On the Capacity of Opportunistic Time-Sharing Downlink With a Reconfigurable Intelligent Surface

We provide accurate approximations of the sum-rate capacity of an opportunistic time-sharing downlink, when a reconfigurable intelligent surface (RIS) assists the transmission from a single-antenna base station (BS) to single-antenna user equipments (UEs). We consider the fading effects of both the direct (i.e., BS-to-UEs) and reflection (i.e, BS-to-RIS-to-UEs) links, by developing two approximations: the former one is based on hardening of the reflection channel for large values of the number of meta-atoms; the latter one relies on the distribution of the sum of Nakagami variates and does not require channel hardening. Our derivations show the dependence of the sum-rate capacity as a function of both the number of users and the number of meta-atoms, as well as to establish a comparison with a downlink without an RIS. Numerical results corroborate the accuracy and validity of the mathematical analysis.

On the Capacity of Opportunistic Time-Sharing Downlink with a Reconfigurable Intelligent Surface Donatella Darsena, Senior Member, IEEE and Francesco Verde, Senior Member, IEEE Abstract-We provide accurate approximations of the sumrate capacity of an opportunistic time-sharing downlink, when a reconfigurable intelligent surface (RIS) assists the transmission from a single-antenna base station (BS) to single-antenna user equipments (UEs).We consider the fading effects of both the direct (i.e., BS-to-UEs) and reflection (i.e, BS-to-RIS-to-UEs) links, by developing two approximations: the former one is based on hardening of the reflection channel for large values of the number of meta-atoms; the latter one relies on the distribution of the sum of Nakagami variates and does not require channel hardening.Our derivations show the dependence of the sumrate capacity as a function of both the number of users and the number of meta-atoms, as well as to establish a comparison with a downlink without an RIS.Numerical results corroborate the accuracy and validity of the mathematical analysis.

I. INTRODUCTION
I N this letter, we consider a downlink channel in which a reconfigurable intelligent surface (RIS) is employed to assist the transmission from a single-antenna transmitter towards K ≫ 1 single-antenna user equipments (UEs).An RIS is a metasurface composed of sub-wavelength meta-atoms, whose reflection coefficients can be designed via software in order to suitably manipulate the impinging signal [1].Relying on the feasibility of engineering the meta-atoms, the wireless propagation environment might be programmed by optimizing on-the-fly the reflecting properties of an RIS to achieve different network-wide aims [2].
When the base station (BS) can track the composite channels of the UEs, opportunistic time sharing is a simple and effective transmission technique, which allows the BS to use time-division multiplexing and transmit to the best user.Such a scheduling strategy has been shown to achieve the sum-rate capacity (maximum throughput) of the singleantenna downlink channel [3].A relevant question to ask is the following: How large of a performance boost does RISaided opportunistic time-sharing downlink provide over its conventional (i.e, without RIS) counterpart in terms of sumrate?The scaling law of the sum-rate capacity of a Gaussian downlink with many users K using opportunistic time sharing has been deeply studied without an RIS [4].A similar study for an RIS-aided downlink with a large number of users K and meta-atoms Q has not been carried out yet.
We focus on the sum-rate capacity achievable using opportunistic time sharing in RIS-aided downlinks.We develop an approximation of the sum-rate capacity by invoking hardening of the reflection channel in the large Q limit, which allows to readily unveil the scaling laws as a function of K and Q.Furthermore, we provide a very accurate approximation of the sum-rate capacity without invoking channel hardening, which is based on the sum of Nakagami variates.We also investigate the interplay between the gain offered by the RIS and the selection diversity among the UEs.

