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Stacked Intelligent Metasurface-Based Transceiver Design for Near-Field Wideband Systems

Qingchao Li, Mohammed El-Hajjar, Chao Xu, Jiancheng An, Chau Yuen, Lajos Hanzo

arXiv:2502.11182v1cs.ITmath.IT

TL;DR

Prior holographic beamforming designs are described as narrowband, ideal-phase, and far-field, motivating a wideband near-field SIM transceiver that accounts for phase tuning errors. The proposed design combines layered phase optimization with digital precoding; simulations report higher spectral efficiency than a single-layer metasurface design, while the analysis predicts high-SNR saturation from phase errors.

Abstract

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Intelligent metasurfaces may be harnessed for realizing efficient holographic multiple-input and multiple-output (MIMO) systems, at a low hardware-cost and high energy-efficiency. As part of this family, we propose a hybrid beamforming design for stacked intelligent metasurfaces (SIM) aided wideband wireless systems relying on the near-field channel model. Specifically, the holographic beamformer is designed based on configuring the phase shifts in each layer of the SIM for maximizing the sum of the baseband eigen-channel gains of all users. To optimize the SIM phase shifts, we propose a layer-by-layer iterative algorithm for optimizing the phase shifts in each layer alternately. Then, the minimum mean square error (MMSE) transmit precoding method is employed for the digital beamformer to support multi-user access. Furthermore, the mitigation of the SIM phase tuning error is also taken into account in the digital beamformer by exploiting its statistics. The power sharing ratio of each user is designed based on the iterative waterfilling power allocation algorithm. Additionally, our analytical results indicate that the spectral efficiency attained saturates in the high signal-to-noise ratio (SNR) region due to the phase tuning error resulting from the imperfect SIM hardware quality. The simulation results show that the SIM-aided holographic MIMO outperforms the state-of-the-art (SoA) single-layer holographic MIMO in terms of its achievable rate. We further demonstrate that the near-field channel model allows the SIM-based transceiver design to support multiple users, since the spatial resources represented both by the angle domain and the distance domain can be exploited.

I. INTRODUCTION

The introduction positions SIM hybrid beamforming as a response to narrowband, ideal-control, far-field limits in prior holographic designs. It presents a wideband, near-field, phase-error-aware multiuser transceiver design.

A. Related Work

  • Unlike a passive RIS that reconfigures propagation, holographic MIMO uses a base-station metasurface as a reconfigurable antenna array.The introduction also notes the RIS two-hop path-loss limitation.Source: A. Related Work · p. 1
  • Related work covers RHS, DMA, and SIM beamformers, including prior SIM approaches using gradient descent, alternating optimization, or statistical CSI.Source: A. Related Work · p. 2

B. Motivation

  • Prior holographic beamforming designs are described as narrowband, assuming ideal phase control and far-field channels.Source: B. Motivation · p. 2
  • The proposed wideband design alternates SIM-layer phase optimization, uses MMSE digital precoding with phase-error statistics, and allocates user power by iterative waterfilling.Source: B. Motivation · pp. 2–3
  • The analysis predicts that phase tuning errors from imperfect SIM hardware limit spectral efficiency, which saturates at high SNR.Source: B. Motivation · p. 3
  • SIM-aided holographic beamforming promises higher spectral efficiency than state-of-the-art single-layer metasurface designs.The introduction also says near-field angle- and distance-domain resources can support multiple users.Source: B. Motivation · p. 3

A. Signal Model

  • The downlink model uses OFDM subcarriers to serve single-antenna users through digital precoding followed by SIM holographic processing.Source: A. Signal Model · pp. 3–4

B. Stacked Intelligent Metasurface Architecture

The SIM combines an input layer with reconfigurable intermediate metasurface layers. Their element phases shape propagation, while phase-tuning errors are represented statistically for robust design.

B. Stacked Intelligent Metasurface Architecture

C. Channel Model

The channel model uses spherical-wave propagation for near-field links and adopts a non-uniform spherical-wave model for the radiative near-field between the SIM and users. It also approximates element-to-user power gain under a practical element-size condition.

