Spatially-Distributed RISs vs Relay-Assisted Systems: A Fair Comparison

This paper aims to make a comparison between decode-and-forward (DF) relays and reconfigurable intelligent surfaces (RISs) in the case where only one relay or RIS is selected based on the maximization of the signal-to-noise-ratio (SNR). Our study accounts for the spatial distribution of RISs and relays, which is assumed to follow a Poisson point process (PPP). It considers two different path loss models corresponding to RIS/relays randomly located in the near-field and the far-field of the transmitter. Based on the Gamma distribution moment matching method and tools from stochastic geometry, we derive approximations for the outage probability (OP) as well as the energy efficiency (EE) of the RISs-assisted system in the near-field and the far-field scenarios separately. Under the same conditions as RIS, the expressions for OP and EE of the half-duplex and the full-duplex DF relays-assisted systems are also derived. Simulation results are presented to corroborate the proposed analysis and compare between the three technologies. Our results show that RIS is the best choice in the near-field case, regardless of the OP or EE criterion. Compared to half-duplex relays and full-duplex relays, the RIS based system is the most energy-efficient solution to assist communication. Moreover, RIS allows for an improvement in both OP and EE when equipped with more reflecting elements or more densely deployed.


I. INTRODUCTION
A S THE 5G standard is starting its implementation phase, it is critical to look for new communication technologies catering to the ever-increasing demand for traffic rate [1] in the future 5G beyond or even 6G generation. Reconfigurable intelligent surfaces (RISs) are now considered among the key enabling technologies for 6G systems. Empowered by the recent advances in metamaterial [2], RISs are equipped with a large number of low-cost passive elements that allow for the modification of the radio waves, in that they reflect, refract, and scatter radio signals in a controllable fashion to counteract the destructive effect of multipath fading. These features can be leveraged to transform the propagation environment into a smart space that can be programmable for the benefit of the communication application. An important research effort is now being devoted to develop hardware architectures based on this technology [3], [4], and to study other related aspects such as channel modeling [5]- [7] and channel estimation [8]- [10], to name a few. Similar to any other technologies, the design of the RIS has been proposed to respond to a variety of criteria such as controlling the signal strength [11], expanding wireless coverage [12], optimizing system capacity [13], [14], maximizing signal-to-interference-ratio [9], minimizing symbol error rate [15], [16], improving physical layer security [17] and increasing energy efficiency (EE) [18].
resources. Particularly, in [19], approximations for the probability that the channel gain of the backscatter link assisted by RIS exceeds that of the direct link was provided and was shown to match simulation results, while exact closed-form expressions for OP and spectral efficiency were provided in [20] and [21] under reciprocal Rayleigh channels and Fox'H fading channels, respectively. The authors in [23] provide a complete RIS-assisted system performance analysis from the perspective of the instantaneous and average endto-end signal-to-noise ratio (SNR), the diversity gain, OP, the symbol error rate (SER), the diversity order as well as the ergodic capacity. The closed-form of the ergodic capacity upper bound and OP approximation for the RIS aided single-input single-output (SISO) communication system were derived in [24]. Moreover, the pairwise error probability of a RIS-assisted non-orthogonal multiple access system was analyzed and investigated in [25] assuming imperfect successive interference cancellation, where approximated expression and asymptotic expression are provided.
It is worth mentioning that RISs should not be confused with relay-aided systems or backscatter communication. The main distinguishable feature is that they do not necessitate the use of analog-to-digital/digital-to-analog converters or power amplifiers and ideally can operate at any frequency. Moreover, they can be deployed anywhere, e.g., on the facades of buildings, the ceiling of indoor spaces, or human clothing and furniture. However, like relays, RISs aim to assist communication by enabling higher efficiency gains. This principally motivated the emergence of a series of works aiming to compare between RIS and relay-assisted technologies from outage performance [23] and EE standpoints [26]. In this line, it was shown in [27] that large reflecting elements are needed for the RIS to outperform decode-and-forward (DF) relaying, both in terms of minimizing the total transmit power and maximizing EE, while in [28], it was shown that RIS-assisted single user system is more cost-efficient than amplify-and-forward (AF) relaying. Similarly, an important gain enabled by the use of RIS assisted systems in terms of spatial throughput was also reported in [29] once the number of reflecting elements exceeds a certain value determined by simulation. Moreover, when resource allocation algorithms are used, it was shown in [30] that RIS-assisted systems can achieve 300% higher EE in comparison with the use of regular multi-antenna AF relaying. Following the same line of research, the works in [31], [32] focused on designing optimal and suboptimal schemes to control the consumed power at the network under different settings. More Specifically, the authors in [23] compared the theoretical framework in terms of SNR, OP, SER and ergodic capacity of RIS-assisted system with the one of the corresponding AF relay-assisted system. The comparison results also verified the superiority of RIS.
The current literature has thus far investigated the deployment of only one RIS to support communication. However, in the future, it is expected that these RISs will be densely deployed, and thus transmission will be assisted not only by one RIS but by a multitude of them. This lies behind the motivation of the present work. Particularly, we consider the setting in which communication is assisted by multiple relays and multiple RISs spatially distributed according to a Poisson point process (PPP) [33] in the ring centered at the user. We propose to compare RIS-assisted systems and decode-andforward (DF) relaying systems of which one RIS/relay is selected based on the maximization of the received SNR. Comparison is carried out in terms of OP and EE and considered both half-duplex and full-duplex relaying systems. Overall, the main differences between our work and the previous works in [23], [27], [30] are: • Multiple RISs and relays are considered in this work, while only one RIS and one relay are considered in [23], [27], [30]. The selection combining (SC) scheme is also adopted in this work to improve the system performance. • We account for the spatial distribution of RISs and Relays in this work, which we model by PPP. Under this situation, the distances between base station (BS) and RISs/relays are random, leading to varying path loss and making the system more practical. • Moreover, our study investigates both near-field and far-field communications, for which expressions for the OP and EE have been provided, while most existing literature investigates only the performance of far-field communications. • Given that RISs operate in full-duplex mode, we carry out a performance comparison with full-duplex relaying systems in addition to half-duplex relaying systems. Such a comparison is more comprehensive and allows us to draw more insights than its predecessors in the literature. To pave the way towards a fair comparison of this system, closed-form expressions for OP and EE are derived by leveraging tools from stochastic geometry. Simulations are then carried out to confirm the accuracy of the analytical results and to draw a fair comparison between both systems. Particularly, we confirm that RIS assisted systems compares favorably with both half-duplex and full-duplex assisted systems. Moreover, higher gains can be reaped from this technology when it is equipped with more elements or densely deployed.
The rest of the paper is organized as follows: In Section II, the system model is described. Section III analyzes the performance of RIS-assisted systems in terms of OP and EE, while in Section IV, a theoretical performance analysis for relay-assisted systems is presented. In Section V, some asymptotic results are derived. Some selected discussions about the performance analysis under more practical assumptions are provided in Section VI. Finally, a set of numerical results comparing both systems are provided in Section VII, and conclusions of this paper are drawn in Section VIII. Table 1 presents the main notions adopted in the following part of this work.

