I Introduction
Reconfigurable intelligent surface (RIS) has been regarded as an emerging costeffective technology for future wireless communication systems. These artificial surfaces are composed of reconfigurable electromagnetic materials that can be controlled and programmed by integrated electronic devices, providing potential gains in terms of spectrum and energy efficiencies [1]. By comparing with traditional relaying technology, RISs do not need high hardware complexity and cost overhead during their operation. In addition, RISs are able to customize the wireless environment through the use of nearlypassive reflecting elements, enabling the system designers to fully control the electromagnetic response of the impinging signals into the environmental objects [2]. Therefore, due to their unique properties, RISs arise as one of the key technologies to realize the futuristic concept of smart radio environment.
Very recently, RISs have been extensively studied in the literature. Specifically, the authors in [3] formulated a beamforming optimization problem of RISaided wireless communication systems under discrete phase shift constraints. In [4], the authors applied RISs in downlink multiuser communications. The authors in [5] proposed a deep learning method for deploying RISs in an indoor environment. In [6], the authors quantitatively analyzed the coverage of RISassisted communication systems through an outage probability (OP) analysis. The authors in [7] conducted a performance analysis of the application of RISs in mixed freespace optical (FSO) and radiofrequency (RF) dualhop communication systems. In [8], it was shown that the application of RISs can effectively improve the coverage and reliability of unmanned aerial vehicles (UAV) communication systems. In [9], the use of RISs for improving physical layer security was examined. On the other hand, along the years, the study of capacity scaling law of wireless communication systems has been of paramount importance. For example, in [10], it was studied the capacity of multiuser multiantenna relay networks with cochannel interference, while the authors in [11] analyzed the scaling rates for the OP and average sumrate assuming a single RISassisted communication system.
Although the aforementioned works have provided interesting contributions, the research field on RIS is still at its infancy. This paper aims to fill out an important gap which exists in the literature, which is the investigation of multiple RISs in dualhop cooperative networks. To the best of the authors’ knowledge, such kind of system setup has not been investigated in the literature yet. Specifically, assuming a multiple RISaided system setup where the RIS with the highest instantaneous signaltonoise ratio (SNR) is selected to assist the communication, an exact analysis for the OP is developed. In addition, relying on the extreme value theory, closedform expressions for the asymptotic OP and asymptotic sumrate are derived, based on which new capacity scaling laws are established. Our results are corroborated through simulations and insightful discussions are provided. In particular, our analysis shows that the number of RISs as well as the number of reflecting elements play a crucial role in capacity scaling law of multiple RISaided cooperative networks. Also, comparisons with relayaided systems are carried out to show that the proposed system setup outperforms relaying schemes both in terms of the OP and average sumrate.
The remainder of this paper is structured as follows. In Section II, the system and channel models are introduced. Section III carries out a detailed performance analysis in terms of the OP and average sumrate, while Section IV presents illustrative numerical results which are followed by insightful discussions and corroborated through simulations. Finally, Section V concludes the paper.
Ii System and Channel Models
As shown in Fig. 1, we consider a wireless communication system consisted of one source (S), one destination (D), and RISs, where each RIS is composed by reflecting elements. Similar to [1], it is assumed full channel state information (CSI). The communication process is briefly described next. Firstly, S sends the signal to the th RIS ()^{1}^{1}1
As will be shown later, we consider that the RIS which provides the highest instantaneous endtoend SNR is selected to assist the communication. However, at this moment, assume that the communication is carried out through the
th RIS. and then it passively reflects the signal to D^{2}^{2}2It is worth noting that RISs work similar to relaying systems, however their working principles are rather different and more detailed explanations can be found in [12].. With this aim, the th RIS (denoted, for simplicity, by ) optimizes the phase reflection coefficient to maximize the received SNR at D, therefore improving the endtoend quality of the communication system. The channels are assumed to undergo independent Rayleigh fading. Thus, the signal received at D can be expressed as(1) 
where denotes the average transmitted energy per symbol, is the transmitted signal, stands for the additive white Gaussian noise (AWGN), represents the adjustable phase produced by the th reflecting element of the , and are the channel gains of the S and D links, respectively, where and denote the distances of the S and D links, respectively, and denotes the path loss coefficient. In addition, and
represent the respective channels’ amplitudes, which are independent distributed Rayleigh random variables (RVs) with mean
and variance
, and and refer to the phases of the respective fading channel gains. As considered in previous works, we assume that has perfect knowledge of and . From (1), the instantaneous endtoend SNR at D can be expressed as(2) 
From [1], in order to maximize , can be smartly configured to fully eliminate the phase shifts by setting . Therefore, the maximized can be written as
(3) 
where and stands for the average SNR.