C. Channel Model

  • When communication distance is below the Rayleigh distance, the model uses spherical rather than plane waves to characterize the BS-to-user channel.The near-field model is selected for short-range, high-frequency links with many metasurface elements.Source: C. Channel Model · p. 5
  • For SIM-to-user links, the non-uniform spherical-wave model applies in the radiative near-field, where radiative components dominate and reactive components are negligible.The stated region satisfies r > 0.62 D^3/λc.Source: C. Channel Model · p. 5
  • Because each reconfigurable element is wavelength-scale, power-gain variation across its area is treated as negligible, enabling an approximate element-to-user gain.The gain depends on the user position and the distance from the element in the outermost SIM layer.Source: C. Channel Model · p. 5

III. HYBRID DIGITAL AND HOLOGRAPHIC DESIGN

This section formulates hybrid digital and SIM-based holographic beamforming for maximizing wideband-system spectral efficiency. The design accounts for uncertain baseband channels caused by SIM phase-tuning errors and uses their statistics in beamformer design.

III. HYBRID DIGITAL AND HOLOGRAPHIC DESIGN

  • The design jointly optimizes per-subcarrier digital precoders and SIM phase shifts to maximize average spectral efficiency across users.The SIM-based holographic beamformer and digital beamformer form separate subproblems in the non-convex optimization.Source: §III · pp. 5–6
  • The RF-chain mutual-coupling model uses a Z-parameter formulation with antenna and load impedances fixed at 50 Ohms.The associated mutual impedance matrix represents coupling between RF-chain ports.Source: §III · p. 6
  • Because phase-tuning errors make the equivalent baseband channel uncertain, the digital beamformer uses MMSE precoding and error statistics to address inter-user interference and tuning error.The channel mean and covariance are derived from the phase-error statistics.Source: §III · p. 6
  • The optimization targets average spectral efficiency for all users, while user-fairness criteria are left for future work.The paper notes fairness as an alternative optimization criterion.Source: §III · p. 6

A. SIM Phase Shift Optimization for Holographic Beamformer

The SIM phase shifts are optimized to maximize the sum of users’ equivalent baseband eigen-channel gains, an objective that tightly upper-bounds achievable sum-rate. A layer-by-layer iterative method optimizes phase shifts alternately, using a central-carrier design for the wideband system.

A. SIM Phase Shift Optimization for Holographic Beamformer

  • Maximizing the sum of equivalent baseband eigen-channel gains across users is used as a tight upper-bound objective for achievable sum-rate.The SIM phase shifts are optimized under the sum-path-gain maximization criterion.Source: A. SIM Phase Shift Optimization for… · p. 7
  • A shared SIM phase design is optimized for the central carrier because a design tuned to one subcarrier may degrade spectral efficiency at others.The passage notes that this frequency-selectivity issue grows more severe as subcarrier spacing increases.Source: A. SIM Phase Shift Optimization for… · p. 7
  • The proposed layer-by-layer iteration alternately updates each layer’s phase shifts, using a principal-eigenvector phase choice as a low-complexity alternative to solving an NP-hard subproblem.The procedure repeats until the objective change falls below a threshold.Source: A. SIM Phase Shift Optimization for… · pp. 7–8
  • Each layer update increases channel gain monotonically, and the bounded gain implies convergence of the algorithm.The update coherently combines equivalent deterministic channels.Source: A. SIM Phase Shift Optimization for… · p. 8

B. Precoding Vector Optimization for Digital Beamformers

The digital-beamformer subproblem optimizes per-subcarrier precoding vectors under power constraints. The design uses channel statistics to account for SIM phase-error effects and iteratively allocates user power in the presence of interference.

B. Precoding Vector Optimization for Digital Beamformers

  • The MMSE-based digital design uses the mean and covariance of the baseband channel to account for random channel uncertainty from SIM phase-tuning errors.The section formulates spectral efficiency using the resulting channel statistics.Source: B. Precoding Vector Optimization for… · pp. 8–9
  • The digital precoding problem is reformulated over per-subcarrier user vectors and solved using a generalized Rayleigh quotient.The optimization accounts for inter-user interference and phase-tuning-error effects.Source: B. Precoding Vector Optimization for… · p. 8
  • Each user’s power-sharing ratio is constrained to sum to one across users, and the method updates these ratios with iterative waterfilling.At each iteration, interference matrices from the previous allocation are treated as noise for the waterfilling update.Source: B. Precoding Vector Optimization for… · p. 9

IV. PERFORMANCE ANALYSIS

The analysis characterizes SIM spectral efficiency in high-power and vanishing inter-layer-distance regimes. Non-ideal SIM hardware limits high-power performance, while vanishing layer spacing degrades spectral efficiency.

