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Rayleigh 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 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 ϕ 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 ∈ 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 = E{|s| } 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 ∈ 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 ] ∈ 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 ] ∈ 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(ϕ)+(j(n)−j(m))d sin(θ))o = e λ H V . (9) space, where a local spherical coordinate system is defined E with ϕ 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(ϕ, θ) = , ϕ ∈ − , , θ ∈ − , . (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(ϕ, θ)dθdϕ row-by-row by n ∈ [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 ϕ 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(ϕ, θ) = ejk(ϕ,θ) u1 , . . . , ejk(ϕ,θ) uN (4) channel h1 from the transmitter to the RIS. As expected, it where k(ϕ, θ) ∈ 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(ϕ, θ) = [cos(θ) cos(ϕ), cos(θ) sin(ϕ), 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 E{kh1sk } = P Aµ1tr (R) = P µ1 · NA (13) L |{z} X cl Total RIS area h1 = √ a(ϕl, θl) (6) l=1 L since tr (R) = N and it is proportional to the total RIS area √ NA. Hence, the propagation conditions are independent of the where c / L ∈ is the complex signal attenuation of the l C wavelength. Since practical elements are sub-wavelength-sized lth component, ϕ is the azimuth angle-of-arrival, and θ is l l A ∝ λ2 [3], [4], the number of elements N needed to achieve the elevation angle-of-arrival. The attenuations c , . . . , c are 1 L a given total area NA is inversely proportional to λ2. i.i.d. with zero mean and variance Aµ1, where A = dHdV is the area of an RIS element and µ1 is the average intensity attenuation. The angles have the PDF f(ϕ, θ) in (2). A. Spatial Correlation As L → ∞, it follows from the central limit theorem that The isotropic scattering environment gives rise to Rayleigh fading, as expected, but it will only be i.i.d. Rayleigh fading h →d N (0, Aµ R) (7) 1 C 1 if R is an identity matrix. Proposition 1 shows that the spatial where the convergence is in distribution and the normalized correlation between two different RIS elements is a sinc- spatial correlation matrix R ∈ CN×N is computed as function of the physical distance between the elements divided by λ/2. Since the sinc-function is only zero for non-zero 1 H H R = E {h1h1 } = E {a(ϕ, θ)a(ϕ, θ) } . (8) integer arguments, all the elements must be separated by λ/2 Aµ1 3

