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Path Loss in Reconfigurable Intelligent Surface-Enabled Channels

Steven W. Ellingson

arXiv:1912.06759v3eess.SPcs.IT

TL;DR

RIS path loss depends on element design, geometry, and control states, but practical discrete models and continuous-surface theory offer different strengths. This paper develops a parameterized element-sum model, shows equivalence to a perfectly conducting plate for a benchmark design, and uses it to compare RIS, direct, and specular-reflection channels. Its main scope boundary is reduced accuracy when a transmitter or receiver is very close to the RIS because polarization variation across the array is omitted.

  • Problem

    RIS path-loss analysis needs both practical element-level modeling and technology-independent surface limits, while comparisons with direct and specular-reflection channels remain important.

  • Method

    The paper calculates path loss by summing controlled fields from parameterizable RIS elements and uses a physically motivated benchmark element pattern with half-wavelength spacing.

  • Results

    The benchmark model is consistent with electromagnetic plate-scattering theory and prior work, and the derived design matches the continuous-surface prediction for a perfectly conducting plate.

  • Takeaways & Limitations

    The model identifies practical reflectarray-type RIS designs and scenarios exhibiting the performance predicted by continuous-surface theory, while clarifying comparisons with direct and specular-reflection channels.

  • Takeaways & Limitations

    Accuracy is reduced when the transmitter or receiver is very close to the RIS because the model omits per-element polarization variation.

Abstract

from arXiv · show

A reconfigurable intelligent surface (RIS) employs an array of individually-controllable elements to scatter incident signals in a desirable way; for example, to facilitate links between base stations and mobile stations that would otherwise be blocked. A principal consideration in the study of RIS-enabled propagation channels is path loss. This paper presents a simple yet broadly-applicable method for calculating the path loss of a channel consisting of a passive reflectarray-type RIS. This model is then used to characterize path loss as a function of RIS size, link geometry, and the method used to set the element states. Whereas previous work presumes either (1) an array of parameterizable element patterns and spacings (most useful for analysis of specific designs) or (2) a continuous electromagnetic surface (most useful for determining scaling laws and theoretical limits), this work begins with (1) and is then shown to be consistent with (2), making it possible to identify specific practical designs and scenarios that exhibit the performance predicted using (2). This model is used to further elucidate the matter of path loss of the RIS-enabled channel relative to that of the free space direct and specular reflection channels, which is an important consideration in the design of networks employing RIS technology.

I. INTRODUCTION

The paper develops a practical path-loss model for passive reflectarray-type RIS channels, addressing how path loss depends on RIS design, geometry, and element states. It connects discrete element-based analysis with continuous-surface theory and compares RIS propagation with direct and specular-reflection channels.

  • I. INTRODUCTION: RISs control scattering by individually changing the phase and possibly magnitude of each reflectarray element.The paper focuses on passive reflectarray-type RISs and their transmitter-RIS-receiver path loss.
  • I. INTRODUCTION: Discrete field summation exposes the effects of element patterns and spacing but requires difficult embedded-element-pattern calculations.Large regularly spaced arrays commonly approximate embedded patterns using isolated-element patterns.
  • I. INTRODUCTION: Continuous-surface models reveal technology-independent performance limits and scaling laws but are less useful for practical design analysis.Their formulation uses electromagnetic boundary conditions and Maxwell’s equations.
  • I. INTRODUCTION: The paper starts from parameterizable element patterns and spacings, then derives a design matching the continuous-surface prediction for a perfectly conducting plate.The matched design uses approximately 5 dBi elements with half-wavelength spacing.
  • I. INTRODUCTION: The paper also clarifies RIS path loss relative to free-space direct and specular-reflection channels.This comparison addresses a source of confusion in prior literature.
  • II. RECEIVED POWER IN THE RIS-ENABLED CHANNEL: The model calculates received power by summing the controlled complex contributions scattered from all RIS elements.The formulation includes transmitter and receiver gains, element gains, propagation distances, phases, and passive efficiency.

A. Path Loss

The path-loss formulation separates RIS-channel propagation from transmitter and receiver antenna gains by assuming those gains are constant across the RIS.

  • A. Path Loss: The model approximates transmitter and receiver antenna gains as constant over the RIS.This makes the antenna factors separable from the propagation-channel expression.
  • A. Path Loss: The approximation is exact for isotropic antennas and broadly applicable when antenna gains vary weakly across the RIS angular span.This includes weakly directional mobile-station antennas and sufficiently distant narrow-beam systems.
  • A. Path Loss: Because the received-power expression has Friis-equation form, it directly defines the RIS path loss.The paper identifies the resulting quantity as LRIS.

B. Element Pattern Model

The paper proposes a simple element-pattern model with a benchmark parameter chosen so that half-wavelength-spaced elements collectively match the RIS physical aperture.

  • B. Element Pattern Model: The element pattern models electrically small, low-gain elements above a conducting ground screen using broadside angle, gain parameter q, and normalization coefficient γ.γ is selected to conserve total radiated power.
  • B. Element Pattern Model: The benchmark condition sets the broadside effective aperture of each element to (λ/2)^2.With λ/2 spacing, the sum of element effective apertures equals the RIS physical area.
  • B. Element Pattern Model: The benchmark parameters are γ = π, q0 = 0.285, and broadside element gain approximately 5 dBi.The gain is consistent with typical patch-element gains of 3 dBi to 9 dBi.
  • B. Element Pattern Model: The pattern model permits other element spacings and q values, allowing it to represent specific element designs.The benchmark choice is not mandatory and can be tuned.
  • B. Element Pattern Model: The model assumes identical element patterns, while edge elements in large RISs may have asymmetric and varying responses.The paper states that edge-to-interior ratios are small for electrically large RISs.

