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Stacked Intelligent Metasurfaces for Multiuser Downlink Beamforming in the Wave Domain

Jiancheng An, Marco Di Renzo, Mérouane Debbah, H. Vincent Poor, Chau Yuen

arXiv:2309.02687v2cs.ITeess.SPmath.IT

TL;DR

Single-layer metasurfaces have limited wave-domain processing capabilities, while discrete phase shifts can impair beam alignment and typical single-layer designs cannot suppress inter-user interference. This paper uses a stacked intelligent metasurface for multiuser MISO downlink beamforming and jointly optimizes transmit power and SIM phase shifts. Its SIM-based system achieves a 200% sum-rate increase over conventional MISO systems.

  • Problem

    Single-layer metasurfaces constrain wave-domain processing, and discrete phase shifts can cause beam misalignment while typical single-layer designs lack inter-user interference suppression.

    Source: §I · p. 1

  • Method

    A stacked intelligent metasurface performs multiuser MISO downlink beamforming in the wave domain, with joint optimization of transmit power and discrete phase shifts.

    Source: §I · p. 2

  • Results

    A SIM-based system achieves a 200% increase in sum rate compared to conventional MISO systems.

    Source: §V · p. 12

  • Takeaways & Limitations

    The paper concludes that SIM technology represents a shift from conventional digital signal processing to wave-based computation, with potential for energy-efficient wireless networks.

    Source: §V · p. 12

Abstract

from arXiv · show

Intelligent metasurface has recently emerged as a promising technology that enables the customization of wireless environments by harnessing large numbers of low-cost reconfigurable scattering elements. However, prior studies have predominantly focused on single-layer metasurfaces, which have limitations in terms of wave-domain processing capabilities due to practical hardware limitations. In contrast, this paper introduces a novel stacked intelligent metasurface (SIM) design. Specifically, we investigate the integration of SIM into the downlink of a multiuser multiple-input single-output (MISO) communication system, where an SIM, consisting of a multilayer metasurface structure, is deployed at the base station (BS) to facilitate transmit beamforming in the electromagnetic wave domain. This eliminates the need for conventional digital beamforming and high-resolution digital-to-analog converters at the BS. To this end, an optimization problem is formulated to maximize the sum rate of all user equipments by jointly optimizing the transmit power allocation at the BS and the wave-based beamforming at the SIM, subject to constraints on the transmit power budget and discrete phase shifts. Furthermore, we propose a computationally efficient algorithm for solving the formulated joint optimization problem and elaborate on the potential benefits of employing SIM in wireless networks. Numerical results are illustrated to corroborate the effectiveness of the proposed SIM-enabled wave-based beamforming design and to evaluate the performance improvement achieved by the proposed algorithm compared to various benchmark schemes. It is demonstrated that considering the same number of transmit antennas, the proposed SIM-based system achieves about 200\% improvement in terms of sum rate compared to conventional MISO systems. The code for this paper is available at \url{https://github.com/JianchengAn}.

I. INTRODUCTION

Single-layer RIS designs offer limited beam-pattern control, and discrete phase shifts can cause beam misalignment and leave inter-user interference unsuppressed. The paper introduces a stacked metasurface for multiuser downlink beamforming in the wave domain and optimizes its phase shifts jointly with transmit power.

I. INTRODUCTION

  • Single-layer RIS designs constrain beam-pattern optimization, while discrete phase shifts can misalign beams and typical implementations lack inter-user interference suppression.These limitations motivate a multilayer alternative.Source: §I · p. 1
  • The proposed SIM supports wave-domain signal processing through stacked programmable metasurfaces, unlike the single-layer RIS counterpart.The paper describes SIM as diffracting rather than reflecting signals, integrated into the BS, and modifying both amplitude and phase.Source: §I · p. 2
  • The paper extends its earlier SIM-based multiuser MISO design by restricting meta-atoms to discrete tuning, addressing the difficulty of arbitrary tuning coefficients in real-world applications.The earlier design assumed that meta-atoms could apply any tuning coefficient.Source: §I · p. 2
  • The SIM-based downlink design performs transmit beamforming in the electromagnetic wave domain, eliminating conventional digital beamforming and high-resolution DACs at the BS.The joint sum-rate objective optimizes BS transmit power and SIM beamforming under a power budget and discrete phase-shift constraints.Source: §I · p. 2
  • The proposed alternating optimization decomposes power allocation and SIM phase-shift design into nested subproblems and is evaluated against benchmark schemes.The power subproblem uses modified iterative water-filling; phase shifts use projected gradient ascent and successive refinement.Source: §I · p. 3