II. SIGNAL MODEL AND PRELIMINARIES
As in [5], the downlink transmission among the BS and K UEs is assisted by a digitally programmable RIS working in reflection mode, which is made of Q = Q x × Q y metaatoms that can be independently and dynamically controlled by digital logic devices [1].The meta-atoms are positioned along a rectangular grid having Q x and Q y elements on the x and y axes, respectively, with constant inter-element spacing d RIS .The channel between the BS and the RIS is assumed to be characterized by a dominant line-of-sight component, which is modeled as g = σ g a RIS , with pathloss σ 2 g and signature where λ 0 = c/f 0 is the wavelength, c is the speed of the light in the medium, θ RIS ∈ [0, 2π) and ϕ RIS ∈ [−π/2, π/2) identify the azimuth and elevation angles, respectively, the corresponding directional cosines are u x ≜ sin θ RIS cos ϕ RIS and u y ≜ sin θ RIS sin ϕ RIS , and ⊗ is the Kronecker product.
All the other relevant links are modeled as narrowband frequency-flat channels.Specifically, for k ∈ {1, 2, . . ., K}, h k ∼ CN (0, σ 2 h k ) models the low-pass equivalent channel response from the BS to UE k, whereas f k ∼ CN (0 Q , σ 2 f k I Q ) represents the low-pass equivalent channel response from the RIS to the k-th UE.The parameters σ 2 h k and σ 2 f k are the largescale geometric path losses of the links seen by the k-th UE with respect to the BS and the RIS, respectively.
We customarily assume that each UE uses standard timing synchronization with respect to its direct link.After matched filtering and sampling at the baud rate, the discrete-time baseband signal received at the k-th user reads as where the overall channel gain seen by the k-th UE is , . . ., K}, with the q-th diagonal entry of Γ ≜ diag (γ 1 , γ 2 , . . ., γ Q ) representing the reflection coefficient of the q-th meta-atom, s u being the information-bearing symbol intended for the u-th user with corresponding transmit power P u , and v k ∼ CN (0, 1) is the noise sample at the output of the matched filter, with v k1 statistically independent of v k2 , for k 1 ̸ = k 2 .The transmitted symbols s 1 , s 2 , . . ., s K are independent and identically distributed (i.i.d.) complex circular zero-mean unit-variance random variables (RVs).The couple (h k , f k ) is independent of both v k and s k , ∀k ∈ {1, 2, . . ., K}.
The sum-rate capacity (in bits/s/Hz) is defined as follows subject to the transmit power constraint K k=1 P k ≤ P TX , with P TX > 0 being the (fixed) maximum allowed transmit power, and the global passivity constraint For a lossless RIS, the latter constraint yields ∥γ∥2 = Q. 1iven the reflection vector γ, the sum-rate capacity is equal to the largest single-user capacity in the system [3], that is, the resource allocation policy is the opportunistic time-sharing strategy: Recalling the expression of c k , the corresponding sum-rate capacity boils down to with where Lemma 2.1: Under the constraint ∥γ∥ 2 = Q, the cost function in (2) can be upper bounded for each k as follows where the equality holds if and only if Proof: The proof comes from: (a) the Rayleigh-Ritz theorem and observing that the maximum eigenvalue of the rank-one matrix B k is ∥diag(f * k ) g∥ 2 .Physically, solution (3) implies that the RIS may introduce local power amplifications (i.e., |γ q | > 1) or local power losses (i.e., |γ q | < 1) for some meta-atoms, while ensuring that the total reradiated power by the lossless RIS is equal to the total incident power (see [6] for implementation details).

III. THEORETICAL PERFORMANCE ANALYSIS
By applying channel coding across channel coherence intervals (i.e., over an "ergodic" interval of channel variation with time), the average sum-rate capacity [3] is given by where f αopt (α) is the probability density function (pdf) of the RV α opt , which, by virtue of Lemma 2.1, can be explicitly written as Under the opportunistic time-sharing strategy, there is just one user transmitting at any time and, thus, we can resort to the encoding and decoding procedures for the code designed for a point-to-point channel [3].We consider the case in which the users approximately experience the same large-scale geometric path loss, i.e., the parameters σ 2 h k and σ 2 f k do not depend on k, i.e., σ 2 , . . ., K}, which will be referred to as the case of homogeneous users. 2 In this case, the RV α opt is the maximum of K i.i.d.RVs and its pdf is computed as where f X k (α) and F X k (α) denote the pdf and the cumulative distribution function (cdf) of the RV Trying to work with (5) for evaluating (4) is numerically difficult even for small values of K. Hence, we apply extreme value theory [7] to calculate the distribution of α opt when K is sufficiently large.Relying on the limit laws for maxima [7], provided that the cdf of X k is a von Mises function, 3 as K → ∞, the RV α opt convergences in distribution [8] to the , where Replacing f αopt (α) with the Gumbel pdf, eq. ( 4) reads as 2 From a physical viewpoint, this happens when the users form a cluster, wherein the distances between the different UEs are negligible with respect to the distance between the transmitter and the RIS. 3 It fulfills [9] lim α→+∞ with x ≍ y indicating that lim K→+∞ x/y = 1. 4  The pdf or cdf of X k must be derived to evaluate C sum by using either (4)-( 5) or (7).To approximate the distribution of X k , we distinguish two cases: in Subsection III-A, we assume that the reflection channel from the RIS to each UE hardens for sufficiently large values of Q; in Subsection III-B, we derive a more general approximation that does not require hardening of the reflection channel.In Subsection III-C, we derive the average receive signal-to-noise ratio (SNR) of the user selected for scheduling, which allows us to discuss the impact of the parameters K and Q on system performance.

