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Near-Field Dual-UPA Communications: A Generalized Geometric Approach
Li Zheng, Xing Hao, Ziru Chen, Yong Liu, Li Chen, Lin X. Cai
TL;DR
Dual-UPA near-field MIMO exposes limits of conventional planar-wave modeling and single-array Rayleigh-distance criteria. The paper develops a geometry-based distance and boundary analysis, then uses Kronecker channel structure for low-complexity beamforming, with numerical results showing validated boundaries and near-optimal rates at reduced computational complexity.
Problem
Dual-UPA near-field MIMO lacks sufficiently developed NF-FF boundary analysis and low-complexity beamforming despite joint transmitter–receiver aperture effects and high-dimensional channels.
Method
The paper derives 3D antenna distances and a closed-form NF-FF boundary, decomposes the NF channel into a Kronecker product, and designs beamforming using two lower-dimensional SVDs with water-filling.
Results
The derived boundary generalizes the Rayleigh distance, while Kronecker-SVD beamforming achieves near-optimal rate performance with reduced computational complexity.
Takeaways & Limitations
Geometric channel structure supports practical NF beamforming that preserves the main channel structure while avoiding full-dimensional SVD.
Abstract
from arXiv · showhide
This paper investigates a near-field (NF) multiple-input multiple-output (MIMO) communication system equipped with dual uniform planar arrays (UPAs). We first develop a generalized geometric model to calculate the 3D distance between arbitrary antenna elements across the transmitter and receiver panels. Leveraging the distance analysis, we derive a closed-form near-field to far-field (NF-FF) boundary for dual-UPA configurations. By exploiting the geometric structure of the UPAs, we further decompose the near-field channel matrix into a Kronecker-product of two lower-dimensional matrices. This decomposition enables a low-complexity NF beamforming design for achievable-rate maximization. Numerical results validate the analysis and demonstrate that the conventional Rayleigh distance is a special case of the generalized model. Furthermore, the proposed beamforming design achieves near-optimal rate performance while significantly reducing the computational complexity compared to state-of-the-art NF beamforming methods.
I. INTRODUCTION
Dual-UPA near-field MIMO requires geometry-aware modeling because enlarged apertures and short wavelengths invalidate planar-wave assumptions. The paper develops a generalized distance model, NF-FF boundary, and structured channel representation for this setting.
- 6G-oriented XL-MIMO and high-frequency systems challenge the conventional far-field planar-wave model because of enlarged apertures and shortened wavelengths.
- Far-field propagation depends mainly on angle, whereas near-field propagation uses spherical waves and depends on both angle and distance.
- The classical Rayleigh distance rRayl = 2D^2/λ is mainly based on single-UPA point-to-array propagation and may not capture joint transmitter–receiver aperture effects.
- Prior work covers point-to-line and point-to-planar geometries, but dual-UPA NF-FF boundaries and beamforming remain underexplored amid high-dimensional channel computations.
- The paper models 3D distances between arbitrary antennas, derives a closed-form dual-UPA boundary, and decomposes the NF channel into two lower-dimensional Kronecker factors.
- The geometric setup places parallel transmitter and receiver UPAs in a 3D Cartesian coordinate system and represents antenna positions with planar coordinates.
B. Communication Model
The communication model represents dual-UPA downlink transmission through a near-field channel whose coefficients are determined by antenna-pair distances. Transmit beamforming maps data streams to antennas, while receiver noise is modeled as Gaussian.
- Each near-field channel coefficient h_m,n is determined by the derived distance d_m,n under the uniform spherical-wave model.
- The near-field channel matrix H ∈ C^{M×N} is formed by arranging the antenna-pair channel coefficients h_m,n.
- The transmitted signal x = Ws uses beamforming matrix W to map the L data streams in s to the N transmit antennas.
- Receiver noise is modeled as additive white Gaussian noise n ∼ CN(0, σ^2I_M).
C. Problem Formulation
The problem formulation evaluates achievable rate for the dual-UPA downlink and optimizes the transmit beamforming matrix under a maximum transmit-power constraint.
- The receiver achievable rate is defined from the received signal model.
- Transmit beamforming is formulated as an achievable-rate maximization problem subject to a transmit-power constraint.
- P_max denotes the maximum transmit power available at the transmitter.
III. BOUNDARY BETWEEN NEAR-FIELD AND FAR-FIELD REGIONS
The NF-FF boundary is derived by comparing spherical- and planar-wave phase differences using a Taylor-expanded geometric distance model. The resulting condition bounds the maximum phase difference by a prescribed threshold.
- Distance and phase analysis: The exact antenna distance is expanded around the center-to-center distance, and a Taylor approximation is applied when the aperture is much smaller than the transmission distance.The approximation uses √1 + x ≈ 1 + x/2 for a fractional term significantly below one.
