Rapidly Time-Varying Reconfigurable Intelligent Surfaces for Downlink Multiuser Transmissions

Amplitude, phase, frequency, and polarization states of electromagnetic (EM) waves can be dynamically manipulated by means of artificially engineered planar materials, composed of sub-wavelength meta-atoms, which are typically referred to as metasurfaces. In wireless communications, metasurfaces can be configured to judiciously modify the EM propagation environment in such a way to achieve different network objectives in a digitally programmable way. In such a context, metasurfaces are typically referred to as reconfigurable intelligent surfaces (RISs). Until now, researchers in wireless communications have mainly focused their attention on slowly time-varying designs of RISs, where the spatial-phase gradient across the RIS is varied at the rate equal to the inverse of the channel coherence time. Additional degrees of freedom for controlling EM waves can be gained by applying a time modulation to the reflection response of RISs during the channel coherence time interval, thereby attaining rapidly time-varying RISs. In this paper, we develop a general framework where a downlink multiuser transmission over single-input single-output slow fading channels is assisted by a digitally controlled rapidly time-varying RIS. We show that reconfiguring the RIS at a rate greater than the inverse of the channel coherence time might be beneficial from a communication perspective depending on the considered network utility function and the available channel state information at the transmitter (CSIT). The conclusions of our analysis in terms of system design guidelines are as follows: (i) if the network utility function is the sum-rate time-averaged network capacity, without any constraint on fair resource allocation, and full CSIT is available, it is unnecessary to change the electronic properties of the RIS within the channel coherence time interval; (ii) if partial CSIT is assumed only, a rapidly time-varying randomized RIS allows to achieve a suitable balance between sum-rate time-averaged capacity and user fairness, especially for a sufficiently large number of users; (iii) regardless of the available amount of CSIT, the design of rapid temporal variations across the RIS is instrumental for developing scheduling algorithms aimed at maximizing the network capacity subject to fairness constraints.


Rapidly time-varying reconfigurable intelligent surfaces for downlink multiuser transmissions
Francesco Verde, Senior Member, IEEE, Donatella Darsena, Senior Member, IEEE, and Vincenzo Galdi, Fellow, IEEE Abstract-Amplitude, phase, frequency, and polarization states of electromagnetic (EM) waves can be dynamically manipulated by means of artificially engineered planar materials, composed of sub-wavelength meta-atoms, which are typically referred to as metasurfaces.In wireless communications, metasurfaces can be configured to judiciously modify the EM propagation environment in such a way to achieve different network objectives in a digitally programmable way.In such a context, metasurfaces are typically referred to as reconfigurable intelligent surfaces (RISs).Until now, researchers in wireless communications have mainly focused their attention on slowly time-varying designs of RISs, where the spatial-phase gradient across the RIS is varied at the rate equal to the inverse of the channel coherence time.Additional degrees of freedom for controlling EM waves can be gained by applying a time modulation to the reflection response of RISs during the channel coherence time interval, thereby attaining rapidly time-varying RISs.In this paper, we develop a general framework where a downlink multiuser transmission over single-input single-output slow fading channels is assisted by a digitally controlled rapidly time-varying RIS.We show that reconfiguring the RIS at a rate greater than the inverse of the channel coherence time might be beneficial from a communication perspective depending on the considered network utility function and the available channel state information at the transmitter (CSIT).The conclusions of our analysis in terms of system design guidelines are as follows: (i) if the network utility function is the sum-rate time-averaged network capacity, without any constraint on fair resource allocation, and full CSIT is available, it is unnecessary to change the electronic properties of the RIS within the channel coherence time interval; (ii) if partial CSIT is assumed only, a rapidly time-varying randomized RIS allows to achieve a suitable balance between sum-rate timeaveraged capacity and user fairness, especially for a sufficiently large number of users; (iii) regardless of the available amount of CSIT, the design of rapid temporal variations across the RIS is instrumental for developing scheduling algorithms aimed at maximizing the network capacity subject to fairness constraints.Index Terms-Capacity, downlink transmission, fair scheduling, multiuser diversity, partial channel state information, reconfigurable intelligent surface (RIS), rapidly time-varying RIS, space-time metasurfaces.

I. INTRODUCTION
S INCE the birth of radio communications in the early 1900s, engineers, mathematicians, and physicists have dealt with the uncontrollable nature of propagation environments.Reconfigurable intelligent surfaces (RISs) are emerging as an innovative technology for the electromagnetic (EM) wave manipulation, signal modulation, and smart radio environment reconfiguration in wireless communications systems [1]- [6].An RIS is a metasurface composed of uniform or nonuniform sub-wavelength metal/dielectric structures, referred to as meta-atoms (or unit cells), which interact with the incident EM fields, whose working frequency can vary from sub-6 GHz to THz.Such meta-atoms can be independently controlled via software to switch among different reflection amplitude and phase responses that collectively shape the wavefront of the impinging wave [7], [8], without the need of bulky phase shifters.Thanks to their ultra-thin profile, square-law array gain, low-power consumption, low noise, and easy fabrication and deployment, RISs yield more freedom and flexibility to create different reflecting patterns for incident waves from diverse directions [9]- [13], compared to conventional phased antenna arrays.RISs can be realized by embedding into metaatoms tunable varactor diodes, switchable positive-intrinsicnegative (PIN) diodes, field effect transistors, or microelectromechanical system switches, which collectively form an inhomogeneous surface with specific phase profiles [14]- [18].
In wireless communications, the channel response may change with time.The channel coherence time is a measure of the expected time duration over which the response of the channel is essentially invariant and it is related to the bandwidth of the Doppler spectrum.Due to the timevarying nature of wireless channels, the reflection coefficient of the meta-atoms has to be varied in both space and time domains, thus leading to the so-called space-time metasurfaces in the EM literature [19]- [35].In slowly time-varying design techniques, the rate at which the reflecting coefficients of the meta-atoms are updated is equal to the inverse of the channel coherence time.For each channel coherence time interval, anomalous reflection/refraction (for beam steering) and reflective/transmissive focusing (for high directivity) are obtained by spatially varying the state of meta-atoms at different positions on the RIS.In this paper, we instead consider rapidly time-varying design techniques: in this case, besides constructing the spatially varying phase profile, the phase responses of the meta-atoms are modulated in time during the channel coherence time interval, by assuming that the timemodulation speed is much smaller than the frequency of the incident EM signal.Rapidly time-varying RISs offer additional degrees of freedom for manipulating signals, such as, e.g., nonlinear harmonic manipulations [32] and spread-spectrum camouflaging [33].
Relying on the feasibility of engineering the meta-atoms on the metasurface, a plethora of papers has been published where the wireless channel is programmed by optimizing on-the-fly the response of built-in RISs to achieve different system objectives, such as capacity gains [36]- [42], coverage extension [43], [44], energy efficiency [45], [46], maximization of minimum signal-to-noise ratio (SNR) [47], joint maximization of rate and energy efficiency [48], quality of services (QoS) targets under imperfect channel state information (CSI) [49], super resolution CSI estimation [50], and high level of localization accuracy [51].When the control of the RIS is performed by the transmitter, the RIS is also equipped with a smart controller that communicates with the transmitter via a separate wireless link for coordinating transmission and exchanging information on the phase shifts of all reflecting elements in real time.In this scenario, CSI at the transmitter (CSIT) is required to derive the optimal response of the RIS.
To the best of our knowledge, almost all the studies within the wireless communications community consider only slowly time-varying RISs. 1 Notable exceptions are represented by [52] and [53].Along the same lines of [21], only point-topoint communication (i.e., the scenario of a single transmitter and a single receiver) has been considered in [52].In [53], the modulation in time of the reflection response of the RIS is used to transmit information over a full-duplex pointto-point link.Nevertheless, the use of rapidly time-varying RISs specifically subordinated to the interest of a network of many users (i.e., RISs do not transmit their own information) remains an unexplored issue and their benefits in multiuser communications have not been studied yet.

