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Communicating with Large Intelligent Surfaces: Fundamental Limits and Models

Davide Dardari

arXiv:1912.01719v2cs.ITeess.SP

TL;DR

The paper addresses how to characterize optimal communication between intelligent surfaces when electromagnetic eigenfunction solutions are difficult to compute. It develops analytical descriptions of link gain and DoF, finding that geometry-normalized factors govern the link and that LIS antennas can support multiple DoF in LOS conditions.

  • Problem

    Optimal communication between LIS/SIS requires understanding link gain and spatial DoF when direct electromagnetic eigenfunction solutions are computationally prohibitive.

  • Method

    The paper formulates LIS/SIS communication as an electromagnetic eigenfunctions problem and derives approximate, accurate analytical expressions for link gain and available DoF.

  • Results

    Achievable DoF and link gain depend on geometric factors normalized to wavelength; Friis’ formula is invalid for LIS links, and LIS antennas can exploit DoF greater than 1 in LOS.

  • Takeaways & Limitations

    LIS-based communication can provide spatial multiplexing in strong LOS conditions at practical distances, where conventional MIMO provides only beamforming gain.

Abstract

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This paper analyzes the optimal communication involving large intelligent surfaces (LIS) starting from electromagnetic arguments. Since the numerical solution of the corresponding eigenfunctions problem is in general computationally prohibitive, simple but accurate analytical expressions for the link gain and available spatial degrees-of-freedom (DoF) are derived. It is shown that the achievable DoF and gain offered by the wireless link are determined only by geometric factors, and that the classical Friis' formula is no longer valid in this scenario where the transmitter and receiver could operate in the near-field regime. Furthermore, results indicate that, contrarily to classical MIMO systems, when using LIS-based antennas DoF larger than 1 can be exploited even in strong line-of-sight (LOS) channel conditions, which corresponds to a significant increase in spatial capacity density, especially when working at millimeter waves.

I. INTRODUCTION

Higher-frequency wireless networks face increased path loss and sparser multipath, weakening conventional MIMO spatial multiplexing. Intelligent surfaces offer a route toward flexible wireless environments, but their communication limits and near-field behavior require new models.

  • Higher frequencies increase channel path loss and make multipath sparse, motivating cell densification, massive MIMO, and millimeter- or THz-band operation.
  • Sparse multipath can replace MIMO’s spatial multiplexing with beamforming gain, causing capacity to grow logarithmically rather than linearly with antenna count.
  • Programmable metasurfaces can act as smart electromagnetic reflectors and configurable antennas embedded in objects such as walls, clothes, and buildings.
  • Smart radio environments jointly optimize wireless devices and the reconfigurable environment instead of adapting devices alone to propagation conditions.
  • Prior RIS studies examine rate enhancement, LOS rank increases through artificial paths, RIS-relay comparisons, metaprisms, distributed LIS assignment, and non-stationary massive-array reception.
  • Existing work does not fully characterize spatial DoF for intelligent surfaces used as transmit and receive antennas, especially in near-field conditions.
  • Classical antenna-array models assume far-field plane waves and do not capture LIS near-field geometry or holographic current-distribution flexibility.

B. Main Contribution

The paper formulates optimal LIS/SIS communication electromagnetically and derives tractable expressions for link gain and spatial DoF. It shows that geometry, wavelength, and LOS conditions determine capabilities beyond classical MIMO and Friis predictions.

  • The paper formulates optimal LIS/SIS communication as an eigenfunctions problem derived from electromagnetic arguments.
  • Extensive electromagnetic-level simulations can be computationally prohibitive for large surfaces and often provide limited general insight.
  • The paper derives approximate but accurate analytical expressions for link gain and available orthogonal communication channels.
  • LIS-based antennas can provide more than one spatial DoF under LOS conditions, unlike classical MIMO, potentially boosting channel capacity.
  • Achievable DoF and link gain depend only on geometric factors normalized to wavelength, while classical Friis’ formula is invalid for LIS links.
  • Asymptotic expressions for very large LISs or large distances expose differences between classical and LIS-based communication systems.
  • The paper presents general and configuration-specific analytical results for link gain and communication DoF, followed by numerical results and discussion.

C. Notation and Definitions

The notation defines vector, matrix, electromagnetic-function, surface, volume, and norm conventions used throughout the formulation.

