Fellow IEEE Marco Di Renzo

Student Member, IEEE Xiaojun Xi

Fellow IEEE Merouane Debbah

On The Optimal Number of Reflecting Elements for Reconfigurable Intelligent Surfaces

This work considers a point-to-point link where a reconfigurable intelligent surface assists the communication between a transmitter and a receiver. The system rate, energy efficiency, and their trade-off are optimized by tuning the number of reflecting elements to be activated and the phase shifts that they apply. Unlike most previous works, the considered resource allocation problem explicitly accounts, at the design stage, for the time and energy that are necessary for channel estimation and for reporting the optimal configuration of the phase shifts to the reconfigurable intelligent surface. Numerical results confirm the optimality of the proposed methods and show the potential gains of reconfigurable intelligent surfaces.

I. INTRODUCTION

Recently, the concept of smart radio environment has emerged as a candidate architecture for future sixth generation (6G) networks [START_REF] Di Renzo | Smart radio environments empowered by reconfigurable intelligent surfaces: How it works, state of research, and road ahead[END_REF], [START_REF] Zappone | Wireless networks design in the era of deep learning: Model-based, AI-based, or both?[END_REF], [START_REF] Renzo | Smart radio environments empowered by reconfigurable AI meta-surfaces: An idea whose time has come[END_REF], [START_REF] Huang | Holographic MIMO surfaces for 6G wireless networks: Opportunities, challenges, and trends[END_REF], [START_REF] Basar | Wireless communications through reconfigurable intelligent surfaces[END_REF], [START_REF] Renzo | Reconfigurable intelligent surfaces relaying: Differences, similarities, and performance comparison[END_REF], wherein reconfigurable intelligent surfaces (RISs) are used to coat environmental objects that are present in the propagation environment. An RIS is a planar structure made of several individually tunable elements, called meta-atoms or passive scatterers, that can be programmed and appropriately reconfigured to control the phase of the incoming electromagnetic signal, by reflecting or refracting it towards specified locations [START_REF] Di Renzo | Smart radio environments empowered by reconfigurable intelligent surfaces: How it works, state of research, and road ahead[END_REF]. In [START_REF] Qian | Beamforming through reconfigurable intelligent surfaces in single-user MIMO systems: SNR distribution and scaling laws in the presence of channel fading and phase noise[END_REF] and [START_REF] Zhou | Spectral and energy efficiency of IRS-assisted MISO communication with hardware impairments[END_REF] the signal-to-noise ratio (SNR) scaling law of RIS-aided transmission is characterized, and the impact of hardware impairments is analyzed, respectively. Experimental testbeds have confirmed the potential gains of RISs [START_REF] Tang | Wireless communications with reconfigurable intelligent surface: Path loss modeling and experimental measurement[END_REF], [START_REF] Tang | Wireless communications with programmable metasurface: Transceiver design and experimental results[END_REF], [START_REF]Programmable metasurface-based RF chain-free 8PSK wireless transmitter[END_REF], [START_REF] Dai | Reconfigurable intelligent surface-based wireless communication: Antenna design, prototyping and experimental results[END_REF].

In the context of radio resource allocation, several works have recently appeared. In [START_REF] Huang | Reconfigurable intelligent surfaces for energy efficiency in wireless communication[END_REF], the rate and energy efficiency (EE) of an RIS-based multiple-input single-output (MISO) downlink system are optimized by means of alternating optimization, fractional programming, and sequential optimization. A similar system setup is addressed in [START_REF] Wu | Intelligent reflecting surface enhanced wireless network: Joint active and passive beamforming design[END_REF], and the problem of power minimization subject to minimum rate constraints is tackled by alternating optimization. In [START_REF] Yang | IRS-enhanced OFDM: Power allocation and passive array optimization[END_REF] and [START_REF] Yu | MISO wireless communication systems via intelligent reflecting surfaces[END_REF], sum-rate maximization in a MISO downlink system with orthogonal frequency division multiplexing is studied by optimizing the transmit beamformer and the RIS phase shifts with the aid of alternating optimization. An RIS with discrete A. Zappone is with the University of Cassino and Southern Lazio, Cassino, Italy (alessio.zappone@unicas.it). M. Di Renzo and X. Xi are with Université Paris-Saclay, CNRS and CentraleSupélec, Laboratoire des Signaux et Systèmes, 91192 Gif-sur-Yvette, France (marco.direnzo@centralesupelec.fr), M. Debbah is with Huawei France R&D, Boulogne-Billancourt, France (merouane.debbah@huawei.com).

