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End-to-End Mutual Coupling Aware Communication Model for Reconfigurable Intelligent Surfaces: An Electromagnetic-Compliant Approach Based on Mutual Impedances
Gabriele Gradoni, Marco Di Renzo
TL;DR
Existing RIS models do not provide a complete operational end-to-end formulation that jointly captures electromagnetic fields, mutual coupling, and unit-cell amplitude-phase behavior. The paper introduces a mutual-impedance-based circuit model with an equivalent channel mapping transmitter port voltages to receiver port voltages, and reports that mutual coupling materially affects received intensity and motivates coupling-aware load optimization.
Problem
Existing EM-compliant RIS studies address specific issues in isolation rather than providing an operational end-to-end model that explicitly identifies EM-field and induced-current effects.
Method
The paper introduces a circuit-based RIS communication model using mutual impedances among transmit antennas, receive antennas, and passive scatterers, with tunable RIS loads representing unit-cell responses.
Results
Mutual coupling among RIS elements has a noticeable impact on received intensity, supporting designs that optimize tunable loads to maximize |HVLOS|2.
Takeaways & Limitations
The model is end-to-end, EM-compliant, mutual-coupling aware, unit-cell aware, and compatible with conventional communication-theoretic frameworks.
Abstract
from arXiv · showhide
Reconfigurable intelligent surfaces (RISs) are an emerging technology for application to wireless networks. We introduce a physics and electromagnetic (EM) compliant communication model for analyzing and optimizing RIS-assisted wireless systems. The proposed model has four main notable attributes: (i) it is end-to-end, i.e., it is formulated in terms of an equivalent channel that yields a one-to-one mapping between the voltages fed into the ports of a transmitter and the voltages measured at the ports of a receiver; (ii) it is EM-compliant, i.e., it accounts for the generation and propagation of the EM fields; (iii) it is mutual coupling aware, i.e., it accounts for the mutual coupling among the sub-wavelength unit cells of the RIS; and (iv) it is unit cell aware, i.e., it accounts for the intertwinement between the amplitude and phase response of the unit cells of the RIS.
I. INTRODUCTION
The paper addresses the lack of operational, end-to-end RIS communication models that jointly capture electromagnetic behavior, mutual coupling, and unit-cell responses. It introduces a circuit-based model founded on mutual impedances and mapping transmitter port voltages to receiver port voltages.
- RIS communication models must be realistic, accurate, tractable, and account for the physics and EM properties of scattering elements.
- Existing EM-compliant studies address specific issues separately but do not provide an operational end-to-end model with explicitly identifiable EM-field and induced-current effects.
- The proposed model is founded on electromagnetism for EM-field generation, propagation, and scattering, while accounting for amplitude-phase intertwinement and mutual coupling.
- The model maps voltages impressed at multi-antenna transmitter ports one-to-one to voltages measured at multi-antenna receiver ports.
- The circuit-based formulation uses mutual impedances among transmit antennas, receive antennas, and passive RIS scatterers, jointly exposing incident fields, generator and load impedances, and tunable RIS loads.
A. Transmitter Modeling
The system formulation models transmitters, receivers, and RIS scatterers as coupled radiating elements driven by voltage generators, receiver loads, and tunable RIS impedances. Electromagnetic fields and induced currents are linked through thin-wire assumptions, boundary conditions, and Pocklington’s equation.
- Transmitter and receiver modeling: Transmit antennas use independent voltage generators with internal impedances, while receive antennas use independent load impedances whose port voltages and currents are solved.
- RIS modeling: The RIS comprises passive scatterers connected to tunable loads, whose optimized impedances adaptively configure the scattered electromagnetic fields.
- RIS modeling: PIN-diode resistance, inductance, and capacitance determine tunable scatterer impedances for forward and negative bias implementations.
- Modeling assumptions: The formulation assumes free-space transmission, interaction, and detection of electromagnetic waves, postponing generalization to random media.
- Electromagnetic methodology: Incident fields induce currents on radiating elements, whose currents generate radiated fields; the total field is their sum.