  • Bold lowercase symbols denote three-dimensional vectors, while bold uppercase symbols denote matrices.
  • The vector r has Cartesian coordinates, magnitude r = |r|, and direction represented by the unit vector r-hat.
  • Italic capital letters represent electromagnetic vector functions, and ∇2, ∇, and divergence notation identify differential operators.
  • Calligraphic surfaces and volumes use measure notation A_T = |S_T|, while L2(S_T) denotes square-integrable functions on S_T.
  • The outer product satisfies {r ⊗ s}_kj = r_k s_j, and ||r|| and ||X|| denote vector and Frobenius norms.
  • The symbols μ, ϵ, η, and c denote free-space permittivity, permeability, impedance, and speed of light, respectively.

II. GENERAL PROBLEM FORMULATION

The formulation models intelligent surfaces as continuous electromagnetic apertures and represents their optimal communication through singular functions. Truncating the coupled operator yields parallel orthogonal channels whose number is the effective DoF.

  • II. GENERAL PROBLEM FORMULATION: Metamaterials permit, in principle, arbitrary current distributions, so the paper asks how many orthogonal channels two intelligent surfaces can establish.
  • II. GENERAL PROBLEM FORMULATION: Each intelligent surface is approximated as a continuous array of infinitely many infinitesimal antennas, corresponding to holographic MIMO.
  • A. Problem Formulation: The electromagnetic kernel maps transmit-surface current functions to receive-surface electric fields through a Hilbert-Schmidt operator.
  • A. Problem Formulation: Coupled eigenfunction problems provide orthonormal transmit and receive bases, with each transmit eigenfunction mapped to a receive eigenfunction by singular value ξ_n.
  • A. Problem Formulation: Truncating the singular-function expansion gives the best D-dimensional approximation, with error controlled by omitted singular values that tend toward zero.
  • A. Problem Formulation: The first eigenfunction produces the largest-intensity received field, while subsequent eigenfunctions generate orthogonal communication modes.
  • A. Problem Formulation: The DoF is conventionally the minimum number D of eigenvalues needed to represent signals to a specified accuracy, such as relative to noise intensity.
  • A. Problem Formulation: The resulting architecture consists of D orthogonal parallel channels, y_n = ξ_n x_n + w_n, and multiplexing can increase capacity relative to D = 1.

B. Maximum Coupling Intensity Between Intelligent Surfaces

The section develops a coupling formulation for intelligent surfaces and motivates geometric approximations because direct eigenfunction solutions are generally elusive and computationally prohibitive.

  • Electromagnetic formulation: The coupling analysis treats orthogonal current-polarization components separately, with x-directed excitation represented through the first column of the Green tensor.The generic source current is decomposed into x, y, and z components, and each excitation direction can be analyzed separately.
  • Geometric assumptions: The transmit and receive surfaces are modeled as separated rectangular regions, with a far-field approximation requiring their distance to exceed their characteristic dimensions.The cited prior solution applies to collinear rectangular volumes when d ≫ Δx_T, Δy_T, Δz_T, Δx_R, Δy_R, and Δz_R.
  • Limiting cases: For very small antennas, only one eigenfunction solution exists, corresponding to a plane wave traveling from the transmit antenna to the receive antenna.This regime yields a single available communication mode.
  • LIS-specific challenge: The prior far-field result is not valid for LIS analysis because the antennas are not sufficiently far apart and the Fraunhofer parallax approximation fails.This motivates a different treatment for large intelligent surfaces operating outside the far-field assumptions.
  • Geometric approximation: Geometric arguments are used to bypass direct eigenfunction derivation and obtain simple expressions for the available spatial DoF in radiating near-field configurations.The stated goal is to derive particular closed-form expressions that remain useful without prohibitive electromagnetic computation.

III. POWER GAIN BETWEEN LARGE AND SMALL-MEDIUM INTELLIGENT SURFACES

The section derives link-gain expressions for a small or medium transmit intelligent surface communicating with a large receive surface, emphasizing geometric dependence and near-field saturation.

  • Geometry and assumptions: The transmit and receive surfaces are represented by planar coordinates and areas A_T = L_xL_y and A_R = S_xS_y, with the transmit surface small relative to distance d.The receive LIS lies in the xy-plane, while the transmit surface may have dimensions comparable to d only on the receive side.
  • Link-gain derivation: The link-gain calculation retains the power-integrand component perpendicular to the receive surface and approximates the transmit coordinates because the transmit antenna is small relative to d.These approximations make the result independent of the SIS orientation and dependent on its area A_T.
  • Closed-form gain: The resulting gain depends on relative geometric quantities, including the wavelength-normalized transmit area and the receive-surface geometry.For a square LIS, the expression is written using the normalized distance-related quantity F = d^2/A_R.
  • Far-field limit: In the Fraunhofer far-field limit, the gain reduces to the classical Friis expression, g(large d) = G_T G_R G_I.The corresponding factors are the isotropic free-space channel gain and the transmit and receive aperture gains.
  • Near-field behavior: Near-field diffraction limits link gain by the normalized area of the smaller antenna, so enlarging both surfaces cannot increase gain without bound.The comparison between large-LIS and large-distance regimes exposes the limitation of classical path-loss formulas for LISs.
  • Design implications: Equations (19), (21), and (22) provide design formulas for LIS link budgets without extensive electromagnetic computations.The formulas are presented as practical approximations for characterizing power gain.

