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Modeling, Optimization and Electromagnetic Validation of Stacked Intelligent Metasurfaces by Using a Multiport Network Model
Giuseppe Pettanice, Andrea Abrardo, Alberto Toccafondi, Marco Di Renzo
TL;DR
Existing SIM optimization often uses simplified models that may omit important electromagnetic effects. This paper develops a multiport S-parameter modeling and optimization framework and validates it with electromagnetic simulations. Its reported communication and sensing results support the approach while showing that SIM model choice involves an accuracy–complexity trade-off.
Problem
Many RIS/SIM optimization models use simplified end-to-end channels that may omit mutual coupling and other electromagnetic effects relevant to practical implementations.
Source: §I · p. 1Method
The paper develops a multiport S-parameter framework that models SIM interactions and non-ideal circuits, then validates models and optimized configurations through electromagnetic simulations.
Source: §I · p. 2, §II · p. 2, §IV · p. 4Results
The authors report that electromagnetic simulations confirm the reliability of the approach, with communication and sensing results showing effective channel manipulation across SIM models with different coupling levels.
Source: §VIII · p. 14Takeaways & Limitations
The considered SIM models expose a trade-off between electromagnetic modeling accuracy and computational complexity.
Source: §VIII · p. 14The simulated device is an open electromagnetic system with power leakage, so reported SNR values are normalized comparison indicators rather than absolute radiated-power and thermal-noise measures.
Source: A. General Simulation Scenario · p. 9
Abstract
from arXiv · showhide
Stacked intelligent metasurfaces (SIMs) extend the concept of reconfigurable intelligent surfaces by cascading multiple programmable layers, enabling advanced electromagnetic wave transformations for communication and sensing applications. However, most existing optimization frameworks rely on simplified channel abstractions that may overlook key electromagnetic effects such as multiport coupling, circuit losses, and non-ideal hardware behavior. In this paper, we develop a modeling and optimization framework for SIMs based on a multiport network representation using scattering parameters. The proposed formulation captures realistic circuit characteristics and mutual interactions among SIM ports while remaining amenable to optimization. The resulting models are validated through electromagnetic simulations, enabling a systematic comparison between idealized and practical SIM configurations. Numerical results for communication and sensing scenarios confirm that the proposed framework provides accurate performance predictions and enables the effective design of SIM configurations under realistic electromagnetic conditions.
I. Introduction
SIMs extend programmable surfaces through stacked layers, while their optimization requires models that are both electromagnetically accurate and tractable. The paper develops and validates an S-parameter framework for that purpose, then introduces a multiport CEO representation.
I. Introduction
- SIMs cascade reconfigurable layers for richer electromagnetic transformations, including beamforming, near-field focusing, and sensing.The paper identifies wave-domain linear transformations as a motivation for this hardware.Source: §I · p. 1
- Existing RIS/SIM optimization models may omit mutual coupling and other electromagnetic effects, leaving a gap between optimization-friendly abstractions and realistic implementations.The paper’s framework targets tractability alongside electromagnetic accuracy and incorporates non-ideal circuit behavior.Source: §I · pp. 1–2
II. Contributions
- The proposed multiport network framework accounts for multiport interactions and coupling while remaining tractable for communication and sensing optimization.It also develops an optimization-oriented S-parameter model incorporating tunable-circuit and return losses, with models and algorithms validated through full-wave simulations.Source: §II · p. 2
III. CEO S-parameter model
- The CEO model treats the transmitter, CEO, and receiver as one linear (L + N + M)-port network described by a scattering matrix.The CEO termination can be reconfigured through a control-dependent matrix Γ(η).Source: §III · pp. 2–3
IV. S-parameters representation and optimization of a SIM
The SIM is represented as a layered CEO whose antenna ports interact through a multiport scattering network. Tunable cells are modeled with general scattering matrices, allowing losses and mismatch while retaining a block structure for optimization.