M. Di Renzo's work was supported in part by the European Commission through the H2020 ARIADNE project under grant agreement 675806.

phase shifts is considered in [START_REF] Liu | Symbol-level precoding design for intelligent reflecting surface assisted multi-user MIMO systems[END_REF], wherein sum-rate maximization in a multi-user MISO system is addressed. In [START_REF] Jiang | Over-the-air computation via intelligent reflecting surfaces[END_REF], a multi-user MISO channel is considered, in which an RIS is used to perform over-the-air computations. Therein, alternating optimization and difference convex programming are used for system optimization. In [START_REF] Li | Joint active and passive beamforming for intelligent reflecting surface-assisted massive MIMO systems[END_REF], multiple RISs are used in a massive multiple-input multiple-output (MIMO) setup, and the problem of maximizing the minimum of the users' signalto-interference-plus-noise-ratio with respect to the transmit precoder and the RISs phase shifts is considered. In [START_REF] Han | Intelligent reflecting surface aided network: Power control for physical-layer broadcasting[END_REF], the problem of power control for physical-layer broadcasting under quality of service constraints for the mobile users is addressed. The downlink of a MIMO multi-cell system is considered in [START_REF] Pan | Multicell MIMO communications relying on intelligent reflecting surface[END_REF], wherein the problem of weighted sum-rate maximization is tackled by alternatively optimizing the base station beamformer and the RIS phase shifts. An RIS-based MISO millimiter-wave system is studied in [START_REF] Wang | Intelligent reflecting surface-assisted millimeter wave communications: Joint active and passive precoding design[END_REF], wherein the transmit beamforming and the phase shifts of multiple RISs are optimized. In [START_REF] You | Intelligent reflecting surface with discrete phase shifts: Channel estimation and passive beamforming[END_REF], channel estimation and sum-rate maximization are investigated for a single-user uplink RIS-based system, by considering an RIS with discrete phase shifts. In [START_REF] Pan | Intelligent reflecting surface aided MIMO broadcasting for simultaneous wireless information and power transfer[END_REF], the sum-rate of a MIMO RIS-based system employing simultaneous wireless information and power transfer is maximized with respect to the transmitter beamforming and the RIS phase shifts. Power control for physical layer broadcasting in an RISbased network is investigated in [START_REF] Han | Intelligent reconfigurable surface aided power control for physical-layer broadcasting[END_REF]. Rate maximization in RIS-based indoor millimeter-waves systems is addressed in [START_REF] Perović | Channel capacity optimization using reconfigurable intelligent surfaces in indoor mmWave environments[END_REF] by adjusting the RIS and the transmitter phase shifts.

The above literature survey shows that most works focus on how to maximize the rate of RIS-based systems with respect to the transmit power/beamforming and the RIS phase shifts, whereas the number of tunable elements at the RIS is not optimized. Moreover, previous works consider only the data transmission phase, while neglecting the overhead for channel estimation and for reporting the optimized configuration to the RIS. In contrast to these research works, this paper addresses the problem of jointly optimizing the RIS phase shifts and the number of tunable elements to be activated at the RIS, in order to optimize the rate, the EE and their trade-off. Notably, this is performed based on the model recently proposed in [START_REF] Zappone | Overhead-aware design of reconfigurable intelligent surfaces in smart radio environments[END_REF], which quantifies the impact of channel estimation and resource allocation feedback on the rate and EE of RIS-based systems.

