Multiport S-Parameters Model

• The multiport S-parameters model is a framework that describes electromagnetic systems by relating incident and reflected waves through a scattering matrix, capturing reflection, transmission, and coupling.
• It finds applications in reconfigurable intelligent surfaces, stacked metasurfaces, microwave imaging arrays, and cryogenic RF systems by modeling complex interactions in multiport networks.
• The model explicitly reveals coupling, mismatch, and structural scattering, enabling effective optimization and calibration strategies in advanced RF and microwave designs.

A multiport S-parameters-based model describes a linear wave system by relating the incident and outgoing waves at all ports through a scattering matrix, $\mathbf{b}=\mathbf{S}\mathbf{a}$. In RF and microwave engineering, this formalism is especially natural because reference planes are easy to define, measurements are naturally expressed in terms of reflected and transmitted waves, and reflection, transmission, mismatch, and coupling are all encoded in the same matrix representation (Caspers, 2012). In contemporary research, the same framework is used not only for conventional $n$-port circuits, but also for reconfigurable intelligent surfaces (RISs), stacked intelligent metasurfaces (SIMs), microwave imaging arrays, cryogenic RF chains, and time-modulated scatterers, where it functions as a physically consistent abstraction of coupled electromagnetic interactions (Abrardo et al., 2023, Yahya et al., 12 Mar 2025, Kuznetsov et al., 17 Jul 2025).

1. Formal network-theoretic structure

For an $n$-port network, the incident and outgoing waves are collected as

$$\mathbf{a}=(a_1,a_2,\ldots,a_n)^T,\qquad \mathbf{b}=(b_1,b_2,\ldots,b_n)^T,$$

and the multiport relation is

$$\mathbf{b}=\mathbf{S}\mathbf{a},\qquad b_i=\sum_{j=1}^n S_{ij}a_j.$$

Here $S_{ii}$ is a reflection coefficient at port $i$, while $S_{ij}$ for $i\neq j$ is a transmission or coupling coefficient. With real reference impedance $Z_0$, usually $n$0, the wave variables are

$n$1

so incident and reflected powers can be tracked directly (Caspers, 2012).

In reconfigurable-surface problems, the matrix is usually block-partitioned by subsystem. For RIS-aided links, one common partition is transmitter ($n$2), RIS ($n$3), and receiver ($n$4): $n$5 The port terminations are then represented through reflection matrices,

$n$6

with diagonal entries

$n$7

When transmitter and receiver are matched, $n$8, the exact end-to-end RIS channel reduces to

$n$9

which is one of the central multiport S-parameter channel models used in recent RIS research (Abrardo et al., 2023, Abrardo et al., 2023).

This same partitioned viewpoint extends to larger electromagnetic systems. In SIM models, the full transmitter-SIM-receiver arrangement is also cast as a single scattering network, but with many more internal ports. In one formulation, if the transmitter has $n$0 ports, the SIM has $n$1 ports, and the receiver has $n$2 ports, then the total system is an $n$3 port network with

$n$4

and the SIM is represented as a reconfigurable termination network attached to the internal port set (Pettanice et al., 15 Mar 2026).

2. Equivalence to impedance models and the question of physical consistency

A recurring theme in the literature is that scattering- and impedance-based descriptions are mathematically equivalent but not equally transparent in all interpretations. The standard conversion is

$n$5

or equivalent identities with the same reference impedance. In this sense, $n$6- and $n$7-parameters are not competing physical theories; they are different coordinate systems for the same multiport network (Nossek et al., 2023, Abrardo et al., 2023).

The principal controversy concerns interpretation, not formal validity. Several RIS papers argue that the widely used “phase-only, unit-amplitude” abstraction is generally physically inconsistent. In a physically consistent multiport model, a RIS element is not an isolated phase shifter; it is a port in a coupled network. Changing its termination changes the network response seen at the receiver in both phase and magnitude (Nossek et al., 2023). In the one-element toy example with a blocked direct link, the normalized transfer is

$n$8

where $n$9. Hence,

$$\mathbf{a}=(a_1,a_2,\ldots,a_n)^T,\qquad \mathbf{b}=(b_1,b_2,\ldots,b_n)^T,$$0

which shows immediately that changing the reactance changes both amplitude and phase (Nossek et al., 2023).