IV. S-parameters representation and optimization of a SIM
- A SIM comprises Q transmissive RIS layers, each with K receive–transmit antenna pairs connected by independent tunable two-port cells.The full SIM therefore has QK tunable cells and 2QK antenna ports.Source: §IV · pp. 3–4
- A linear multiport scattering network models electromagnetic coupling among all SIM antenna ports through propagation in the surrounding medium.Layer and cell indices identify each paired port and its tunable parameter.Source: §IV · pp. 3–4
- Each tunable cell uses a 2×2 scattering matrix that may represent loss and mismatch, rather than only an ideal phase shift.Its transmission magnitudes may be below one, while diagonal entries may be nonzero due to return loss and other non-idealities.Source: §IV · p. 4
- The global termination matrix is block-diagonal, with one non-overlapping block per layer and each control affecting only its cell’s 2×2 submatrix.This structure lets the inverse be computed by independently inverting each cell block.Source: §IV · p. 4
A. SIM optimization problem
SIM configuration is posed as minimizing output error over control parameters for given excitations and desired responses. The paper derives gradients using an adjoint system and notes that the dominant linear solves determine computational cost.
A. SIM optimization problem
- The SIM configuration minimizes a loss measuring mismatch between outputs for given input excitations and desired responses.A complex scaling factor is included in defining the error matrix.Source: A. SIM optimization problem · p. 4
- The loss is minimized with gradient-based iterations and an Armijo backtracking line search, with model-specific gradient expressions defining the optimization procedure.The method derives efficient gradients for the different SIM models considered.Source: A. SIM optimization problem · p. 4
B. Gradient evaluation
- For a fully coupled SIM, gradient evaluation uses an adjoint system and derivatives of each cell’s termination matrix with respect to its control parameter.The derivative inherits the two-port architecture’s sparsity, with at most four nonzero entries per cell.Source: B. Gradient evaluation · pp. 4–5
- The fully coupled model avoids assuming layer structure, sparsity, or electromagnetic isolation, while exploiting layered structure can reduce computational complexity.The dominant linear-system solves make the overall cost scale cubically with antenna count.Source: B. Gradient evaluation · p. 5
V. Simplified SIM models and optimization
The simplified SIM formulation recasts an isolated-layer model in the S-parameter domain under assumptions that restrict coupling to intended links and neighboring arrays. Its multiport structure preserves bidirectional interactions and reciprocity without representing the SIM as a cascade.
A. S-parameters representation of a layered isolated SIM
- Ground planes between layers and absorbing materials can help approximate the isolated model by suppressing unwanted electromagnetic leakage.The assumptions are idealized but described as physically approachable through these design strategies.Source: A. S-parameters representation of a… · p. 5
- Under the isolation assumptions, coupling occurs only between adjacent arrays, with the transmitter linked to the first layer and the receiver to the last.Interactions between non-adjacent layers are neglected.Source: A. S-parameters representation of a… · p. 5, Figure 4 · p. 5
- These assumptions produce a block-diagonal or nearest-neighbor block-banded internal scattering matrix, allowing the algebraic framework and computational strategy of to be reused.The structure follows from ordering ports by SIM layer.Source: A. S-parameters representation of a… · p. 5
- The S-parameter formulation is not a cascade: its global scattering matrix retains bidirectional port coupling and naturally preserves reciprocity.This distinguishes it from cascade-based abstractions often used in SIM studies.Source: A. S-parameters representation of a… · p. 5
B. Gradient evaluation for a layered isolated SIM
The isolated SIM’s structured sparsity supports efficient gradient evaluation, with complexity cubic in ports per array and linear in the number of layers. The weakly coupled model admits residual inter-block interactions as small perturbations, retaining tractability while requiring a full inverse matrix.