IV. PERFORMANCE ANALYSIS

  • As transmit power tends to infinity, non-ideal SIM hardware limits average spectral efficiency, which saturates in the high-power region.Theorem 2 characterizes the achievable sum-rate in this limit.Source: §IV · p. 9
  • When SIM layers are extremely close (dl → 0 for l = 0, 1, …, L), the achievable sum-rate R(d→0) can be expressed, and optimal beamforming matrices are not unique.Source: §IV · pp. 9–10
  • As inter-layer distance tends to zero, spectral efficiency degrades with more SIM layers because the fully connected structure and optimization freedom are reduced, while phase-tuning-error effects worsen.Increasing the number of layers cannot compensate for the lost optimization freedom in this regime.Source: §IV · p. 10

V. NUMERICAL AND SIMULATION RESULTS

The simulations use a wideband setup with specified carrier, bandwidth, subcarrier, RF-chain, and SIM-element settings. The multi-user scenario places three user pairs at shared angles but different distances.

V. NUMERICAL AND SIMULATION RESULTS

  • Unless otherwise specified, simulations use a 10 GHz carrier, 600 MHz bandwidth, 64 subcarriers, and 256 × 256 SIM elements per layer.The RF-chain count is 4×4 for single-user simulations and 8×8 for multi-user simulations.Source: §V · p. 10
  • The setup assumes 5λ_c inter-layer spacing, four beamforming iterations, evenly distributed feeds, and neglected mutual coupling between RF-chain ports.Neglecting mutual coupling simplifies analysis to evaluate fundamental performance trends.Source: §V · p. 10

A. Performance Comparison of SIM-aided Transceiver Design and SoA Single-layer Metasurface

The simulations compare SIM-aided transceivers with a single-layer metasurface and vary SIM layers, element counts, subcarriers, and layer spacing. The results show higher rates over the single-layer baseline, alongside trade-offs and operating conditions that affect performance.

A. Performance Comparison of SIM-aided Transceiver Design and SoA Single-layer Metasurface

B. Effect of Metasurface Phase Tuning Error

Phase-tuning errors limit high-SNR spectral efficiency, while adding SIM layers can help at low SNR but can worsen distortion when hardware is imperfect. The layer count should therefore reflect both the channel environment and hardware quality.

B. Effect of Metasurface Phase Tuning Error

C. Achievable Rate Comparison in Near-field and Far-field

The near-field model uses both angle and distance to support multiple users, while the far-field model relies on angular separation. The reported comparisons show higher achievable sum-rate for the SIM in the stated multi-user settings, and the conclusion describes its hybrid beamforming design.

C. Achievable Rate Comparison in Near-field and Far-field

  • As the number of intermediate layers increases, the SIM achieves higher sum-rate than the SoA metasurface while supporting multiple users.Fig. 10 compares achievable sum-rate across layer counts and different numbers of UEs.Source: Figure 10 · p. 13, C. Achievable Rate Comparison in… · p. 13
  • For UE-1 and UE-2 at the same angle, the near-field SIM supports both simultaneously and achieves higher sum-rate than far-field TDMA, especially with more layers.The far-field model cannot simultaneously support the same-angle pair; near-field access exploits spatial distance.Source: C. Achievable Rate Comparison in… · pp. 12–13, Figure 11 · p. 13

VI. CONCLUSIONS

  • The proposed wideband SIM design combines layer-by-layer holographic phase optimization with MMSE digital precoding and phase-error statistics.The conclusion reports higher spectral efficiency than a SoA single-layer metasurface and notes high-SNR saturation from phase-tuning errors.Source: §VI · p. 13

APPENDIX A PROOF OF THEOREM 1

The appendices present derivation steps for the stated theorem results, including an iterative expectation and covariance calculation and substitutions used to obtain equation (45).

APPENDIX A PROOF OF THEOREM 1

  • The Appendix A proof uses an iterative operation to obtain E[h(k)] and then expresses the covariance matrix of eh(k).The proof passages also refer to the SIM phase-tuning error.Source: Appendix A · p. 14

APPENDIX B PROOF OF THEOREM 2

  • The Appendix B derivation states that cross-layer links between different element indices are zero under its stated limit condition.It then represents the link from an element in layer l−1 to an element in layer l.Source: Appendix B · p. 14
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