II. SYSTEM MODEL
We consider a scenario in which a user communicates through a multiple RIS-assisted system with a single antenna BS. The BS is located far from the user while the direct link is blocked and as such is ignored. All RISs are composed of N = N x × N y passive and low-cost reflecting elements installed on the wall of a surrounding high-rise building. We assume each RIS to be composed of elements with size L×L equally spaced on a N x × N y grid. As shown in Fig. 1, RISs are assumed to be spatially distributed according to a homogeneous PPP with density ρ in a circle centered at the user with inner radius R 1 and outer radius R 2 . For the sake of simplicity, the planar array of the RIS is perpendicular to the direction of the propagation from the user and that the user, while the centers of all RIS and BS are in the same plane. 1 We consider both near-field and far-field communication 2 between the RISs and the user, which are distinguished by the inner radius R 1 max{N x , N y }L [26]. More specifically, we assume near-field communication between the RISs and the user when the RISs are located within R 1 , and far-field communication when the RISs are located between R 1 and R 2 . Specifically, in the far-field scenario, the spherical wave generated by the transmitter can be approximately regarded as a plane wave at the antenna array side when the distance between the transmitter and the center of the RIS is bigger than R 1 . Thus, the probability of RISs to be located in the near-field of the user is P near = , while the probability of RISs located in the far-field of the user is P far = The distance between the user and BS denoted by r U is fixed. To reduce the complexity and improve the diversity through multiple RISs, the SC scheme [34] is considered, by which the RIS achieving the largest received SNR is always selected. The RIS selection process is conducted 1. The aim of these assumptions is to simplify the calculation of the intervening distances between the user and the RIS and between the RIS and BS. The first assumption facilitates the calculation of the near-field distances in the first hop, while the second one simplifies the calculation of distances in the second hop. Without these assumptions, the distance between all devices is hard to statistically characterize due to the PPP distributed RISs. Although the considered scenario in this work may be thought of as an ideal case, the closed-form analytical expression derivations for the system performance can be performed to yield important insights into the behavior of the intervening parameters.
2. The far-field refers to the propagation range at which the direction and thus the channel gains are approximately the same from all elements in the array to the transmitting/receiving antenna. On the other hand, the near-field is described in this work by considering that the distance between the transmit antenna to the elements vary over the array. at the BS by comparing the SNRs estimated based on the received pilot signals transmitted from the user and reflected by each RIS. Other complex combining techniques, e.g., maximum ratio combining (MRC), can also be similarly applied and analyzed based on the proposed analytical method. As the combining technique is not the focus of this work, we omit further details of this question. The fading channels between the user and the n-th reflecting element of RIS m and that between the n-th reflecting element of RIS m and BS are denoted by h m,n = α m,n e −jθ m,n and g m,n = β m,n e −jϕ m,n , where m = 1, . . . , M, n = 1, . . . , N. Let m = diag(e jφ m,1 , . . . , e jφ m,N ) be the phase shifts applied by the m-th RIS. To facilitate analysis, we assume that the phase resolution is infinite [30]. Since the estimation and feedback process is not the focus of this work, we ignore the delay and power consumption for this process and assume perfect channel state information (CSI) at the BS and perfect feedback information to the RIS. More details regarding this question can be found in [35]. In the next subsection, we will discuss the system performance in the near-field and the far-field respectively. Finally, it is worth mentioning that although this work investigates the performance of uplink RIS-assisted system, where signals transmitted from the user to BS, the analytical method and results can also be applied in the downlink RIS-assisted system, due to signals experiencing the same channel fading and path loss. Specifically, the distance between the user and RISs and the distance between RISs and BS in the downlink RISassisted system follow the same distribution as the uplink RIS-assisted system.