B. Gradient evaluation for a layered isolated SIM
- Gradient evaluation for the isolated SIM scales cubically in ports per array and linearly in the number of layers.The recursive algorithms from the impedance-domain formulation can compute the quantities needed for the gradient.Source: B. Gradient evaluation for a layered… · p. 6
C. Weakly-coupled SIM
- The weakly coupled SIM models residual inter-block interactions as a small perturbation to a block-diagonal dominant term, approximating the inverse with a first-order Neumann-series correction.The model is relevant when isolation is only approximate, such as with ground planes, absorbing materials, or sufficient inter-layer spacing.Source: C. Weakly-coupled SIM · p. 6
- Unlike the isolated formulation, SIM-W requires the full inverse matrix to account for residual inter-block coupling, but its complexity remains significantly below that of a non-isolated model requiring dense-matrix inversion.The dominant block-structured term can use block-recursive techniques; the first-order correction adds multiplications by ∆S.Source: C. Weakly-coupled SIM · p. 6
VI. Optimization with discrete cell-state sets
The discrete-state optimization models each tunable cell using a finite codebook that can represent hardware constraints and non-ideal effects. A continuous phase-only solution can initialize the discrete search, which then updates one cell at a time and converges finitely to a local optimum in general.
A. Discrete scattering-cell model
- Each 2×2 scattering cell is assigned one of a finite set of admissible states, which can capture circuit non-idealities while respecting applicable hardware constraints.Examples include insertion losses, residual reflections, amplitude variations, passivity, and reciprocity.Source: §VI · p. 6, A. Discrete scattering-cell model · p. 7
B. Initialization via projection onto the codebook
- A continuous phase-only solution provides an initialization by projecting each cell’s scattering matrix onto its codebook.For phase-only cells, the projection reduces to standard phase quantization.Source: B. Initialization via projection onto… · p. 7
1) Coordinate-wise discrete descent
- Coordinate-wise discrete descent exhaustively searches the selected cell’s finite state set while holding other cells fixed, and each sweep updates all QK cells.Because the loss is non-increasing at each update and the search space is finite, the method converges in finitely many steps to a local optimum in general.Source: 1) Coordinate-wise discrete descent · p. 7
2) Fast loss evaluation via low-rank updates
The section develops a low-rank update for evaluating candidate SIM cell states and discusses its computational cost. It then describes the electromagnetic simulation setup used to assess communication and estimation scenarios.
2) Fast loss evaluation via low-rank updates
- Updating one cell state changes only its corresponding 2 × 2 block, enabling rank-2 candidate updates and O(N) evaluation per candidate.The update requires inverting a 2 × 2 matrix once the needed columns are available.Source: 2) Fast loss evaluation via low-rank… · p. 7
3) Complexity discussion
- For each coordinate, evaluating all |P| candidates scales as O(|P| N), while obtaining the required portions of A^-1 dominates the computational burden.The cost of computing or factorizing A depends on the adopted SIM model.Source: 3) Complexity discussion · p. 7
VII. Simulation Results
- Communication and estimation scenarios are assessed using electromagnetic simulations to represent the underlying physical system.The section presents representative results for both task types.Source: §VII · p. 7
A. General Simulation Scenario
- Infinite ground planes isolate SIM layers and external ports, yielding a band-diagonal S-parameter matrix compatible with SIM-I; finite planes model nonzero coupling and require SIM-NI or SIM-W.The infinite-plane model also reduces computational time, while the finite-plane configuration is more computationally demanding.Source: A. General Simulation Scenario · p. 8
- The finite-ground setup is an extreme non-isolated case, not a practically engineered SIM enclosure.Shielding structures or absorbing panels were not included because of their modeling and computational complexity.Source: A. General Simulation Scenario · p. 8
- The simulations compare ideal continuous phase control with measured, lossy phase-shifter networks using discrete states, followed by full-wave co-simulation.Continuous control is treated as an upper bound; the practical networks use Qorvo TGP2100 measured responses and account for insertion and return loss.Source: A. General Simulation Scenario · p. 9
B. Communication-oriented scenarios
The communication scenarios use a SIM to orthogonalize wireless channels, with one setup for multiple spatially separated transmitters and another for a near-field MIMO channel. Both are studied with a five-layer SIM under several phase-control resolutions.