C. Alternative Form of the Path Loss Equation

The paper rewrites the path-loss expression using the proposed element pattern and expresses angular dependence through dot products with the RIS broadside normal.

  • C. Alternative Form of the Path Loss Equation: The alternative path-loss form substitutes the proposed element pattern into the earlier path-loss equation under the benchmark choice q = q0.The resulting expression is introduced as a useful alternative form.
  • C. Alternative Form of the Path Loss Equation: The cosine of an element’s pattern angle is represented by the dot product between its outward direction and the RIS unit normal.The unit normal defines RIS broadside, corresponding to ψ = 0.

IV. FAR CASE

The far case assumes nearly constant transmitter- and receiver-side directions and distances across the RIS, while retaining exact element-dependent phases.

  • The RIS is far from the transmitter when element-specific directions and distances are approximately constant across the surface.
  • The RIS is far from the receiver under the analogous approximation that receiver-side directions and distances are nearly constant across elements.
  • The far approximation does not simplify the phases, which remain exact rather than being assumed independent across elements.

A. Expressions for Path Loss in the Far Case

Under the far approximation, the model gives path-loss expressions that favor phase alignment and ultimately depend on RIS physical area rather than frequency.

  • Path loss in the far case is proportional to r_s^2 regardless of the selected element coefficients.
  • Path loss is minimized by setting each coefficient phase to −φ_n; with phase-only control, this gives b_n = e^−jφ_n.
  • With λ/2 element spacing, the RIS area satisfies A = N(λ/2)^2, allowing the path-loss expression to be written in terms of physical area.
  • The resulting far-case path loss depends only on RIS physical area and not on frequency.

B. Consistency with Plate Scattering Theory

The far-case RIS model reproduces the path loss of a perfectly conducting plate in monostatic broadside geometry, but this equivalence depends on the chosen element pattern and spacing.

  • The plate comparison replaces the far-case RIS with a flat perfectly conducting plate of area A and assumes monostatic geometry.
  • Plate scattering is related to path loss through the radar range equation and its broadside monostatic radar cross section.
  • For ˆr_i = −ˆn, ˆr_s = +ˆn, and ϵ_p = 1, Equation 22 gives precisely the plate-scattering result.
  • The RIS–plate equivalence is not universal; here it follows from the q = q_0 element pattern with λ/2 spacing, although other combinations may also work.

C. Comparison to Specular Reflection

The paper argues that far-case RIS scattering should not generally be interpreted as specular reflection. It compares RIS and specular-reflection path loss and identifies RIS sizes needed for equality.

  • Specular reflection models scattering from electrically large smooth surfaces when edge diffraction is negligible, whereas far-case RIS path loss depends on RIS size.
  • The comparison replaces the RIS with an infinitely large conducting plate, whose flat surface preserves incident phasefront curvature and produces path loss equivalent to a free-space path of length ri + rs.
  • The RIS-to-specular path-loss ratio depends on physical RIS area and frequency, so specular reflection can have lower, equal, or higher path loss than the RIS channel.
  • The effective focal length fe summarizes the geometry: it is ri/2 when ri = rs and approaches the shorter link distance when the two distances are highly unequal.
  • Under typical disadvantaged geometry, LRIS = LS for RIS side lengths from meters to tens of meters, depending on frequency and effective focal length.
  • Table II requires RIS side lengths of tens to hundreds of wavelengths for equality; larger RISs increasingly outperform specular reflection because they focus the scattered field.
  • The required electrical size increases with frequency in proportion to λ^-1/2, producing nearly a tenfold increase from 0.8 GHz to 60 GHz.

V. GENERAL CASE

The general-case study compares focusing, beamforming, and the far approximation across RIS distances and aperture sizes. Focusing can substantially outperform beamforming when the aperture is large relative to the link distances, while the far approximation fails in the near case.

  • ri = rs = 104λ: For ri = rs = 104λ, focusing closely agrees with the far approximation, and an aperture side-length of at least 70λ reaches 0 dB relative path gain.Here, 0 dB denotes path gain equal to that of an equal-length free-space path.
  • ri = rs = 10λ: For ri = rs = 10λ, the far approximation diverges from focusing beyond side-lengths of about ri, while focusing approaches 45 dB at 100λ and beamforming remains 0 ± 6 dB.The results illustrate focusing’s large gain over beamforming when path distances are shorter than the aperture side-length.
  • Model limitation: The near-case results should be used with caution because the model omits variations in per-element polarization and antenna gains across the RIS.These omissions may introduce significant error when ri or rs is small compared with the RIS size, as in Fig. 4.

VI. CONCLUSIONS

The paper develops a physical RIS-scattering model for path-loss expressions and evaluates it using a benchmark element pattern with half-wavelength spacing. The resulting calculations are consistent with electromagnetic plate-scattering theory and prior work, but accuracy decreases when a transmitter or receiver is very close to the RIS.

  • The paper presents a physical model for RIS scattering and uses it to derive path-loss expressions.
  • A benchmark element pattern with λ/2 spacing makes the sum of effective element apertures equal to the RIS physical area.Other RIS designs can be represented by varying the element parameter q and element spacing.
  • The benchmark-pattern model agrees with electromagnetic plate-scattering theory and prior work.
  • Model accuracy is reduced when either transmitter or receiver is very close to the RIS because per-element magnitude, phase, and polarization responses can vary across the array.The presented model accounts for magnitude and phase variation but not polarization variation.
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