A. NEAR-FIELD COMMUNICATION
When RISs are distributed within the inner radius R 1 , the number of available RISs M 1 is Poisson distributed with probability Pr(M 1 ) = (μ M 1 1 /M 1 !)e −μ 1 , where μ 1 = R 2 1 πρ denotes the mean measure. The instantaneous SNR at BS is given by: where P t is the transmit power, N 0 is the variance of the noise, P L = 16π 2 λ 2 , λ = c f is the wavelength, c is the speed of light, f is the carrier frequency, and d m,n,1 and d m,2 are the distance between the user and reflecting element n of RIS m and that between RIS m and the BS respectively, while η 1 and η 2 denote the path-loss exponents. Since the distance between the user and RISs are less than R 1 , the near-field communication with traditional path loss model is assumed. The distance between reflecting element n to the center of RIS m can be calculated as Therefore, d m,n,1 can be calculated as where d m,1 is the distances between the user and the center of RIS m. We can easily see that the optimal reflection coefficients are φ m,n = θ m,n + ϕ m,n . In practice, some estimation errors on the channel phases may exist. In this case, the derived results should be thought of as upper bounds on the achieved performances. Under optimal phases, the SNR associated with m-th RIS is: Considering that there is no line-of sight between the BS and RISs, as well as between RISs and the user, we assume that h m,n and g m,n are drawn from a standard Gaussian distribution independently, and hence their modulus follow a standard Rayleigh distribution. In other words, α m,n and β m,n are independent Rayleigh distributed random variables. 3 Under this assumption, the exact distribution for could not be easily derived. To overcome this issue, we propose to approximate it with a Gamma distribution, whose shape and sacle parameters are tuned based on the moment matching technique [19], [20]. This technique stands out among many different techniques for approximating non-tractable distributions [36]. Pursing the moment matching technique, the following Lemma gives the Gamma approximation for the probability density function (PDF) and cumulative distribution function (CDF) of S N .
Lemma 1: The distribution of S N conditioned on d m,1 and d m,2 can be approximated with a Gamma distribution whose PDF and CDF are given by: and  (8) represent the shape parameter and scale parameter of Gamma distribution, respectively, (·) and γ (·, ·) are the Gamma function and the incomplete Gamma function defined in [37, (8.310 Proof: The proof of Lemma 1 is provided in Appendix A.
The accuracy of the Gamma approximation was verified in Section VII. Based on Lemma 1, the CDF of the SNR can be derived as: Corollary 1: The CDF of SNR associated with m-th RIS γ m,N = P t S 2 N P L N 0 can be expressed as whereγ = P t P L N 0 . Proof: The distribution of γ m,N =γ S 2 N can be obtained easily through variable substitution based on the distribution of S N shown in (6).

B. FAR-FIELD COMMUNICATION
For the far-field case, RISs are located within the ring with inner radius R 1 and outer radius R 2 . The number of available RISs M 2 is Poisson distributed with probability Pr( πρ denotes the mean measure. In this case, the distances between the user or the BS and each element of the RIS can be approximated to be same. Thus, the instantaneous SNR with optimal phase shifts at BS is given by: Similar to the near-field, We firstly derive the distribution of S F = N n=1 α m,n β m,n in the following lemma and then derive the CDF of γ m,F accordingly.
Lemma 2: The distribution of S F conditioned on d m,1 and d m,2 can be approximated with a Gamma distribution which has the PDF and CDF and where F = Nπ 2 16−π 2 and F = 16−π 2 4π represent the shape parameter and scale parameter of Gamma distribution, respectively.  16 ). Based on the moment matching method, we find the parameters of the Gamma distribution that has the same mean and variance as S F . The expectation and variance of a Gamma distribution with shape parameter F and scale parameter F are F F and F 2 F , respectively. We thus obtain the following system of two equation for F and F : 16 ), from which we can easily deduce F = Nπ 2 16−π 2 , F = 16−π 2 4π . Therefore, according to the moment matching method, the distribution of S F = N n=1 α m,n β m,n can be approximated by a Gamma distribution with shape parameter F and scale parameter F . Similarly to the near-field communication, we derive the CDF for γ m,F as: The CDF of γ m conditioning on d m,1 and d m,2 can be expressed as: can be obtained easily through variable substitution based on the distribution of S F shown in (12).

III. RISS-ASSISTED SYSTEM PERFORMANCE ANALYSIS
In this section, we analyze the system performance in terms of OP and EE for the RISs-assisted system in the nearfield and the far-field cases separately. We assume that a SC scheme is adopted by which the BS chooses the RIS that achieves the maximum end-to-end SNR to assist communication.