- Electromagnetic methodology: For perfectly conducting thin wires, vanishing tangential surface fields determine the current distribution through Pocklington’s equation, approximated by a sinusoidal current under stated assumptions.
III. IMPEDANCE-BASED MUTUAL COUPLING MODELING
The paper develops impedance-based mutual-coupling modeling by computing radiated fields, defining self and mutual impedances, and evaluating voltages for coupled antennas.
- The impedance-based development introduces isolated-antenna radiated fields, self and mutual impedances, and port voltages for coupled transmit and receive antennas.
- For arbitrary transmit or receive antennas, the surface current is represented with a sinusoidal thin-wire profile parameterized by the port current.
A. Isolated Antenna: Radiated Electric Field
The isolated-antenna analysis computes the tangential radiated electric field from the antenna current distribution using Green-function-based expressions derived from Maxwell’s equations.
- The analysis focuses on the tangential component of the electric field radiated by an isolated antenna at an observation point.
- The Green function represents propagation from points on the antenna to the observation point through distance- and wavenumber-dependent terms.
- The radiated field expression is derived from Maxwell’s equations and integrates the antenna current distribution with a Green function.
B. Self and Mutual Impedances
Mutual coupling describes energy exchange between nearby radiating elements and is especially relevant for closely spaced RIS scatterers. Self and mutual impedances quantify this interaction from antenna geometry, enabling precomputation for later system analysis.
- Mutual coupling is the energy interchange in which radiation from one nearby element excites currents on another.The induced currents generate additional scattered fields that influence the other element’s boundary field.
- The mutual impedance Zqp quantifies the coupling induced by element p on element q, while Zpp is the corresponding self impedance.The self impedance is obtained by setting q = p in the mutual-impedance definition.
- The mutual impedance is formulated through the radiated field from p observed on q and an integral involving the current distributions and Green-function-related terms.
- By reciprocity in vacuum, the mutual impedances satisfy Zpq = Zqp.
- Zqp depends only on the geometry of the radiating elements, not on port currents, voltage generators, or tunable loads.Its geometry-only dependence allows Zqp to be precomputed and reused across different RIS-assisted wireless systems.
C. Pair of Antennas: Two Transmit Antennas
For two transmit antennas, mutual coupling links port voltages and currents through constitutive linear equations derived from electromagnetic fields and delta-gap boundary conditions.
- The two-transmit-antenna analysis seeks an explicit relation between port voltages VTp, VTq and currents ITp, ITq as a function of mutual coupling.
- The constitutive equations interweave the voltages and currents of transmit antennas p and q through self and mutual impedances.The coupling term uses the mutual impedance Zqp.
- The derivation starts from Maxwell’s equations for the radiated field on q in the presence of p.
- The delta-gap model represents the voltage generator as an incident electric field localized outside the antenna gap, with perfectly conducting material filling the gap.
- Comparing the impedance-based current expression with the boundary-condition expression yields the constitutive equations for the coupled transmit antennas.
D. Pair of Antennas: One Transmit and One Receive Antenna
For one transmitting and one receiving antenna, the port relations remain linear but distinguish a driven transmit port from a receive port terminated by a load impedance.
- The mixed transmit–receive case relates VTp and VLq, or VSmn, to the corresponding transmit, receive, or RIS port currents.
- A transmit antenna is driven by a voltage generator, whereas the receive antenna is connected to a load impedance.
- The system of linear equations uses Vq = VLq and Iq = ILq for the receiver, or Vq = VSmn and Iq = ISmn for an RIS scatterer.The mutual impedance Zqp is the same coupling quantity used in the two-antenna formulation.
- The proof models the load voltage drop as an equivalent incident electric field localized around the load gap.The remaining derivation follows the two-transmit-antenna proof with the receive-mode boundary condition.
IV. END-TO-END COMMUNICATION MODEL
The paper constructs an end-to-end equivalent channel from mutual-impedance matrices for transmit antennas, RIS scatterers, and receive antennas. This channel maps transmitter port voltages to receiver port voltages while incorporating tunable RIS loads and coupling effects.