Iii Performance Analysis
In this section, we analyze the OP and average sumrate of the considered system setup. Along the analytical derivations, it will be considered that one out of RIS is selected to aid the communication. Specifically, the choice of the suitable RIS is performed to maximize the received signal at the destination. Therefore, the selection principle of the RIS can be expressed as
(4) 
where is the average channel SNR measured by in the past window of length . Note that is introduced in the denominator to maintain the longterm fairness. Next, similar to [13] ,we focus on the investigation of smallscale channel fading. Thus, the selection principle in (4) simplifies to
(5) 
In what follows, the analysis is developed assuming a clustered configuration for the deployment of the RISs, which implies that the endtoend links undergo independent identically distributed (i.i.d.) Rayleigh fading^{3}^{3}3Nonclustered configuration analysis arises as an interesting study for future works. Therefore, the results obtained in this paper can serve as a benchmark for these new investigations..
Iiia Outage Probability Analysis
IiiA1 Exact Analysis
Based on the clustered deployment of the RISs, it follows that , which yields . Let
, then the probability density function (PDF) of
can be readily obtained as , where is the modified Bessel function of the second kind with zero order [14]. According to [15], one can attest that the PDF of is a special case of the distribution. In [15], the authors stated that the PDF of the sum of multiple RVs can be wellapproximated by the PDF of with , in which the PDF of is approximated by a squared distribution. Therefore, the PDF of can be represented by a squared distribution. Thus, the PDF of can be written as [16](6) 
where and are the shaping parameters, denotes the gamma function, is the mean power, and .
Relying on the idea presented in [17], we make use of the mixed gamma (MG) distribution to rewrite the PDF given in (6) as
(7) 
where denotes the number of terms of the sum, , , , and , with and
representing, respectively, the weight factor and the abscissas of the GaussianLaguerre integration [18]. From probability theory, the cumulative distribution function (CDF) of
can be derived as(8) 
where represents the lower incomplete gamma function [14].
According to the order statistics theory, the CDF of , which is given in (5), can be formulated as
(9) 
which yields the following PDF
(10) 
From [19], the OP can be defined as the probability that the effective received SNR is less than a given threshold , which is mathematically written as . Thus, by replacing (8) into (9), the system OP can be expressed as
(11) 
IiiA2 Asymptotic Analysis
According to [14], can be rewritten in the form . Based on this identity, can be asymptotically expressed as
(12) 
At SNR regime, one can ignore the high order term . Thus, an asymptotic outage expression for (11) can be expressed as
(13) 
The above expression indicates that the achievable diversity order of the proposed system setup is , which can also be written as since equals to , and this latter is determined by .
IiiB Asymptotic SumRate Analysis
To analyze the asymptotic sumrate, we depart from the idea presented in [20, Lemma 2]. More specifically, let , …, i.i.d. RVs with a common CDF and PDF , satisfying the property that is less than one for all finite and is twice differentiable for all , which implies that
(14) 
for some constant . Then, the expression converges in distribution to a limiting RV with CFD given by . It is worth noting that the CDF of is given by .
The above result allows to say that the maximum of i.i.d. RVs grows like . Therefore, in the sequel we derive the asymptotic sumrate assuming a high number of RISs, i.e., as . Due to the intricacy in carrying out such kind of analysis departing from the MG distribution presented in the previous section, here, for sake of tractability, the CDF of
will be written by using the noncentral chisquare distribution [6], i.e.,
(15) 
where denotes the Marcum function, , and . At high SNR regime, based on [22], the Marcum function can be asymptotically expressed as
(16) 
where stands for the Chernoff parameter . By replacing (16) into (15), the CDF of can be asymptotically expressed as
(17) 
and its corresponding PDF is
(18) 
Then, we can show that
(19) 
Also, by solving , it follows that
(20)  
Therefore, for a large number of RISs, the maximum SNR grows as in (20), which is the function of and for fixed . Accordingly, the asymptotic sumrate can be approximated by
(21)  
The above expression indicates that and play a significant role in increasing the sumrate.