A. OUTAGE PERFORMANCE
The overall received SNR based on the SC scheme could be derived as γ RIS = max m∈S {γ m }, where S denotes the set of PPP distributed RISs in the ring. From Fig. 1, the relationship between d m,1 and r U is where r U is the fixed distance from the user to BS and ω m is the randomly distributed angle between the direct link and the first reflect link. Since d m,2 is function of d m, 1 and ω m , the CDF of the overall SNR conditioned on d m, 1 and ω m at the BS is 1 , ω m ). Hence, the CDF associated with the SNR is given as and E d m, 1 ,ω m denotes the expectation with respect to the random variables d m, 1 and ω m . The above integral is hard to solve for both near-field and far-field communication scenarios. The difficulty stems from the randomness of d m,2 which is not easy to handle. To overcome this issue, we assume from now on that the outer radius of the ring is much smaller than the distance between the user and the BS, which allows us to approximate d m,2 by r U .

1) NEAR-FIELD COMMUNICATION
As a consequence, the OP of the proposed system can be shown as P out, where Proof: Using the representation of the incomplete Gamma function in terms of Meijer's G-Function in [38, 8.4.16.1] and approximating d m,2 by r U , the CDF of γ m,F conditioning on d m,1 is given by: Then, applying (14) to (9), we calculate F γ RIS ,F (x) as By using [39,Eq. (26)], the expression of F γ RIS ,F (x) can be finally expressed as (16). This completes the proof of Lemma 4.
Hence, the OP of the end-to-end SNR in the far-field communication scenario is given by P out,F = F γ RIS,F (γ th ).

B. ENERGY EFFICIENCY
The EE is defined as η = BR P total where B is the transmission bandwidth, R is the achievable capacity in bps and P total is the total power consumption in Joule. The total power consumption of the RIS-assisted system is P RIS = ξ −1 P t + P B +P U +NP b [27], where ξ is the efficiency of the transmit power amplifier, P B and P U denote the hardware-dissipated power consumption at the user and BS, P b is the power consumed by each element at RIS.

Lemma 5:
The EE of RIS-assisted system in near-field can be approximated by (19) shown at the bottom of the page, where 2) FAR-FIELD COMMUNICATION Lemma 6: Based on the distribution of γ RIS,F , we can obtain EE for far-field as where m,2 P L N 0 and using the expression , the EE in far-field can re-expressed through the same way shown in Appendix C as: Also, using the relation The approximated expression of η RIS,F can be obtained using the Chebyshev-Gauss quadrature approximation.

IV. COMPARISON WITH RELAY-ASSISTED SYSTEMS
In this section, we analyze the system performance when the RIS are replaced by DF relays to assist communication. For the sake of fairness, we assume that DF relays are geometrically placed and distributed as the RISs, and are equipped similarly to RIS with N receive and transmit antennas while employing MRC. Similar to the RIS-assisted system, the SC scheme is also adopted in the relay-assisted system and consists in selecting the relay achieving the largest received SNR to assist the transmission. Under this setting, we compare the OP and EE in near-field and farfield scenarios with those obtained by the RIS assisted system. Both half-duplex and full-duplex relaying will be considered. , and h mB ∈ C N denotes the channel between m-th DF relay and BS and follows a standard normal distribution. Based on the CSI of both links, the optimal source-relaydestination link (OSRDL) is selected [40]. Its SNR is given by γ DF = max m∈S {min{γ Um,N , γ mB }}. Considering MRC at each relay, the CDF of γ Um,N conditioned on d m,1 can be given by [41] as

Near-Field
while the distribution of γ mB conditioned on d m,2 is expressed as Thus, the CDF of γ UmB,N = min{γ Um,N , γ mB } conditioning on d m,1 and d m,2 is Lemma 7: Assuming relays are PPP distributed in the inner circle, the CDF of γ DF,N in the near-field communication can be approximated as: where Proof: Based on the approximation r U ≈ d m,2 and applying (14) to (26), the CDF of γ DF,N can be calculated as The proof of Lemma 7 can be completed through the Chebychev-Gauss quadrature approximation. The OP of the relay-assisted system for the near-field communication is thus given by F γ DF,N (γ th ).
Far-Field Communication: The received SNR at the output of the MRC scheme of the m-th relay in the far-field communication can be expressed as γ Um,F = P t |h Um,F | 2 P L N 0 , where h Um,F follows a standard Gaussian distribution with mean 0 and covariance 1 d η 1 m,1 I N . Therefore, the CDF of γ UmB,F = min{γ Um,F , γ mB } in the far-field scenario conditioning on d m,1 and d m,2 is given by: Lemma 8: Assuming relays are PPP distributed in the outer circle, we obtain CDF of γ DF,F in the far-field communication as where . Proof: Based on the approximation r U ≈ d m,2 and applying (14) to (29), the CDF of γ DF,F can be calculated as By using [37, Eq. (3.381.1)], we can easily derive F γ DF,F (x) as (30). The OP of the relay-assisted system is thus given by F γ DF,F (γ th ).

2) ENERGY EFFICIENCY
As compared to a RIS assisted system, the DF relaying system consumes a transmit power P r at the relay, but it is active half of the time. The power consumed by the DF relay is given by [30], [42]: Hence, the EE of the relaying system is , where P H denotes the dissipated power at each half-duplex DF relay transmit-receive antenna. Setting P r = P t , the transmit power at the user and DF relay that is required to achieve a capacity R is: where i ∈ {N, F}. Near-Field Communication: Lemma 9: Based on the distribution of γ DF,N , we can obtain EE of relay-assisted system for near-field as where m,2 P L N 0 }} and using the expression of P t and P r required to obtain fixed capacity R, the EE can be simplified as: Using again the Chebyshev-Gauss quadrature approximation to solve the above integral, we can obtain the expression of η DF,N . Far-Field Communication: Lemma 10: Based on the distribution of γ DF,F , we can obtain EE of relay-assisted system for far-field as where F F ( D , D , P D,H ) is given by where w D,F = π W D,F , ν p,F = cos( 2p−1 2W D,F π). Proof: The proof follows along the same lines as in the near-field communication. It is based on setting P t and P r as (32) and using the Chebyshev-Gauss quadrature approximation.