A second issue is structural scattering. In the S-parameter RIS formulation, $$\mathbf{a}=(a_1,a_2,\ldots,a_n)^T,\qquad \mathbf{b}=(b_1,b_2,\ldots,b_n)^T,$$1 is not merely a direct-path term. One explicit decomposition is

$$\mathbf{a}=(a_1,a_2,\ldots,a_n)^T,\qquad \mathbf{b}=(b_1,b_2,\ldots,b_n)^T,$$2

where

$$\mathbf{a}=(a_1,a_2,\ldots,a_n)^T,\qquad \mathbf{b}=(b_1,b_2,\ldots,b_n)^T,$$3

This shows that $$\mathbf{a}=(a_1,a_2,\ldots,a_n)^T,\qquad \mathbf{b}=(b_1,b_2,\ldots,b_n)^T,$$4 already contains the RIS structural scattering. Consequently, even when the RIS ports are matched, so that $$\mathbf{a}=(a_1,a_2,\ldots,a_n)^T,\qquad \mathbf{b}=(b_1,b_2,\ldots,b_n)^T,$$5, the received signal need not vanish; the structure can still reradiate energy and produce a specular component (Abrardo et al., 2023). The same point appears in a different form in the simpler impedance-based analysis: even with a blocked direct physical path, the effective scattering from transmitter to receiver is not necessarily zero once the full multiport conversion is done consistently (Nossek et al., 2023).

These results are often presented as a correction to communication-theoretic channel models of the form

$$\mathbf{a}=(a_1,a_2,\ldots,a_n)^T,\qquad \mathbf{b}=(b_1,b_2,\ldots,b_n)^T,$$6

which neglect structural scattering and assume that zero programmable reflection implies zero RIS reradiation. The literature cited here rejects that implication for general coupled electromagnetic systems (Abrardo et al., 2023).

3. Cascaded and stacked structures

The multiport S-parameters-based model becomes more intricate when several programmable surfaces are stacked. In one SIM formulation, a stack of $$\mathbf{a}=(a_1,a_2,\ldots,a_n)^T,\qquad \mathbf{b}=(b_1,b_2,\ldots,b_n)^T,$$7 transmissive RIS layers, each with $$\mathbf{a}=(a_1,a_2,\ldots,a_n)^T,\qquad \mathbf{b}=(b_1,b_2,\ldots,b_n)^T,$$8 cells, is modeled as a balanced $$\mathbf{a}=(a_1,a_2,\ldots,a_n)^T,\qquad \mathbf{b}=(b_1,b_2,\ldots,b_n)^T,$$9-port network. The overall scattering matrix is obtained recursively by repeated application of a cascade operator: $$\mathbf{b}=\mathbf{S}\mathbf{a},\qquad b_i=\sum_{j=1}^n S_{ij}a_j.$$0 Here $$\mathbf{b}=\mathbf{S}\mathbf{a},\qquad b_i=\sum_{j=1}^n S_{ij}a_j.$$1 is the $$\mathbf{b}=\mathbf{S}\mathbf{a},\qquad b_i=\sum_{j=1}^n S_{ij}a_j.$$2-matrix of the $$\mathbf{b}=\mathbf{S}\mathbf{a},\qquad b_i=\sum_{j=1}^n S_{ij}a_j.$$3-th RIS layer, and $$\mathbf{b}=\mathbf{S}\mathbf{a},\qquad b_i=\sum_{j=1}^n S_{ij}a_j.$$4 is the $$\mathbf{b}=\mathbf{S}\mathbf{a},\qquad b_i=\sum_{j=1}^n S_{ij}a_j.$$5-matrix of the propagation medium between layers $$\mathbf{b}=\mathbf{S}\mathbf{a},\qquad b_i=\sum_{j=1}^n S_{ij}a_j.$$6 and $$\mathbf{b}=\mathbf{S}\mathbf{a},\qquad b_i=\sum_{j=1}^n S_{ij}a_j.$$7, with

$$\mathbf{b}=\mathbf{S}\mathbf{a},\qquad b_i=\sum_{j=1}^n S_{ij}a_j.$$8

where $$\mathbf{b}=\mathbf{S}\mathbf{a},\qquad b_i=\sum_{j=1}^n S_{ij}a_j.$$9 and $S_{ii}$0 captures self-impedance and mutual coupling (Yahya et al., 12 Mar 2025).