A. Contributions
We focus on a single-input single-output downlink of a wireless communication multiuser system operating over a slow fading channel with coherence time T c , which is assumed to be much longer than the symbol period of the transmit digital modulator.We assume a delay-tolerant scenario, where the user rates can be adapted according to their instantaneous channel conditions.The scheduler function in the mediumaccess-control sublayer of the transmitter consists of dividing each channel coherence time interval in time slots and introducing a scheme for resource assignment among the users.The downlink transmission is assisted by a digital rapidly timevarying RIS, whose reflection response is fixed during each time slot, but it can be digitally varied in both time and space domains from one time slot to another in a controlled manner.Besides channel conditions, the total system performance and the data rate of each user are affected by the scheme used for scheduling and the states of the reflection coefficients of the RIS meta-atoms.The contributions of this study are summarized as follows.1) We prove that maximization of the sum-rate time-averaged capacity, with respect to transmit powers allocated at the users and reflection coefficients of the meta-atoms, can be achieved by a slowly time-varying RIS in the case of full CSIT.In this case, the reflection coefficients can be optimized with a manageable computational complexity by resorting to the block-coordinate descent (or Gauss-Seidel) method [54], which accounts for discrete variations of the reflection response of the RIS.
2) The scheduling deriving by the maximization of the sumrate time-averaged capacity with a slowly time-varying RIS would result in a very unfair sharing of the channel resource, by letting only the user with the best channel condition to transmit during each channel coherence time interval.Moreover, in many applications, it is not reasonable to assume that full CSIT can be made available.To partially overcome such shortcomings, we propose to use a rapidly time-varying randomized RIS where the reflection coefficients of the meta-atoms are randomly varied from one time slot to another in both space and time.This induces random fluctuations in the overall channel seen by each user even if the physical channel has no fluctuations over a time interval of duration T c .In this case, maximization of the sum-rate time-averaged capacity with respect to transmit powers allocated at the users requires only partial CSIT and more than one user may be scheduled during each channel coherence time interval.3) To ensure a proportionally fair allocation of channel resource [55], we show that, when the aim is to maximize the sum of logarithmic exponentially weighted moving average user rates, a rapidly time-varying RIS is the natural solution in the case of both full and partial CSIT.

B. Notations
Upper-and lower-case bold letters denote matrices and vectors; the superscripts * , T and H denote the conjugate, the transpose, and the Hermitian (conjugate transpose) of a matrix; [x] + stands for max{x, 0}; C, R and Z are the fields of complex, real and integer numbers; C n [R n ] denotes the vector-space of all n-column vectors with complex [real] coordinates; similarly, C n×m [R n×m ] denotes the vector-space of all the n × m matrices with complex [real] elements; j ≜ √ −1 denotes the imaginary unit; ℜ{x} is the real part of x ∈ C; * denotes the (linear) convolution operator; Π(t) is the basic rectangular window, i.e., Π(t) = 1 for |t| ≤ 1/2 and zero otherwise; ⌊x⌋ rounds x ∈ R to the nearest integer less than or equal to that element; ⊗ is the Kronecker product; 0 n , O n×m and I n denote the n-column zero vector, the n×m zero matrix and the n×n identity matrix; matrix A = diag(a 0 , a 1 , . . ., a n−1 ) ∈ C n×n is diagonal; {A} i,ℓ indicates the (i, ℓ)th element of A ∈ C n×m , with i ∈ {1, 2, . . ., n} and ℓ ∈ {1, 2, . . ., m}; ∥a∥ denotes the Frobenius norm of a ∈ C n ; E[•] denotes ensemble averaging; x ∼ CN (µ, C) is a circularly symmetric complex Gaussian vector x ∈ C n with mean µ ∈ C n and covariance C ∈ C n×n .