B. COMPARISON WITH FULL-DUPLEX RELAY-ASSISTED SYSTEM
Under the same setting as before, we analyze the OP and EE in near-field and far-field scenarios for a full-duplex DF relay-assisted system. For simplicity, we assume perfect relay interference mitigation at each relay. The analysis follows along the same lines as the half-duplex, the only difference being that the rate is twice as the one achieved by the half-duplex and that a certain amount of power is required to mitigate the inter-relay interference. The power consumed by the full-duplex DF relay is given by: P DF,F = ξ −1 (P r + P t ) + P B + P U + NP F , where P F composed of the consumed hardware-dissipated power and the power used for mitigating self-interference at each transmitreceive antenna of full-duplex DF relay. Hence, the EE of the relaying system is η DF, F} is used to either refer to the near-field or the far-field communication. Setting P r = P t , the transmit power at the user and DF relay that is required to achieve a capacity R is: Therefore, the EE of full-duplex DF relaying system in near-field communication and far-field communication is given by the following Lemmas.
Lemma 11: The EE for the full-duplex relay in near-field and far-field can be expressed as and where P D,F = P B + P U + NP F , F = 2R +1 −2 ξ , and F = BR F P D,F . Proof: The EE for the full-duplex relay-assisted system is obtained by replacing P D,H by P D,F , D by F , and D by F in the expressions obtained for the half-duplex relay-assisted system.
Obviously, the EE of full-duplex relay is better than the half-duplex relay because full-duplex relay can achieve the double of the rate that can deliver half-duplex relays while using the same transmit power.

V. ASYMPTOTIC RESULTS
In the following, we will derive approximations for the OP in high SNR regime to provide more insights.

A. RIS-ASSISTED SYSTEM
The incomplete Gamma function in (9) possesses the following Taylor expansion: Based on the approximation r U ≈ d m,2 and plugging (14) to (41), the CDF of γ RIS,N can be approximated as (42) shown at the bottom of the page, where i = R 1 (ϑ i +1)

2
. Similarly, the CDF of SNR γ m,F in the far-field can be approximated as Therefore, the asymptotic expression for F A γ RIS ,F (x) can be derived as where (42) and (44) reveal that their distributions go to constants when γ → ∞, which are given by

Remark 1: A careful investigation of the end-to-end SNR for near-field and far-field communication shown in
and These two terms equal to the probabilities that there are no RIS candidates either in the far-field and the near-field, which means that in the high SNR, the outage occurs when there is no RIS that can be used to assist the transmission. This suggests that the outage performance can be significantly improved when RIS are densely deployed, leading to increased RIS selection diversity. We further note that since the outage probabilities do not decrease asγ increases, the diversity order calculated through DO = − limγ →∞ for the near-field communication and F A γ RIS,F for the far-field communication.

B. RELAY-ASSISTED SYSTEM
The exponential matrix exp( x γ A 1 (d m,1 )) in (21) can be re-presented by power series as exp( Keeping the first two terms and neglecting the higher order terms in high SNR regime, the CDF of γ Um,N achieved by m-th relay conditioned on d m,1 can be given by The CDF of γ mB can be approximated through the same way as Thus, the asymptotic CDF of γ UmB,N = min{γ Um,N , γ mB } conditioning on d m,1 is from which we obtain the asymptotic CDF of γ DF in the near-field as Similarly, the asymptotic CDF of γ UmB,F = min{γ Um,F , γ mB } in the far-field scenario conditioning on d m,1 is given by: from which we obtain the asymptotic CDF of γ DF,F in the far-field setting as: where Remark 2: Similar to the asymptotic results of RIS-assisted system, the end-to-end outage probabilities for near-field and far-field communication shown in (50) and (52) go to constants shown in (45) and (46) whenγ → ∞, while their corresponding diversity orders are zero.
The above derived approximation results in high SNR regime have much simpler forms than the exact ones. The approximation expression does not involve the incomplete Gamma function and the exponential function, which can simplify numerical computation.

VI. EXTENSIONS AND DISCUSSIONS FOR RIS-ASSISTED SYSTEM
In this section, some selected discussions about the performance analysis of the RIS-assisted system under more practical assumptions are provided.

