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Path Loss Modeling and Measurements for Reconfigurable Intelligent Surfaces in the Millimeter-Wave Frequency Band

Wankai Tang, Xiangyu Chen, Ming Zheng Chen, Jun Yan Dai, Yu Han, Marco Di Renzo, Shi Jin, Qiang Cheng, Tie Jun Cui

arXiv:2101.08607v2eess.SPcs.IT

TL;DR

The paper addresses limits in RIS path-loss modeling relevant to configuring and deploying RISs. It further improves a prior model to make it more accurate and easier to use, and examines unit-cell size and frequency-related power requirements.

  • Problem

    Existing RIS path-loss modeling has limits relevant to optimizing RIS configuration and deployment.

  • Method

    The paper further improves a prior path-loss model to make it more accurate and easier to use.

  • Results

    At 30 GHz, an RIS consumes 100 times more power than at 3 GHz to achieve the same performance.

  • Takeaways & Limitations

    The proposed model is presented as a simple, sufficiently accurate basis for analyzing RIS-assisted wireless communications.

  • Takeaways & Limitations

    The prior path-loss model still requires improvement to become more accurate and easier to use.

Abstract

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Reconfigurable intelligent surfaces (RISs) provide an interface between the electromagnetic world of wireless propagation environments and the digital world of information science. Simple yet sufficiently accurate path loss models for RISs are an important basis for theoretical analysis and optimization of RIS-assisted wireless communication systems. In this paper, we refine our previously proposed free-space path loss model for RISs to make it simpler, more applicable, and easier to use. The impact of the antenna's directivity of the transmitter, receiver, and the unit cells of the RIS on the path loss is explicitly formulated as an angle-dependent loss factor. The refined model gives more accurate estimates of the path loss of RISs comprised of unit cells with a deep sub-wavelength size. Based on the proposed model, the properties of a single unit cell are evaluated in terms of scattering performance, power consumption, and area, which allows us to unveil fundamental considerations for deploying RISs in high frequency bands. Two fabricated RISs operating in the millimeter-wave (mmWave) band are utilized to carry out a measurement campaign. The measurement results are shown to be in good agreement with the proposed path loss model. In addition, the experimental results suggest an effective form to characterize the power radiation pattern of the unit cell for path loss modeling.

I. INTRODUCTION

RISs are presented as a promising architecture for controlling wireless environments, while simple and accurate path-loss models remain an open requirement for analysis and deployment optimization.

  • mmWave and THz bands offer abundant spectrum and high peak data rates for future 6G services, but suffer from large path loss and blockage sensitivity.
  • RISs use sub-wavelength unit cells to modify incident-signal phase and amplitude, enabling electromagnetic-wave manipulation and wireless-channel shaping.
  • RISs can perform analog beamforming by co-phasing reflected signals, enhancing received signals and helping address distance and blocking problems.
  • RIS-assisted communications support applications including wireless power transfer, environmental sensing and positioning, and secure communications.
  • Simple but sufficiently accurate RIS path-loss models are needed for link-budget analysis, performance-limit assessment, and configuration and deployment optimization.
  • Prior studies developed analytical path-loss models using scattering theory, diffraction, Green’s theorem, mutual impedances, and measurements, but left limitations in applicability or unit-cell characterization.

B. Main Contributions

The paper refines RIS path-loss modeling, analyzes single-unit-cell scaling and efficiency, and validates the resulting model through mmWave measurements.

  • The refined model explicitly incorporates transmit-antenna, receive-antenna, and RIS-unit-cell radiation patterns into an easier-to-use path-loss formulation.
  • Single-unit-cell evaluation shows energy efficiency is inversely proportional to the square of operating frequency, while power consumption per unit area is proportional to it.
  • These scaling results identify fundamental challenges for deploying RISs at high frequencies.
  • A mmWave measurement campaign validates the path-loss model and experimentally supports the metal-plate scaling relation for unit-cell scattering gain.
  • The analysis relates unit-cell scattering gain to cell size and uses a metal plate benchmark to model this relation for deep sub-wavelength cells.
  • Measurements suggest cos θ as an effective expression for the power-radiation-pattern shape of a sub-wavelength unit cell, with results agreeing well with the analytical model.

B. Previously Proposed Path Loss Models for RIS-Assisted Transmission

The prior framework provides a general RIS path-loss model applicable across configurations and propagation regions, with specialized models for far-field beamforming and near-field focusing.