The rest of the paper is organized as follows. Section II describes the system model, Sections III, IV, and V develop the proposed algorithm for the optimization of the rate, EE, and their trade-off, respectively. Section VI provides a numerical performance analysis, while Section VII concludes the work.

II. SYSTEM MODEL AND PROBLEM STATEMENT

We consider a point-to-point system wherein a singleantenna transmitter and a single-antenna receiver communicate through an RIS equipped with N individually tunable elements. This scenario models, for example, a device-to-device communication link, or a cellular network in which the base stations employ antenna selection to serve each user by a different antenna, and multi-user interference is suppressed through any orthogonal signaling technique. Also, [START_REF] Zappone | Overhead-aware design of reconfigurable intelligent surfaces in smart radio environments[END_REF] numerically shows that RISs are especially useful when few antennas are employed, since RIS optimization can compensate for the lack of multiple transmit and/or receive antennas.

We assume that the direct path between the transmitter and the receiver is not available, and denote by h " th n u N n"1 and g " tg n u N n"1 the fast-fading channel vectors from the transmitter to the RIS and from the RIS to the receiver, respectively, and by δ the overall propagation path-loss. Before data communication can start, channel state information must be acquired. This can be accomplished by means of pilotbased methods which estimate the product channels h n g n , for all n " 1, . . . , N , with ˚denoting complex conjugate. The specific estimation algorithm is inessential for the purpose of this work. We only make the baseline assumption that the product channels are estimated sequentially. Based on the estimated channels, the RIS configuration can be optimized and deployed. Specifically, the optimized RIS configuration must be communicated to the RIS. This is done by sending a configuration signal, which is processed at the RIS by a controller with minimal signal processing, transmission/reception, and power storage capabilities. The controller is a key component of RIS, which enables their reconfiguration capability in order to dynamically adapt to changes of the propagation channels [START_REF] Di Renzo | Smart radio environments empowered by reconfigurable intelligent surfaces: How it works, state of research, and road ahead[END_REF]Fig. 4]. However, transmitting the control signal introduces a non-negligible overhead to the communication phase, especially for large N , since N b F bits must be encoded in the configuration signal, with b F the number of bits used to represent the phase applied by each RIS element. Denoting by T the total duration of the time slot comprising the channel estimation phase of duration T E , the control feedback phase of duration T F , and the data communication phase of duration T ´TE ´TF , and defining β " pδ{pBN 0 q, with B the communication bandwidth, N 0 the receive power spectral density, and p the transmit power, the system rate and EE are expressed as

RpNq " ˆ1 ´TE `TF T ˙B log `1 `β|g H Φh| 2 ˘(1) EEpN q " RpN q{P tot (2)
where Φ " diagpe jφ1 , . . . , e jφ N q is the RIS phase matrix and P tot is the total power consumption in the whole timeframe T . Following the model developed in [START_REF] Zappone | Overhead-aware design of reconfigurable intelligent surfaces in smart radio environments[END_REF], T E " T 0 pN `1q, with T 0 the duration of each pilot tone, and

T F " N b F B F logp1 `pF |h F | 2 {pN 0 B F qq (3)
P tot " P E `p1´T E {T qµp`T F {T pµ F p F ´µpq`N P c,n `Pc,0 ,
since a power p is used for T ´TE ´TF seconds, with transmit amplifier efficiency 1{µ, a power p F is used for T F seconds, with transmit amplifier efficiency 1{µ F , while hardware static power is consumed during the whole interval T , where P c,n is the hardware power required for each RIS element, P c,0 is the hardware power for all the other system components, and P E " T E P 0 {T is the power consumption during the channel estimation phase, with P 0 the power of each pilot tone. The aim of this work is to optimize the RIS phase shift matrix Φ and the number of tunable elements N at the RIS, in order to optimize the system rate and EE in (1), and their trade-off. Note that the approach in this work differs from robust resource allocation methods which assume imperfect channel state information [START_REF] Zhou | A framework of robust transmission design for IRS-aided MISO communications with imperfect cascaded channels[END_REF], [START_REF] Hong | Robust transmission design for intelligent reflecting surface aided secure communication systems with imperfect cascaded CSI[END_REF], [START_REF] Zhou | Robust beamforming design for intelligent reflecting surfaceaided MISO communication systems[END_REF]. Indeed, we assume that reliable channel estimation is performed and the resulting overhead is appropriately accounted for in our system model.