- The model accounts for arbitrary numbers of coupled transmit and receive antennas together with passive RIS scatterers.
- The end-to-end channel HE2E is defined by VL = HE2EVG, mapping the transmitter voltage vector VG to the receiver voltage vector VL.HE2E has dimensions Nr × Nt.
- HE2E incorporates transmit and receive elements, passive RIS scatterers, receiver load impedances, and tunable RIS load impedances.
- Theorem 1 builds HE2E from block mutual-impedance matrices ZXY for transmitter, RIS, and receiver elements.The matrices use T, S, and R to identify the transmitter, RIS, and receiver, respectively.
- The derivation first applies the coupled-port lemmas to the full system, then solves the resulting V = ZI equations using generator, receiver-load, and RIS-load relations.
- HE2E can quantify RIS advantages and limitations through rank, eigenvalues, and eigenvectors, while optimizing ZRIS can target a desired multiplexing–diversity tradeoff.
B. Design Insights
The model provides design insights by expressing RIS-assisted transmission through mutual impedances, while preserving end-to-end voltage relationships and unit-cell response coupling. It recovers conventional path-loss behavior and enables optimization of mutually coupled tunable loads.
- Model structure: HE2E accounts for mutual coupling among all Nr + Nris + Nt radiating elements in the RIS-assisted system.The model is formulated using the available transmit antennas, receive antennas, and RIS scattering elements.
- Far-Field Path-Loss: In the far-field scalar case, HE2E is the sum of the direct LOS link HLOS and the virtual LOS link HVLOS.The virtual link is formed through the RIS and complements the transmitter-receiver LOS contribution.
- Far-Field Path-Loss: The received-power scaling follows |HLOS|^2 ∝ rRT^-2 and |HVLOS|^2 ∝ rST^-2 rRS^-2.These relations recover the expected dependence on the transmitter-receiver, transmitter-RIS, and RIS-receiver distances.
- Reconfigurability of the RIS: HE2E depends explicitly on tunable RIS load impedances, which can be optimized while inherently capturing each unit cell’s coupled amplitude-phase response.The amplitude-phase intertwinement depends on the circuital model of the tuning circuit.
- Conventional vs. Mutual Coupling Modeling: Mutual coupling enters HVLOS through the full matrix Φ = (ZRIS + ZSS)^-1, whereas diagonal ZSS yields the no-coupling formulation.The model therefore preserves mutual-impedance interactions while retaining analytical similarity to conventional communication models.
V. NUMERICAL RESULTS
The numerical study evaluates mutual coupling in RIS-assisted transmission at 28 GHz by varying element spacing and RIS size. In the considered setup, coupling noticeably changes received intensity, supporting coupling-aware load optimization.
- Study design: The study varies inter-element distance d and the number Nris of RIS scattering elements to examine mutual-coupling effects.Fig. 2 reports the received-intensity quantity from the scalar far-field expression as these parameters change.
- Study design: The RIS is arranged as an M × N grid with M = N = √Nris, using identical thin-wire antennas and passive scatterers.The element dimensions and circuit parameters are specified as a = λ/500, l = λ/32, RSmn = 1 Ω, and LSmn = 1 nH.
- Observed impact: Mutual coupling has a noticeable impact on the received intensity |HVLOS|^2 in the considered setup.The reported results motivate designs that optimize the tunable loads ZSmn to maximize |HVLOS|^2.
VI. CONCLUSION
The paper concludes that its communication model is end-to-end, EM-compliant, mutual-coupling aware, and unit-cell aware. It is compatible with conventional communication frameworks for physics-compliant RIS modeling, analysis, and optimization.
- Conclusion: The proposed model jointly provides end-to-end, EM-compliant, mutual-coupling-aware, and unit-cell-aware communication modeling for RIS-assisted systems.These properties are stated as the paper’s concluding contribution.
- Conclusion: The model can support physics-compliant modeling, analysis, and optimization of RIS-assisted communications while remaining compatible with conventional frameworks.
- Conclusion: Hardware-platform and experimental-measurement validation is identified as a relevant continuation of the research.