This S-domain cascade is physically correct but becomes increasingly unwieldy. The number of matrix inversions reported for the recursive SIM S-model is 11 for $S_{ii}$1, 30 for $S_{ii}$2, 67 for $S_{ii}$3, and 145 for $S_{ii}$4 (Yahya et al., 12 Mar 2025). That observation motivates a reformulation in transfer scattering parameters. The same paper introduces a $S_{ii}$5-parameter description in which cascaded networks multiply directly,

$S_{ii}$6

and the end-to-end channel becomes

$S_{ii}$7

The purpose of this reformulation is tractability, not a change of physics: the derivation remains grounded in full multiport theory, mutual coupling, and reciprocity/losslessness constraints (Yahya et al., 12 Mar 2025).

For lossless reciprocal RIS layers, the S-domain constraints are

$S_{ii}$8

while the corresponding $S_{ii}$9-domain constraints become a pseudo-unitary condition,

$i$0

and a complex-conjugate persymmetry condition,

$i$1

For transmissive single-connected RIS layers, the design variables reduce to phase shifts $i$2 (Yahya et al., 12 Mar 2025).

A parallel SIM line of work retains the S-parameter description directly, but reorganizes it for optimization. One formulation writes

$i$3

so the tunable part enters through a structured inverse that combines the cell terminations with the SIM internal coupling matrix $i$4 (Pettanice et al., 15 Mar 2026). In the stage-isolated nonlinear extension, the same architecture is preserved with explicit nonlinear terminations and fixed-point forward evaluation, while the stated complexity remains $i$5 (Abrardo et al., 22 May 2026).

4. Optimization and design

Because the multiport S-parameter model exposes coupling and mismatch explicitly, it supports optimization strategies that are unavailable or unreliable in simplified diagonal channel models. For RISs, one S-parameter formulation optimizes

$i$6

under unit-modulus reflection constraints, using small phase perturbations and a Neumann-series linearization. The corresponding papers state that small perturbations of the step size produce larger changes in the S-parameter matrix than in the Z-parameter matrix, leading to a faster convergence rate. The same work generalizes the optimization to suppress the specular reflection due to structural scattering while maximizing received power toward a direction of interest; in the reported example, $i$7 suppresses the specular component by about $i$8 dB while reducing the desired beam by about $i$9 dB (Abrardo et al., 2023).

In SIM-aided communication, the optimization objective is often the sum rate

$S_{ij}$0

with

$S_{ij}$1

One reported workflow is a two-stage design: first optimize RIS phase shifts for fixed power allocation, then optimize power allocation for fixed RIS phases. The phase stage uses a gradient descent algorithm with Armijo step size and numerical first-order approximation of the gradient; initialization is obtained from a simplified channel model and an MRT-based design (Yahya et al., 12 Mar 2025).

The numerical conclusions in the same SIM study are notable because they contradict a common heuristic. Accounting for the exact channel improves sum rate, and the improvement becomes more significant when elements are packed more closely, that is, when mutual coupling is stronger. The exact model also captures inter-layer feedback and concludes that this feedback is important for accurate performance prediction. However, when the total number of SIM elements $S_{ij}$2 is fixed, increasing the number of layers generally does not improve sum rate for the exact model, because the transmission medium becomes more lossy as the stack deepens (Yahya et al., 12 Mar 2025). By contrast, a simplified channel model based on Rayleigh-Sommerfeld diffraction coefficients may predict the opposite trend, which the authors describe as physically questionable (Yahya et al., 12 Mar 2025).

Discrete RIS hardware introduces a separate optimization regime. In a 1-bit setting, the reflection coefficients satisfy

$S_{ij}$3

and the exact end-to-end channel is

$S_{ij}$4

This full model captures multiple scattering and mutual coupling through $S_{ij}$5. In the reported comparison, coordinate descent with the full multiport-network model yields the most favorable trade-off in terms of performance, execution time and memory usage except when mutual coupling is negligible, and the cascaded affine approximation becomes inadequate as coupling grows (Hammami et al., 3 Aug 2025).

