Source-linked AI summary
Rayleigh Fading Modeling and Channel Hardening for Reconfigurable Intelligent Surfaces
Emil Björnson, Luca Sanguinetti
TL;DR
The paper addresses the mismatch between the widely used i.i.d. Rayleigh model and the physical behavior of rectangular RISs in isotropic scattering. It derives a spatially correlated Rayleigh model based on RIS geometry, then revisits correlation rank and channel hardening. The analysis concludes that RIS channels are always spatially correlated and that properly configured RISs can exhibit channel hardening, with convergence depending on the direct path and RIS size.
Problem
The i.i.d. Rayleigh model is commonly used for RIS analysis, but the paper identifies that it does not physically appear for rectangular RISs in isotropic scattering.
Method
The paper derives a physically feasible spatially correlated Rayleigh fading model from the rectangular RIS geometry and isotropic-scattering assumptions.
Results
The derived model shows that RIS channels are always spatially correlated, with correlation-matrix rank and channel-hardening behavior determined by physical geometry and propagation conditions.
Takeaways & Limitations
i.i.d. Rayleigh fading should not be used for RIS-aided communications; a geometry-based correlated model is needed for realistic performance assessment.
Abstract
from arXiv · showhide
A realistic performance assessment of any wireless technology requires the use of a channel model that reflects its main characteristics. The independent and identically distributed Rayleigh fading channel model has been (and still is) the basis of most theoretical research on multiple antenna technologies in scattering environments. This letter shows that such a model is not physically appearing when using a reconfigurable intelligent surface (RIS) with rectangular geometry and provides an alternative physically feasible Rayleigh fading model that can be used as a baseline when evaluating RIS-aided communications. The model is used to revisit the basic RIS properties, e.g., the rank of spatial correlation matrices and channel hardening.
I. INTRODUCTION
RIS technology uses controllable two-dimensional surfaces to shape electromagnetic interactions, but the commonly used i.i.d. Rayleigh model is physically feasible only in specific array and scattering conditions. This letter shows that the model does not physically appear for rectangular RISs and develops an alternative correlated model.
- I. INTRODUCTION: RISs use many controllable sub-wavelength elements to synthesize scattering and absorption properties and shape constructive or destructive interference.The intended effects include enhancing received power at desired locations and suppressing interference elsewhere.
- I. INTRODUCTION: The i.i.d. Rayleigh model is tractable and can occur for a half-wavelength-spaced ULA in an isotropic scattering environment.It has supported foundational analyses of Massive MIMO before extensions to spatially correlated channels.
- I. INTRODUCTION: The RIS geometry consists of NH elements per row and NV elements per column in a three-dimensional arrangement.The geometry is illustrated in Fig. 1.
- I. INTRODUCTION: For a rectangular RIS in isotropic scattering, the paper proves that i.i.d. Rayleigh fading does not physically appear and derives a valid spatially correlated alternative.The alternative model is intended as a baseline for RIS-aided communication analysis.
II. SYSTEM MODEL
The system considers a single-antenna transmitter and receiver communicating through an RIS in isotropic scattering. The model explicitly represents the rectangular element grid, propagation geometry, phase configuration, direct path, and independently distributed transmitter–RIS and RIS–receiver channels.
- II. SYSTEM MODEL: The setup uses a single-antenna transmitter, a single-antenna receiver, and an RIS with N reconfigurable elements in an isotropic scattering environment.The received signal is modeled at the receiver under RIS assistance.
- II. SYSTEM MODEL: The RIS phase configuration is represented by the diagonal matrix Φ = diag(e−jφ1, . . . , e−jφN ), while the direct path follows Rayleigh fading.The direct channel has variance βd, and the noise is modeled as circularly symmetric complex Gaussian with variance σ2.
- II. SYSTEM MODEL: The paper characterizes h1 between the transmitter and RIS and h2 between the RIS and receiver using the two-dimensional surface geometry.The channels are represented as N-dimensional vectors.
- II. SYSTEM MODEL: The rectangular RIS contains N = NHNV edge-to-edge elements arranged on a two-dimensional grid, with uniformly distributed multipath components over the forward half-space.The geometry uses azimuth and elevation angles in a local spherical coordinate system.
- II. SYSTEM MODEL: Element indices are mapped to horizontal and vertical grid coordinates using modulus and truncation operations.The mapping defines the location of each element in the rectangular array.
- II. SYSTEM MODEL: A plane wave arriving from azimuth ϕ and elevation θ is represented through an array response vector.This response connects the propagation angles to the RIS channel representation.
III. RAYLEIGH FADING MODELING
The paper derives the Rayleigh fading distribution and spatial correlation of channels between a transmitter, rectangular RIS, and receiver under isotropic scattering. The resulting model links received power and correlation structure to the RIS’s physical geometry and area.
- III. RAYLEIGH FADING MODELING: The channel derivation begins with L impinging plane waves and then characterizes the fading distributions and spatial correlations of h1 and h2.The transmitter and receiver are assumed well separated, so their channels are independently distributed.
- III. RAYLEIGH FADING MODELING: Each multipath attenuation is modeled as zero-mean with variance Aµ1, while arrival angles follow the isotropic-scattering PDF.Here A = dHdV is the area of one RIS element and µ1 is the average intensity attenuation.
- III. RAYLEIGH FADING MODELING: Proposition 1 gives the spatial correlation matrix R for a rectangular RIS under isotropic scattering in the half-space in front of the surface.The proof evaluates correlations between element locations and uses coordinate rotation for elements not on the same row.