IV. COMMUNICATION DOF BETWEEN INTELLIGENT SURFACES

The section estimates communication DoF by mapping source-induced spatial variation on the receive surface into local two-dimensional wavenumber bandwidth.

  • DoF approximation: The DoF approximation follows two-dimensional sampling theory by converting spatial bandwidth and receive-surface area into the number of Nyquist-rate samples.The method is applied specifically to a transmit SIS and receive LIS configuration.
  • Projected wavenumber: The observed wavenumber on the receive surface is the projection of the propagation wavenumber onto the surface, expressed through k(r, s).The projection depends on the propagation direction p̂ and the receive-surface normal n̂.
  • Local bandwidth: Because the projected wavenumber can vary across a two-dimensional surface, the local spatial bandwidth changes slowly with the observation point.For an infinitesimal region dr, the bandwidth is approximated as locally constant.
  • Bandwidth construction: The local bandwidth is obtained from the maximum wavenumber spread generated by all point sources on the transmit surface.The area operator measures the region spanned by k(r, s) as s varies over the transmit surface.
  • Specialization and validation: The resulting analytical expressions are specialized to relevant LIS configurations to obtain simple DoF formulas and qualitative insights.Their accuracy relative to eigenfunction-based DoF values is assessed in numerical results.

A. DoF of Communicating Parallel LIS and SIS

For parallel LIS and SIS, the section derives DoF behavior from geometric approximations and shows that large line-of-sight surfaces can support multiple spatial modes.

  • Geometric approximation: The general DoF expression is approximated geometrically by limiting the relevant source-region curve with a quadrilateral.The construction uses corner points of the transmit surface and Gauss' formula to evaluate the approximation.
  • Analytical tractability: The general expression lacks a closed form, but its two-fold integral can be computed quickly compared with direct eigenfunction-based analysis.The paper therefore seeks closed-form expressions for important configurations under assumptions such as L_x, L_y ≪ d.
  • Maximum DoF: The maximum DoF depends only on the transmit-surface area normalized by the square half-wavelength.The expression identifies the relevant area as that of the smaller of the two antennas.
  • Geometric limit: The ultimate DoF limit is independent of distance and is determined by the normalized area of the smallest antenna.This geometric limit contrasts with distance-dependent classical link behavior.
  • LOS spatial multiplexing: Unlike classical LOS MIMO, LIS configurations can provide DoF larger than 1 even under line-of-sight conditions.Classical LOS channel matrices may have rank 1, whereas the LIS result permits multiple exploitable spatial modes.
  • Capacity consequence: Large LOS DoF can significantly increase link capacity, especially at millimeter-wave and THz frequencies where multipath may be weak or LOS-dominated.The stated benefit is particularly relevant when rich multipath is unavailable.

B. DoF of Communicating Perpendicular LIS and SIS

For perpendicular intelligent surfaces, the achievable DoF depends on the surface geometry, including the ratio S_y/d, and can differ from the parallel-surface case.

  • Geometry: The perpendicular link uses a transmit surface in the xz plane and a receive LIS in a perpendicular plane.The transmit coordinates are s = (s_x, y_0, s_z), while the receive coordinates are r = (r_x, r_y, 0).
  • Geometry: The transmit and receive surface areas are A_T = L_x L_z and A_R = S_x S_y.Their centers and dimensions are specified by s_0 = (x_0, y_0, d), (L_x, L_z) and (0, 0, 0), (S_x, S_y).
  • DoF dependence: The perpendicular-surface expression contains a dependence on the ratio S_y/d that is absent from the parallel-surface expression.This additional term reflects the receive surface’s height relative to separation.
  • DoF dependence: A tall LIS can increase the DoF because the additional geometric term reflects its height.The text states that the term contributes to increasing DoF when the LIS is tall.
  • Comparison: The perpendicular configuration achieves less DoF than parallel surfaces, which represent the best case.The limiting behavior for very large surfaces is reported as the same as for parallel surfaces.