II. DOWNLINK WITH RAPIDLY TIME-VARYING RIS
With reference to Fig. 1, we consider a wireless system including a single-antenna transmitter, whose aim is to transfer information in downlink to K > 1 single-antenna user equipments (UEs).The transmission is assisted by a digitally programmable RIS working in reflection mode, which is installed on the face of a surrounding high-rise building, indoor walls, or roadside billboards.The RIS is made of Q = Q x × Q y spatial meta-atoms, which are positioned along a rectangular grid having Q x and Q y elements on the x and y axes, respectively.The meta-atoms are independently controlled to switch among different EM responses that collectively shape the spatial and spectral characteristics of the illuminating wave in a time-varying fashion [19]- [30].The functionality of such a metasurface is determined by the instantaneous digital reflection response of each metaatom, which can be dynamically controlled by digital logic devices [7], [8].We consider a transmission system with baud rate 1/T s .Throughout the paper, we assume that the communication bandwidth W ∝ 1/T s is much smaller than the carrier frequency f 0 (narrowband assumption), i.e., f 0 T s ≫ 1. Narrowband frequency-flat channels 2 are considered with coherence time T c spanning L c symbol intervals, i.e., T c = L c T s .These channels are modeled by a single-tap filter, as most of the multipaths arrive during one symbol time, which is approximately constant within frame intervals of length T c , but is allowed to independently change from one coherence period to another (block fading model).Each channel tap coefficient 2 Wideband channels can be turned into flat fading ones by multicarrier techniques that are widely applied in broadband wireless standards [56], [57].
is formed by the superposition of a large number of microscattering components (e.g., due to rough surfaces) having (approximately) the same delay.By virtue of the central limit theorem, it is customary to model the superposition of such many small effects as Gaussian [58].
The signal received by each user consists of two components (see Fig. 1): the direct ray, which corresponds to the signal propagating from the transmitter to the k-th UE with propagation delay τ 0,k , and a reflected ray, which is the signal reflected off the RIS with (overall) propagation delay τ 1,k .The delay spread ∆ k ≜ τ 1,k − τ 0,k of the composite k-th channel (i.e., direct and reflected rays) equals the difference between the delay of the direct ray and that of the reflected one.We assume that the transmitted signal is also narrowband relative to the delay spread, 3 i.e., ∆ k ≪ T s , such that the effects of τ 0,k and τ 1,k can be ignored at the k-th UE.

A. Signal emitted by the transmitter
Each channel coherence time interval is divided into M time slots of duration T r ≜ T c /M .Each slot spans P symbols intervals, i.e., T r = P T s .Consequently, it results that L c = M P .Let s k (n) be the information-bearing symbol intended for the k-th user within the n-th symbol interval, for k ∈ {1, 2, . . ., K} and n ∈ Z.The transmitted symbols {s k (n)} n∈Z are modeled as mutually independent sequences of zero-mean unit-variance independent and identically distributed (i.i.d.) complex circular random variables.For each k ∈ {1, 2, . . ., K}, the data stream {s k (n)} n∈Z of the kth user at rate 1/T s is converted at the transmitter side into M parallel substreams s where is the transmit power allocated to the k-th user in the m-th slot, subject to the time-averaged sum-power constraint 1

M K k=1
M −1 m=0 P (m) k ≤ P TX , where P TX > 0 denotes the (fixed) maximum allowed transmit power.With reference to a generic coherence interval, the continuous-time baseband signal emitted by the transmitter reads as where ψ(t) is the unit-energy square-root Nyquist pulseshaping filter [58].In this case, the communication bandwidth is W = (1 + ϱ s )/T s , with 0 ≤ ϱ s ≤ 1 being the rolloff factor.

B. Rapidly time-varying RIS
The RISs widely studied in existing works [7]- [30] consume minimal power, which can be as low as a few µW per meta-atom, and the introduced thermal noise is also negligible.By assuming that the reflection-response of the RIS is frequency flat, i.e., it is constant over the whole signal bandwidth W , the far-field baseband signal reflected by a rapidly time-varying RIS reads as where according to the time-switched array theory [22], [23], the complex parameter γ q (t) is the time-modulated reflection coefficient of the q-th meta-atom, ∀q ∈ {1, 2, . . ., Q}, whereas g ∈ C Q denotes the low-pass equivalent channel response between the transmitter and the RIS within the considered channel coherence time.RISs have the capability of controlling the amplitude, phase, and polarization of incident EM waves [59].Design methods of programmable RISs have been proposed in [60]- [62] where the reflection magnitude and phase can be independently controlled.We focus on an RIS with phase-only modulation, by deferring the case of RISs with amplitudephase modulated elements to future work.If R(t) does not vary during each channel coherence time interval, the effect of the RIS consists of multiplying the impinging signal g x(t) by a constant matrix R(t) ≡ R, i.e., slowly time-varying RIS, which is the configuration considered in previous works [1]- [6], [36]- [51]: in this case, each UE sees a time-invariant overall channel (see also Subsection II-D).In this paper, relying on the results of recent experimental studies [19]- [30], [34], [35], we consider the case where R(t) is time-varying over the channel coherence time interval.In its simplest form, the reflection coefficient of the generic q-th meta-atom is subject to a linear modulation as where Tr is a pulse-shaping filter of duration T r , 4 the complex coefficients γ q , . . ., γ (M −1) q are at the designer's disposal.For an b-bit digital meta-atom with phaseonly modulation capability, each coefficient γ By considering the balance between the system cost, design complexity, and overall performance, it has been found that bbit meta-atoms are suitable for RIS implementation [63], for small-to-moderate values of b.
According to [19]- [30], [34], [35], eq. ( 3) is approximately valid when the reflection coefficients vary at a very slow pace compared to the carrier frequency of the impinging signal, i.e., the switching frequency f r ≜ 1/T r is much smaller than f 0 or, equivalently, f 0 T r ≫ 1.This operative condition is surely satisfied in the narrowband regime since T r > T s by assumption.The main system parameters and the time-scale of change of key quantities are summarized in the first two columns of Table I, whereas their values are reported in the third column for an exemplifying scenario of practical interest.
In particular, the values of the carrier frequency f 0 and the communication bandwidth W are consistent with Frequency Range 1 (FR1) of 5G New Radio (NR) [64].The value of the channel coherence time is typical of a pedestrian scenario.In current implementations [19]- [30], [35], the switching frequencies of the meta-atoms can reach tens of MHz.Larger values of f r can be achieved by exploiting faster switching mechanisms based on innovative materials, such as graphene or vanadium dioxide [65].However, the time variation induced by the RIS should be slow enough and should happen at a time scale allowing the reflection response R(t) to be unaffected by the receiving filter at the UEs (see Subsection II-C).Further, the variation should be slow enough to ensure that CSI is reliably acquired (see Subsection II-F).