Iv Numerical Results and Discussions
In this section, illustrative numerical examples are presented to verify the impact of the key system parameters on the overall performance. Our analysis is corroborated through Monte Carlo simulations, in which simulation points are generated. Moreover, comparisons with relayaided systems are carried out to demonstrate that the proposed system setup outperforms relaying schemes both in terms of the OP and average sumrate.
In Fig. 2, the OP of RISaided systems and relaying systems is plotted assuming , , and . From this figure, one can see that RISaided systems have better performance than relaying ones. Moreover, the system performance improves as increases, in which the slope of the curves changes according to
, corroborating the presented asymptotic analysis. In Fig. 3, the OP of RISaided systems for different values is plotted for different values of
and . It can be clearly seen that increasing the value of can significantly improve the system performance. In both figures, one can observe that the analytical results match perfectly with the simulation results. Also, at high SNRs, the asymptotic results are close to the exact values.Fig. 4 depicts the OP for different combinations of and , but keeping the product with the same value. It can be clearly seen that the four curves have the same slopes, which verifies the correctness of the derived diversity order, i.e., . Furthermore, the slight change of and can lead to an improved system performance, however such performance is mostly dependent on the value of .
In Fig. 5, the asymptotic sumrate is plotted for different values of and assuming . From this figure, one can attest that has a great impact on the system performance. Our results are also compared with the relay schemes (assuming the same number as considered for RISaided case, i.e., ) to show the performance gain of the proposed system setup. In Fig. 6, the average sumrate versus is depicted by setting and . It can be clearly seen that the asymptotic curves well reflect the scaling law of the considered RISaided system, although they do not coincide with simulation results. However, the asymptotic value becomes tighter to the simulation values when increases. Finally, it can be observed that the proposed system setup is significantly better than the relay schemes in terms of average sumrate.
V Conclusion
In this work, we investigated dualhop cooperative network assisted by multiple RISs. Assuming that one of out RISs (which one having elements) is selected to add the communication process, an exact analysis for the OP was presented and closedform expressions for the asymptotic OP and asymptotic sumrate were derived, based on which the capacity scaling law was determined. Our results showed that the achievable diversity order of the considered system equals to . In addition, we have compared the proposed scheme with a DF relay scenario and it is shown that the former one outperforms considerably the latter one both in terms of OP and average sumrate. Our results are not only novel but can be used as a benchmark for future studies. Potential new works include nonclustered configuration for the RISs’ deployment as well as the proposal of other techniques of RIS selection.
References
 [1] E. Basar, M. Di Renzo, J. De Rosny, M. Debbah, M. S. Alouini and R. Zhang, “Wireless communications through reconfigurable intelligent surfaces,” IEEE Access, vol. 7, pp. 116753116773, Aug. 2019.
 [2] C. Liaskos, S. Nie, A. Tsioliaridou, A. Pitsillides, S. Ioannidis, and I. Akyildiz, “A new wireless communication paradigm through software controlled metasurfaces,” IEEE Commun. Mag., vol. 56, no. 9, pp. 162 169, Sep. 2018.
 [3] Q. Wu and R. Zhan, “Beamforming optimization for wireless network aided by intelligent reflecting surface with discrete phase shifts,” IEEE Trans. Wireless Commun., vol. 68, no. 3, pp. 18381851, Mar. 2020.
 [4] C. Huang, A. Zappone, G. C. Alexandropoulos, M. Debbah and C. Yuen, “Reconfigurable intelligent surfaces for energy efficiency in wireless communication,” IEEE Trans. Wireless Commun., vol. 18, no. 8, pp. 41574170, Aug. 2019.
 [5] C. Huang, G. C. Alexandropoulos, C. Yuen, and M. Debbah, “Indoor signal focusing with deep learning designed reconfigurable intelligent surfaces,” in Proc. IEEE IWSPAWC, Cannes, France, 2019, pp. 15.