A. EXISTENCE OF THE DIRECT USER-BS LINK
The above performance analysis is under the assumption that the direct link between the user and BS is fully blocked. However, the existence of the direct user-BS link in some scenarios motivates us to analyze the performance of the RISassisted system considering the direct link between the user and BS. Accounting for the direct link between the user and the BS, the instantaneous SNR at BS can be re-expressed as: where α d represents the amplitude of the direct user-BS link and is assumed to follow a Rayleigh distributed random variable, θ d denotes the phase shift of the direct user-BS channel, and η d is its associated path-loss exponent. Therefore, the optimal reflection coefficients can be set as φ m,n = −θ d + θ m,n + ϕ m,n . Under the use of optimal phases, the SNR associated with communication through the m-th RIS is: The expectation and variance of Similar to (9), the CDF of γ D m,N conditioned on d m,1 , d m,2 , and α d can be approximated as Therefore, the CDF of γ D RIS,N based on the approximation r U ≈ d m,2 can be calculated as Similarly, the instantaneous SNR of the RIS-assisted system with direct user-BS link in the far-field communication with optimal phase shifts is given by  One can observe that the performance analysis of RISassisted with the direct user-BS link is much complex than the one without the direct user-BS link. Actually, it is reasonable to ignore the direct user-BS link because RIS provides much higher SNR gain than the SISO direct link, especially when using massive reflecting elements and under high deployment density.

B. MORE COMPLEX CHANNEL MODELS
Our analysis, relying on Gamma approximation, is valid also for more complex channel models including Nakagami-m fading and Rician fading. Indeed, even when other distributions are considered, the sum of the product of channel amplitudes involved in the SNR expression can be approximated by a Gamma distribution using the moment matching method. The single difference with the Rayleigh fading channels is that the computation of the moments will involve the parameters of the considered distributions. For example, assume that α n,m is drawn from a Nakagamim distribution with average value α and Nakagami-m fading parameter m α , and β n,m follows a Nakagami-m distribution with average value β and Nakagami fading parameter m β , the expectation and variance of the quantity and where Then, S N can be approximated through the Gamma distribution with shape parameter and scale parameter obtained by matching the expectation of S N and its variance to those of a Gamma random variable. Based on this Gamma approximation, the performance analysis can be performed in the same way by replacing the parameters of the Gamma distribution by those obtained under the desired channel models.

VII. NUMERICAL RESULTS
In this section, we compare the OP and EE of the RIS assisted system, the half-duplex relaying system, and the full-duplex relaying system based on both simulations and analytical results. It should be noted that the OP for the  full-duplex relaying system is equal to the one for the halfduplex relaying system. Hence, both relaying systems will be referred to as 'DF' in the figures representing the OP. In the figures regarding EE, the half-duplex DF relaying system is denoted by 'HD-DF', while the full-duplex DF relaying system is denoted by 'FD-DF'. Following the parameters adopted in [27], [30], [40], unless otherwise mentioned, the main parameters in our simulations are set as follows: N 0 = −124 dBW, γ th = 1 dB, R 1 = 8 m, R 2 = √ 2R 1 m, r U = 150 m, ξ = 0.5, P B = P U = 100 mW, P b = 5 mW, P H = 10 mW, P F = 20 mW, η 1 = η 2 = η d = 2, f = 3 GHz and L = λ 4 . Prior to carrying out the system performance comparison, we shall first check the accuracy of the Gamma approximation provided in Lemma 1 and Lemma 2. For that, we plot in Fig. 2 the empirical CDF of S N in the case of nearfield communication along with the Gamma approximation presented in (6) for different numbers of reflecting elements. One can see from this figure that the exact CDF matches the Gamma distribution in (6), which validates the proposed  approximation. Similarly, in Fig. 3, we represent the empirical PDF of S F along with its approximation given in (11) in the case of far-field communication. Again, we note the perfect match between both density functions, which supports the validity of the Gamma approximation. To quantify the accuracy of this approximation, we invoke the Kullback-Leibler divergence, defined as where f Ext (x) denotes the exact PDF and f App (x) represents the PDF approximation. We plot in Fig. 4 the Kullback-Leibler divergence metric as a function of the number of reflecting elements N for the far-field communication. In this case f App (x) denotes the Gamma approximation in (11) while f Ext (x) is the empirical PDF of the exact distribution. As can be noted, the value of the Kullback-Leibler divergence is smaller than 10 −3 which again supports the accuracy of the Gamma approximation. Moreover, this accuracy becomes better as the number of reflecting elements N increases, which confirms the usefulness of the Gamma approximation, especially for large reflecting surfaces.
The derived expressions are also based on the Chebyshev-Gauss quadrature approximation, which we shall verify the accuracy. Considering the near-field communication, we plot in Fig. 5 the OP of RIS assisted systems as derived in Lemma 3 versus the Chebyshev-Gauss parameter W R . As can be shown, the accuracy of the Chebyshev-Gauss quadrature approximation becomes even better as the number of reflecting elements increases. It also improves with W R , going to the exact value as W R increases. Moreover, from this experiment, it appears that a tight approximation is obtained for as much as 20 elements in the sum involved in the Chebyshev-Gauss quadrature approximation. Based on that, in all forthcoming simulations, we set the Chebyshev-Gauss parameter to 20.