C. Signal received by each user
We customarily assume that each UE uses standard timing synchronization with respect to its direct link.To decode the data signal, each UE performs matched filtering with respect to the symbol pulse ψ(t).After matched filtering and sampling with rate 1/T s at time instants t (m) p ≜ p T s + m T r , with p ∈ {0, 1, . . ., P −1} and m ∈ {0, 1, . . ., M −1}, the discrete-time baseband signal received at the k-th user reads as for k ∈ {1, 2, . . ., K}, where h k ∈ C models the low-pass equivalent channel response from the transmitter to user k, the vector f k ∈ C Q denotes the low-pass equivalent channel response from the RIS to the k-th user, and v (m) k (p) ∼ CN (0, 1) is the noise sample at the output of the matched filter, with v (m1) and m 1 ̸ = m 2 .Without loss of generality, we have assumed that the noise has unit variance.Hence, the parameter P TX in Subsection II-A takes on the meaning of total transmit signal-to-noise ratio (SNR).
By substituting (2)-( 3) in ( 5), and accounting for (4), the block r k (P − 1)] T ∈ C P received by the k-th user within the m-th slot is given by5 where the overall channel gain seen by the k-th UE during the m-th slot can be written as with In eq. ( 6), we have used which holds if R(t) varies at a very slow pace compared to the effective duration of ϕ(t), i.e., f r ≪ 1/T s or, equivalently, P ≫ 1.In many applications, the channel coherence time is of the order of hundreds to thousands symbols, and therefore sufficiently large values of P can be chosen in these operative scenarios (see Table I

for instance).
A key observation is now in order.The received signal (6) shows that, even though the channels h k , g, and f k are time invariant over each coherence interval spanning L c consecutive symbol periods, the overall channel (7) seen by each user varies in time on a slot-by-slot basis.The rate of variation of c (m) k in time is dictated by M , as well as the diagonal matrices Γ (0) , Γ (1) , . . ., Γ (M −1) , which can be suitably controlled to dynamically schedule resources among the users and, thus, they are design parameters of the system.Strictly speaking, the overall effect of the rapidly time-varying RIS is to increase the rate and dynamic range of channel fluctuations.

D. The conventional case of a slowly time-varying RIS
In many recent works [1]- [6], [36]- [51], the RIS does not introduce time variability during the transmission of an information-bearing symbol block, i.e., it is slowly timevarying.This can be regarded as a particular case of our general model, where M = 1 and T r = T c (see Fig. 2).In this special case, one has γ q (t) = γ (0) q , for q ∈ {1, 2, . . ., Q}.Therefore, if the reflection process of the RIS is time invariant over each channel coherence time interval, that is, Q ], the overall channel vector (7) from the transmitter to each user becomes constant over each channel coherence time interval.

E. Assumptions regarding the channels
It is common to model the channels between the transmitter and the meta-atoms of the RIS by assuming that the received wavefront is planar (see, e.g., [66]).In this case, the transmitter-to-RIS channel vector g can be modeled as where the first summand represents the (deterministic) line-ofsight (LoS) component, whereas the second one is the (random) diffuse (i.e., non-LoS) component, with g DIF ∼ CN (0, 1).The variance σ 2 g describes the large-scale geometric pathloss and κ ≥ 0 is the Ricean factor.It is worthwhile to note that κ = 0 yields Rayleigh fading, Rician fading is obtained when κ > 0, and, in the limiting case κ → +∞, there is only the LoS channel component.The spatial signature a RIS of the RIS in ( 11) is given by where λ 0 = c/f 0 is the wavelength, with c being the speed of light in the medium, d RIS is the inter-element spacing at the RIS, θ RIS ∈ [0, 2π) and ϕ RIS ∈ [−π/2, π/2) identify the azimuth and elevation angles, respectively, whose directional cosines are u x ≜ sin θ RIS cos ϕ RIS and u y ≜ sin θ RIS sin ϕ RIS .
It should be noted that

F. High-level overview of the CSI acquisition process
In the sequel, we assume that, for each k ∈ {1, 2, . . ., K}, the k-th UE has perfect knowledge of its channel gain c (m) k , ∀m ∈ {0, 1, . . ., M − 1}, referred to as CSI at the receiver (CSIR), which can be estimated through standard training algorithms, by using N down-full downlink pilot symbols repeated at the beginning of each slot.In this case, if the power spent by the transmitter for pilot symbol transmission is significantly greater than the noise variance, a single downlink pilot per slot (i.e., N down-full = 1) is sufficient to ensure that the channel estimation error is quite small [67].On the other hand, we consider two different amount of CSIT: 1) In the case of full CSIT, the transmitter has perfect knowledge of all the relevant channel parameters h k and f k , ∀k ∈ {1, 2, . . ., K}, and g. 2) In the case of partial CSIT, ∀m ∈ {0, 1, . . ., M − 1}, the transmitter knows only the user index for which |c For acquisition of full CSIT, a viable strategy consists of resorting to a time-division duplexing (TDD) protocol.In this case, where channel reciprocity can be exploited, acquisition of full CSIT additionally requires a non-trivial uplink training session of N up-full pilot symbols at the beginning of each channel coherence time (see Fig. 2), which might be unrealistic in many scenarios of practical interest, especially for large values of K and Q.Indeed, the CSIT estimation process requires N up-full ≥ K(Q + 1) pilot symbols [68].On the other hand, partial CSIT involves only the discovery of the UE having the "best" end-to-end path, which can be identified in a centralized or decentralized fashion.In the centralized case, each UE estimates its own channel coefficient c (m) k , through, say, a common downlink pilot and, then, feeds back the (quantized version of) instantaneous channel quality |c (m) k | to the transmitter. 6In such a centralized scenario, the duration N par (in symbol intervals) of the CSI acquisition phase in Fig. 3 also depends on the delay in feeding |c If feedback control channels from each UE to the transmitter are not available, a decentralized CSI acquisition scheme can be implemented [70], according to which, after receiving a downlink training packet, each UE will start its own timer with an initial value that is inversely proportional to a local estimate of channel quality |c (m) k |.The timer of the UE with the strongest end-to-end channel will expire first and such an UE will transmit an uplink flag packet to the transmitter, communicating its identification number.In this decentralized scenario, the amount of overhead N par involved in selecting the best UE also depends on the statistics of the wireless channel (see [70] for a probabilistic analysis).
Since the capacity results reported in this paper are valid only when the required CSI is perfectly known at the transmitter and at the UEs, we do not account for channel estimation imperfections hereinafter, by deferring CSI estimation issues to a future work.

