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Multi-User Holographic MIMO Surfaces: Channel Modeling and Spectral Efficiency Analysis
Li Wei, Chongwen Huang, George C. Alexandropoulos, Wei E. I. Sha, Zhaoyang Zhang, Merouane Debbah, Chau Yuen
TL;DR
MU-HMIMOS communication needs tractable multi-user channel models that account for coupling among densely packed patch antennas. This paper extends an EM-compliant Fourier plane-wave model, analyzes MRT and ZF spectral efficiency, and proposes a Neumann-series ZF precoder that achieves similar performance to conventional ZF at lower hardware cost.
Problem
Multi-user HMIMOS channel modeling is challenging because conventional independent Rayleigh models do not capture mutual coupling from very small antenna spacing and the resulting continuous-aperture structure.
Method
The paper extends an EM-compliant MU-HMIMOS channel model in the wavenumber domain using Fourier plane-wave sampling, then analyzes MRT and ZF and replaces ZF matrix inversion with Neumann-series expansion.
Results
Theoretical performance expressions approximate the simulated schemes sufficiently well, while the NS-based ZF precoder achieves similar performance to conventional ZF at lower hardware cost.
Takeaways & Limitations
The model and NS-based ZF design provide an EM-aware and hardware-efficient basis for analyzing and implementing MU-HMIMOS communications.
Abstract
from arXiv · showhide
The multi-user Holographic Multiple-Input and Multiple-Output Surface (MU-HMIMOS) paradigm, which is capable of realizing large continuous apertures with minimal power consumption, has been recently considered as an energyefficient solution for future wireless networks, offering increased flexibility in impacting electromagnetic (EM) wave propagation according to the desired communication, localization, and sensing objectives. The tractable channel modeling in MU-HMIMOS wireless systems is one of the most critical research challenges, mainly due to the coupling effect induced by the excessively large number of closely spaced patch antennas. In this paper, we focus on this challenge for the downlink of multi-user MIMO communications and extend an EM-compliant channel model to multiuser case, which is expressed in the wavenumber domain using the Fourier plane wave approximation. Based on the presented channel model, we investigate the spectral efficiency of maximumratio transmission and Zero-Forcing (ZF) precoding schemes. We also introduce a novel hardware efficient ZF precoder, leveraging Neumann series (NS) expansion to replace the required matrix inversion operation, which is very hard to be computed in the conventional way due to the extremely large number of patch antennas in the envisioned MU-HMIMOS communication systems. In comparison with the conventional independent and identical Rayleigh fading channels that ignore antenna coupling effects, the proposed EM-compliant channel model captures the mutual couplings induced by the very small antenna spacing. Our extensive performance evaluation results demonstrate that our theoretical performance expressions approximate sufficiently well ...
I. INTRODUCTION
MU-HMIMOS targets energy-efficient, flexible wireless environments through densely packed holographic surfaces, but their multi-user channels require EM-compliant modeling because conventional models overlook coupling and continuous-aperture effects. The paper extends Fourier plane-wave channel modeling to multiple users, analyzes MRT and ZF spectral efficiency, and introduces a Neumann-series ZF alternative.
- Motivation: Multi-user HMIMOS channel modeling is challenging because densely packed antennas create coupling, massive element counts, and surface-to-surface transmission effects that traditional Rayleigh models do not capture.The paper identifies these issues as central obstacles to tractable modeling for spatially continuous apertures.
- Contributions: The paper extends a Fourier plane-wave EM-compliant channel model from single-user to MU-HMIMOS systems by sampling the continuous channel according to antenna spacing and surface size.The sampled channel is represented through finite spatial information points tied to the transmitter and receiver surfaces.
- Contributions: The paper derives achievable spectral-efficiency expressions for MRT and ZF precoding using the proposed MU-HMIMOS channel model.These analyses exploit the model’s EM and wavenumber-domain structure.
- Contributions: The proposed Neumann-series ZF precoder replaces matrix inversion, which is impractical with the extremely large number of patch antennas in MU-HMIMOS systems.This targets lower hardware and computational burden while retaining the ZF design objective.
- Evaluation: The paper evaluates the model and precoding methods through simulations, including the effects of antenna spacing and patch-antenna count on spectral efficiency.The stated evaluation also verifies the theoretical capacity expressions.
A. System Model
The system model considers a downlink from a BS HMIMOS surface to multiple users, with densely spaced patch antennas at both ends. It defines the surface dimensions, antenna indexing, user locations, and transmit/receive wave-vector representations.
- System configuration: The downlink system contains a BS with Ns patch antennas and M users, each equipped with Nr patch antennas.The BS and user surfaces use patch spacings below half the wavelength and ∆r, respectively.
- Transmitter surface: The BS surface is formed from Ns = NV NH unit cells whose metamaterial elements adjust reflection coefficients.Its antenna locations are determined by horizontal and vertical indices obtained from the row-by-row antenna index.
- Wave representations: The transmit vector is parameterized by azimuth and elevation angles, with k = 2π/λ defining the wavenumber.The receive representation uses user positions and receive wave vectors in R3.