A. Optimization of Φ and Upper-Bound of N Since Φ does not affect the power consumption P tot , the optimal Φ for both the rate and the EE is obtained by maximizing the term |g H Φh| 2 . This is accomplished by setting φ n " ´=g n h n for any n. With this choice, the received power at the destination is pδ

´řN n"1 α n ¯2, with α n " |h n g n |.
On the other hand, since the received power can not be larger than the transmit power, it must hold that δ ´řN n"1 α n ¯2 ď 1. Defining α max " max n α n , a sufficient condition for this to hold is N α max ? δ ď 1, which provides a simple upperbound on the maximum number of elements that can be placed on the RIS in order for the considered path-loss model to be physically meaningful. In addition, in order to prove the mathematical results to follow, we assume βα 2 max ě 1. This is considered a practical assumption, since βα 2 max is the receive SNR over the reflection path with the largest gain. Thus, a wireless system in which βα 2 max ă 1, would have little practical value, because it could not provide the data-rate and reliability that are necessary in modern wireless systems. Since N α max ? δ ď 1, in order to enforce βα 2 max ě 1, we must have N 2 ď β{δ, which finally allows us to bound the maximum number N max of elements at the RIS as

N max ď min ! pα max ? δq ´1, a β{δ
) . (4)

III. RATE OPTIMIZATION

Using the optimal Φ, the rate maximization problem is

max 1ďN ďmintNmax,tc{duu pc´dN qB log ¨1`β ˜N ÿ n"1 α n ¸2' (5) with c " 1 ´T0 T , d " T0 T `bF T B F log ´1`p F |h F | 2 B F N 0
¯, and where, without loss of generality, we consider that the coefficients α n are arranged in decreasing order of magnitude, i.e. n ě α n`1 for all n " 1, . . . , min N max , X c d

\(

, which also means that α max " α 1 . The constraint in [START_REF] Basar | Wireless communications through reconfigurable intelligent surfaces[END_REF] ensures that the sum of the durations of the channel estimation and feedback phases is shorter than the total length of the frame, and that N is smaller than its maximum feasible number N max . The challenge in solving (5) lies in the fact that the first factor of the objective decreases with N , while the second factor increases with N , which makes it difficult to determine the behavior of the rate R as a function of the discrete variable N . In order to globally solve (5), the following result is instrumental.

Proposition 1: RpN q in (5) is a unimodal function, i.e. it is either increasing with N or, if there exists an N such that Rp N q ě Rp N `1q, RpN q is decreasing for N ě N .

Proof: If N does not exist, the rate is increasing with N .

If N exists, then defining f pN q " B log ˆ1`β ´řN n"1 α n ¯2˙, the condition Rp N q ě Rp N `1q is equivalent to

pc ´d N qpf p N `1q ´f p N qq ď d f p N `1q . (6)
Thus, the result holds if we can show that (6) implies that Rp N `1q ě Rp N `2q, i.e.

pc ´dp N `1qqpf p N `2q ´f p N `1qq ď d f p N `2q . ( 7
)
At this point, let us show that, for any N , it holds f pN `1q ´f pN q ě f pN `2q ´f pN `1q .