III. SUM-RATE TIME-AVERAGED CAPACITY: FULL CSIT
In this section, we assume full CSIT (see Subsection II-F).The achievable data rate transmitted to the k-th UE is affected by the interference from the signals intended for the other users u ̸ = k ∈ {1, 2, . . ., K} on the same slot.Assuming that the transmitter encodes the information for each user using an i.i.d.Gaussian code, the time-averaged rate of user k can be written as (in bits/s/Hz) (13) for k ∈ {1, 2, . . ., K}, with the signal-to-interference-plusnoise ratio (SINR) at UE k given by It should be observed that R 13) represents the achievable rate of user k over slot m (depending on the current channel conditions), which is referred to as the instantaneous per-user rate.Accordingly, R k is the timeaveraged per-user rate over a given channel coherence time.
The (net) sum-rate time-averaged capacity can be expressed as follows , . . ., p (M −1) γ (0) , γ (1) , . . ., γ (M −1) 2 , . . ., P K ] T ∈ R K , for m ∈ {0, 1, . . ., M − 1}, and we recall again that the channel gain c (m) k explicitly depends on the RIS reflection vectors Q ] H ∈ C Q of the rapidly timevarying RIS through (7), whereas accounts for the pilot overhead required to acquire both CSIT and CSIR (see Fig. 2).We perform the optimization (15) in the following two steps, by resorting to an approach that is called concentration in the literature of estimation theory [71].
Theorem 3.1: Given the reflection response Γ of the rapidly time-varying RIS, the slot assignment policy maximizing Proof: This result can be proven by using information theoretic arguments [72] or, equivalently, by resorting the principle of mathematical induction [73].
Theorem 3.1 states that, for fixed Γ, the sum-rate timeaveraged capacity is equal to the largest single-user capacity in the system.The slot assignment policy is a simple opportunistic time-sharing strategy: at each slot, transmit to the UE with the strongest channel.This result comes from the fact that, in information theory jargon, the model ( 6) falls into the class of degraded broadcast channels [74], for which users can be ordered in terms of the quality of signal received in each time slot m, i.e., on the basis of |c In this case, there is always one UE experiencing a stronger channel than any other UE, which is able to decode codewords intended for other UEs, and thus the long-term average sumrate is maximized by only transmitting to such a user.
To derive the sum-rate time-averaged capacity, given the reflection response Γ of the RIS, we have to determine the amount of transmit power to be allocated to the slots in order to maximize the sum rate, s.t.constraint (18).When the slot assignment for UEs is carried out according to Theorem 3.1, the multiuser downlink can be viewed as an opportunistic time-sharing system with dynamic slot allocation, where UEs receive data through a number of slots assigned to their own independently.Therefore, given Γ, the sum-rate time-averaged capacity assumes the expression reported in (20), with The solution of problem ( 20) is water-filling [74] over the slots with the best channel gains among multiple users, i.e, for m ∈ {0, 1, . . ., M − 1}, where λ > 0 is a threshold to be determined from the total transmit power constraint by solving In conclusion, the sum-rate time-averaged capacity with fixed Γ is given by B. Second step: optimization with respect to reflection coefficients of the rapidly time-varying RIS At this point, we determine the values of the reflection coefficients of the rapidly time-varying RIS in order to maximize (24), s.t.constraint C 2 .If channels are constant within a coherence block and their realizations are perfectly known at the transmitter, it is expected that one can just optimize the reflection response of the RIS given the perfectly known channel realizations and keep the optimal reflection coefficients constant for the whole coherence block.Lemma 3.2 provides a simple proof of this intuition.
In light of Theorem 3.1 and (24), the optimization problem (15) amounts to solving (25), where we have also used wellknown properties of the logarithm function.
Lemma 3.2: The sum-rate time-averaged capacity can be achieved by employing a slowly time-varying RIS, i.e, M = 1 and T r = T c , and, hence, problem (25) simplifies as follows with Θ ≜ 2π 2 b ℓ, for ℓ ∈ {0, 1, . . ., 2 b − 1} .Proof: By virtue of the inequality of arithmetic and geometric means [75], it holds that with equality if and only if γ ≜ γ (0) = γ (1) Lemma 3.2 infers that, if there is no constraint on the time-averaged rate of each user, exploitation of the temporal C sum (Γ) = max P (0) ,P (1) ,...,P (M −1) C sum = max γ (0) ,γ (1) ,...,γ (M −1) = max γ (0) ,γ (1) ,...,γ (M −1) dimension of the rapidly time-varying RIS is unnecessary, in the sense that time-domain variations of the reflection response do not improve the sum-rate time-averaged capacity, compared to a slowly time-varying RIS.However, as we will show later on, Lemma 3.2 does not hold if other network requirements are taken into account, e.g., optimization without full CSIT or fairness for the efficient utilization of scarce radio resources.The mathematical programming ( 26) can be decomposed into the following K simpler problems: F k (e j θ1 , e j θ2 , . . ., e j θ Q ) (28) for k ∈ {1, 2, . . ., K}, where, according to (7), we can write F k (e j θ1 , e j θ2 , . . ., e where Q×Q is a Hermitian matrix.Accordingly, the sum-rate time-averaged capacity reads as where α opt ≜ max k∈{1,2,...,K} F k (e j θ k,1 , e j θ k,2 , . . ., e j θ k,Q ).Optimization ( 28) is known to be a hard non-convex NPhard problem.By exploiting its structure, problem (28) can be transformed into an integer linear program [76], for which the globally optimal solution can be obtained by applying the branch-and-bound method.Some methods have been developed in [76]- [78] to reduce the computational complexity for the optimal solution, which are based on the block-coordinate descent (or Gauss-Seidel) method [54, Sec.2.7].Herein, we leverage the successive refinement algorithm proposed in [76, III-B], which is shown to achieve close-to-optimal performance (see, e.g., the numerical results reported in [76]).
{B k } ν,q e −j θ k,q (i+1) (37) The optimization procedure of the reflection response of the slowly time-varying RIS is summarized as Algorithm 1.It is noteworthy that such an optimization algorithm explicitly accounts for the discrete nature of the reflection coefficients of the RIS.Moreover, it involves only simple operations and does not require any step size selection.
As a system performance metric, by applying channel coding across channel coherence intervals (i.e., over an "ergodic" interval of channel variation with time), the ensemble average of the sum-rate time-averaged capacity ( 30) is given by where f αopt (α) is the probability density function (pdf) of the non-negative random variable α opt .Analytical evaluation of ( 39) is complicated by the fact that there is no closedform solution of the optimization problem (28).Therefore, the dependence of E[C sum ] on the main system parameters (e.g., K and Q) will be numerically studied in Section VI.