- Receiver model: Each user is assigned a location r_m and a receive vector whose entries describe the user-side patch responses.The model indexes users from m = 1 to M and represents the receive vector in C^M×1.
B. Channel Modeling for Individual Users
The individual-user channel model represents EM propagation in the wavenumber domain and converts it into a finite spatial channel through Fourier-plane-wave sampling. Its coefficients depend on scattering, array geometry, and integration regions defined by the transmitter and receiver surfaces.
- Wavenumber-domain channel: The spatial channel H is obtained from a wavenumber-domain channel through a space–wavenumber transformation.The channel connects transmitter and receiver patch responses through angular-domain propagation components.
- Angular channel structure: The equivalent angular channel depends on the scattering environment and array geometry, with its eigenvalues indicating the number of strongly connected channels.The spectral density includes a scattering-dependent factor and a random complex Gaussian term.
- Finite sampling: Because the angular channel is sparse, the Fourier plane-wave expansion can be approximated using finite sampling points inside an effective lattice ellipse.The source and receiver sampling sets have cardinalities n_s and n_r, while the patch counts satisfy Ns ≥ n_s and Nr ≥ n_r.
- Fourier coefficients: The variance σ^2(ℓ_x, ℓ_y, m_x, m_y) of each sampling point is separable under scattering separability and is computed over receive and transmit integration regions.The receive integration region is divided into three subregions, with the transmit region treated analogously.
C. Channel Modeling for Multiple Users
The multi-user channel model decomposes user channels in the wavenumber domain and approximates the spatially continuous EM channel through finite sampling points. Each user’s spatially correlated channel is represented through a lower-dimensional equivalent channel and associated correlation structure.
- C. Channel Modeling for Multiple Users: The multi-user channel matrix is decomposed into separate channel matrices for independently distributed users in an isotropic scattering environment.The model characterizes the BS and user surfaces through transmit and receive sampling points.
- C. Channel Modeling for Multiple Users: The transmit and receive response matrices collect the variances associated with the sampled plane-wave components.The receive response matrix is an identity matrix because its columns describe discretized receive plane-wave harmonics.
- C. Channel Modeling for Multiple Users: Finite spatial sampling points approximate the continuous channel at the patch antennas and received antenna responses.The sampling representation uses the effective bandwidth of the wavenumber-domain channel, whose nonzero components lie within a lattice ellipse.
- C. Channel Modeling for Multiple Users: The multi-user correlation matrix is organized from the individual users’ correlation matrices.The supplied formulation presents the correlation structure in block form.
- C. Channel Modeling for Multiple Users: Each user’s complex spatially correlated channel is equivalent to a lower-dimensional channel matrix because the physical antenna counts exceed the sampling dimensions.The construction uses Ns ≥ ns and Nr ≥ nr.
III. ACHIEVABLE RATE WITH LINEAR PRECODING
The paper derives spectral-efficiency expressions for MRT and ZF in MU-HMIMOS while addressing the computational burden caused by very large, coupled antenna arrays. It also proposes a Neumann-series-based ZF precoder as a lower-cost hardware-efficient alternative to conventional matrix inversion.
- III. ACHIEVABLE RATE WITH LINEAR PRECODING: The analysis derives MU-HMIMOS spectral efficiency for MRT and ZF using the equivalent wavenumber-domain channel.The system assumes perfect receiver CSI and a perfectly configured phase matrix.
- III. ACHIEVABLE RATE WITH LINEAR PRECODING: A Neumann-series-based ZF precoder replaces conventional matrix inversion to reduce the computational burden of extremely large patch-antennas arrays.The proposed scheme is motivated by the operational cost of matrix inversion in MU-HMIMOS.
- III. ACHIEVABLE RATE WITH LINEAR PRECODING: The analysis accounts for nonidentical entry variances in the EM-compliant channel rather than assuming unit-variance channel elements.This difference prevents direct use of traditional large-MIMO analysis methods.
A. Ergodic Rate
The ergodic-rate formulation models the received multi-user signal through the equivalent wavenumber-domain channel, HMIMOS phase matrix, linear precoder, transmitted symbols, and Gaussian noise. It defines user- and sampling-point-specific achievable rates from this system model.
- A. Ergodic Rate: The MU-HMIMOS received signal is modeled with channel matrix H, phase matrix Φ, precoder V, transmitted signal x, and Gaussian noise w.The phase matrix is diagonal and contains the HMIMOS patch-antenna phase vector.
- A. Ergodic Rate: The equivalent wavenumber-domain channel incorporates the HMIMOS phase matrix and maps the precoded signal to the received signal.The transformed received signal and equivalent channel are used in the rate analysis.
- A. Ergodic Rate: The precoding matrix is partitioned into user-specific sub-blocks, with each sub-block providing the transmitted signal components for one user.The m-th sub-block has dimensions Ns × Nr and its columns correspond to received signal components.
- A. Ergodic Rate: The achievable rate is specified for each user at each received sampling point.The formulation uses the received signal of the m-th user and its i-th received response point.
B. MRT Precoding
The MRT analysis normalizes the precoder to satisfy the transmit-power constraint and derives a theoretical capacity bound using the equivalent channel. The derivation uses Jensen’s inequality and channel variance expressions associated with the EM-compliant model.