A. OUTAGE PROBABILITY
Then, we conduct several experiments to compare the outage performance of the RIS-assisted system and the one of the relay-assisted system. In a first experiment, we compare the OP of the RIS-assisted system with that of DF relaying under the near-field scenario. Note that as perfect-inter-relay interference is considered, both full-duplex and half-duplex relaying system correspond to the same OP. As can be seen in Fig. 6, the RIS-assisted system shows better performance than that of the DF relaying system in the low SNR regime and goes to the same OP as the DF relaying system in the high SNR regime. The degradation in performance of the DF relaying system in the low SNR regime is because the transmit power at the relay is not sufficient to compensate for the path-loss in the second hop transmission. Moreover, it is noted that OP can be improved by increasing the deployment density ρ, leading to more RISs/relays candidates from which the best one is selected.
In the second experiment, we compare the OP of the RIS-assisted system with that of DF relaying in the farfield scenario. As shown in Fig. 7, DF relaying outperforms RIS with the same number of transmit antennas, which is opposite to the near-field. Indeed, the DF relaying system with N = 16 transmit antennas achieves 5 − 6 dB gain in transmit power compared to RIS. This can be explained by the fact that the SNR of the first hop achieved by the RISassisted system is low in the far-field scenario due to a higher path loss. However, the relay-assisted system is not affected by this higher path-loss since contrary to the RIS assisted system, the SNR in the second hop is independent of the path loss in the first hop due to the processing carried out at the relay, which consists in re-encoding, and retransmitting the message with the same power used to transmit it [43]. However, the performance gap can be mitigated by increasing the number of reflecting elements at RIS. About a 6 dB power gain can be reaped by adding 16 reflecting elements. As a matter of fact, compared to the DF relaying system when N = 8, the RIS-assisted system achieves much better performance in the low SNR regime when N = 32 and N = 64. This shows that to reduce power consumption while achieving the same or better performance, increasing the number of reflecting elements at RIS is a promising strategy. Finally, it can be noted that the OP for both near-field and far-field saturates in the high SNR regime. This behavior stems from the fact that there is a non-vanishing probability that none RIS or relays are surrounding the user to assist communication.
In the third experiment, we plot the asymptotic OP derived in Section V. As depicted in Fig. 8, the asymptotic OP matches the empirical ones in the high SNR regime. One can also observe a saturation of the OP lines in the high SNR region. The reason for such a behavior lies in that the OP is mainly related to how many RIS/relays are deployed. Since RISs and relays are distributed with the same density, the asymptotic behavior of the OP for both RISs and relays is exactly the same. This phenomenon was predicted by the high SNR asymptotic analysis (see Remark 1 and Remark 2). As can also be seen, when using RISs, the OP in the far-field communication becomes better in the high SNR regime than that in the near-field communication. The reason for that is that the far-field communication is associated with area R 2 2 − R 2 1 , which is higher than that of the near-field communication (R 2 1 ) when R 2 > R 1 . As a result, with more RISs present, the far-field communication benefits from a higher selection diversity, and hence presents better performance in the high SNR regime. In Fig. 9, we investigate the impact of the carrier frequency. As can be seen from (1) and (10), on the one hand, a high carrier frequency has a negative impact on the received SNR. On the other hand, as the carrier frequency increases, the RIS size decreases. As a result, for a fixed total area for the RIS, it is possible to pack more elements when operating at a high frequency than when operating at a low frequency. Consequently, the RIS operating at a high frequency is expected to provide more diversity gain. The question is whether this diversity gain enabled by the possibility of using more elements would allow for compensating the severe path loss attenuation experienced by radio high frequency waves. To answer this question, we have performed some experiments under different carrier frequencies assuming RISs have the same size S. Assuming that the area of each element is L 2 = ( λ 4 ) 2 , the number of elements that can be packed in a surface of size S is N = S L 2 . We represent in Fig. 9 the OP under a fixed total RIS size (S = 1m 2 ) for different operating frequencies. As can be seen, operating over high frequencies improves the performances of both near-field and far-field communications, providing the evidence that the diversity gain offered by high frequencies enables to compensate the incurred high loss attenuation. Moreover, one can see that the outage performance in near-field communication is better than the one in far-field communication. This can be explained by the fact that near-field communication undergoes shorter distances between the transmitter and the receiver than far-field communication.
Lastly, we compare the OP achieved by the RIS-assisted system with the direct user-BS link and the one achieved by RIS-assisted without the direct user-BS link. As shown in Fig. 10, the OP of the direct user-BS link without RIS is much higher than the one of the RIS-assisted system without the direct user-BS link. Moreover, there is a small gap between the OP lines for the RIS-assisted system with the direct user-BS link and the RIS-assisted system without the direct user-BS link. This suggests that the direct link does not bring an important improvement, and as a consequence the direct SISO user-BS link can be reasonably ignored to simplify performance analysis while keeping the same performance trends. Particularly, one can see that the OP achieved in the near-field communication is lower than the OP achieved in the far-field communication, which is the same as the results obtained in Fig. 9. This again confirms the possibility of ignoring the direct user-BS link for SISO RIS assisted systems even in near field communication.