  • The prior work derived a general free-space RIS path-loss model based on the physics and electromagnetic nature of RISs.
  • The framework produced specialized path-loss models for three typical RIS-assisted transmission scenarios.
  • The general model can be applied to arbitrarily configured RISs in both near-field and far-field regions.
  • 2) RIS-Assisted Far-Field Beamforming: In far-field beamforming, the RIS directs incident signals toward a specific direction to enhance received signal power.
  • 2) RIS-Assisted Far-Field Beamforming: The far-field model assumes identical reflection-coefficient amplitudes across all RIS unit cells.
  • In near-field focusing, the RIS focuses reflected signals toward a user while accounting for non-planar-wave distances and differing incidence and reflection angles.

4) RIS-Assisted Near-Field Broadcasting:

The paper addresses limitations in RIS path-loss modeling by explicitly formulating geometry- and directivity-dependent losses and examining unit-cell scaling, efficiency, and radiation patterns.

  • 1) Joint Radiation Pattern of Antennas and Unit Cells: Existing models often assume isotropic transmit and receive antennas, thereby neglecting antenna directivity.
  • 1) Joint Radiation Pattern of Antennas and Unit Cells: Prior formulations did not explicitly express link-geometry effects or validate the unit-cell power-radiation pattern through measurements.
  • 2) Relation Between the Scattering Gain of the Unit Cell and Its Size: With fixed RIS area, decreasing unit-cell area while increasing cell count could make predicted received power grow without bound unless scattering gain and cell size are related.
  • 2) Relation Between the Scattering Gain of the Unit Cell and Its Size: The paper addresses the scaling inconsistency through refined path-loss models and further discussion of unit-cell properties.
  • 1) Joint Radiation Pattern of Antennas and Unit Cells: The refined joint radiation pattern accounts for transmit-antenna, unit-cell, and receive-antenna directivities through an angle-dependent loss factor.
  • 1) Joint Radiation Pattern of Antennas and Unit Cells: The power sensed and reflected by a unit cell depends on its location relative to the transmitter and receiver, with edge cells receiving less power.

B. Explicit Relation Between the Scattering Gain of the Unit Cell and Its Size

The section derives a path-loss model that explicitly relates RIS unit-cell scattering gain to cell size, while accounting for incidence and reception angles. It further characterizes how RIS area, operating frequency, and unit-cell properties affect propagation loss.

  • Far-field beamforming: Beamforming gain in the far field depends on the square of the RIS geometric area, (MNdxdy)^2.The result follows from Proposition 2 for identical unit-cell reflection amplitudes.
  • Angular dependence: Path loss depends on incidence angle θt and reception angle θr, increasing as either angle increases and satisfying channel reciprocity.This angular dependence is consistent with previously reported findings.
  • Unit-cell characterization: The scattering gain of a unit cell is G = 4πdxdy, linking cell gain directly to its dimensions.This relation is used to refine the general RIS path-loss model.
  • General path-loss model: Theorem 1 expresses the general free-space RIS path-loss model by explicitly relating scattering gain to unit-cell size.The model is presented as applicable to RIS-assisted wireless communication systems.
  • Frequency dependence: Because unit-cell area is inversely proportional to the square of operating frequency, RIS path loss is proportional to the fourth power of frequency.This frequency dependence follows from Theorem 1 and the unit-cell size relation.

SUMMARY OF THE REFINED PATH LOSS MODELS FOR RIS-ASSISTED WIRELESS COMMUNICATIONS

The refined RIS path-loss formulation relates unit-cell performance to geometry, distance, frequency, antenna gains, and an angle-dependent loss factor. It also exposes tradeoffs among scattering performance, power consumption, and area, motivating careful high-frequency system design.

  • The unit-cell sub-channel path loss scales with the square of the transmitter/receiver distances and inversely with unit-cell area and the loss factor.The model captures distance, cell size, and combined angle-dependent effects.
  • Scattering performance, power consumption, and area: Scattering performance is proportional to unit-cell area and inversely proportional to the square of operating frequency.For sub-wavelength dimensions, reducing cell dimensions with increasing frequency lowers single-cell scattering performance.
  • Scattering performance, power consumption, and area: Average unit-cell power consumption is frequency-independent and often on the order of mW, including tunable components and external control circuits.Reported examples are 2.8 mW, 6.2 mW, and 3.75 mW for RISs operating in different frequency bands.
  • Scattering performance, power consumption, and area: Unit-cell area is inversely proportional to f 2, while energy efficiency is inversely proportional to f 2 and area efficiency is frequency-independent.Energy efficiency is scattering performance divided by power consumption; area efficiency is scattering performance divided by cell area.
  • High-frequency deployment tradeoffs: 100 times higher energy efficiency at 3 GHz than 30 GHz means a 30 GHz RIS consumes 100 times more power to achieve the same performance.The higher-frequency RIS needs 100 times more unit cells because each cell has 100 times lower scattering performance.
  • High-frequency deployment tradeoffs: High-frequency RIS deployment requires a performance–power tradeoff, potentially addressed through higher transceiver gains or fixed-coded passive RISs.Fixed-coded metasurfaces have zero power consumption and extremely low hardware cost even when their area is large.
  • Angle-dependent factor: The angle-dependent factor keeps received power finite as RIS size approaches infinity and supports reciprocal path-loss models when the reflection coefficient is reciprocal.Reciprocity holds for reciprocal incidence and reception angles.

