1 Rayleigh Fading Modeling and Channel Hardening for Reconfigurable Intelligent Surfaces Emil Björnson, Senior Member, IEEE, Luca Sanguinetti, Senior Member, IEEE NH Abstract—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 z 1 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 NV surface (RIS) with rectangular geometry and provides an alter- y native physically feasible Rayleigh fading model that can be used Multipath as a baseline when evaluating RIS-aided communications. The component model is used to revisit the basic RIS properties, e.g., the rank of spatial correlation matrices and channel hardening. θ Index Terms—Reconfigurable intelligent surface, channel mod- 1 ' eling, channel hardening, isotropic scattering, spatial correlation. x I. INTRODUCTION Reconfigurable intelligent surface (RIS) is an umbrella term Fig. 1. The 3D geometry of an RIS consisting of NH elements used for two-dimensional surfaces that can reconfigure how per row and NV elements per column. they interact with electromagnetic waves [1], to synthesize the scattering and absorption properties of other objects. This Several recent works have analyzed RIS-aided communi- feature can be utilized to improve the wireless physical-layer cations under the assumption of i.i.d. Rayleigh fading [2], channel between transmitters and receivers; for example, to [7], [8]. In this letter, we prove that this fading distribution enhance the received signal power at desired locations and is not physically appearing when using an RIS in an isotropic suppress interference at undesired locations [2]. The RIS scattering environment, thus it should not be used. Motivated technology can potentially be implemented using software- by this observation, we derive a spatially correlated Rayleigh defined metasurfaces [3], which consist of many controllable fading model that is valid under isotropic scattering. We sub-wavelength-sized elements. The small size makes each analyze the basic properties of the new model, including how element act as an almost isotropic scatterer and the RIS assigns the rank of the correlation matrices depends on the physical a pattern of phase-delays to the elements to create constructive geometry. We also define a new channel hardening concept and destructive interference in the desired manner [4]. and prove when it is satisfied in RIS-aided communications. When analyzing new physical-layer technologies, it is a Reproducible Research: The simulation code is available at: common practice to consider the tractable independent and https://github.com/emilbjornson/RIS-fading identically distributed (i.i.d.) Rayleigh fading channel model. For example, the basic features of Massive MIMO (multiple- II. SYSTEM MODEL input multiple-output) were first established using that model We consider a single-antenna transmitter communicating arXiv:2009.04723v2 [cs.IT] 2 Jan 2021 [5] and later extended to spatially correlated channels [6]. Only physically feasible channel models can provide accurate with a single-antenna receiver in an isotropic scattering en- insights, but the i.i.d. Rayleigh fading model can be observed vironment, while being aided by an RIS equipped with N in practice if a half-wavelength-spaced uniform linear array reconfigurable elements. The received signal r 2 C is [8] (ULA) is deployed in an isotropic scattering environment [5]. T r = (h2 Φh1 + hd) s + w (1) ©2020 IEEE. Personal use of this material is permitted. Permission from 2 IEEE must be obtained for all other uses, in any current or future media, where s is the transmitted signal with power P = Efjsj g 2 including reprinting/republishing this material for advertising or promotional and w ∼ NC(0; σ ) is the noise variance. The configuration purposes, creating new collective works, for resale or redistribution to servers of the RIS is determined by the diagonal phase-shift matrix or lists, or reuse of any copyrighted component of this work in other works. −jφ1 −jφN E. Björnson was supported by ELLIIT and the Wallenberg AI, Autonomous Φ = diag(e ; : : : ; e ). The direct path hd 2 C has Systems and Software Program (WASP). L. Sanguinetti was supported by the a Rayleigh fading distribution due to the isotropic scattering University of Pisa under the PRA 2018-2019 Research Project CONCEPT, and assumption [5]: hd ∼ N (0; βd) where βd is the variance. by the Italian Ministry of Education and Research (MIUR) in the framework C of the CrossLab project (Departments of Excellence). A main goal of this paper is to characterize the fading T N E. Björnson is with the Department of Computer Science, KTH Royal distribution of the channel h1 = [h1;1; : : : ; h1;N ] 2 C Institute of Technology, 10044 Stockholm, Sweden, and Linköping University, between the transmitter and RIS and of the channel h2 = 58183 Linköping, Sweden ([email protected]). L. Sanguinetti is with the T N University of Pisa, Dipartimento di Ingegneria dell’Informazione, 56122 Pisa, [h2;1; : : : ; h2;N ] 2 C between the RIS and receiver. To this Italy ([email protected]). end, we need to utilize the two-dimensional surface geometry. 