Enabling User Grouping and Fixed Power Allocation Scheme for Reconfigurable Intelligent Surfaces-Aided Wireless Systems

This paper considers a two-user downlink transmission in reconfigurable intelligent surface (RIS)-aided network over fading channels. Particularly, by employing user grouping and fixed power allocation scheme to enable non-orthogonal multiple access (NOMA) approach, RIS-aided network can significant benefit from NOMA. To highlight the advantages of RIS, we compare the system performance of NOMA-RIS and traditional orthogonal multiple access (OMA) based RIS. Main practical circumstances are carefully analyzed, such as, RIS with direct link, system without RIS, and imperfect phase shifts. More specifically, we consider a RIS-aided downlink network, where the base station communicates with a group of two users under assistance of RIS, which acts equivalently as relay. As key expectation, the RIS is efficiently designed to improve the performance of users. To evaluate the system performance, two main system performance metrics including outage probability and average capacity are studied by deriving new closed-form expressions. The goal is to find out which system parameters need to be adjusted to achieve the expected performance. The numerical results reveal that: i) the outage probability and average capacity of considered NOMA-RIS aided wireless system outperforms the conventional NOMA network over fading channels; ii) with different power allocation factors assigned to users, the performance gap among two users can be adjusted to guarantee the fairness characteristic; iii) the number of reflecting elements in RIS has significant impact on the system performance of the considered NOMA-RIS system, which shows advantage of both RIS and NOMA compared with conventional OMA system.


I. INTRODUCTION
In recent years, reconfigurable intelligent surface (RIS) is recognized as a promising candidate to achieve high spectrum and energy efficiency in the wireless communications [1]. RIS contains reconfigurable reflecting elements in its planar surfaces which are passive and low-cost. The phase and/or amplitude for the incident signal in each element can be The associate editor coordinating the review of this manuscript and approving it for publication was Prakasam Periasamy . separately adjusted. For the highly efficient deployment of emerging techniques in sixth generation (6G) wireless networks, RIS has received focused attention owing to its significant capability of introducing a smart and controllable signal propagation environment with respect to enhance the spectrum utilization [2], [3], broadband connectivity and radio coverage [4]- [6].
In recent studies [7], [8], the weighted sum rate of all cell-edge users can be greatly enhanced by collaboratively designing the RIS elements in a downlink multicell scenario VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ and precoders at multiple base stations (BSs). The work in [9] explored a downlink multiple-input-single-output (MISO) system with self-sustainable RIS in terms of optimal sum-rate as well as energy harvesting and power consumption (due to reflecting elements' phase shifter) at the RIS. In [10], the authors evaluated the minimal rate of cell-edge users by significantly improving the cooperative beamforming in joint processing coordinated multipoint downlink transmissions. The works [11], [12] studied downlink simultaneous wireless information and power transfer (SWIPT) coupled with transmit beamforming optimization in RIS-enabled network, without negotiating the qualityof-service (QoS) requirements. Besides, these results regarding wireless networks under various settings pointed RIS as a technology with great potentials. Besides, the brand-new RIS technology has also showed its potential in various wireless techniques, such as, non-orthogonal multiple access (NOMA), orthogonal frequency division multiplexing (OFDM), physical layer secure transmission, cognitive radio (CR), and mobile edge computing (MEC) [13]- [19].