IV. MEASUREMENTS IN THE MMWAVE BAND TO VALIDATE THE PATH LOSS MODELS

The paper validates its free-space RIS path-loss models through mmWave measurements using two fabricated, 1-bit phase-programmable metasurfaces and two complementary measurement systems. System A measures angular reflected-power distributions, while system B enables flexible distance and angle configurations.

  • Experimental objective: The measurement campaign is designed to validate the proposed free-space path-loss models in the mmWave frequency band.The paper introduces the fabricated RISs, experimental setup, and calibration method for this purpose.
  • Fabricated RISs: Two fabricated metasurfaces, mmWave RIS1 and mmWave RIS2, are used for mmWave path-loss measurements; both employ 1-bit phase coding.Their detailed specifications are reported in Table III.
  • Measurement system A: Measurement system A uses fixed RIS and transmitter positions with normal incidence and records reflected received power as a function of reception angle.A horn receiver and RF signal analyzer measure power at different reception angles for comparison with the model.
  • Measurement system A: System A’s limitation is that its fixed transmitter position prevents flexible variation of transmission distance d1 during measurements.This motivates system B for studying model validity across d1 and d2.
  • Measurement system B: Measurement system B uses movable antennas in a large absorber-lined room, enabling configurations with different d1, d2, θt, and θr.Unlike system A, system B does not use a rotation platform; the RIS is placed on a stable tripod.

3) Calibration Method:

The paper calibrates two free-space measurement systems to account for cable losses and antenna-gain factors when comparing RIS measurements with the path loss model. The calibrated setup is then used to compare RIS reflection with an equal-size metal plate.

  • Calibration procedure: The measured calibration parameters replace G_tG_r in the path loss model with G_tG_rG_line to include RF-cable losses.All theoretical curves based on the model use the calibration parameters listed in Table IV.
  • Calibration procedure: Calibration aligns the transmit and receive antennas while the signal propagates directly through free space.The received calibration power is used to determine the effective antenna and cable factors for each system and operating frequency.
  • Metal-plate benchmark: The RIS and an equal-size, equal-shape metal plate are measured without changing the measurement settings.The RIS is covered with the metal plate to provide a direct reflection benchmark.
  • Measurement systems: Measurement systems A and B characterize power reflected from the RIS under different experimental configurations.System A measures reflected power as a function of reception angle, while system B permits flexible variation of distances and angles.
  • Metal-plate benchmark: A uniformly coded, high-efficiency RIS has reflection characteristics comparable to an equal-size metal plate, although its main-lobe scattered power is slightly lower.The lower power is attributed to the unit-cell reflection-coefficient amplitude of 0.9, below that of a metal plate.
  • Model validation: The measured RIS1 reflection results agree with the proposed path loss model after practical attenuation and fluctuation near θ_r = 0° are considered.Near zero reception angle, the transmit antenna blocks the received link.

C. Normalized Power Radiation Pattern of the Unit Cell

The paper models each RIS unit cell’s normalized power radiation pattern and tests the resulting angle dependence experimentally. Measurements with two mmWave RISs support the assumed cosine-shaped pattern and the proposed path loss model.