5. Measurement, calibration, and identification

The multiport S-parameters-based model is also a measurement framework. In microwave imaging systems with many antennas, one calibration procedure uses one ECal-equipped port to establish a reflection calibration at the antenna connector, transfers that calibration to the remaining ports under the assumption that all antennas have the same reflection coefficient when measuring homogeneous phantoms, and finally calibrates pairwise transmission paths with the unknown-thru technique. The only requirement for the transmission step is reciprocity,

$S_{ij}$6

and the paper states that the method is typically viable if path attenuation is not more than $S_{ij}$7 dB. The resulting objective is a full multiport calibrated S-matrix referenced at the antenna connectors, obtained without perturbing the RF chain (Kasper et al., 2019).

Once multiport S-parameter datasets are available, validation becomes another modeling problem. One approach represents each complex S-parameter trace as a point cloud in a real-imaginary-frequency space and compares two datasets with the modified Hausdorff distance. For an $S_{ij}$8 S-matrix, the overall distance is the worst element-wise match,

$S_{ij}$9

and the distance is mapped to a similarity percentage SPS. The method is designed for measured and simulated datasets that do not share the same frequency grid, do not need interpolation, and may exhibit slightly shifted resonances (Shlepnev, 2021). In the reported examples, a $i\neq j$0 GHz normalization frequency is used, and suggested application-specific tiers are “Good” $i\neq j$1, “Acceptable” $i\neq j$2, “Inconclusive” $i\neq j$3, and “Bad” $i\neq j$4 (Shlepnev, 2021).

A more ambitious identification setting arises when not all ports are directly accessible. “Virtual VNA 2.0” addresses the recovery of the full scattering matrix $i\neq j$5 when only $i\neq j$6 ports are connected to the VNA and the remaining $i\neq j$7 ports can only be terminated. The key refinement over the original Virtual VNA is the introduction of a known multi-port load network, experimentally realized by a simple coaxial cable as a two-port load network, which removes the sign and phase ambiguities that remain when only individual loads are used. The paper states that at most $i\neq j$8 additional measurements are required, and reports $i\neq j$9 dB at 771 MHz for the final ambiguity-free phaseless experiment (Hougne, 2024).

6. Extensions beyond static linear channel modeling

The same S-parameter formalism extends naturally to problems in which deterministic scattering must be combined with other physical effects. In cryogenic RF systems, one generalized cascade model augments the multiport scattering equation to

$Z_0$0

where $Z_0$1 is an internal noise-wave vector with covariance matrix

$Z_0$2

This matrix-based description captures return loss, isolation, coupling, temperature differences between components, and operation near the quantum noise limit. The cited work emphasizes that the method reduces to Friis cascade theory in the ideal matched two-port limit, but remains applicable to circulators, switches, and other genuinely multiport networks (Grando et al., 2022).

Time-modulated scatterers require a different extension because an input tone can generate harmonic outputs. A multifrequency multiport S-parameters-based model handles this by replicating radiation and load ports across harmonics and polarizations and assembling a harmonic-domain system matrix

$Z_0$3

Here $Z_0$4 contains structural scattering, $Z_0$5 captures coupling among load ports, and $Z_0$6 is a load-network matrix built from Fourier coefficients of the periodic modulation. The reported validation uses a fabricated space-time-modulated structure of 9 monopoles, an RF input frequency of $Z_0$7 GHz, a modulation frequency of $Z_0$8 kHz, and $Z_0$9 harmonics, with strong agreement between measured and predicted harmonic-dependent bistatic patterns (Kuznetsov et al., 17 Jul 2025).

At the network-composition level, recent work has also shown that arbitrarily complex interconnections of multi-port subsystems can be evaluated in closed form in the scattering domain, and updated efficiently through the Woodbury identity when only a small subsystem changes. The same work argues that scattering parameters are fundamentally better suited than impedance or admittance parameters for connection schemes involving ideal delayless, lossless, reflectionless, reciprocal two-port links, because the corresponding $n$00-connections remain exact in S-parameters while Z/Y formulations require quasi-$n$01-approximations (Prod'homme et al., 2024).

Taken together, these developments define the multiport S-parameters-based model less as a single equation than as a research program: a common wave-based language for static and dynamic scattering, exact and reduced-order composition, physically constrained optimization, and measurement-driven validation. The cited literature consistently treats its main strength as the ability to preserve electromagnetic constraints—reciprocity, losslessness, passivity, structural scattering, mutual coupling, and feedback—while remaining close to the data structures delivered by full-wave simulation and network analysis instrumentation (Abrardo et al., 2023, Yahya et al., 12 Mar 2025, Kuznetsov et al., 17 Jul 2025).