IV. RAPIDLY TIME-VARYING RIS: PARTIAL CSIT
Achievement of the sum-rate time-averaged capacity (30) requires optimization of the slowly time-varying RIS reflection response at a rate 1/T c approximately equal to the Doppler spread of the underlying channels, which in turn involves acquisition of full CSIT.In this section, we propose a scheme that constructs random reflections at the RIS in both space and time domains in order to take advantage of the presence of the rapidly time-varying RIS in the system and to perform multiuser diversity scheduling by assuming partial CSIT (see Subsection II-F).It should be observed that the performance of randomized slowly time-varying RIS has been evaluated in [79].However, since the reflection coefficients of the RIS are randomly varied at a rate 1/T c , the beneficial effects of randomized rapidly time-varying reflections in terms of multiuser diversity and network fairness have not been studied.
Herein, we choose the reflection coefficients γ q in (4), for q ∈ {1, 2, . . ., Q} and m ∈ {0, 1, . . ., M − 1}, as a sequence of i.i.d.random variables with respect to both the space index q and the time index m, where each random variable Θ (m) q assumes equiprobable values in Θ (defined in Lemma 3.2).In this case, besides constructing the spatially varying phase profile, the phase responses of the meta-atoms are randomly varied in time from one slot to another.After each coherence interval, we independently choose another sequence of reflection coefficients, and so forth.In this scenario, Theorem 3.1 still holds and the optimal scheduling strategy consists of transmitting in each slot to the user with the strongest channel.Since only partial CSIT is assumed, we allocate equal powers at all slots, i.e., we set P (m) = P TX in (18), ∀m ∈ {0, 1, . . ., M − 1}.This is particularly relevant for the downlink, where the base station can operate always at its peak total power.Such a strategy is almost optimal at high SNR [74], i.e., for sufficiently large values of P TX .Therefore, the corresponding (net) sum-rate time-averaged capacity is with α max (γ (m) ) defined in (21), where the reflection vector 1 , e j Θ (m) 2 , . . ., e j Θ (m) Q ] H is chosen according to the proposed rapidly time-varying randomized scheme, whereas accounts for the overhead required to acquire CSI (see Fig. 3).As expected, it results that C rand ≤ C sum .It is noteworthy that, by varying the phases of the meta-atoms in both space and time from one slot to another, the RIS randomly scatters multiple reflected beams, whose number, amplitudes, and directions change on a slot-by-slot basis.Therefore, with partial CSI only, the transmitter can schedule to the user currently closest to one of such beams that ensures maximum signal power.With many users, there is likely to be a user very close to a "strong" reflected beam at any slot.Such an intuition is formally justified in the next subsection.

A. Asymptotic performance analysis
Herein, we evaluate the ensemble average E[C rand ], which is defined as the expected value of ( 46), over the sample space of the 4-tuple {h, g, F, Γ}, and Γ has been defined at the beginning of Subsection III-A.By virtue of the conditional expectation rule [58] and according to the proposed random reflection scheme, one has (42), where For the sake of analysis, we assume that the channel between the transmitter and the RIS is characterized by a dominant LoS component, i.e., g = g LOS .The LoS assumption is reasonable if the distance between the BS and the RIS is small compared to their height above ground, without any sort of an obstacle between them.The case in which the channels between the transmitter and the meta-atoms of the RIS have a diffuse component will be numerically studied in Section VI.For simplicity, we also 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 h k ≡ σ 2 h and σ 2 f k ≡ σ 2 f , ∀k ∈ {1, 2, . . ., K}, which will be referred to as the case of homogeneous users.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.A numerical performance analysis when the users have heterogeneous path losses will be provided in Section VI.Under the above assumptions, it is readily seen that the random variable α rand turns out to be the maximum of K i.i.d.exponential random variables Y k with mean σ 2 h +σ 2 f σ 2 g Q and, thus, the pdf of α rand is given by where f Y k (x) and F Y k (x) denote the pdf and the cumulative distribution function (cdf) of the random variable Y k .Since the distribution of the random variables Y 1 , Y 2 , . . ., Y K does not depend on the indexes ℓ 1 , ℓ 2 , . . ., ℓ Q , the average sum-rate capacity (42) simplifies to Trying to work with (45) for evaluating (46) is difficult even for small values of K.For such a reason, we apply extreme value theory [80] to calculate the distribution of α rand when K is sufficiently large (i.e., in practice greater than 10).It can be shown that, as K → +∞, the random variable α rand convergences in distribution to the Gumbel function [81], i.e., where F αrand (α) is the cdf of α rand , and the normalizing constants b K and a K are given by The proof of this result relies on the limit laws for maxima [80] and the fact that the cdf of the exponential random variable Y k is a von Mises function [82], i.e., On the basis of ( 47), the average sum-rate capacity (46) can be explicated as with x ≍ y indicating that lim K→+∞ x/y = 1.By using the Maclaurin series of the exponential function, the integral (51) can be rewritten as an absolutely convergent series [82, Appendix C], which can be approximately evaluated by using a finite number of terms.Let ρ rand ≜ P TX α rand be the receive SNR, whose average value can be calculated for large K by using the Gumbel distribution (47), thus obtaining with C ≈ 0.5772 being the Euler-Mascheroni constant.For sufficiently large values of K, the mean overall channel logaritmically scales with the number of user K: this implies that E[C rand ] increases double logarithmically in K, and, thus, asymptotically, our randomized scheme does not suffer of a loss in multiuser diversity [74].However, there is only a linear growth of E[ρ rand ] with the number Q of meta-atoms: this implies that E[C rand ] increases logarithmically in Q.

V. FAIR SCHEDULING ALGORITHMS
The scheme discussed in Section III focuses on maximizing the sum rate time-averaged capacity, i.e., finding the solution of the optimization problem (15).However, fairness among users is an important practical issue when the users experience different fading effects.A simple yet informative index to quantify the fairness of a scheduling scheme was proposed in [83], which can be expressed as follows If all the users get the same time-averaged rate, i.e., the rates R k are all equal, then the fairness index is F = 1 and the scheduler is said to be 100% fair.On the other hand, a scheduling scheme that favors only a few selected users has a fairness index F → 0. The scheme developed in Section III, which exploits full CSIT and employs a slowly time-varying optimized RIS, exhibits a fairness index F sum = 1/K over the time-scale T c , since it supports only one user at each channel coherence time.As expected, in this case, the scheduler becomes increasingly unfair as the number K of users increases.On the other hand, the suboptimal scheme proposed in Section IV, which relies on partial CSIT and makes use of a randomized rapidly timevarying RIS, might ensure a fairness index F rand over the timescale T c that is greater than or equal to 1/K, i.e., F rand ≥ F sum .This behavior will be numerically shown in Section VI and, intuitively, it comes from the fact that more than one user might be scheduled for each channel coherence time interval, due to the time variability induced by the rapidly time-varying RIS.Motivated by such an intuition, our goal is herein to demonstrate that exploitation of the temporal dimension of the rapidly time-varying RIS is useful for developing scheduling strategies that explicitly take into account fairness over a time horizon equal to the channel coherence time.