102 is small, which is line with the proposition. The i.i.d. Rayleigh fading is also reported as reference. We can see that none of the considered cases resembles it. The case dH = dV = λ/2 is the closest one, but there are major differences: 25% of the 0 10 eigenvalues are larger than one, while 20% of the eigenvalues are much smaller than one. Since an RIS is envisioned to be implemented with element sizes d ∈ [λ/8, λ/4] [3], [4], we 10-2 should expect the spatial correlation to be far from i.i.d. fading. Remark 1. Proposition 2 states that the eigenvectors asso- ciated with the πNA/λ2 largest eigenvalues of R span the 10-4 eigenspace where all the channel realizations reside. This is a 0 200 400 600 800 1000 1200 1400 1600 useful property during channel estimation. If R is known, the pilot signals can be transmitted along its eigenvectors and we Fig. 2. The eigenvalues of R in decreasing order for an RIS can ignore those associated with the smallest eigenvalues [12]. with N = 1600 and dH = dV = d ∈ {λ/8, λ/4, λ/2}. Hence, it is sufficient to transmit approximately πNA/λ2 pilot signals to estimate h1. While the received power in (13) only times different integers to achieve i.i.d. fading. This is satisfied depends on the total area NA, the rank also depends on the for any one-dimensional ULA with λ/2-spacing [5] or a two- wavelength λ. The rank increases with the carrier frequency dimensional triangular array with the right spacing between the and, thus, the pilot resources must also increase. three elements. None of these setups match with an RIS, which has correlation along all the diagonals and sub-λ/2 spacing. B. Comparison With the Kronecker model Corollary 2. Any RIS deployed on a rectangular grid is A so-called Kronecker model has been utilized to analyze subject to spatially correlated fading if NH > 1 and NV > 1. the spatial correlation of planar arrays in previous works (e.g., [13]). We will now compare it with the exact characterization This property holds for any practical RIS since these are provided by Proposition 1. To this end, we enrich the notation by definition two-dimensional. It also holds for other surface by letting R(N ,N ) denote the exact correlation matrix with shapes than rectangles. The strength of the spatial correlation H V NH elements per row and NV columns. The Kronecker model depends on the configuration. The eigenvalue spread of R is a creates an approximate correlation matrix Rapprox as [13] common way to quantify the spatial correlation. In particular, (NH,NV) one can consider its rank, i.e., rank(R). All the eigenvalues approx R(N ,N ) = R(1,NV) ⊗ R(NH,1) (15) are equal in i.i.d. Rayleigh fading and the rank is maximum, H V i.e., rank(R) = N. In correlated channels, however, the rank where ⊗ denotes the . This is a combination NV×NV can be smaller and the eigenvalues are non-identical. of the spatial correlation matrix R(1,NV) ∈ C of a NH×NH vertical ULA and R(NH,1) ∈ C of a horizontal ULA. Proposition 2. As N → ∞ and A → 0 such that NA → ∞, The numerical results of [13], [14] show that the eigenvalue we have that spectrum of Rapprox matches quite well with R rank(R) (NH,NV) (NH,NV) → 1. (14) in a few simulation setups with small arrays. However, the πNA/λ2 approximate equivalence breaks down immediately if an RIS Proof: Define the degrees of freedom (DoF) of the RIS as with dH = dV = λ/2 is considered. In this case, both the the number of non-zero eigenvalues of R. Hence, we have that vertical and horizontal ULAs have elements with λ/2−spacing DoF = rank(R). As N → ∞ and A → 0 such that NA → and thus the spatial correlation between their respective ele- ∞, the RIS becomes an infinitely large spatially-continuous ments is zero, i.e., R(1,NV) = INV and R(1,NH) = INH . We electromagnetic aperture of rectangular geometry. Therefore, approx have that R(N ,N ) = IN , thus the approximation in (15) the proposition follows from [10], [11], which prove that the H V gives rise to i.i.d. Rayleigh fading if dH = dV = λ/2. 2 DoF per m in an isotropic environment is asymptotically (as This is in contrast to Corollary 2 and makes the Kronecker 2 the aperture size grows) equal to π/λ . model inappropriate in these conditions. Notice also that The practical interpretation of Proposition 2 is that rank(R) rank(R ) = rank(R ) ≈ 2NA/λ as the ULAs 2 (1,NV) (NH,1) can be approximated by πNA/λ for a sufficiently large and grow large and inter-element spacing becomes smaller.1 Under 2 dense RIS. This means that the πNA/λ largest eigenvalues the conditions of Proposition 2, we then have that of R almost sum up to tr(R), which is the sum of all the eigenvalues. This is illustrated in Fig. 2, which shows the approx 4NA rank(R ) = rank(R(1,N ))rank(R(N ,1)) ≈ eigenvalues of R in decreasing order in a setup with N = 1600 (NH,NV) V H λ2 (16) elements (N = N = 80). We consider square elements of H V while rank(R ) ≈ πNA/λ2; that is, the rank is different size: d = d = d ∈ {λ/8, λ/4, λ/2} with A = d2. (NH,NV) H V miscalculated by a factor π/4 < 1 [11]. Hence, the Kronecker The approximate rank πN(d/λ)2 is indicated by circles on the model does not capture the basic properties of a large RIS. curves. The figure shows that the first πN(d/λ)2 eigenvalues are large but non-identical. After that, the eigenvalues quickly 1This can be proved following the same steps of the proof in Proposition 2 approach zero. The approximation is particularly good when d using the results in [11, Sec. III.A]. 4

2 0.25 10

0 0.2 10

-2 0.15 10

0.1 10-4

0.05 10-6

0 10-8 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40

Fig. 3. The distance between the exact correlation matrix and (a) Without direct path: βd = −∞ dB Kronecker model in (15) for varying N = N ∈ [1, 40]. H V 102

Moreover, the eigenvectors are not matching, thus 0 R and Rapprox are more different than their eigen- 10 (NH,NV) (NH,NV) value spectra reveal. This is illustrated in Fig. 3 that compares the exact and approximate matrices using the correlation 10-2 matrix distance [15], which is a value between 0 and 1.2 There are curves for dH = dV ∈ {λ/8, λ/4} considering either the full matrices or only diagonal matrices containing the ordered 10-4 eigenvalues. Fig. 3 shows that the distance (i.e., approximation error) increases with the RIS size, and that the full matrices 10-6 are much more different than their eigenvalue spectra. 0 5 10 15 20 25 30 35 40