- III. RAYLEIGH FADING MODELING: The derived correlation matrix coincides with Clarke’s 3D model, and h2 has the same distribution as h1 with a different average intensity attenuation µ2.The two channels are independent under the stated propagation conditions.
- III. RAYLEIGH FADING MODELING: The average received signal power at the RIS is Pµ1 · NA, proportional to the total RIS area NA and independent of wavelength under the model.Because practical elements satisfy A ∝ λ2, the element count required for a fixed total area is inversely proportional to λ2.
A. Spatial Correlation
For a rectangular RIS in isotropic scattering, fading is spatially correlated rather than i.i.d. Rayleigh, with correlation and effective rank determined by the physical aperture geometry. The largest approximately πNA/λ^2 eigenvalues capture nearly all channel realizations, making element size and spacing central to modeling and estimation.
- Spatial correlation: The correlation between distinct RIS elements follows a sinc function of their physical distance divided by λ/2.Achieving i.i.d. fading requires separations equal to λ/2 times different integers, which does not match rectangular RIS geometry with sub-λ/2 spacing.
- Spatial correlation: A rectangular-grid RIS with NH > 1 and NV > 1 necessarily experiences spatially correlated fading.The property applies to practical two-dimensional RISs, although correlation strength depends on configuration.
- Rank and eigenvalues: The degrees of freedom equal rank(R), the number of non-zero eigenvalues, and asymptotically scale as πNA/λ^2 for a sufficiently large and dense RIS.This follows from the asymptotic degrees of freedom per square meter, π/λ^2, for a rectangular aperture in isotropic scattering.
- Rank and eigenvalues: For N = 1600 square-element RISs, the first approximately πN(d/λ)^2 eigenvalues are large but non-identical, after which they rapidly approach zero.The approximation is particularly good for small d; none of the considered cases resembles i.i.d. Rayleigh fading.
- Channel estimation: Approximately πNA/λ^2 pilot signals suffice to estimate h1 when R is known, by transmitting pilots along the dominant eigenvectors.The associated eigenvectors span the eigenspace containing all channel realizations, while eigenvectors for the smallest eigenvalues can be ignored.
B. Comparison With the Kronecker model
The Kronecker model can approximate exact RIS correlation spectra in some small-array setups, but it fails for large rectangular RISs and for λ/2 element spacing. It can incorrectly produce i.i.d. fading, miscalculate rank, and miss substantial differences between the exact and approximate eigenvectors and matrices.
- Model construction: The Kronecker model combines the vertical and horizontal ULA correlation matrices through a Kronecker product.It has matched the exact eigenvalue spectrum in a few small-array simulation setups.
- Failure at λ/2 spacing: At dH = dV = λ/2, the Kronecker model produces i.i.d. Rayleigh fading because both constituent ULA correlation matrices become identity matrices.This contradicts the rectangular-RIS result that two-dimensional grids remain spatially correlated.
- Rank mismatch: The Kronecker model miscalculates the large-RIS rank by a factor π/4 < 1, using an approximation inconsistent with rank(R) ≈ πNA/λ^2.Consequently, it does not capture the basic properties of a large RIS.
- Approximation error: The exact and Kronecker models have mismatched eigenvectors, so agreement between their eigenvalue spectra can conceal larger differences between the full matrices.This limits spectral agreement as a sufficient validation of the approximation.
- Approximation error: The exact and approximate correlation matrices become more distant as RIS size increases, with full matrices differing more than their ordered eigenvalue spectra.Figure 3 measures this difference using correlation matrix distance for dH = dV ∈ {λ/8, λ/4}.
IV. CHANNEL HARDENING
The paper defines asymptotic channel hardening through convergence of RIS-aided SNR toward a deterministic N^2-scaled quantity, and shows this occurs under optimized RIS phases but not random phases. A direct path delays the approximation, while the asymptotic limit remains unchanged.
- Definition and interpretation: Channel hardening means the random SNR becomes approximately N^2 times a deterministic constant for large N.The definition concerns convergence of the relevant random-variable sequences, with convergence in probability used in the analysis.
- Proposition 3: The SNR can be approximated by a deterministic term when the RIS is sufficiently large.This convergence is in probability and follows from bounded variance and diminishing covariance between sufficiently separated elements.
- Direct-path effect: Although the asymptotic expression does not depend on βd, the direct-path strength determines how many RIS elements are needed before it applies.The RIS path must be much stronger than the direct path for the deterministic approximation to become accurate.
- Numerical illustration: With optimal phases, the instantaneous SNR approaches the deterministic approximation, whereas random phases retain large variations and show no hardening.The comparison is made with and without a direct path as the square RIS grows from 1 to 40 elements per dimension.
- Numerical illustration: NH ≥10 gives close agreement without a direct path, while NH ≥25 is needed when a direct path is present.The direct path is observed in the figure only once the RIS path becomes stronger, occurring around NH ≥12 under the stated setup.
V. CONCLUSIONS
The conclusions discourage i.i.d. Rayleigh modeling for RIS channels because fading is always spatially correlated. The paper provides an isotropic-scattering model and characterizes its rank and channel-hardening properties.
- V. CONCLUSIONS: RIS-aided channel fading is always spatially correlated, so the paper discourages using the i.i.d. Rayleigh fading model.The asymptotic SNR limit is equal under the two models, but their convergence rates and spatial-correlation ranks differ.
- V. CONCLUSIONS: The derived channel properties also apply to holographic MIMO arrays with the same form factor.The stated properties include spatial-correlation rank and channel hardening.