A. Approximation 1: Hardening of the reflection channel
The k-th reflection channel hardens [10] if k ] converges in probability to 1, as Q → +∞, for each k ∈ {1, 2, . . ., K}.Based on the Markov inequality [13], a sufficient condition for the hardening of k ] → 0, as Q → +∞.By virtue of ( 14) and (15), one gets VAR[Z given by ( 14).Accordingly, relying on the results of the transformations of RVs [11], the pdf of X k is approximated by ) k ].
After algebraic manipulations, the cdf of X k is correspondingly approximated: k ] .(8) For any value of K, the above distributions can be substituted in (5) in order to obtain an approximation of the pdf of α opt , which is involved in the calculus of the average sum-rate capacity (4).On the other hand, since the distribution (8) is a von Mises function, one can resort to (7) for sufficiently large values of K, where (6) ends up to

B. Approximation 2: Sum of Nakagami variates
The RV Z k is the sum of the two independent and nonidentically distributed (i.n.i.d.) Nakagami RVs Z k , which does not admit a closed-form expression.Following [12], we propose to approximate the pdf where Z (1) k and Z (2) k are i.i.d.Nakagami RVs with shape parameter m and scale parameter Ω, which are determined 4 By using the Maclaurin series of the exponential function, the integral (7) can be rewritten as an absolutely convergent series, which can be approximately evaluated by using a finite number of terms [9, Appendix C].
such that f Z k (α) be an accurate approximation of f Z k (α).The choice of m and Ω is discussed in Appendix B.
At this point, we would like to point out that the main advantage of using f Z k (α) in lieu of f Z k (α) stems from the fact that the square of Z k is the sum of two i.i.d.gamma RVs [13].Indeed, the square of a Nakagami RV with shape parameter m and scale parameter Ω turns out to be a gamma RV with shape parameter m and scale parameter Ω/ m.Moreover, the sum of the two i.i.d.gamma RVs [ Z k ] 2 is a gamma RV, too, with shape parameter 2 m and scale parameter Ω/ m.Thus, we can conclude that the pdf of the RV X k ≜ Z 2 k is given by (see, e.g., [13]) whose cdf reads as 5 where x 0 t a−1 e −t dt is the (regularized) lower incomplete gamma function and Γ(a) is defined in Appendix A.
According to ( 6), the constant b K is the solution of where P −1 (y, a) is the inverse of the lower incomplete gamma function for y ∈ [0, 1], i.e., P −1 (P (x, a), a) = x.Substituting ( 11) in ( 6) and replacing

C. Average receive SNR
The asymptotic Gumbel distribution allows one to statistically characterize the receive SNR of the user selected for scheduling, which is defined as ρ sum ≜ P TX α opt .The average receive SNR ρ sum ≜ E[ρ sum ] = P TX E[α opt ] can be derived from the mean of the Gumbel distribution, thus yielding with C ≈ 0.5772 being the Euler-Mascheroni constant.
In a downlink without an RIS, the RV In this case, it follows from (6) that b K = σ 2 h ln(K) and a K = σ 2 h .From (13), the average receive SNR in an RIS-unaided opportunistic time-sharing downlink is  For a RIS-aided downlink, let us first consider the case when K → +∞ and the reflection channel hardens.In this situation, using (9), the average receive SNR in ( 13) is approximated as which shows that, compared to the RIS-unaided downlink, the SNR gain provided by the presence of the RIS depends on the relationship between K and Q.Indeed, if Q approaches infinity at the same rate as K, i.e., Q = χ K, where χ ̸ = 0 is a constant independent of K, it results that lim K,Q→+∞ ρ sum /Q 2 = P TX σ 2 f σ 2 g , i.e., the average receive SNR scales like Q 2 as K and Q grow to infinity.In this case, the downlink performance is dominated by the reflection channel and the reflection process of the RIS becomes predominant with respect to multiuser diversity effects.On the other hand, if Q approaches +∞ as Q = χ ln(K), one has lim K,Q→+∞ ρ sum /ln(K) = P TX (σ f σ g χ + σ h ) 2 + C σ f σ g σ h χ .In this case, the sumrate capacity also depends on the direct channel and the effect of the RIS becomes negligible when σ f σ g χ ≪ σ h , thus approaching the performance of the RIS-unaided downlink.
When K grows to infinity and hardening of the reflection channel does not hold, we resort to (10) for approximating the distribution of X k .In this case, the average receive SNR can be obtained by substituting (11) and ( 12) in ( 13), whose dependence on K and Q will be shown numerically in the forthcoming Section IV.