To see this, expanding the square in f pN `1q leads to f pN `1q ´f pN q " (9)

B log ¨1`β ˆřN n"1 α n ¯2`α 2 N `1`2α N `1 ř N n"1 α n 1 `β ´řN n"1 α n ¯2 ‹ ‹ ' " Blog 1 `βα N `1 ´αN`1 `2 ř N n"1 α n 1 `β ´řN n"1 α n ¯2 ‹ '
Similarly, it holds

f pN `2q´f pN `1q " Blog 1 `βα N `2´α N `2 `2 ř N `1 n"1 α n 1`β ´řN`1 n"1 α n ¯2 ‹ '.
Then, the condition in (8) becomes

α 2 N `1 1 `β ´řN n"1 α n ¯2 `2α N `1 ř N n"1 α n 1 `β ´řN n"1 α n ¯2 ě α 2 N `2 1 `β ´řN`1 n"1 α n ¯2 `2α N `2 ř N `1 n"1 α n 1 `β ´řN`1 n"1 α n ¯2 . ( 10
)
Since α n ě α n`1 for any n " 1, . . . , N , the first summand at the left-hand-side of ( 10) is larger than the first summand at the right-hand-side. Then, for [START_REF] Tang | Wireless communications with programmable metasurface: Transceiver design and experimental results[END_REF] to hold it is sufficient that:

ř N n"1 α n 1 `β ´řN n"1 α n ¯2 ě ř N `1 n"1 α n 1 `β ´řN`1 n"1 α n ¯2 . (11)
Defining z " ř N n"1 α n , it can be seen that ( 11) is equivalent to showing that the function gpzq " z 1`βz 2 is decreasing. Computing the first-order derivative of gpzq, and setting it to be non-positive, yields βα 1 ě 1, which holds on the feasible set of (5). Finally, exploiting [START_REF] Renzo | Reconfigurable intelligent surfaces relaying: Differences, similarities, and performance comparison[END_REF], and observing that f pN q is increasing in N since each α n is positive, it follows that pc ´dp N `1qqpf p N `2q ´f p N `1qq ď (12)

pc ´d N qpf p N `1q ´f p N qq ď d f p N `1q ď d f p N `2q ,
and hence the proof follows.

Equipped with this result, Problem (5) can be solved by a greedy approach in which the tunable elements of the RIS are activated one at a time, following the decreasing order of magnitude of the coefficients tα n u n , until a decrease in RpN q is observed, or N reaches its maximum allowed number. The procedure is stated in Algorithm 1, with ApN q " RpN q.

Algorithm 1 Optimization of N for rate maximization N " 1; A0 " 0; A1 " ApN q; while An ě An´1 and N ď min Nmax,

X c d \( do N " N `1;
An " ApN q; n " n `1; end while

IV. ENERGY EFFICIENCY OPTIMIZATION

Defining γ " P c,0 `P0 T0 T `µpc and ψ " pP 0 ´µp´1q T0 T d `Pc,n , the EE maximization problem is stated as

max N pc´dN qf pN q γ `ψN (13a) s.t. 1 ď N ď min tN max , tc{duu . (13b)
Proposition 2 shows that the EE is unimodal, and thus (13a) can be globally solved by Algorithm 1 with ApN q " EEpN q.

Proposition 2: EEpN q in (13a) is a unimodal function, i.e. it is either increasing with N or, if there exists an N such that EEp N q ě EEp N `1q, EEpN q is decreasing for N ě N .

Proof: Proceeding as in the proof of Proposition 1, if N does not exist, then EEpN q is increasing with N . Instead, if N exists, the result follows if we can prove that EEp N q ě EEp N `1q implies that EEp N `1q ě EEp N `2q. First of all, let us observe that if N is such that Rp N q ě Rp N `1q, then already know from Proposition 1 that N falls in the range in which the rate function RpN q is decreasing. This implies that the EE is decreasing for any N ě N , since increasing N yields a lower numerator and a larger denominator. As a result, the non-trivial case to be considered is when N belongs to the range in which the rate is still increasing, i.e. the first decrease in the EE happens when the rate function is still increasing with N . Thus, without loss of generality, in the rest of this proof we assume Rp N `2q ą Rp N `1q ą Rp N q. Next, let us observe that EEp N q ě EEp N `1q is equivalent to