II. SYSTEM MODEL

The system places a multilayer SIM at the base station to perform transmit beamforming as signals propagate through the metasurface. Its model specifies the layer-to-layer propagation, user channels, power allocation, and practical assumptions that constrain the design.

II. SYSTEM MODEL

  • The SIM has L metasurface layers with N meta-atoms each, and its controller independently adjusts transmitted-wave phase shifts to shape downlink signals.Forward propagation resembles a fully connected ANN; each meta-atom is assumed to have unit amplitude and a phase selected from a finite quantized set.Source: §II · pp. 3–4
  • The model assumes negligible diffraction between non-adjacent layers and negligible mutual coupling, while noting that practical transmission coefficients may require calibration.Penetration loss is ignored for simplicity, and the authors state that a holistic system-level comparison with conventional transceivers requires further investigation.Source: §II · pp. 4–5
  • The model uses a correlated Rayleigh-fading channel from the final SIM layer to each user, with path loss and spatial correlation determined by the channel model.The channel is quasi-static flat-fading, and the spatial correlation matrix represents correlation among meta-atoms on the last layer.Source: §II · p. 4
  • Wave-based beamforming performs transmit-precoding matrix multiplication as signals propagate through the SIM, allowing phase design to mitigate inter-user interference.Unlike digital SDMA, each stream is transmitted from a corresponding BS antenna; the paper assumes M = K and leaves joint antenna selection for future study.Source: §II · pp. 4–5

A. Problem Formulation

The paper formulates sum-rate maximization for a multiuser SIM-aided downlink by jointly choosing BS power allocation and SIM phase shifts. The resulting mixed-integer nonlinear problem is difficult because the variables are coupled and phase shifts are discrete.

A. Problem Formulation

  • The objective is to maximize all users’ sum rate by jointly optimizing BS transmit powers and SIM phase shifts under a total-power budget and discrete-phase constraints.The formulation assumes the BS perfectly knows the channel state information of all channels.Source: A. Problem Formulation · p. 5
  • The joint problem is an NP-hard mixed-integer nonlinear program because power and phase variables are coupled in a non-convex objective and phase shifts take discrete values.The paper proposes an alternating-optimization algorithm to obtain a suboptimal solution.Source: A. Problem Formulation · p. 5
  • Unlike prior SIM-aided MIMO work framed as channel fitting, this paper optimizes SIM phases for system sum rate while accounting for practical hardware constraints.The cited prior work minimizes error between the end-to-end channel and a desired diagonal channel.Source: A. Problem Formulation · pp. 5–6

B. The Proposed Alternating Optimization Algorithm

The proposed alternating-optimization method separates power allocation from SIM phase optimization and iterates between the two. It uses modified iterative water-filling for power and either projected gradient ascent or successive refinement for phase shifts.

B. The Proposed Alternating Optimization Algorithm

  • Alternating optimization decomposes the joint design into power allocation and phase-shift subproblems, updating one variable while holding the other fixed.The power subproblem is conventional allocation across K interference channels; its iterative water-filling updates use damping to address instability when K > 2.Source: B. The Proposed Alternating… · p. 6
  • The power subproblem uses modified iterative water-filling, while the phase subproblem can use projected gradient ascent or successive refinement.Successive refinement updates one of the LN discrete phase shifts at a time using a one-dimensional search; its objective is non-decreasing and bounded.Source: B. The Proposed Alternating… · p. 6, 2) Optimization of the SIM Phase… · p. 6
  • Projected gradient ascent updates phase shifts using the sum-rate gradient, an Armijo step size, and projection to the nearest feasible discrete phase value.It converges to a local maximum under the stated boundedness and non-decreasing-rate conditions, while multiple initializations address performance impacts from quantization.Source: 2) Optimization of the SIM Phase… · pp. 6–7
  • The outer alternating-optimization loop repeats the power and phase updates until the sum rate converges, returning the resulting power allocation and SIM phases.The power update uses modified iterative water-filling, and the phase update uses either projected gradient ascent or successive refinement.Source: 2) Optimization of the SIM Phase… · p. 7

C. Computational Complexity Analysis

The complexity analysis accounts for the nested iterations in alternating optimization and for the separate costs of power and phase updates. It notes that successive refinement is computationally efficient when the discrete phase set is small, while gradient ascent does not scale with that set’s cardinality.

