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Physically Consistent Modeling of Dispersive Time-Modulated Reconfigurable Intelligent Surfaces for Wideband OFDM

Ivan Iudice, Donatella Darsena, Giacinto Gelli, Vincenzo Galdi

arXiv:2609.18360v1eess.SPeess.SY

TL;DR

Wideband RIS models typically separate frequency selectivity from periodic time variation, although practical time-modulated elements exhibit both. The paper develops a resonant LPTV model and derives its OFDM input-output relation, showing that a generalized CP removes ISI and out-of-grid leakage while deterministic on-grid harmonic coupling remains. Full-wave and TDL simulations support the model and highlight the relevance of RIS memory for high-QL elements.

  • Problem

    Existing signal-processing models do not provide a wideband OFDM input-output description that jointly captures RIS frequency selectivity and periodic reconfiguration.

  • Method

    The paper combines a physically based single-resonance equivalent circuit with harmonic-transfer-function analysis of an LPTV RIS to derive a closed-form per-subcarrier OFDM relation.

  • Results

    A generalized CP accommodating propagation delay spread and RIS memory suppresses ISI and out-of-grid leakage, but deterministic on-grid harmonic coupling between subcarriers remains.

  • Takeaways & Limitations

    RIS memory and dispersion must be included in wideband OFDM modeling because periodic reconfiguration produces structured subcarrier coupling that CP removal does not eliminate.

  • Takeaways & Limitations

    Design of transceiver and RIS methods to mitigate or exploit harmonic coupling is beyond the scope of this work.

Abstract

from arXiv · show

The elements of a reconfigurable intelligent surface (RIS) are commonly modeled either as frequency-selective time-invariant reflectors or as instantaneous time-varying reflection coefficients. In practice, however, time-modulated metasurfaces exhibit both frequency selectivity and periodic time variation. We develop a physically consistent linear periodically time-varying (LPTV) model that jointly captures these effects and characterizes their impact on wideband orthogonal frequency-division multiplexing (OFDM) communications. From a canonical equivalent circuit of a generic RIS element, we derive a single-resonance model whose physically meaningful parameters determine both the frequency-selective reflection coefficient and the effective impulse-response duration, i.e., the finite memory of the element. The periodically switched dispersive responses are then represented through harmonic transfer functions, leading to a closed-form per-subcarrier OFDM input-output relation. The resulting coupling is generally non-diagonal: each received subcarrier collects contributions from multiple transmitted subcarriers through the RIS harmonics, each weighted by the element response at the corresponding absolute input frequency. We further derive a generalized cyclic-prefix (CP) condition requiring the guard interval to accommodate both the propagation-channel delay spread and the RIS memory. Under this condition, intersymbol interference and out-of-grid spectral leakage are suppressed, while deterministic on-grid harmonic coupling remains. Full-wave simulations of an OpenRIS unit cell designed for a 5G NR channel validate the proposed resonant model and reveal appreciable in-band dispersion despite nearly ideal binary phase switching. Simulations over 3GPP tapped-delay-line (TDL) channels confirm the generalized CP condition and show its relevance for high-quality-factor RIS elements.

I. INTRODUCTION

RIS elements are often idealized as frequency-flat instantaneous phase shifters or frequency-selective time-invariant reflectors, but physically feasible time-modulated metasurfaces combine dispersion with periodic variation. This paper introduces an LPTV model that captures both effects and derives its wideband OFDM implications.

  • Motivation: Physically feasible time-modulated RIS elements jointly exhibit finite-memory frequency selectivity and periodic time variation, which conventional abstractions treat separately.The paper identifies the missing signal-processing input-output model for their combined behavior in wideband OFDM.
  • Proposed framework: The proposed model represents each RIS element as an LPTV system with a two-dimensional reflection kernel depending on frequency and time.This formulation is used to derive harmonic transfer functions and a closed-form per-subcarrier OFDM relation.
  • Proposed framework: A physics-based circuit model links each element’s frequency-selective reflection coefficient with its finite impulse-response duration.The contribution is intended to preserve physical meaning while supporting signal-processing analysis.
  • OFDM implication: The resulting OFDM coupling is generally non-diagonal because RIS harmonics mix transmitted subcarriers, with weights determined by frequency-dependent element responses.The harmonic representation therefore captures both spectral translation and dispersion.
  • OFDM implication: The paper derives a generalized CP condition that accounts for both propagation delay spread and RIS memory.The stated contribution is a closed-form condition for wideband OFDM transmission through the time-varying RIS.