N ď ´γ ψ `Rp N q Rp N `1q´Rp N q . ( 14
)
Similarly, EEp N `1q ě EEp N `2q is equivalent to

N ď ´γ ψ ´1 `Rp N `1q Rp N `2q´Rp N `1q . (15)
At this point, we note that ( 14) can be written as

N ď ´γ ψ ´1 `Rp N `1q Rp N `1q´Rp N q , ( 16
)
which implies [START_REF] Yang | IRS-enhanced OFDM: Power allocation and passive array optimization[END_REF] if we can show that Rp N `2q´Rp N `1q ď Rp N `1q ´Rp N q. To prove this, we observe that we have

Rp N `2q ´Rp N `1q " (17) `c ´dp N `1q ˘`f p N `2q ´f p N `1q ˘´df p N `2q ď `c ´dp N `1q ˘`f p N `2q ´f p N `1q ˘´df p N q ď `c´dp N `1q ˘`f p N `1q´f p N q ˘´df p N q " Rp N `1q´Rp N q,
where the two equalities follow recalling that RpN q " pc dN qf pN q, while the two inequalities follow since f pN q is non-negative, increasing, and such that f pN `1q ´f pN q ě f pN `2q ´f pN `1q, as proved in Proposition 1.

V. RATE-EE MAXIMIZATION

Rate-EE maximization is cast as the bi-objective problem

max N RpN q, EEpN q ( , s.t. 1 ď N ď min tN max , tc{duu (18)
By virtue of [START_REF] Miettinen | Nonlinear Multiobjective Optimization[END_REF]Th. 3.4.5], all Pareto-optimal points of ( 18) can be obtained by solving

max N mintwpRpNq´R opt q,p1´wqpEEpNq´EE opt qu (19a) s.t. 1 ď N ď min ! N max , Y c d ]) (19b)
for w P p0, 1q, with R opt and EE opt the individual maximizers of RpN q and EEpN q, as derived in Sections III and IV. The following proposition shows that GpN q " mintwpRpNq Ŕopt q,p1´wqpEEpNq´EE opt qu unimodal, and thus Problem (19a) can be solved by Algorithm 1 with ApN q " GpN q.

Proposition 3: GpN q is a unimodal function, i.e. it is either increasing with N , or, if there exists N such that Gp N q ě Gp N `1q, GpN q is decreasing for N ě N .

Proof: Propositions 1 and 2 ensure that RpN q and EEpN q are unimodal functions, which implies that G 1 " wpRpN q Ŕopt q and G 2 " p1 ´wqpEEpN q ´EE opt q are unimodal, too. Let us consider first that both G 1 pN q and G 2 pN q have a finite maximizer, which we denote by N 1 and N 2 , respectively. Without loss of generality, let us assume that N 1 ď N 2 . Then, GpN q is increasing for N ď N 1 and decreasing for N ě N 2 . As for the range N 1 ă N ă N 2 , let us consider two cases:

(a) If G 1 pN 1 q ď G 2 pN 1 q, then GpN q " G 1 pN q for N 1 ă N ă N 2 , because in this range G 2 is increasing while G 1 is decreasing. As a result, GpN q is increasing for N ď N 1 and decreasing for N ą N 1 and we have N " N 1 .

(b) If G 1 pN 1 q ą G 2 pN 1 q, two cases can be considered: if G 1 pN q ě G 2 pN q for N 1 ď N ď N 2 , then GpN q " G 2 pN q for N 1 ď N ď N 2 and the thesis holds with N " N 2 ; if instead G 1 pN q ğ G 2 pN q for N 1 ď N ď N 2 , then we can define N 3 as the smallest number such that N 1 ă N 3 ď N 2 and G 1 pN 3 q ď G 2 pN 3 q. Thus, GpN q " G 2 pN q for N 1 ď N ă N 3 , while GpN q " G 1 pN q for N 3 ď N ď N 2 . Then, it follows that GpN q is increasing for N ă N 3 and decreasing for N ě N 3 , and the thesis follows with N " N 3 .