  • Radiation-pattern model: The unit-cell normalized power radiation pattern is modeled with a cos θ shape.The received power is consequently proportional to the squared normalized pattern, (cos θ)^2, when θ_t = θ_r = θ.
  • Experimental validation: Measurements with mmWave RIS1 and RIS2 are conducted in the far-field using d_1 = d_2 = 2 m and 5 m, respectively.The operating frequencies are 27 GHz for RIS1 and 33 GHz for RIS2.
  • Experimental validation: The experimental measurements confirm the assumed relationship between received power and the squared unit-cell radiation pattern.The measurements reported in Fig. 9 support the modeling assumption.
  • Reflection configurations: Specular reflection uses a common coding state, whereas intelligent reflection designs unit-cell coding states to produce a different reflection angle.A stripe coding pattern is given as an example for dual-beam intelligent reflection.
  • Model accuracy: For deep sub-wavelength RIS1 unit cells, the refined model provides a better estimate than the earlier model.The earlier model overestimates received power in the considered specular-reflection setup.
  • Angle dependence: The received power decreases as θ_t and/or θ_r increase, demonstrating angle-dependent behavior.Measurements across distances and angles agree with the proposed general path loss model and the far-field beamforming model for RIS1.

2) Intelligent Reflection via the mmWave RIS1:

Measurements with mmWave RIS1 validate the general path loss model for both specular and intelligent reflection. Stripe coding successfully produces dual beams at the predicted reflection angles, while the refined model improves accuracy for deep sub-wavelength cells.

  • Intelligent reflection setup: A stripe coding pattern configures mmWave RIS1 for dual-beam intelligent reflection.The binary pattern alternates coding states across unit-cell stripes.
  • Intelligent reflection results: The two reflected beams are experimentally realized at |θ_r| = 34°.The measurements in Fig. 11(a) agree with the proposed general path loss model.
  • Intelligent reflection results: Measurements versus d_2 at θ_t = 0, θ_r = 34°, and d_1 = 0.5 m agree with the general model, which is more accurate than the earlier model.This experiment uses measurement system B.
  • RIS2 validation: For mmWave RIS2, far-field specular measurements agree with both the general model and the far-field beamforming model.The far-field setup uses d_1 = 5 m and d_2 ≥ 5 m.
  • RIS2 validation: For mmWave RIS2 near-field broadcasting, the general model agrees with measurements, while the earlier model is sufficiently accurate within an absolute error difference below 2 dB.The RIS2 unit-cell size is approximately half a wavelength, explaining the smaller gap between the models.
  • RIS2 validation: RIS2 intelligent reflection produces two beams at |θ_r| = 37° and agrees with the general path loss model.The result is reported for θ_t = 0 and d_1 = 5 m.

F. Validation via the Measurements Reported in [22]

Additional measurements from previously studied RISs further test the general path loss model across near-field, intelligent-reflection, and deep sub-wavelength configurations. Overall, the model gives sufficiently good estimates of scattered power across key RIS parameters and unit-cell configurations.

  • Large RISs: For large RIS1, near-field specular-broadcasting measurements at 10.5 GHz agree with the general path loss model.The comparison uses d_1 = 1 m and θ_t = θ_r = 45°.
  • Large RISs: For large RIS2, intelligent-reflection measurements agree with the general model, which is more accurate than the earlier model.The unit cells are approximately half a wavelength at 10.5 GHz, helping explain the relatively good predictions of the earlier model.
  • Small RIS: The small RIS has deep sub-wavelength unit cells of approximately λ/6 × λ/6 at 4.25 GHz.It is configured for far-field specular beamforming.
  • Small RIS: Removing the small RIS’s plastic frame and metal support yields good agreement with the general model and a better fit than the earlier model.The earlier discrepancy is attributed to the frame and support used in the previous measurement campaign.
  • Overall validation: The general model provides sufficiently good estimates of RIS-scattered power across several design parameters and unit-cell configurations.The conclusion synthesizes the measurement results reported for the considered RISs and reflection modes.
  • Overall validation: The paper introduces a simple free-space path loss model, validates it with mmWave metasurface RIS measurements, and supports its unit-cell scattering and radiation-pattern relationships.The results also identify tradeoffs relevant to deploying RISs at high frequencies and assessing their performance limits.

APPENDIX A - PROOF OF PROPOSITION 1

The appendix derives Proposition 1 by combining antenna radiation-pattern expressions with geometric relations and RIS path-loss assumptions. It also supports the analytical formulation as an effective, practical choice for RIS-assisted path-loss modeling.

  • The transmit and receive antenna radiation patterns are formulated explicitly in terms of antenna gains and can accommodate isotropic or other experimental patterns.
  • The model assumes transmit/receive antennas radiate or sense signals over only half of space, while commonly used antennas can satisfy the underlying power-efficiency assumption.
  • For a far-field RIS with unit reflection efficiency and normal incidence and reflection, the surface acts as a rectangular perfect electric conducting plate.
  • The resulting formulation relates RIS scattering gain and unit-cell surface area and is supported as an effective, practical path-loss model.
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