B. Communication-oriented scenarios
- Both communication scenarios use a five-layer SIM with 320 tunable parameters and compare continuous phase control against discrete state sets P ∈ [32, 16, 8, 4].The simulations consider both infinite and finite ground-plane configurations.Source: B. Communication-oriented scenarios · p. 9
COMM-MU-SIMO
- COMM-MU-SIMO uses a reciprocal setup with four spatially separated transmitters under strong far-field conditions, targeting channel orthogonalization.The transmitters are placed on a circle at 220λ from the origin and separated by 12°.Source: B. Communication-oriented scenarios · p. 9
COMM-MIMO
- COMM-MIMO uses one SIM to diagonalize a near-field MIMO channel, with zero-forcing-type configuration possible when the SIM-assisted channel has full rank.The scenario is described as more challenging than MU-SIMO because it requires MIMO channel orthogonalization.Source: B. Communication-oriented scenarios · p. 10
C. Unified formulation and feasibility conditions
The unified formulation uses canonical excitations and geometry-induced operators to express SIM outputs, with channel diagonalization as the communication objective. Communication evaluation combines channel-orthogonalization plots with per-channel capacity, and results compare isolated, weakly isolated, and fully coupled models.
C. Unified formulation and feasibility conditions
- The unified formulation collects L canonical input excitations and uses geometry-induced operators F and G for the TX→SIM and SIM→RX links.The communication objective is channel diagonalization, suppressing inter-stream interference up to a complex scaling factor.Source: C. Unified formulation and feasibility… · p. 10
D. Performance metrics
- Communication performance is assessed through channel-orthogonalization plots and average per-channel capacity, with residual inter-stream interference modeled as additive white noise.SNR is the per-stream signal power at the SIM output divided by noise power.Source: D. Performance metrics · p. 10
- Aggregate spectral efficiency uses each stream’s SINR, treating residual interference from imperfect diagonalization as noise alongside thermal noise.The useful signal is the aggregate received signal power, and noise is normalized to average received signal power per stream at the SIM output.Source: D. Performance metrics · p. 10
- With continuous phase control, the SIM-I model nearly perfectly orthogonalizes the COMM-MU-SIMO channel, and P = 4 still performs well except for capacity loss at high SNR.This is the only reported scenario where satisfactory performance is achieved with coarse quantization.Source: Figure 9 · p. 11, D. Performance metrics · p. 11
- In COMM-MIMO, SIM-I nearly ideally diagonalizes the channel with continuous phase control, but P = 32 leaves residual off-diagonal leakage that grows more relevant at high input SNR.MIMO orthogonalization is more sensitive to phase perturbations because zero-forcing requires high phase precision and resolution.Source: Figure 10 · p. 11, D. Performance metrics · p. 11
- The reported SIM-I communication results begin with COMM-MU-SIMO under infinite ground planes, with P denoting the number of admissible discrete states per unit cell.The reference measures distinguish SIM-I, SIM-W, and SIM-NI optimization models.Source: D. Performance metrics · pp. 10–11
- In COMM-MU-SIMO, SIM-W achieves almost the same channel orthogonality as SIM-NI under continuous-phase control, while SIM-W degrades relative to SIM-NI in COMM-MIMO.With phase quantization, SIM-W and SIM-NI results are very similar, and both degrade significantly relative to continuous-phase control.Source: Figure 12 · p. 11, Figure 11 · p. 11, D. Performance metrics · pp. 11–12
E. Sensing-oriented scenarios
The sensing study separates joint range–angle estimation into range and angle subproblems, optimizing SIM outputs on candidate-location grids and evaluating estimates over broader test locations. With the SIM-I model, continuous phase control gives the best reported range accuracy, while angle estimation remains effective with coarser quantization.
