming gain through separate wideband and geometry loss factors. It shows that sub-array size and user geometry govern the remaining loss, with a representative configuration achieving more than 93% average gain.

  • The average-gain analysis approximates Dirichlet sinc terms with a two-variable quadratic fit before separating wideband and near-field factors.For P = 32, B = 5 GHz, and fc = 100 GHz, the fit is reported as close to the Dirichlet sinc function.
  • PDF beamforming gain loss arises from wideband loss γ(B, fc, P) and geometry loss ξ(r, θ, D).The two factors separately capture intra-sub-array beam split and location-dependent near-field geometry effects.
  • Geometry loss ξ(r, θ, D) increases with |θ| when r > 1/2D, so larger arrival angles produce greater gain loss.The geometry factor accounts for both far-field and near-field regions through the distance parameter r.
  • A smaller sub-array antenna count P reduces intra-array beam-split loss γ(B, fc, P).The analysis states that γ increases with P within the specified range and can approach zero for an appropriate P.
  • 93.93% average beamforming gain is achievable for P = 32, B = 5 GHz, fc = 100 GHz, r = 10 m, D = 0.5 m, and θ = π/3.Under these parameters, γ(B, fc, P) ≈ 0.081 and ξ(r, θ, D) ≈ 0.7496.

D. Extension to Multi-User Hybird Beamforming

The PDF method extends to multi-user hybrid beamforming by applying the DPP architecture per RF chain. Analog beamformers align users individually, while digital zero-forcing precoding removes inter-user interference and enforces power normalization.

  • The multi-user extension employs the same DPP architecture for each RF chain and makes the analog beamformer frequency-dependent.The multi-user analog beamformer is denoted F_m = [f_0,m, f_1,m, · · ·, f_U−1,m].
  • The PDF algorithm aligns each analog beamforming vector with its corresponding user in the analog domain.Distance parameters are determined from −r_u,k, while phase parameters are determined from −sin θ_u,k.
  • The algorithm shifts distance parameters using L = min_k{r′_u,k} to ensure nonnegative TTD distance parameters.The resulting beamforming vectors and analog beamformer are constructed subcarrier by subcarrier.
  • Zero-forcing digital precoding eliminates inter-user interference after analog beamformer construction.The digital precoder is subsequently normalized to total transmit power ρ per subcarrier.

IV. EFFECTIVE RAYLEIGH DISTANCE

The paper contrasts classical Rayleigh distance, based on phase error, with an effective Rayleigh distance based on far-field beamforming-gain loss. The latter incorporates a loss threshold and arrival angle to identify the practical near-field boundary.

  • Classical Rayleigh distance defines the near-field boundary using the maximum phase error between spherical and planar wave models.The conventional criterion requires phase error E(r) no greater than π/8 beyond the Rayleigh distance.
  • The effective Rayleigh distance R_eff is defined where far-field beamforming-gain loss 1 − μ(R_eff, θ) equals a threshold Δ.Users with 0 < r ≤ R_eff satisfy 1 − μ(r, θ) ≥ Δ.
  • Beamforming coherence μ(r, θ) represents achievable gain when far-field beamforming serves a user at (r, θ).The coherence declines as the user approaches the base station and near-field effects become more pronounced.
  • G(β) decreases with β, so the threshold condition 1 − G(β) ≥ Δ requires β ≥ β_Δ.The threshold point satisfies G(β_Δ) = 1 − Δ.
  • For Δ = 5%, β_Δ = 0.8257, yielding R_eff = 0.367 cos^2 θ · 2D^2/λ.The value is obtained by solving the threshold equation numerically with Newton’s method.
  • R_eff adds the beamforming-loss constant C_Δ and arrival angle θ to the classical distance expression.These variables allow the metric to capture where far-field beamforming is not applicable for communications.

A. Discussion on the Fresnel Approximation

The effective Rayleigh distance is derived using the Fresnel approximation, and the resulting bound is shown to exceed the Fresnel distance for extremely large arrays.

  • A. Discussion on the Fresnel Approximation: The derivation employs the Fresnel approximation, which is accurate when distance r exceeds the Fresnel distance.The cited passage identifies the Fresnel-distance condition used in the derivation.
  • A. Discussion on the Fresnel Approximation: For typical cell-sector angles, the effective Rayleigh distance is evaluated against the Fresnel-distance condition.The sector restricts θ to between −π/3 and π/3.
  • A. Discussion on the Fresnel Approximation: N > 15.8490 is sufficient for the effective Rayleigh distance to be much longer than the Fresnel distance.Extremely large arrays with hundreds or thousands of antennas satisfy this condition.

B. Discussion on the Piecewise-Far-Field Approximation

The piecewise-far-field approximation is validated by requiring each user-to-sub-array distance to exceed the effective Rayleigh distance of an individual sub-array.