IV. CHANNEL HARDENING (b) With direct path: βd = −130 dB Channel fading has a negative impact on the communi- Fig. 4. The SNR achieved with an optimized RIS and with cation performance due to the signal-to-noise ratio (SNR) random phase-shifts for varying N = N ∈ [1, 40]. Conver- variations that it creates. MIMO channels generally provide H V gence to the asymptotic expression in (23) is illustrated with spatial diversity that can reduce such variations. In particular, or without the direct path. i.i.d. Rayleigh fading channels give rise to so-called channel hardening, where the SNR variations average out (in relative terms) as the number of antennas increases [6], [16]. We will can be defined differently (e.g., convergence in probability now provide a new general definition of channel hardening or almost surely [6]). The type of convergence is irrelevant that can be utilized in RIS-aided communications. in this context since it is the behavior for large but finite N that matters; the channel models break down if we physically With the optimal phase-configuration φn = arg(h1nh2n) − let N → ∞ [17]. The practical interpretation of channel arg(hd) [4], [8], the instantaneous SNR of the system in (1) SNR is hardening is that the random h1,h2,hd is approximately N !2 equal to N 2 times a deterministic constant when N is large. P X SNR = |h h | + |h | . (17) The quadratic scaling makes the behavior very different from h1,h2,hd σ2 1n 2n d n=1 Massive MIMO and is called the “square law” [2]. We will This SNR plays a key role in fast fading scenarios, where prove that channel hardening appears with the new fading model. the ergodic rate is E {log2 (1 + SNRh1,h2,hd )}. It is important also in slow fading scenarios, where the outage probability for Lemma 1. Let {Xn} be a sequence of random variables a rate R is Pr {log (1 + SNR ) < R}. The random- 2 h1,h2,hd with mean value A, bounded variances, and covariance ness of h1, h2, hd determines the performance in both cases. Cov{Xi,Xj} → 0 when |i − j| → ∞, then Definition 1. Asymptotic channel hardening occurs in an RIS- N aided communication system if 1 X X → A (19) N n SNRh1,h2,hd n=1 2 → constant as N → ∞. (18) N with convergence in probability. This definition involves convergence of sequences of the random variables {hd, h1n, h2n : n = 1,...,N}, which Proof: This is a special case of [18, Ex. 254].

2For two correlation matrices A, B of matching size, the correlation matrix Proposition 3. In an isotropic scattering environment with h1 distance in [15] is 1 − tr(AB)/(kAkkBk) using the Frobenius norm. and h2 being independent and distributed as in Corollary 1, 5 it holds that dotted curve shows the deterministic approximation in (23). 2 PN  The instantaneous SNR matches well with the dotted curve n=1 |h1nh2n| + |hd| π2 for N ≥ 10 in Fig. 4a, in the sense that the median becomes → A2µ µ as N → ∞ H N 2 1 2 16 closer and the random variations reduce (in relative terms). A (20) larger number NH ≥ 25 is needed in Fig. 4b because of the where the convergence is in probability. presence of the direct path. These are examples of the channel p Proof: Corollary 1 implies |h1n| ∼ Rayleigh( Aµ1/2) hardening proved by Proposition 3. In contrast, the random p and |h2n| ∼ Rayleigh( Aµ2/2). Due to their mutual inde- SNR variations in the case with random phases remain large √ in both cases since there is no hardening. When the direct path pendence, it follows that E{|h1nh2n|} = Aπ µ1µ2/4 and that the variance is bounded. A consequence of Proposition 1 is present, the SNR increases very slowly with N. Hence, an is that the covariance between h1n and h1m goes to zero as RIS must be properly configured to benefit from the channel |n − m| → ∞, thus we can invoke Lemma 1 to obtain hardening and SNR gain. N √ 1 X Aπ µ1µ2 V. CONCLUSIONS |h h | → (21) N 1n 2n 4 n=1 The channel fading in RIS-aided communications will al- ways be spatially correlated, thus we discourage from using with convergence in probability. It then follows that N !2 the i.i.d. Rayleigh fading model. The asymptotic SNR limit 1 X |h h | + |h | is equal, but the convergence rate and rank of the spatial cor- N 2 1n 2n d n=1 relation matrices are different. We have provided an accurate 2 channel model for isotropic scattering and characterized its N !  √ 2 1 X 1 Aπ µ1µ2 = |h h | + |h | → (22) properties, including rank and channel hardening. The derived N 1n 2n N d 4 n=1 channel properties also apply for Massive MIMO arrays with the same form factor, called holographic MIMO [10], [20]. by exploiting that |hd|/N → 0. This is equivalent to (20). This proposition shows that the random SNR in an isotropic REFERENCES scattering environment can be approximated by a deterministic term as [1] M. D. Renzo et al., “Smart radio environments empowered by recon- P π 2 figurable intelligent surfaces: How it works, state of research, and road ahead,” IEEE J. Sel. Areas Commun., vol. 38, no. 11, pp. 2450–2525, SNRh1,h2,hd ≈ 2 µ1µ2 AN (23) σ 4 2020. when the RIS is sufficiently large. This deterministic approxi- [2] Q. Wu and R. 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