- B. MRT Precoding: The MRT precoder is normalized so that E{Tr(VVH)} = 1.The normalization coefficient αMRT is used to enforce the power constraint.
- B. MRT Precoding: The MRT capacity analysis applies Jensen’s inequality to obtain a theoretical capacity bound.The bound is derived after substituting the MRT expressions into the achievable-rate formulation.
- B. MRT Precoding: The derivation expresses desired-signal and precoder norms through channel variance terms for the EM-compliant model.These terms include σ2_r,i and σ2_s,j-related quantities in the supplied expressions.
C. ZF Precoding
The ZF precoder is designed to suppress multi-user interference, and its theoretical spectral-efficiency analysis uses the proposed MU-HMIMOS channel structure and normalization constraints.
- C. ZF Precoding: ZF precoding is formulated to eliminate interference among different users through the precoding matrix.
- C. ZF Precoding: ZF normalization uses αZF to satisfy the transmit-power constraint E{Tr(VVH)} = 1.
- C. ZF Precoding: ZF uses vector normalization because analytical numerical results indicate it provides better achievable-rate bounds than matrix normalization.The cited passage contrasts this choice with MRT, for which matrix normalization performs better.
- C. ZF Precoding: The theoretical ZF capacity is derived from average channel observations under the separable scattering environment.
D. NS-Based ZF Precoding
The NS-based ZF scheme replaces exact Gram-matrix inversion with matrix multiplications to reduce practical hardware difficulty, while convergence depends on matrix structure.
- D. NS-Based ZF Precoding: Exact ZF requires inversion of an nr × nr matrix, which becomes impractical as the number of patch antennas grows.
- D. NS-Based ZF Precoding: When the Gram matrix is not strongly diagonally dominant, Neumann iteration may converge slowly or diverge.The method works well in the special case of single-ended correlation.
- D. NS-Based ZF Precoding: The Gram matrix is decomposed into diagonal and off-diagonal components before applying the Neumann approximation.
- D. NS-Based ZF Precoding: Neumann-series expansion replaces matrix inversion with a summation of matrix powers, making the operation more suitable for hardware.
- D. NS-Based ZF Precoding: Setting the iteration number to 4 provides a low-cost precision setting, although more iterations increase computational cost.
IV. NUMERICAL RESULTS
The simulations validate the theoretical spectral-efficiency expressions and show how antenna count, spacing, and precoding affect MU-HMIMOS performance. Smaller spacing strengthens correlation and generally reduces spectral efficiency, while NS-based ZF closely matches conventional ZF with fewer iterations.
- The MU-HMIMOS correlation eigenvalues decay more steeply at smaller received-antenna spacing, and even λ/2 spacing remains correlated rather than i.i.d. Rayleigh.The λ/2 curve is closest to the i.i.d. reference but still differs substantially.
- The theoretical ZF and MRT analyses closely predict their corresponding simulated precoding schemes across the evaluated SNR conditions.Theoretical ZF nearly coincides with ZF, especially at lower SNR, while theoretical MRT predicts MRT across all SNR regions.
- MRT outperforms ZF at low SNR, whereas ZF becomes better at high SNR; MMSE remains best across the evaluated SNR range.The reported crossover is attributed to noise dominance at low SNR and weaker noise impact at high SNR.
- More transmit or receive patch antennas increase spectral efficiency, consistent with the larger surface area they provide.The receive-antenna comparison identifies Nr = 288 as best among Nr = 72, 144, and 288 under fixed spacing.
- Smaller antenna spacing produces stronger spatial correlation and lower spectral efficiency, while the theoretical analyses remain accurate in highly correlated cases.With fixed patch count, reduced spacing also means a smaller surface area; the reported comparison finds Δs = λ/15 worse than Δs = λ/6 for ZF.
- Four NS iterations nearly coincide with seven iterations and provide a practical balance between computational cost and spectral-efficiency performance.The NS-based ZF scheme achieves similar performance to conventional ZF while avoiding expensive matrix inversion.
V. CONCLUSIONS
The paper extends EM-compliant channel modeling to multi-user HMIMOS and derives spectral-efficiency analyses for MRT and ZF precoding. Simulations show that more antennas improve performance, tighter spacing degrades it through stronger correlation, and NS-based ZF approaches conventional ZF at lower hardware cost.
- The paper extends an EM-compliant MU-HMIMOS channel model in the wavenumber domain using the Fourier plane wave approximation.The model accounts for mutual coupling caused by closely spaced patch antennas.
- Analytical spectral-efficiency expressions are derived for MRT and ZF precoding, alongside a hardware-efficient NS-based ZF alternative to matrix inversion.The NS-based approach addresses the impracticality of conventional inversion for extremely large patch arrays.
- More patch antennas improve achievable spectral efficiency at fixed spacing, whereas decreasing antenna spacing strengthens correlation and degrades spectral efficiency.These trends are reported for both transmitter and receiver surfaces.
- NS-based ZF achieves performance similar to conventional ZF while reducing hardware cost.The result supports using Neumann-series expansion to avoid the involved matrix inversion.