2 The RIS is a surface consisting of N = NHNV elements From (4), the (n; m)th element of R can be expanded as which are deployed on a two-dimensional rectangular grid n T o jk(ϕ,θ) (un−um) [R]n;m = e with NH elements per row and NV elements per column [1]. E The setup is illustrated in Fig. 1 in a three-dimensional (3D) n j 2π ((i(n)−i(m))d cos(θ) sin(')+(j(n)−j(m))d sin(θ))o = e λ H V : (9) space, where a local spherical coordinate system is defined E with ' being the azimuth angle and θ being the elevation Proposition 1. With isotropic scattering in the half-space in angle. Since the RIS is deployed in an isotropic scattering en- front of the RIS, the spatial correlation matrix R has elements vironment, the multipath components are uniformly distributed 2ku − u k over the half-space in front of it, which is characterized by the [R] = sinc n m n; m = 1;:::;N (10) n;m λ probability density function (PDF) cos(θ) h π π i h π π i where sinc(x) = sin(πx)=(πx) is the sinc function. f('; θ) = ;' 2 − ; ; θ 2 − ; : (2) 2π 2 2 2 2 Proof: Consider two RIS elements n and m located on We assume each element has size dH × dV, where dH is the same row, such that i(n) = i(m) and (j(n) − j(m))dV = the horizontal width and dV is the vertical height. Hence, the kun − umk. The expression in (9) then simplifies to area of an element is A = dHdV. The elements are deployed π=2 π=2 Z Z 2π j kun−umk sin(θ) edge-to-edge so the total area is NA. The elements are indexed [R]n;m = e λ f('; θ)dθd' row-by-row by n 2 [1;N], thus the location of the nth element −π=2 −π=2 Z π=2 with respect to the origin in Fig. 1 is j 2π ku −u k sin(θ) cos(θ) = e λ n m dθ T 2 un = [0; i(n)dH; j(n)dV] (3) −π=2 2π sin kun − umk where i(n) = mod(n − 1;NH) and j(n) = b(n − 1)=NHc are λ = 2π (11) the horizontal and vertical indices of element n, respectively, λ kun − umk on the two-dimensional grid. Notice that mod(·; ·) denotes the using Euler’s formula. This expression is equal to (10). If the modulus operation and b·c truncates the argument. elements are not on the same row, we can rotate the coordinate Suppose a plane wave with wavelength λ is impinging on system so that un −um becomes a point on the new y-axis. By the RIS from the azimuth angle ' and elevation angle θ. The integrating over isotropic scatterers in the half-space in front array response vector is then given by [6, Sec. 7.3] of the RIS, we get the same result as above. h T T iT Proposition 1 characterizes the correlation matrix for the a('; θ) = ejk(ϕ,θ) u1 ; : : : ; ejk(ϕ,θ) uN (4) channel h1 from the transmitter to the RIS. As expected, it where k('; θ) 2 R3×1 is the wave vector coincides with the Clarke’s model for 3D spaces [9]. Since the channel h2 from the RIS to the receiver is subject to the 2π T k('; θ) = [cos(θ) cos('); cos(θ) sin('); sin(θ)] : (5) same propagation conditions, it has the same distribution as λ h1, except for a different average intensity attenuation µ2. III. RAYLEIGH FADING MODELING Corollary 1. In an isotropic scattering environment, h ; h In this section, we will derive the fading distribution for the 1 2 are independent and distributed as channels h1; h2 and characterize their spatial channel corre- lation. The transmitter and receiver are assumed to be well hi ∼ NC(0; AµiR) i = 1; 2 (12) separated so that their channels are independently distributed. where the (n; m)th element of R is given by (10). We begin with analyzing h1. There are infinitely many mul- tipath components in an isotropic scattering environment, but The average received signal power at the RIS is we begin by considering L impinging plane waves: 2 Efkh1sk g = P Aµ1tr (R) = P µ1 · NA (13) L |{z} X cl Total RIS area h1 = p a('l; θl) (6) l=1 L since tr (R) = N and it is proportional to the total RIS area p NA.