IV. NUMERICAL PERFORMANCE ANALYSIS
We consider a 2-D Cartesian system, wherein the BS and the RIS are located at (0, 0) and (10, 0) (in meters), respectively, whereas the users form a circular cluster centered in (40, −10) (in meters).The inter-element spacing is fixed to d RIS = λ 0 /4, whereas the azimuth and elevation angles at the RIS are uniformly distributed in [0, 2π) and [−π/2, π/2), respectively.All the other channel links are independently generated by assuming a carrier frequency f 0 = 25 GHz, with variance the distance of the link and η = 1.6 is the path loss exponent.The variance σ 2 f of the channel link between the RIS and the UEs is derived from the ratio ϱ ≜ (σ 2 f σ 2 g )/σ 2 h , which assumes the values in {0, ±5, ±10} dB.The effective isotropic radiated power of the BS is set to 33 dBm and the noise power at the UEs is equal to −100 dBm.
In The corresponding analytical curves (7) are plotted by using the parameters (11) and (12) (Approximation 2).Besides corroborating the noticeable accuracy of the proposed approximation, which does not require hardening of the reflection channel, it is seen from Fig. 1 that ∆C sum decreases with K.This behavior is due to the fact that the increase in K of C sum is partially hidden by the larger SNR gain due to reflection process of the RIS, while C w/o RIS sum scales like ln(ln(K)).Fig. 2 and 3 depict the performance of the RIS-aided downlink as a function of the number of meta-atoms Q for different values of ϱ, with K = 10.In this case, C w/o RIS sum = 25.26 bits/s/Hz.In Fig. 2, the rate C sum is compared with the analytical curve (7) by using the parameters (9) (Approximation 1), whereas (11) and ( 12) (Approximation 2) are used in Fig. 3. Results confirm the precision of Approximation 2 and show that the approximation based on the hardening of the reflection channel (Approximation 1) is inaccurate for small values of Q, especially when the reflection channel is stronger than the direct one.From the comparison among Figs. 1, 2, and 3, it can be inferred that, as predicted, the sum-rate capacity increases much faster with respect to Q than K.

V. CONCLUSIONS
We derived two approximations of the sum-rate capacity of an opportunistic time-sharing downlink with an RIS.The approximation based on the hardening of the reflection channel allows to show the asymptotic scaling laws in terms of the number of users K and the number of meta-atoms Q.A more accurate approximation was derived by approximating the overall channel seen by each user as a gamma RV.Both multiuser diversity and reflection process of the RIS provide increased channel magnitudes.However, the SNR gain increases faster with respect to Q than K and, thus, the multiuser diversity effect becomes negligible even for moderate values of Q. Herein, we have considered single-antenna BS and receivers.An interesting research subject consists of extending the proposed framework to the case of multi-antenna terminals.does not depend on the number Q of meta-atoms: it is a Rayleigh-distributed RV or, equivalently, it can be seen as a Nakagami RV with shape parameter m = 1 and scale parameter Ω = σ 2 h [11].On the other hand, the distribution of Z (2) k strongly depends on Q.Indeed, it can be readily verified that Z

APPENDIX A DISTRIBUTIONS OF RVS Z
(2) k is a Nakagami RV with shape parameter m = Q and scale parameter Ω = σ 2 f σ 2 g Q 2 , whose mean and variance are given by (see, e.g., [11]) where Γ(x) ≜ +∞ 0 t x−1 e −t dt, with x > 0, is the gamma function, whereas the approximations come from the Stirling's series of the quotient Γ Q + 1 2 /Γ(Q) [14] for very large Q.

APPENDIX B CHOICE OF THE PARAMETERS m AND Ω
To obtain an accurate approximation of f Z k (α), we resort to the moment matching method.Specifically, the parameters m and Ω are chosen such that to match the second and fourth moments of Z k and Z k , i.e., (i) with the mean of the gamma distribution (10) being the ratio between its shape and scale parameters, i.e., 2 m/( Ω/ m) = 2 Ω, condition (i) yields Ω = with the 2-nd moment of the gamma RV X k given [13] by , where we have also used Γ(x + 1) = x Γ(x), which is in general valid for all complex numbers x except the non-positive integers, condition (ii) leads to m = 1 2 . By observing that the moments of the Rayleigh RV |h k | and the chi-distributed RV ∥f k ∥ can be expressed as (see, e.g., [11]) 2 ) Γ(Q) , for n ∈ N, it can be shown that the first two moments of X k are where Γ(3/2) = √ π/2 and Γ(5/2) = 3 √ π/4 have been used.
pdf of Z k is the convolution of the pdfs of Z

TX σ 2 h
which benefits by a factor of ln(K) asymptotically for large K (so-called multiuser diversity effect), and C w/o RIS sum increases double logarithmically in K[4].

Fig. 1 ,
we report the difference ∆C sum ≜ C sum − C w/o RIS sum between the average sum-rate capacity of the RIS-aided and RIS-unaided downlinks, as a function of the number of users K for different values of ϱ.The rates C sum and C w/o RIS sum are numerically obtained by averaging (1) over 1000 independent Monte Carlo runs, by setting Q = 30 for the RIS-aided downlink and Q = 0 for the RIS-unaided one, respectively.