B. ENERGY EFFICIENCY
In this subsection, we analyze the behavior of EE versus a fixed achievable rate R. As can be seen in Fig. 11, the halfduplex DF relaying becomes the worst choice, while the RIS is the best option in the near-field. We can also note that the path loss factor in the second hop plays an important role in system performance, with EEs associated with the three technologies with η 2 = 2.5 being almost half of those with η 2 = 2. It also can be seen that RIS counteract better  the effect of the path-loss: RISs experiencing higher path loss provides much better performance than half-duplex DF relaying with lower path loss.
On the other hand, in the far-field case, as illustrated in Fig. 12 it is the full-duplex relaying that becomes the best choice in the high data rate regime, while RIS is still the best option in the low data rate regime. However, over the whole range of required achievable rates, the maximum EE achieved by RIS is higher than the one achieved by fullduplex relaying. Moreover, it can be noted that increasing the deployment density of RISs/relays plays a positive role in EE, with a higher improvement in the performance for the RIS-assisted system than for the relay-assisted system. As a conclusion, it clearly appears that the RIS-assisted system is the most energy-efficient solution with the same number of receiving and transmitting antennas as relays. The system performance can be greatly improved when RISs are sufficiently densely deployed. Moreover, compared to the far-field case, all technologies achieve higher EE in the nearfield scenario because signals experience more serious path  loss in far-field communication with the same deployment area.
To investigate the impact of the hardware dissipated power, we plot in Fig. 13 the EE with respect toR for various combinations of P B and P U . It can be easily understood that these powers have a negative impact on the EE as they correspond to inevitable powers dissipated by the hardware. As can be easily predicted from theoretical expressions for EE, the EE curve with P B = 200 mW and P U = 100 mW overlaps with that with P B = 100 mW and P U = 200 mW, which shows that the hardware dissipated power consumption at the user and the base station have the same impact.
The good performance of RIS assisted systems in terms of EE compared to relay assisted systems can be explained as follows: 1) The RIS is composed of passive elements that dissipate little power (P b ) as compared to that of the full-duplex DF relay (P F ) and the half-duplex DF relay systems (P H ) 2) It directly reflects the impinging signal and does not require any transmission power contrary to the relay assisted system. Both of these factors contribute to the RIS outperforming relay assisted systems. In all previous experiments, the values of P F , P H , and P b are chosen such that P F > P H > P b , since full-duplex relays require more power than half-duplex relays to mitigate self-interference, and RISs are composed of passive elements that dissipate little power. However, to further understand the role of P b in the performance of RIS assisted systems, we set it in Figs. 14-15 such that P H = P F = P b . Such an assumption is unrealistic, but it allows us to see whether in this case the RIS can still bring an improvement in terms of EE by only dispensing with the transmission power used at relays. As shown in Fig. 14, the RIS-assisted system still outperforms the half-duplex DF relaying system and full-duplex DF relaying system in near-field communication when P H = P F = P b = 5 mW. Different from the near-field communication, the full-duplex DF relaying becomes the best choice in far-field communication with P b = P H = P F , as given in Fig. 15. It indicates that the full-duplex relay requires the lowest total transmit power to achieve the same target rate. Investigating the impact of P F on EE, we can see that the EE for the full-duplex DF relaying system decreases as P F increases. It becomes lower than that of the half-duplex relaying system and the RIS-assisted system (with P H = 5 mW and P b = 5 mW)in the low data rate regime, which agrees with the results in Fig. 12. The maximum value of the EE achieved by the full-duplex DF relaying system is even smaller than the one of the half-duplex DF relaying system when P F is set to 30 mW. Furthermore, the half-duplex relay is always the worst choice compared to the other two technologies in the high data rate regime for both near-field and far-field communication.
Finally, we note that as can be seen from these figures, there is a perfect match between the theoretical results and those obtained by numerical simulations.

VIII. CONCLUSION
This work is among the first studies to make a fair comparison between spatially-distributed RISs and relay-assisted systems in both near-field and far-field communication scenarios. Both technologies have the common goal of forwarding the received signal to the destination with the difference that, unlike relays, RISs are composed of passive elements that consume little amounts of power. As a major outcome, we prove that RISs achieve the best performance in terms of EE, especially when deployed near the transmitter. Moreover, RISs allows for improved system performance in terms of OP and EE when equipped with more reflecting elements or densely deployed. In conclusion, it clearly appears that the use of RIS is a promising technology that allows for a better EE than relay-assisted systems.

APPENDIX A PROOF OF LEMMA 1
Since α m,n and β m,n are independent Rayleigh distributed random variables, the mean value of their product is (71) Therefore, based on the Gamma distribution ( N (d m,1 , d m,2 ), N (d m,1 , d m,2 )), the PDF and CDF of S N conditioned on d m,1 and d m,2 can be obtained in (5) and (6). This completes the proof of Lemma 1.

APPENDIX B PROOF OF LEMMA 3
Based on the approximation r U ≈ d m,2 and applying (14) to (9), the CDF of γ RIS,N can be calculated as In the near-field communication, the expressions of N (d m,1 ) and N (d m,1 ) are complex, making the simplification of the above integral rather hard. As a solution, we invoke the Chebyshev-Gauss quadrature approximation, which is presented as: 2W π) and the weight w = π W . Therefore, through variable substitution, the integration can be re-expressed as Thus, the CDF of γ RIS,N can be finally calculated as (15) through the Chebyshev-Gauss quadrature approximation. This completes the proof of Lemma 3. , the necessary transmit power P t in near-field to achieve a fixed channel capacity R as: Plugging (74) into η, the EE for a RIS-assisted system can be expressed as: where P R = P B + NP b + P U , R = 2R−1 ξ and R = BR R P R . Using the relation E X (x) = ∞ 0 Pr{X > t}dt which holds for X positive random variable, we have whereγ = P L N 0 . Finally, using again the Chebyshev-Gauss quadrature approximation, the EE of RIS-assisted system in near-field can be approximated as (19). This completes the proof of Lemma 5.