  • B. Discussion on the Piecewise-Far-Field Approximation: Each user-to-sub-array distance r_k must exceed the sub-array’s effective Rayleigh distance for the far-field model to apply.This condition is used to verify the piecewise-far-field approximation.
  • B. Discussion on the Piecewise-Far-Field Approximation: For P = 32 and f_c = 100 GHz, the effective Rayleigh distance per sub-array is upper bounded by 0.5286 m.Distances larger than 0.5286 m are described as common in mobile communications.
  • B. Discussion on the Piecewise-Far-Field Approximation: Under this small-sub-array configuration, distances above 0.5286 m allow each sub-array channel to be precisely modeled as far-field.The passage concludes that the piecewise-far-field approximation is accurate in this setting.

C. Discussion on the Number of Antennas per Sub-Array

The number of antennas per sub-array P is selected to satisfy phase-error, far-field-distance, and beamforming-gain requirements while controlling TTD deployment.

  • C. Discussion on the Number of Antennas per Sub-Array: P is designed by combining requirements from three lemmas governing phase error, sub-array far-field operation, and minimum average beamforming gain.The target gain must exceed a predefined threshold δ.
  • C. Discussion on the Number of Antennas per Sub-Array: The user activity range is modeled as r ∈ [ρ_l, ρ_h] and θ ∈ [−θ_h, θ_h], with ρ_l the least allowable user-to-BS distance.The sector bound θ_h is used when enforcing the far-field requirement.
  • C. Discussion on the Number of Antennas per Sub-Array: The worst-case ξ occurs at the sector edge because ξ decreases with |θ|, enabling maximization over r at θ = θ_h.Gradient ascent and Newton’s method are then used to obtain P_δ.
  • C. Discussion on the Number of Antennas per Sub-Array: The design satisfies P ≤ P_δ, linking the antenna partition size to the prescribed beamforming-gain threshold.The monotonic behavior of the relevant gain function supports obtaining P_δ from the threshold equation.
  • C. Discussion on the Number of Antennas per Sub-Array: For N = 400, B = 5 GHz, f_c = 100 GHz, and ρ_l = 1 m, P can be set to 40, requiring K = 10 TTDs.The reported values are P_δ ≈ 42 and the lower bound P = 1; reducing P increases TTD deployment and improves beamforming gain.

V. SIMULATION RESULTS

Simulations show that PDF mitigates near-field beam split across direction and bandwidth, while its analytical gain remains close to simulated performance under the tested conditions.

  • A. Beamforming Gain: At (r, θ) = (2 m, π/8), PDF focuses f_L, f_c, and f_H beams at the desired location, achieving more than 95% gain at f_L and f_H.Traditional near-field narrowband beamfocusing instead focuses different frequencies at different locations.
  • A. Beamforming Gain: At r = 10 m, PDF maintains more than δ = 90% average beamforming gain across θ ∈ [−θ_h, θ_h].Both PDF and narrowband beamfocusing decline with increasing |θ|, while the analysis closely matches PDF simulations.
  • A. Beamforming Gain: When B = 5 GHz, PDF achieves around 3 times higher average beamforming gain than narrowband beamfocusing.The bandwidth sweep spans 100 MHz to 10 GHz, and PDF provides near-optimal average gain in the simulation.
  • A. Beamforming Gain: At bandwidth around 10 GHz, the analytical and real PDF performances differ slightly because the relevant quantity is reduced, causing analytical gain error.The supplied passage identifies this as a limitation of the analytical result at the widest tested bandwidth.

B. Spectral Efficiency

The PDF method is evaluated against several beamforming benchmarks across distance, antenna count, and SNR. It maintains strong spectral efficiency in near-field wideband settings, while the effective Rayleigh distance better identifies the practical near-field boundary than the classical metric.

  • Distance: PDF outperforms all compared methods across user distances and approaches optimal beamforming by jointly addressing near-field and beam-split effects.The comparison uses fixed 10 dB SNR with large-scale fading compensated by transmit power control.
  • Distance: At θ = π/8, N = 256, fc = 100 GHz, and ∆ = 5%, the effective Rayleigh distance is Reff ≈31 m, matching the onset of far-field rate decline.The classical Rayleigh distance is around 98 m under the same antenna and carrier settings, whereas TTD-DPP declines below approximately 31 m.
  • Antenna count: When N < 50, all algorithms perform well; for 50 < N < 100, far-field beam split severely degrades narrowband beamfocusing and PE-AltMin.The passage further contrasts these methods with TTD-DPP and PDF in the intermediate antenna-count region.
  • SNR: At SNR = 10 dB, PDF improves spectral efficiency by more than 3 bit/s/Hz over TTD-DPP.The spectral-efficiency curves are obtained through 10000 Monte-Carlo simulations.
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