A. RELATED WORK
To integrate RISs with existing wireless communications, multiple access techniques should also be considered. Since non-orthogonal multiple access (NOMA) can support multiple users in the same resource block (i.e., time, frequency, and code), therefore it is regarded as a promising technology to be integrated with RIS, forming a new approach referred as NOMA-RIS. To deploy NOMA in wireless networks, the transmitter sends superimposed signals to the receiver, and the receiver adopts a successive interference cancellation (SIC) technique for successful decoding of the signal [20]- [26]. As main benefits reported in [27]- [33], NOMA exhibits advances in terms of user fairness, sum rate, secrecy rate and outage probability, which can achieve considerable performance gains over traditional orthogonal multiple access (OMA). The authors in [34] presented a two-user NOMA group without deteriorating the performance of the NOMA cell-center user. In particular, RIS approach and the joint transmission coordinated multipoint jointly deployed in order to improve the cell-edge user's performance. While joint transmission coordinated multipoint is implemented to eliminate the effects of inter-cell interference (ICI), the RIS is adopted to construct a strong combined channel gain at the cell-edge user. In such framework, a closed-form expression is derived for the ergodic capacity at the cell-edge user, and then the network spectral efficiency can be achieved. The study in [35] conducted NOMA downlink network by using a multi-cluster MISO setup, in which the BS needs RISs to assist the communication while all the destinations by exploiting passive beamforming. The main result is that the optimal transmit power can be obtained by jointly optimizing the reflection coefficient vector at the RISs and the active beamforming matrices at the BS. In [36], a MISO NOMA network was considered to maximize the energy efficiency of the users by designing joint deployment of phase shift and power allocation. The authors in [37] designed MIMO NOMA with a novel passive beamforming weight at RISs for simultaneously serving paired users. A signal cancellation based (SCB) design is adopted and the minimal number of RISs in both reflector and the diffuse scattering scenarios were evaluated. Table 1 summarizes and compares some of the discussed works.
We can point out many advances of such NOMA-RIS systems. Firstly, destination nodes in non-RIS NOMA networks rely on their channel conditions to usually order them. Secondly, the channels associated with RIS and the phase-shift matrix of RIS configuration are coupled with each other in RIS-aided NOMA networks. Thirdly, by intelligently reconfiguring the reflected signal propagation, RIS experiences a new way to improve the performance of current NOMA systems. Fourthly, by adjusting the RIS's phase shifts, the user fairness of RIS-aided NOMA systems can be highly maintained. More importantly, power-domain NOMA technology can be considered as an effort to empower user connectivity and the spectrum efficiency of non-NOMA RIS wireless networks. These prominent advances motivate us to explore system performance analysis with respect to downlink NOMA-RIS systems.

B. OUR CONTRIBUTIONS
Motivated by the aforementioned studies [34]- [36], in this paper, we consider the two main performance metrics normally used in the NOMA-RIS systems to determine the necessary parameters, so that the performance gap among the two users can be adjusted. As mentioned, RIS is able of configuring the channels of the users by intelligently controlling the reflected signal amplitudes and phase shifts, and thus the user decoding order of non-RIS NOMA can be permuted by adjusting the RIS refection parameters to achieve better users' fairness performance in RIS-NOMA based systems. Specifically, theoretical closed-form expressions of outage probability and average capacity that can be used to efficiently configure these parameters so as to adjust the performance gap among the users, and thus increasing the user fairness. These expressions have not been derived yet for the considered downlink RIS-NOMA scenario and thus, in this work we aim to fill some of the gaps and derive closed-form expressions for both outage probability and ergodic capacity that allow to evaluate the impact of some parameters on system performance.
• We assume a RIS-NOMA aided downlink communication system, where the BS communicates with a dedicated group of users. To reduce the complexity of NOMA configuration, we adopt a fixed power allocation scheme and focus on a group of two users 1 since it was reported with acceptable performance [14]. Moreover, we consider system performance gap among the two users by allocating different power levels to the near and the far users.
• We derive the new closed-form expressions of outage probability and average capacity of the considered system. The aim is to show the trade-off between system performance and main parameters, such as, transmit signal to noise ratio (SNR) at the BS, power allocation coefficients, phase shift coefficient and channel gains.
• Three scenarios related to direct link, normal relay and RIS are studied to provide guidelines for implementation of NOMA-RIS in practice. The main results indicate that system benefits from the unique architecture with RIS and with normal relay, even with or without direct links.
In particular, we consider several practical schemes in this study. We address how system parameters should be set to NOMA-RIS system to outperform the counterpart (cooperative NOMA). Although it is still hard to convey how NOMA-RIS cost in real deployment, but the advances obtained from RIS are enough to recommend RIS in real applications. The rest of this paper is organized as follows. Section II presents the NOMA-RIS system model. Section III describes 1 Note that this work can readily be extended for the multiple users scenario. two main system performance metrics including outage probability and average capacity at Scheme I (NOMA with RIS) while Scheme II (NOMA without RIS) is considered as necessary benchmark represented in Section IV. The direct links are considered in term of two main system performance metrics in Section V (Scheme III). Simulation results are presented in Section VI to support and verify the mathematical derivations. We conclude this study by presenting main results in Section VII.