A. Proportional fair scheduling with rapidly time-varying RIS
We focus on proportional fair scheduling (PFS) originally proposed in the network scheduling context by [55] as an alternative for a max-min scheduler, since it offers an attractive trade-off between the maximum time-averaged throughput and user fairness. 7Specifically, we start from introducing the exponentially weighted moving average rate of user k, which can be written as (in bits/s/Hz) (54) 7 The usefulness of rapidly time-varying RIS can be more generally shown by considering other network utility functions that achieve a desired balance between network-wide overall performance and user fairness [84].
for i ∈ {1, 2, . . ., m}, with m ∈ {1, 2, . . ., M −1}.Therefore, recursion (54) assigns lower weights to the older instantaneous rates that are achievable over the time-scale T c .Such a temporal smoothing introduces a memory in the resource allocation procedure over each channel coherence time.To achieve a desired balance between network-wide overall performance and user fairness, we consider the following per-time-slot PFS optimization problem: for m ∈ {0, 1, . . ., M − 1}, where the constraints C 1 and C 2 have been reported in (15).It is noteworthy that, since problem (56) has to be solved on a slot-by-slot basis, its solution is inherently time varying over the channel coherence time interval, i.e., the optimal values of the power vector p (m) and the reflection vector γ (m) might vary from one time slot to another.Hence, a slowly time-varying RIS (see Subsection II-D) could not be useful and exploitation of the temporal dimension of the RIS may be necessary.Under the assumption that R is sufficiently small, by invoking Taylor's theorem, one can resort to the linear approximation of where the equality comes from the application of (54).Capitalizing on (57), the optimization problem (56) where we have also used the high-SNR approximation with SINR (m) k defined in (14).The optimal values of P (m) k in (17) and γ (m) , which is embedded in |c (m) k | [see (7)], depend on the available CSIT.
1) Full CSIT: In this case, along the same lines of Section III, the power allocated at the user k = k (m) pfs during the m-th time slot is given by ( 22) with α max (γ (m) ) replaced by In such a case, the values of the reflection coefficients of the rapidly time-varying RIS can be obtained by a straightforward modification of Algorithm 1 (details are omitted).
2) Partial CSIT: When there is knowledge of partial CSIT only, as in Section IV, we allocate power uniformly over the time slots by setting P (m) = P TX in (18), ∀m ∈ {0, 1, . . ., M − 1}.Moreover, since the reflection coefficients of the rapidly time-varying RIS cannot be optimized without full CSIT, we choose γ (m) according to the same randomized algorithm proposed in Section IV.For each time slot, the users are scheduled through (61).
It is noteworthy that, by virtue of the proposed algorithms, a rapidly time-varying RIS allows to achieve fairness over the time-scale T c .PFS can be also implemented with a slowly time-varying RIS.However, in this case, since the reconfigurability rate of the RIS is equal to 1/T c , fairness is achieved on a time horizon that is greater than T c and, consequently, users experience longer delays between successive transmissions.

VI. NUMERICAL RESULTS
In this section, with reference to the considered RIS-aided multiuser downlink, we report Monte Carlo numerical results aimed at validating the proposed designs and completing the developed performance analysis.With reference to Fig. 4, we consider a two-dimensional Cartesian system, wherein the transmitter and the center of the RIS are located at (0, 0) and (10, 0) (in meters), respectively, whereas the positions of the UEs are generated as random variables uniformly distributed within a circular cluster centered in (40, −10) (in meters), whose radius is set to 10 m in the case of homogeneous users and 100 m in the case of heterogeneous users (see Subsection IV-A).All the ensemble averages (with respect to the relevant random channel parameters, spatial positions of the UEs, and, wherever applicable, the randomized reflection coefficients of the RIS) are evaluated through 200 independent runs.For the main system parameters, we refer to the exemplifying scenario reported in Table I.The remaining parameters of the simulation setting are reported in Table II.The channel variances σ 2 g (from the transmitter to the RIS), σ 2 h k (from the transmitter to the k-th UE), and σ 2 f k (from the RIS to the k-th UE) are chosen as 0 for the RIS [85], with A RIS = Q (λ 0 /2) 2 denoting the physical area of the RIS, and G α = 5 dBi for the UEs, d α is the distance of the considered link, and η is given in Table II.Regarding the training overhead for CSI acquisition, we set N up-full = K (Q + 1) and N down-full = 1 in the case of full CSIT (see Fig. 2), whereas N par = 2 in the case of partial CSIT (see Fig. 3).
In all the forthcoming examples, we compare the (statistical) average performance of the following RIS designs: (i) the sumrate time-averaged capacity (30), referred to as "STV opt", which has been derived by assuming full CSIT and by using Algorithm 1 for the optimization of the reflection coefficients of the slowly time-varying RIS over each channel coherence time interval; (ii) the sum-rate time-averaged capacity (42), referred to as "RTV rand", which has been derived by assuming partial CSIT and randomized slot-by-slot variations in both space and time of the reflection coefficients of the rapidly time-varying RIS; (iii) the sum-rate time-averaged capacity of the solution derived in Subsection V-A1, referred to as "RTV opt w/ PFS", which has been derived by assuming full CSIT and PFS, where the reflection coefficients of the rapidly time-varying RIS are optimized for each time slot by using Algorithm 1; (iv) the sum-rate time-averaged capacity of the solution derived in Subsection V-A2, referred to as "RTV rand w/ PFS", which has been derived by assuming partial CSIT and PFS, where the reflection coefficients of the rapidly timevarying RIS are randomized slot-by-slot in both space and time.Furthermore, we plot the average performance of the downlink in the absence of an RIS without fairness constraints, referred to as "w/o RIS". 8For all the scheduling algorithms, we also show the fairness index (53).
A. Example 1: Homogeneous users Fig. 5 shows the sum-rate time-averaged capacity (left-side plot) and fairness (right-side plot) performance of the schemes under comparison in the case of homogeneous users as a function of the number of users K, with Q = 10 × 10 metaatoms.We observe that, except for the schemes requiring full CSIT, the sum-rate time-averaged capacity increases double logarithmically with the number of users K in the system: this is the multiuser diversity effect [74].The same effect is not visible for the "STV opt" and "RTV opt w/ PFS" downlinks since the logarithmic growth of the mean overall channel with K is contrasted by the pilot overhead required to acquire full CSIT, which also depends on K.It is clear that the presence of an RIS leads to significant sum-rate gains with respect the "w/o RIS" case.
As expected, even thought the "STV opt" downlink allows to achieve the best sum-rate capacity performance, it requires full CSIT and, moreover, as in the scenario of "w/o RIS", it becomes fairer and fairer as the number of users increases.In contrast, from the curves of "RTV rand", one can argue that, compared to "STV opt", a much fairer system can be obtained by introducing a rapidly time-varying randomization of the reflection response of the RIS, which also allows to highlight an interesting trade-off between performance and required amount of CSIT for downlink optimization.A 100% fair system can be designed by using PFS.Obviously, such a fairness comes at the price of a sum-rate reduction: indeed, the sum-rate curves of the "RTV opt w/ PFS" and "RTV rand w/ PFS" systems are worse than those of their corresponding counterparts "STV opt" and "RTV rand", respectively.Interestingly, although it involves partial CSIT only, the "RTV rand" downlink pays a mild performance gain with respect to "RTV opt w/ PFS" -which however requires full CSITby ensuring a fairness of nearly 100% up to 18 users and no smaller than 94% up to 32 users.Fig. 6 shows the sum-rate time-averaged capacity (left-side plot) and fairness (right-side plot) performance of the schemes under comparison in the case of homogeneous users as a function of the number of meta-atoms Q, with K = 16 users.Two additional interesting trends can be inferred from such figures.First, in accordance with our analysis in Subsection IV-A, the sum-rate time-averaged capacity of the schemes employing partial CSIT increases logarithmically with the number of meta-atoms Q at the RIS, whereas the downlinks optimized with full CSIT exhibit a faster growth rate.Second, as the number of meta-atoms increases, the sum-rate capacity of "RTV opt w/ PFS" tends to that of "STV opt" by ensuring at the same time 100% fairness, while F = 100/16 = 6.25% for the "STV opt" scheme.