A. Harmonic Representation

The harmonic representation models periodic switching through Fourier coefficients of the frequency-selective steady-state responses. A canonical equivalent circuit supplies physically realizable resonant responses while preserving the model’s frequency and time dependence.

  • Periodic control: Each RIS element is driven by a periodic, piecewise-constant control signal whose slot response is approximated by the corresponding steady-state transfer function.The approximation requires switching transients to decay essentially before the next state transition.
  • Harmonic representation: Fourier expansion separates harmonic generation from dispersion: the harmonic index captures periodic switching, while each coefficient retains dependence on absolute frequency.This is the core LPTV representation of the periodically switched dispersive response.
  • Special cases: Frequency-flat time-modulated RISs and frequency-selective LTI RISs arise as special cases when, respectively, state responses are frequency independent or nonzero harmonics vanish.The formulation therefore contains both conventional model classes.
  • Equivalent circuit: The equivalent circuit combines grounded-substrate, metallic-patch, and tunable-load impedances to obtain a physically realizable reflection coefficient for each control state.The model assumes a resonant unit cell under normal-incidence, narrowband conditions and related circuit assumptions.
  • Scope: Full-wave characterization is restricted to normal incidence; oblique-incidence parameters depend on angle and polarization and are left for future work.The single-resonance structure and LPTV formulation are stated to remain applicable after reevaluation of those parameters.
  • Equivalent circuit: The single-pole response approximates the physical Hermitian cell response with relative error O(1/QL) at positive in-band frequencies.A mirror resonance restores Hermitian symmetry, while its positive-band contribution can be absorbed into the background term.

C. Impulse Response and Finite Memory

The resonant impulse response has a carrier-scale oscillation and exponential decay, making the decay rate determine an effective finite memory. That memory governs steady-state switching validity and contributes directly to CP design.

  • Finite memory: An effective memory Tγ is defined as the time after which the resonant impulse response falls below a prescribed amplitude fraction χ.For χ = 10^-3, the retained tail threshold is −60 dB in amplitude.
  • Finite memory: The effective memory depends only on the total decay rate ξ and is independent of operating detuning Δ.A worst-case RIS memory includes the slowest-decaying element across all control states.
  • Steady-state condition: Switching transients must decay before the next state transition, so the finite-memory condition implies the classical adiabatic condition.This requirement connects the physical resonator dynamics to the validity of the piecewise-constant control model.
  • OFDM implication: RIS memory acts like an additional delay spread and therefore enters the overall OFDM cyclic-prefix requirement.The surface’s temporal spreading is treated analogously to propagation-channel delay spread.

III. OFDM SIGNAL ANALYSIS

The OFDM analysis combines static two-hop tapped-delay-line propagation with the RIS’s LPTV response. Under a CP and synchronization condition, ISI and out-of-grid leakage disappear, while harmonic subcarrier coupling remains.

  • System model: The source–RIS and RIS–destination links are modeled as static LTI tapped-delay channels, with periodic RIS reconfiguration as the only source of time variation.The direct link is assumed blocked or severely attenuated, and no mobility-induced Doppler shift is present.
  • Receiver model: The receiver removes the CP and demodulates each OFDM subcarrier using timing aligned to the earliest two-hop propagation path.The exact relation is then transformed into a tractable per-subcarrier expression.
  • Generalized CP condition: When the CP accommodates the two-hop propagation delay spread plus RIS memory and T = Tu, demodulated samples are free of ISI and out-of-grid spectral leakage.This is the generalized CP condition established by Theorem 1.
  • Harmonic coupling: The OFDM relation remains generally non-diagonal: each received subcarrier can contain contributions from all transmitted subcarriers through periodic RIS reconfiguration.Thus, CP removal does not diagonalize the channel even when it completely suppresses ISI.

A. Discussion and Special Cases

The model produces deterministic, on-grid harmonic coupling between OFDM subcarriers while retaining frequency-selective RIS weighting and memory effects. Special cases recover conventional static or frequency-flat models, but CP extension removes ISI rather than modulation-induced ICI.