C. Computational Complexity Analysis

  • The alternating-optimization complexity depends on outer-loop iterations and the iteration counts of its power and phase subproblems.The power-update complexity is expressed in terms of the number of iterative-water-filling iterations; phase-update costs depend on the selected phase algorithm.Source: C. Computational Complexity Analysis · p. 7
  • Successive refinement is computationally efficient when the discrete phase-shift set is small, while gradient-ascent complexity is independent of the set’s cardinality.The analysis gives the gradient method’s complexity in terms of user count, layers, meta-atoms, and its iteration count.Source: C. Computational Complexity Analysis · p. 7

IV. NUMERICAL RESULTS

The numerical-results section evaluates the proposed wave-based beamforming design and the sum rate of the SIM-assisted multiuser downlink MISO system.

IV. NUMERICAL RESULTS

  • The numerical evaluation tests the proposed wave-based beamforming design and measures system sum rate.The stated scope is an SIM-assisted multiuser downlink MISO system.Source: §IV · p. 7

A. Simulation Setup

The setup places a multilayer SIM at the BS to beamform toward multiple single-antenna UEs, then evaluates the design under specified channel and simulation conditions.

A. Simulation Setup

  • The BS uses a ULA, while each SIM metasurface uses a parallel UPA; the SIM performs transmit beamforming in the wave domain.The setup places the SIM at the BS and aligns the center antenna and meta-atom along the z-axis.Source: A. Simulation Setup · pp. 7–8
  • The SIM thickness is 5λ, with layer spacing set to 5λ/L and half-wavelength spacing between adjacent antennas and meta-atoms.Square metasurfaces have Nx = Ny and N = NxNy; UE locations are spaced 10 m apart.Source: A. Simulation Setup · p. 8
  • Inter-layer attenuation is computed from the spacing and transmission geometry, while last-layer-to-UE channels use a spatial correlation model.The stated path-loss setup uses a 28 GHz indoor scenario and exponent 3.5.Source: A. Simulation Setup · p. 8
  • The simulations use 28 GHz carrier frequency, 10 MHz bandwidth, and effective UE noise power of −104 dBm.The setup averages results over 100 independent channel realizations with small-scale fading.Source: A. Simulation Setup · p. 8
  • Four benchmarks compare continuous phase shifts, nearest-value phase quantization, average power allocation, and random phase-shift codebook search.The codebook size is Q = 10LN, and the algorithms use a fractional sum-rate increase threshold of 10^-6 with a 100-iteration maximum.Source: A. Simulation Setup · p. 8

B. Sum Rate Performance Evaluation of the Proposed Wave-Based Beamforming Design

The SIM-based system improves sum rate across the evaluated settings, with performance depending on layer count, meta-atoms, phase quantization, user load, and power allocation. In the same-antenna comparison, it substantially outperforms conventional MISO, while the paper notes that system-level evaluation remains future work.