Finally, we observe that the reasoning above includes as special cases the situations in which either G 1 or G 2 is monotonically increasing for all N , whereas if both G 1 and G 2 are monotonically increasing, then GpN q is monotonically increasing and N does not exist. In our numerical simulations we set p F " 30 dBm, B " 5 MHz, B F " 1 MHz, δ " 110 dB, N 0 " ´174 dBm/Hz, µ " µ F " 1, P c,0 " 45 dBm, P c,n " 10 dBm, b F " 16, T 0 " 1 ms, P 0 " 10 dBm. Moreover, h n " CN pv h , 1q and g n " CN pv g , 1q. Thus, |h n | and |g n | are Rice distributed, where v h and v g are chosen so that the power of the line-of-sight path is four times larger than the power of all the other paths1 . A similar model is used for the feedback channel h F . Also, N max " 300 and all presented results have been averaged over 10 4 independent realizations of the channel vectors h and g. (e) N " N max and random φ n in r0, 2πs for all n " 1, . . . , N . As for Schemes (c) and (e), we set T F " 0 and T E " T 0 , since no RIS optimization is performed. The results show that, accounting at the design stage for the overhead due to channel estimation and optimal RIS configuration outperforms the other schemes. In particular, Scheme (a) significantly outperforms Schemes (c) and (e), which employ suboptimal allocations of both N and Φ, despite the fact that in these cases no overhead for RIS configuration is required. Thus, despite the overhead, an optimized RIS improves the performance.

Table I reports the average rate maximizer N R and the EE maximizer N EE versus p, that correspond to the performance of Scheme (a) in Fig. 1. The results confirm that both the rate and the EE have a finite maximizer with N .

Finally, Fig. 2 shows the Rate-EE Pareto-frontier of the considered RIS-based system for p " 20 dBm, p " 30 dBm, VII. CONCLUSIONS This work maximized the rate, EE, and their trade-off in a RIS-based system, as a function of the number and the phase shifts of the RIS elements, and by accounting for the overhead due to channel estimation and RIS configuration. Numerical results confirmed that overhead-aware resource allocation may significantly outperform sub-optimal methods. Future studies are needed to address more complex network topologies.

Fig. 1 :

 1 Fig. 1: Rate (left) and EE (right) versus p for: (a) Optimal N and Φ; (b) Random N and optimal Φ; (c) Random N and Φ; (d) N " Nmax and optimal Φ; (e) N " Nmax and random Φ

Fig. 1

 1 shows the average rate and EE versus the transmit power p achieved by the following schemes: (a) Optimization of N by Algorithm 1 and optimization of Φ as described in Section II-A. (b) Random N in r1, N max s and optimization of Φ as in Section II-A. (c) Random N in r1, N max s and random φ n in r0, 2πs for all n " 1, . . . , N . (d) N " N max and optimization of Φ as in Section II-A.

p

  " 40 dBm. For each value of p, the setups with P c,n " 10 dBm and P c,n " 15 dBm are shown. As expected, a larger P c,n yields a wider Pareto-region, because the larger P c,n is, the more the rate and the EE are contrasting objectives.

Fig. 2 :

 2 Fig. 2: Rate-EE Pareto region for p " 20 dBm (left), 30 dBm (middle), 40 dBm (right), with Pc,n " 10 dBm and 15 dBm

TABLE I :

 I Average number of the RIS elements that maximize the rate and energy efficiency versus p

		0 dBm 10 dBm 20 dBm 30 dBm 40 dBm
	N R	198.89	197.32	193.38	184.74	172.04
	N EE 187.05	167.29	147.01	133.92	145.99

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In the numerical results, we did not enforce the condition βαmax ě 1, which is only needed at the design stage.