 [6] L. Yang, Y. Yang, M. O. Hasna, and M. Alouini, “Coverage, prob ability of SNR gain, and DOR analysis of RISaided communication systems,” IEEE Wireless Commun. Lett., Early Access, DOI: 10.1109/L WC.2020.2987798.
 [7] L. Yang, W. Guo, and I. S. Ansari, “Mixed dualhop FSORF communi cation systems through reconfigurable intelligent surface,” IEEE Commun. Lett., vol. 24, no. 7, pp. 15581562, Jul. 2020,
 [8] L. Yang, F. Meng, J. Zhang, M. O. Hasna, and M. Di Renzo, “On the performance of RISassisted dualhop UAV communication systems,” IEEE Trans. Veh. Technol., Early Access, DOI:10.1109/TVT.2020.3004598.
 [9] L. Yang, J. Yang, W. Xie, M. O. Hasna, T. Tsiftsis, M. Di Renzo, “Secrecy performance analysis of RISaided wireless communication systems,” IEEE Trans. Veh. Technol., Early Access, DOI: 10.1109/TVT.2020.3007521.
 [10] I. Trigui, S. Affes and A. St phenne, “Capacity scaling laws in interferencelimited multipleantenna AF relay networks with user scheduling,” IEEE Trans. Commun., vol. 64, no. 8, pp. 32843295, Aug. 2016.
 [11] I. Trigui, W. Ajib and W. Zhu, “A comprehensive study of reconfigurable intelligent surfaces in generalized fading,” [Online]. Available: https://arxiv.org/abs/2004.02922.
 [12] K. Ntontin, M. Di Renzo, J. Song, F. Lazarakis, J. D. Rosny, D. T. Phan Huy, O. Simeone, R. Zhang, M. Debbah, G. Lerosey, M. Fink, S. Tretyakov, S. Shamai, “Reconfigurable intelligent surfaces vs. relaying: Differences, similarities, and performance comparison,” [Online]. Available: https://arxiv.org/abs/1908.08747.
 [13] C. Chen and L. Wang, “A unified capacity analysis for wireless systems with joint multiuser scheduling and antenna diversity in Nakagami fading channels,” IEEE Trans. Commun., vol. 54, no. 3, pp. 469478, Mar. 2006.
 [14] I. S. Gradshteyn, and I. M. Ryzhik, Table of integrals, series, and products, 7th ed. San Diego, CA, USA: Academic, 2007.
 [15] K. P. Peppas, “Accurate closedform approximations to generalised sum distributions and applications in the performance analysis of equal gain combining receivers,” IET Commun., vol.5, no.7, pp. 982989, May 2011.
 [16] L. Yang, F. Meng, Q. Wu, D. B. da Costa and M. Alouini, “Accurate closedform approximations to channel distributions of RISaided wireless systems,” IEEE Wireless Commun. Lett., Early Access, DOI:10.1109/LWC.2020.3010512.

[17]
S. Atapattu, C. Tellambura, and H. Jiang, “A mixture gamma distribution to model the SNR of wireless channels,”
IEEE Trans. Wireless Commun., vol. 10, no. 12, pp. 41934203, Dec. 2011.  [18] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions: With formulas, graphs, mathematical tables. New York, USA: Dover, 1965.
 [19] M. K. Simon and M.S. Alouini, Digital communication over fading channels: A unified approach to performance analysis, 1st Edition. John Wiley, 2000.
 [20] P. Viswanath, D. N. C. Tse, and R. Laroia, “Opportunistic beamforming using dumb antennas,” IEEE Trans. Inform. Theo., vol. 48, pp. 12771294, Jun. 2002.
 [21] V. M. Kapinas, S. K. Mihos, and G. K. Karagiannidis, “On the monotonicity of the meneralized marcum and nuttall Qfunctions,” IEEE Trans. Inform. Theo., vol. 55, no. 8, pp. 37013710, Aug. 2009.
 [22] M. K. Simon and M. S. Alouini, “Exponentialtype bounds on the generalized marcum Qfunction with application to error probability analysis over fading channels,” IEEE Trans. Commun., vol. 48, no. 3, pp. 359366, Mar. 2000.
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