V. NUMERICAL RESULTS

Numerical results validate the analytical models for link gain and DoF across near- and far-field configurations. LIS geometry strongly affects performance: square parallel surfaces maximize gain and DoF, while near-field operation enables many LOS spatial degrees of freedom.

  • Geometric effects: The square LIS shape, AR = 1 : 1, provides the best geometric configuration for both link gain and DoF.For parallel surfaces, the square shape is identified as optimal; perpendicular surfaces achieve fewer DoF than parallel surfaces.
  • Link gain: Friis’ formula fails to model the link budget for LISs, especially at low F, because diffraction invalidates the usual geometric-area aperture scaling.The link gain approaches a limit value, and the commonly used antenna aperture formula is no longer valid in this regime.
  • Parallel surfaces: 78 DoF are available at low F for the evaluated parallel-surface configuration, whereas DoF tends to 1 as the Fraunhofer far-field is approached.The continuous DoF curve is plotted for readability, although values from the analytical expression should be rounded to integers no smaller than 1.
  • Model validation: The analytical DoF model agrees well with numerical eigenfunction solutions, especially for small F, while large-F discrepancies may reflect numerical issues.The numerical computation discretizes surfaces into λ/16 square patches, but the resulting singular-value decomposition becomes difficult for electrically large surfaces.
  • Eigenfunction structure: Eigenfunctions can produce orthogonal waves despite substantial spatial overlap, so spatially non-overlapped beamforming or focusing is not generally optimal for LISs.Their phase distributions guarantee orthogonality even when the received waves are almost overlapped; implementing them requires flexible antenna configurations and dedicated architectures.
  • Geometric effects: Perpendicular surfaces provide fewer DoF than parallel surfaces, and the prior expression cannot capture DoF along the SIS z direction.The perpendicular-surface result is independent of wavelength and absolute distance under the stated parameterization.

VI. CONCLUSION

The paper formulates optimal LIS/SIS communication electromagnetically and derives analytical descriptions of link gain and spatial DoF. It finds that these limits are geometry-dependent, differ from Friis-based intuition, and permit spatial multiplexing in strong LOS conditions.

  • VI. CONCLUSION: The optimal LIS/SIS communication problem is formulated as an electromagnetic eigenfunctions problem, while analytical expressions avoid extensive and potentially prohibitive electromagnetic-level simulations.The resulting expressions describe link gain and orthogonal communication modes between transmitter and receiver.
  • VI. CONCLUSION: The analytical expressions provide insights into intelligent-surface communication and design guidelines for future LIS-based wireless networks.They characterize both coupling gain and available communication modes.
  • VI. CONCLUSION: The achievable DoF and link gain are determined only by geometric factors normalized to the wavelength.The fundamental limits depend on the normalized area of the smallest antenna involved in communication.
  • VI. CONCLUSION: Classical Friis’ formula is no longer valid for this scenario, where transmitter and receiver can operate in the near-field regime.The paper therefore characterizes link behavior through geometry and wavelength-normalized quantities rather than the classical far-field expression.
  • VI. CONCLUSION: LISs can exploit spatial multiplexing with DoF larger than 1 in LOS conditions at practical distances, unlike conventional MIMO systems in strong LOS.Conventional MIMO systems can only exploit SNR enhancement through beamforming in strong LOS, according to the paper.
  • VI. CONCLUSION: Several practical issues remain, including implementing the required eigenfunctions with affordable-complexity holographic metasurfaces and defining regulatory emission masks for LISs.The paper discusses whole-antenna masks versus ad hoc masks related to ERP per square meter.

APPENDIX A

Appendix A derives closed-form analytical expressions from the electromagnetic formulation using orthogonality, integration, and approximations valid when surface dimensions are much smaller than the separation distance.

  • APPENDIX A: The appendix derives the final result by expressing tensor G(r − s), applying orthogonality, and integrating over the transmit and receive surfaces.The derivation proceeds through intermediate relations before obtaining the final result (37).
  • APPENDIX A: When Lx, Ly ≪ d, the derivation sets x0 = y0 = 0 to obtain a simplified expression.A more general expression for nonzero x0 and y0 exists but is not reported because of space constraints and limited expected insight.
  • APPENDIX A: A first-order Taylor double-series expansion in Lx and Ly approximates the integrand and leads to a closed-form solution for equation (31).Similar arguments are used to derive another closed-form expression.
  • APPENDIX A: The appendix uses the same approximation strategy to obtain closed-form expressions from the expanded integral.The derivation explicitly connects the approximated integral to equation (31).
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