B. Sum Rate Performance Evaluation of the Proposed Wave-Based Beamforming Design

  • At L=7, the SIM sum rate peaks, attaining a 130% enhancement over a single-layer SIM under the stated N=49, b=2, M=K=4, and PT=10 dBm setup.The sum rate tends to converge as the number of layers increases; discrete phase shifts lose less than 0.5 bps relative to continuous shifts.Source: B. Sum Rate Performance Evaluation of… · p. 8, Figure 3 · p. 9
  • At N=100, successive refinement achieves a 125% rate increase over the quantization scheme, while b=2 discrete phases incur only a 0.3 bps penalty versus continuous phases.The gap between modified iterative water-filling and average power allocation diminishes as N increases.Source: Figure 4 · p. 9, B. Sum Rate Performance Evaluation of… · p. 9
  • When b≥6, the proposed phase-optimization schemes approach continuous-phase performance; the quantization scheme loses 0.2 bps, and average power allocation loses 1.2 bps versus iterative water-filling.The codebook method's rate is unaffected by the number of quantization bits in the reported settings.Source: Figure 5 · p. 9, B. Sum Rate Performance Evaluation of… · pp. 9–10
  • For L=7 and N=49, the SIM effectively mitigates interference for at most K=4 users; at K=10, iterative water-filling yields a 400% rate improvement over the average benchmark scheme.Sum rate initially rises with user count, then falls beyond a threshold as residual interference becomes harder to mitigate.Source: Figure 6 · p. 9, B. Sum Rate Performance Evaluation of… · p. 10
  • Across moderate and high transmit powers, successive refinement and projected gradient ascent improve on the codebook method by 4.5 bps and approach continuous-phase performance.The average-power allocation gap versus iterative water-filling is negligible at higher transmit powers.Source: Figure 7 · p. 10, B. Sum Rate Performance Evaluation of… · p. 10
  • With M=4 transmit antennas, the SIM-assisted system achieves about 200% sum-rate improvement over conventional MISO, which requires M=7 to outperform it.The comparison uses N=49 meta-atoms.Source: B. Sum Rate Performance Evaluation of… · p. 10
  • The SIM optimization adds computational complexity, and a more precise comparison with conventional digital precoding requires further system-level simulations.The authors identify system-level simulations as needed for a more precise performance evaluation.Source: B. Sum Rate Performance Evaluation of… · p. 11

C. Convergence of the Proposed Algorithms

The proposed algorithms converge within bounded iteration counts in the reported setups, though phase-optimization complexity and runtime vary with the SIM configuration. Projected gradient ascent is faster than successive refinement but can yield a slightly lower sum rate because quantization may trap it in a local optimum.

C. Convergence of the Proposed Algorithms

  • The modified iterative water-filling algorithm reaches its maximum within 20 iterations across five channel realizations with random phase shifts.The sum rate varies substantially across those realizations.Source: C. Convergence of the Proposed… · p. 11
  • Successive refinement converges within 17 iterations for N=49 and N=100, with a non-decreasing objective; optimizing more phase shifts at larger N increases per-iteration complexity.The reported convergence is for average power allocation.Source: C. Convergence of the Proposed… · p. 11
  • Projected gradient ascent converges within 10 iterations, but quantization can trap it in a local optimum and produce slightly lower sum rates than successive refinement.Multiple initial phase-shift sets can improve this outcome.Source: Figure 11 · p. 11, C. Convergence of the Proposed… · p. 11
  • The AO outer loop converges within seven iterations for N=49 and N=100; larger N requires more iterations because it expands the optimization variables and search space.The reported cases use 10 independent channel realizations.Source: Figure 12 · p. 11, C. Convergence of the Proposed… · pp. 11–12
  • The iterative water-filling algorithm has the lowest computational complexity independent of SIM parameters; phase-shift optimization complexity increases linearly with L and N, while projected gradient ascent reduces simulation time versus successive refinement.Runtime results average 1,000 experiments across four SIM setups using MATLAB R2023b on an AMD Ryzen 9 5950X processor.Source: C. Convergence of the Proposed… · p. 12

V. CONCLUSIONS

The paper concludes that SIM-based wave-domain beamforming can improve sum rate over conventional MISO systems, with layer count and phase quantization affecting performance. Its joint design uses alternating optimization for power allocation and SIM phase shifts.

V. CONCLUSIONS

  • The proposed alternating-optimization algorithm combines modified iterative water-filling for power allocation with projected gradient ascent and successive refinement for SIM phase shifts.Source: §V · p. 12
  • Increasing the number of metasurface layers can increase sum rate through the SIM’s interference cancellation ability.Source: §V · p. 12
  • The SIM-based system provided a 200% increase in sum rate compared to conventional MISO systems.Source: §V · p. 12
  • The SIM performed best with seven layers, while more than four quantizing bits made discretely tuned meta-atoms almost identical to the continuous case.Source: §V · p. 12

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