B. Example 2: Heterogeneous users
Figs. 7 and 8 report the sum-rate time-averaged capacity (left-side plot) and fairness (right-side plot) performance of the schemes under comparison in the case of heterogeneous users.In Fig. 7, we set Q = 10 × 10 meta-atoms, whereas we fix K = 16 users in Fig. 8.Besides substantially confirming the above trends evidenced in Example 1, a behavior that is worth mentioning is the reduction in fairness of the "RTV rand" downlink, compared to the homogeneous case.Hence, we might argue that the randomization in both space and time of the reflection coefficients of the RIS leads to a fair resource allocation if there is no severe "near-far effect", i.e., users that experience significantly different fading effects.It is apparent that, to cope with such an heterogeneous situation, the schemes employing PFS are forced to meaningfully cut down their sumrate performance in order to fulfill the fairness constraint.

VII. CONCLUSION AND DIRECTIONS FOR FUTURE WORK
In this article, we have investigated the impact of a rapidly time-varying RIS on the resource allocation in a downlink multiuser system operating over slow fading channels.
The following results have been obtained: 1) Joint optimization of transmit powers and reflection coefficients of the RIS in the case of full CSIT leads to a slowly time-varying RIS, whose reflection response is updated with a rate that is (approximately) the inverse of the channel coherence time.In this case, only one user is scheduled within a channel coherence interval.

2)
The assumption of full CSIT can be relaxed by randomizing the reflection coefficients of a rapidly time-varying RIS in both space and time domains, without paying a penalty in terms of multiuser diversity gain.The resulting resource allocation scheme requires partial CSIT only and ensures a certain fairness, since the randomization in time allows the transmitter to serve more than one user for each channel coherence time interval.3) If the main design concern is to meet fairness among users while at the same time exploiting the multiuser      diversity gain, then a rapidly time-varying RIS has to be employed in the downlink.In such a case, the temporal dimension of the RIS is beneficial not only in the partial CSIT scenario, but also in the full CSIT one.
This study demonstrates that, if acquisition of full CSIT is not possible in practice and/or the communication system has stringent requirements in terms of fairness, the exploitation of the temporal dimension of a rapidly time-varying RIS is instrumental in designing the downlink of a multiuser systems even in slow fading scenarios.We have considered singleantennas transmitter and receivers.In this respect, a first interesting research subject consists of extending the proposed framework to the case of multi-antenna terminals.In this case, the downlink is in general a nondegraded broadcast channel and sum capacity is achieved by using more sophisticated multiple access strategies than opportunistic time sharing, i.e., dirty-paper coding.An additional research direction involves investigating the benefits of a rapidly time-varying RIS in fast fading environments.

Fig. 1 .
Fig. 1.A pictorial view of a downlink transmission in a multiuser system assisted by a rapidly time-varying RIS.In the figure, the direct and reflected rays of the generic k-th user are depicted only, with k ∈ {1, 2, . . ., K}.

Fig. 2 .
Fig. 2. According to a TDD protocol, acquisition of full CSIT requires an uplink training phase at the beginning of each channel coherence time interval, in addition to downlink training phases for CSIR acquisition.

Fig. 3 .
Fig.3.Acquisition of partial CSIT can be performed in a centralized or decentralized manner by providing at the beginning of each slot an initial phase devoted to both CSIR acquisition and discovery of the UE with the strongest path for downlink information transmission.
for k ∈ {1, 2, . . ., K} and m ∈ {1, 2, . . ., M − 1}, whereas R k .Eq. (54) is recursive, since the current average rate R (m) k at the time slot m is calculated using the average achievable data rate that has been benefited from user k till the time slot m.One might arrange R (m) k to show that it is the weighted average of all the preceding m instantaneous peruser rates R k , s.t.C 1 and C 2(58) for m ∈ {0, 1, . . ., M −1}.Strictly speaking, under scheduling (58), users compete in each time slot not directly based on their instantaneous per-user rates but based on such rates normalized by their respective average rates.For a fixed reflection vector γ (m) , following the same information-theoretic arguments invoked in Subsection III-A (see, in particular, Theorem 3.p (m) , s.t.constraint C 1 , leads to the opportunistic time-sharing power allocation scheme(17), s.t.(18), with k

TABLE I SUMMARY
OF THE MAIN SYSTEM PARAMETERS AND THE TIME-SCALE OF CHANGE OF THE KEY PHYSICAL QUANTITIES.

TABLE II PARAMETERS
OF THE SIMULATION SETUP.