  • Harmonic coupling: Each received subcarrier generally combines multiple transmitted subcarriers through deterministic RIS harmonics, unlike diagonal LTI-OFDM coupling.Under T = Tu, transmitted subcarrier m reaches received subcarrier m + h through the h-th harmonic.
  • CP implications: A sufficient CP suppresses ISI and out-of-grid leakage, but modulation-induced off-diagonal ICI remains independent of CP duration and multipath.Periodic reconfiguration synchronized to the useful symbol duration keeps translations on integer OFDM-grid multiples.
  • Design implications: Structured harmonic coupling can in principle be mitigated or exploited through joint equalization, precoding, guard-subcarrier allocation, control-sequence design, index modulation, or waveform shaping.Developing these transceiver and RIS designs is outside the paper’s scope.
  • Dispersive weighting: The coupling coefficient jointly depends on harmonic-order difference and the absolute frequency of the transmitted subcarrier.Thus, frequency selectivity and harmonic mixing cannot generally be treated as independent effects.
  • Special cases: If the RIS is static, only the zeroth harmonic remains, so harmonic coupling disappears while RIS memory still belongs in the CP condition.The resulting model is consistent with frequency-selective time-invariant RIS behavior.
  • Special cases: If each control-state response is frequency flat, absolute-frequency weighting disappears, harmonic coupling remains, and RIS memory vanishes.In this idealized case, the CP only needs to cover the two-hop propagation-channel delay spread.

IV. NUMERICAL RESULTS

The numerical study evaluates a simplified OpenRIS unit cell with full-wave simulations and a single-pole resonant approximation. It uses two varactor bias states selected to provide binary phase-shifting operation across a 100-MHz n78 channel.

  • Simulation setup: The study models an OpenRIS varactor-tuned unit cell operating in the 5G NR n78 band using full-wave finite-element simulations.The simplified model omits biasing vias and RF chokes and replaces varactors with lumped ports.
  • Simulation setup: The computational model uses doubly periodic Floquet boundaries, a top Floquet port, specular-mode retention, and adaptive frequency sweeps from 3 to 4 GHz.Responses are computed on a 10-MHz grid and interpolated to 1 MHz for processing.
  • Simulation assumptions: Higher-order Floquet modes are evanescent and attenuated by more than 140 dB at the port plane across the modeled band.This supports retaining only the fundamental specular mode in the reported simulations.
  • Model validation: The full-wave and single-pole responses are compared for co-polar reflection of both control states over n78, with the selected 100-MHz channel highlighted.The figure directly assesses the resonant approximation against the electromagnetic response.
  • Control states: The two control states use 4 V and 19.25 V reverse-bias voltages, corresponding to approximately 1.68 pF and 0.83 pF junction capacitances.The states and channel placement are selected to realize binary 1-bit phase-shifting over the full 100-MHz channel.

A. Single-Pole Fitting Over the Full Band

The single-pole model is fitted over the full n78 band and improved with a common static phase renormalization, yielding close agreement with full-wave responses.

  • A. Single-Pole Fitting Over the Full Band: The fitting experiment evaluates the canonical single-pole model against full-wave reflection coefficients for both control states across the n78 band.Parameters are estimated by least-squares minimization of complex-valued error over the full band.
  • A. Single-Pole Fitting Over the Full Band: The intrinsic decay rate remains approximately 8.9–9.0 MHz across bias voltages, while bias mainly shifts the resonance frequency.Both control states operate in the strongly overcoupled regime.
  • A. Single-Pole Fitting Over the Full Band: The joint-fit renormalized model reduces RMS errors to 0.017 and 0.023 for the two control states.The fitted common phase is φ0 = −10.7◦.
  • A. Single-Pole Fitting Over the Full Band: The bare model has RMS errors of 0.104 and 0.044, corresponding approximately to relative errors of 10.4% and 4.4%.Their scale is consistent with the O(1/QL) accuracy floor for the fitted quality factors.

B. In-Channel Dispersion and Binary Switching

The selected 100-MHz channel exhibits nearly ideal binary switching but appreciable state-wise dispersion, which the LPTV OFDM model incorporates through frequency-dependent harmonic coupling.