- Distance and phase analysis: The distance reformulation separates horizontal and vertical coordinate contributions into algebraic components used to characterize the near-field phase.The corresponding propagation phase shift is referenced to the phase between the array centers.
- Distance and phase analysis: The far-field model uses a planar propagation distance, enabling a distance difference and relative phase difference to be defined against the spherical-wave model.These phase quantities compare the exact near-field propagation with the planar-wave approximation.
- Boundary condition: The NF-FF boundary is defined where the maximum spherical-versus-planar received-signal phase difference does not exceed threshold ϕ.The resulting threshold condition leads to the boundary expression, with geometric bounds supported by coordinate definitions and the triangle inequality.
IV. BEAMFORMING DESIGN
The proposed beamforming design uses the geometric structure of dual UPAs to reduce achievable-rate maximization complexity. It combines a Kronecker channel representation with low-dimensional SVD-based processing.
- Beamforming design: The geometry-based design expresses the near-field channel in Kronecker-product form before constructing the beamformer.The method targets achievable-rate maximization and replaces full-dimensional processing with two lower-dimensional SVDs followed by water-filling.
A. Kronecker Product-Based Channel
Algebraic distance decoupling factorizes each near-field channel coefficient into center-distance attenuation and independent horizontal and vertical terms. This yields a Kronecker product of two lower-dimensional component matrices.
- Channel factorization: Each channel coefficient factors into a center-to-center term, a y-coordinate term, and an x-coordinate term.The center-distance factor γ0 contains complex attenuation and phase shift determined by d0.
- Channel factorization: The element-wise factorization becomes a matrix-level Kronecker product of lower-dimensional exponential matrices.The component matrices correspond to the vertical and horizontal UPA dimensions.
- Component matrices: Hy and Hx have dimensions My × Ny and Mx × Nx, respectively, and represent the vertical and horizontal channel components.Their tractable structure is derived from the algebraic decoupling of the distance expression.
B. Near Field Beamforming Design
The beamforming method performs separate SVDs on the horizontal and vertical channel components, combines their singular structures through Kronecker products, and allocates power by water-filling.
- Component SVDs: Separate SVDs of Hx and Hy provide unitary factors and singular-value matrices for the two array dimensions.The factors are denoted by Ux, Vx, Uy, and Vy, while Σx and Σy contain the respective singular values.
- Equivalent channel decomposition: The full channel has equivalent singular factors Uy ⊗ Ux and Vy ⊗ Vx, with singular values formed by pairwise products.The equivalent singular-value matrix is Σeq = γ0(Σy ⊗ Σx).
- Beamformer construction: The method selects the L largest pairwise singular values and uses the corresponding columns of Vy ⊗ Vx for beamforming.The selection is based on descending order of λi,j = |γ0| σy,iσx,j.
- Power allocation: Water-filling determines the optimal power allocation across the selected beamforming directions under the transmit-power constraint.The allocation matrix is diagonal with entries p1 through pd.
C. Complexity Analysis
The proposed Kronecker-SVD beamforming method reduces complexity by replacing one full-dimensional channel SVD with two lower-dimensional SVDs.
- The conventional SVD beamforming design operates on the full channel matrix and has complexity O(MxMyNxNy min(MxMy, NxNy)).
- The proposed method computes SVDs for the two lower-dimensional channel matrices Hx and Hy.These matrices have dimensions Mx×Nx and My×Ny, respectively.
- The proposed Kronecker-SVD method has complexity O(MxNx min(Mx, Nx) + MyNy min(My, Ny)).
V. SIMULATION RESULTS
The simulations evaluate the derived NF-FF boundary and the proposed Kronecker-SVD beamforming design under a 60 GHz dual-UPA downlink configuration.
- The simulations validate the NF-FF boundary across transmitter antenna counts, phase-difference thresholds, and carrier frequencies.
- The evaluation considers achievable rate and CPU running time as the number of transmitter antennas varies.
- The system operates at 60 GHz with wavelength λ = 0.005 m and half-wavelength spacing at both UPAs.
B. Numerical Results
Numerical results show how carrier frequency, array size, and phase-difference threshold affect the NF-FF boundary, while the proposed beamforming preserves rate with lower computational cost.
- Higher carrier frequency produces a larger phase difference at the same propagation distance, with 90 GHz requiring the largest boundary distance and 30 GHz the smallest.
- The NF-FF boundary increases with transmitter antenna count because the transmitter aperture becomes larger.The derived boundary scales with (DT + DR)^2.
- At threshold ϕ̄ = π/8, the proposed NF-FF boundary is equivalent to the conventional Rayleigh distance.
- A smaller phase-difference threshold yields a larger NF-FF boundary distance.
- The proposed Kronecker-SVD design approaches full-dimensional SVD with water-filling, outperforms equal-power and far-field SVD, and requires much less CPU time.Its running-time advantage becomes more evident as the number of transmitter antennas increases.