Sensing method
- The sensing approach induces distinctive SIM output signatures for transmitter locations and optimizes a desired response at each grid location.The optimization minimizes mismatch between the SIM output matrix and a prescribed target, consistent with the communication design approach.Source: E. Sensing-oriented scenarios · p. 12
Performance metrics and noisy observations
- Grid-based estimation selects the candidate location whose output signature best matches the noisy observation, with interpolation available for positions between grid points.The evaluation adds white noise at a prescribed SNR and measures the standard deviation of range or angle estimation error across test locations.Source: E. Sensing-oriented scenarios · pp. 12–13
Scenario setup
- The joint range–angle task is divided into separate subproblems to keep FEKO electromagnetic-simulation effort manageable.The range case fixes θ = 0 and uses a radial grid; the DoA case estimates angle using transmitter locations on an angular span.Source: E. Sensing-oriented scenarios · p. 12
Scenario scope
- Range accuracy is studied in the near field at distances below 1 m, whereas angular estimation can be examined over a wide angular span.Near-field conditions are used because accurate range estimation requires pronounced wavefront curvature across the aperture.Source: E. Sensing-oriented scenarios · pp. 12–13
Performance metrics and noisy observations
- The SIM is trained on a finite polar grid, but performance is evaluated over many test points spanning the area of interest.A polynomial interpolator maps grid-point output signatures continuously to estimated transmitter positions between grid locations.Source: E. Sensing-oriented scenarios · p. 13
Evaluation scope
- The sensing study uses infinite ground planes and the SIM-I model, reporting range and angle error standard deviations across phase-discretization levels.Finite ground-plane models are excluded because of their substantially higher computational burden.Source: E. Sensing-oriented scenarios · p. 13
Range estimation
- Quantized phase control supports acceptable range-estimation performance when P is at least 8, while continuous phase control gives the lowest σd across the SNR range.Eight pronounced low-error bands for P = ∞ correspond to the SIM training-grid locations.Source: E. Sensing-oriented scenarios · p. 13, Figure 15 · p. 13
Angle estimation
- Angle estimation is less challenging than range estimation, with good performance reported even at P = 4.The reported σθ curves use the same phase-quantization levels and SNR sweep as the range study; eight angular training directions appear in the false-color representation.Source: E. Sensing-oriented scenarios · p. 13, Figure 16 · p. 13
VIII. Conclusion
The paper presents an electromagnetically consistent multiport framework for SIM modeling and optimization, validated through electromagnetic simulations. Communication and sensing results show channel orthogonalization and localization functionality with practical phase quantization.
VIII. Conclusion
- The framework uses an S-parameter representation to capture multiport interactions, tunable-circuit effects, and hardware non-idealities in SIM optimization.Three models with different coupling levels illustrate the trade-off between modeling accuracy and computational complexity.Source: §VIII · p. 14
Appendix A Computation of the Full Inverse Matrix
The appendix computes the full inverse matrix by exploiting the layered SIM’s block structure. A block-Thomas factorization is reused to solve for each block-column of the inverse.
Appendix A Computation of the Full Inverse Matrix
- The layered SIM topology and dominant coupling pattern produce a block-tridiagonal scattering matrix.Its diagonal blocks are Bℓ, with lower and upper off-diagonal blocks Aℓ and Cℓ.Source: Appendix A · p. 14
- A block-Thomas factorization is performed once, then reused to solve 2Q block linear systems, one for each block-column of T.The recursion is well defined when all forward Schur-complement matrices Fℓ are nonsingular.Source: Appendix A · p. 14
- The procedure constructs each block-column t(r) using the corresponding block vector e(r), repeating the recursion for r = 1, …, 2Q.Together, the block-columns yield the full inverse matrix.Source: Appendix A · pp. 14–15