II. SYSTEM MODEL
In this paper, we consider a RIS-enabled NOMA system wherein the base station (S) serves two users, D i (i ∈ {1, 2}), with the assistance of a RIS. We design a RIS with N reflecting metasurfaces and all remaining nodes are facilitated with single antenna. In this circumstance, we assume that the direct-link transmission does not exist due to the blockage.
Then, S transmits the superimposed signal 2 j=1 P j x j to D i . The main parameters can be shown in Table 2. Therefore, the received signal at D i can be expressed as where ε n = n (φ n ) e jφ n is the reflection coefficient produced by the n-th reflector of the RIS, n (φ n ) = 1 is the ideal phase shifts (n = 1, 2, . . . , N ), g n and h ni are the channel gain. Moreover, we have g n = f n e −jθn to the result reported in [41], we assume that RIS has perfect knowledge of θ n and ϕ n . Then, the signal to interference plus VOLUME 9, 2021 noise ratio (SINR) is decode x 2 at D 1 is given by As per the result given [42], we can obtain the maximal SINR by setting φ n = θ n + ϕ n . 2 Then, we can rewrite (2) as Moreover, (3) can be simplified as where η = P S N 0 denotes the average SNR at the base station, Then, by employing SIC procedure, the SINR at D 1 used to decode the own signal x 1 is given as Furthermore, the SINR at D 2 used to decode x 2 is computed by In the next sections, we examine the system's performance for several scenarios under investigation. These expected 2 Although the practical phase shift approach can be considered, we limit our study by ignoring the effect of nonlinear dependency between phase shift and reflection amplitude on the system performance. In particular, we consider the ideal phase shift approach for simplicity. In this regard, the magnitude of reflection factor at each meta-surface is independent of its phase shift, which was widely implemented in [34]- [36].
results are enough to provide guidelines for implementation of RIS in practical environment.

III. SCHEME I: NOMA-RIS SYSTEM WITHOUT THE DIRECT LINK A. OUTAGE PERFORMANCE
In this section, the formulas of outage probability for each NOMA destination, D 1 and D 2 , are derived. To provide sevices with different QoS requirement, the outage events depend partly on the SNR thresholds relative to the target quality of service of each user.

1) CHANNEL DISTRIBUTION
First, let's denote C = C 1 = C 2 and the probability density function (PDF) of C is given as [46] where λ = N π 4 2 , σ 2 = N 1 − π 2 16 and I v (x) denotes the modified Bessel function of the first class with order v.

2) OUTAGE PROBABILITY OF D 1
An outage event occurs at user D 1 when its instantaneous received SINR, given as in (5), falls below the threshold SINR γ 1 . Consequently, the outage probability of D 1 can be given as where γ i = 2 ν i − 1. Then, with the help of (5), it can be rewritten as 92266 VOLUME 9, 2021 Then, by the use of (7), we can express (9) as By using the fact that [48,Eq.8.445], v+2k , we can express the outage probability of D 1 as Finally, using [48, Eq.3.381.3], OP I D 1 can be evaluated as We can express the outage probability of D 2 as Then, the closed-form outage probability of D 2 can be obtained as where P out,1 can be expressed as where ϑ = P 2 − γ 2 P 1 . Next, it can be calculated as With the help (7), we can obtain P out,1 as Similarly, P out,2 can be expressed as Remark 1: Firstly, due to main results reported in (12), (18) containγ D 1 ,γ D 2 which depend on η, such outage probabilities depend on the average SNR at the base station η. We expect to look the trends of outage probability as varying value of η in the section of numerical simulation. Further, the closed-form expressions of outage probability derived in (12), (18) point out that they are an increasing function with respect to the number of meta-surface N at high SNR regime. Such finding is important result to improve performance of cell-edge users in cellular network since RIS provide higher gain of order N . It can be explained that the RIS can attain the inherent aperture gain of order N by collecting more signal power.
Based on explicit results from computations of outage probability, another metric should be considered, i.e., average capacity. We expect to compare system performance of two NOMA users under the context of RIS deployment. We first examine closed-form expressions of average capacity in the next section below.