  • B. In-Channel Dispersion and Binary Switching: The two states maintain a phase difference within 180◦± 6.5◦, magnitude imbalance below 0.75 dB, and reflection loss below 1.1 dB across the occupied band.The channel occupies 98.28 MHz over 3.5449–3.6431 GHz.
  • B. In-Channel Dispersion and Binary Switching: Individual state phases vary by 47◦ and 50◦, while each state has approximately 0.6 dB in-band magnitude ripple.The near-constant binary phase difference arises because the states operate on corresponding resonance flanks with similar group delays.
  • B. In-Channel Dispersion and Binary Switching: Each transmitted subcarrier experiences the RIS response at its own absolute frequency, while harmonic coefficients depend on the state-dependent pole parameters.Thus, binary differential behavior does not remove the frequency selectivity of the individual states.
  • B. In-Channel Dispersion and Binary Switching: With T = Tu and Tcp ≥∆τmax + Tγ, the OFDM relation reduces to the linear per-tone mapping y[n] = H s[n] + v[n].For K = 2, Tr = Tu/2 = 16.67 µs, exceeding Tγ by more than three orders of magnitude.
  • B. In-Channel Dispersion and Binary Switching: The single-element simulation extends to multiple elements by linear superposition, with the slowest-decaying element determining effective RIS memory and the CP requirement.The complete numerical parameter set is reported in Table II.
  • B. In-Channel Dispersion and Binary Switching: For K = 2, nonzero even-order harmonics vanish and odd-order harmonics decay, producing an approximately banded on-grid subcarrier-coupling structure.The reduced-system visualization uses M = 64 active subcarriers and ideal propagation hops.

D. Validation of the Generalized CP Condition

The validation shows that CP length must account for both propagation delay spread and RIS memory, while RIS dispersion and harmonic coupling persist beyond CP sizing.

  • CP-length validation: For CP durations shorter than the channel delay spread, residual ISI decreases rapidly; beyond that spread, the RIS resonance tail governs the remaining interference.The higher-QL response decays five times more slowly and therefore requires a proportionally longer guard interval.
  • CP-length validation: The generalized bound Δτmax + Tγ identifies the CP duration at which the resonant tail falls below threshold χ, unlike the channel-only bound Δτmax.This comparison isolates the additional guard duration associated with RIS memory.
  • Practical regime: For the considered n78 unit cell, Tγ = 13.8 ns is far shorter than the approximately 2.3-µs 5G NR CP, so propagation remains the dominant CP constraint.The RIS-memory term becomes relevant for higher-QL elements or shorter OFDM symbols.
  • Frequency selectivity: The RIS can remain strongly frequency selective across the signal bandwidth even when its memory is negligible relative to the CP.For χ = 10^-4, Tγ = 13.8 ns corresponds to a characteristic frequency scale of approximately 106 MHz, comparable to the 98.28-MHz occupied bandwidth.
  • Residual coupling: When the generalized CP condition is satisfied, ISI and out-of-grid leakage are suppressed, but deterministic on-grid harmonic coupling remains.Each transmitted subcarrier is weighted by the dispersive response at its absolute frequency while periodic reconfiguration redistributes energy among harmonic bands.

APPENDIX A CANONICAL DISPERSIVE UNIT CELL

The appendix reduces a canonical RIS unit-cell circuit to a physically interpretable resonant reflection model, then characterizes its passband response, memory, and approximation limits.

  • Circuit reduction: The reflection coefficient consists of grounded-substrate reflection plus a Lorentzian resonant contribution centered at ω0 with half-power half-width ξ.Their superposition produces the resonance notch in reflection magnitude.
  • Operating point: The resonance is placed near, but not necessarily at, the carrier; operation on a resonance flank preserves a steep phase slope while keeping reflection magnitude near unity.At exact resonance, the reflection magnitude can collapse under critical coupling.
  • Impulse response and memory: The real passband impulse response combines rapid oscillation at f0 with exponential decay time 1/ξ, and the loaded quality factor sets their separation.Adding the mirror pole restores Hermitian symmetry without introducing another relaxation timescale.
  • In-band approximation: Across the signal band, the near positive-frequency pole determines in-band selectivity and finite memory, while the distant conjugate pole contributes only O(B/fc) variation.The single-pole model therefore has relative error O(1/QL) for the in-band response, with neglected terms absorbed into a static renormalization factor.

APPENDIX B PROOF OF THEOREM 1

The proof establishes the CP condition by showing that the OFDM windowing integrals eliminate contributions from neighboring symbols when propagation and RIS-memory delays fit within the guard interval.

  • Windowing argument: The inner integral in the OFDM relation represents the cascade of the transmit window, DFT window, and RIS impulse response.The RIS response contributes temporal spreading through its finite memory.
  • Guard-interval condition: Imposing the most stringent timing inequality ensures that the combined propagation delays and RIS-memory variable remain inside the CP-supported interval.The condition is derived by bounding the relevant delayed response for all τ ∈ [0, Tγ).
  • Consequence: Under this condition, all contributions from symbols other than the aligned symbol vanish, and substitution yields the closed-form OFDM input-output relation.The proof directly connects the timing bound to elimination of intersymbol interference.
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