B. AVERAGE CHANNEL CAPACITY
This section presents the analytical framework that quantifies the average capacity of NOMA-RIS systems, which to the best of authors' knowledge is yet to be studied in the literature. In this sense, we initially present novel a expression for cumulative distribution function (CDF), F γ x 1 D 1 , which is computed based on the end-to-end SINR of the NOMA-RIS system. By this regard, F γ x 1 D 1 is PDF which need be examined. To this end, we try to find the closed-form formulas of the average capacity which can be applied to the two cases, i.e. NOMA system with and without RIS.
The average channel capacity of D 1 can be evaluated as Proposition 1: The closed-form expression of average capacity R I D 1 can be expressed as Proof: See Appendix for the detailed analysis. The average channel capacity of D 2 is given as

VOLUME 9, 2021
Proposition 2: The closed-form expression to point out average capacity of user D 2 , i.e.n R I D 2 is expressed by Proof: The CDF of γ x 2 D 2 can be expressed by . (23) Substituting (23) into (21), we have Now, using Gaussian-Chebyshev property with ζ m = cos 2m−1 2M π and solving the integral, we can obtain (22). This completes the proof. Remark 2: Since main results reported in (20), (22) contain common term which depends on the number of meta-surface N , we expect that the average capacity performance relies mainly on values of N . Therefore, it is likely important to enhance capacity performance by increasing meta-surface in the RIS. However, two users exhibit different SINRs, and hence performance gap among two users exist once we increase η to look how the average performance of two user can improve. This finding lead to further confirmation by comparing RIS system with traditional relaying system.

IV. THE BENCHMARK-SCHEME II: RELAY SYSTEM
In this scheme, we replace a RIS by a normal relay (denoted as R). 3 For the relaying system with the Decode-and-Forward (DF) protocol, the SINR of R when detected x 1 and x 2 in the 3 The total power dissipated to operate the considered NOMA-RIS system or NOMA relaying system is composed of the base station transmit power, the hardware static power consumed in the base station, mobile users, and RIS or relay. Since RIS's reflectors are passive elements that do not directly alter the magnitude of the incoming signal, the RIS does not require any transmit power. However, RIS provides amplification gain by achieving a suitable adjustment of the phase shifts of the reflecting elements with a phase coherence. In contrast, a relay need separated power for its operation. As a result, NOMA relaying system has significant difference compared to NOMA-RIS architectures, which points out the important benefit of RIS. first phase are given, respectively, as [27], [28] γ II R,x 2 = ηP 2 |g| 2 and In the second phase, the SINR at D 1 when detected x 2 and then the SINR achieved by employing SIC to detect x 1 are given, respectively, as and Moreover, the SINR at D 2 to detect x 2 is given as A. OUTAGE PERFORMANCE The outage probability of D i can be given as where γ II i = 2 2ν i − 1. The closed-form expression of the outage probability for user D 1 is calculated as Similarly, the closed-form expression of the outage probability for user D 2 can be obtained as B. AVERAGE CHANNEL CAPACITY expression of the average channel capacity of D 1 by where Next, the average channel capacity of D 2 is expressed as Similarly, the closed-form expression of (35) can be obtained as

V. SCHEME III: NOMA-RIS SYSTEM WITH THE DIRECT LINK
For the scenario, when the direct links between S and D i are present, the received signal at D i is given as where . Then, the SINR to decode x 2 at D 1 is given by To simpler manipulation, (38) can be rewritten as where Next, the SINR at D 1 used to decode its own signal x 1 is formulated by Considering signal at D 2 , the SINR used to decode x 2 is expressed as We continue to examine the system performance metrics as the next subsections.

A. OUTAGE PERFORMANCE 1) CHANNEL STATISTICS
By employing results reported in [43], the CDF of U i can be obtained as where a and b are calculated by The outage probability of D 1 can be computed as Then, the outage probability of D 2 can be expressed as As further steps, P out,1 is given as .
Similarly, P out,2 can be formulated as .

VI. NUMERICAL RESULTS
In this section, we evaluate several scenarios with a set of the parameters, shown in Table 3, which are similar to [4], [45], except for some specific cases. In the following figures, Monte-Carlo simulations are performed to validate the analytical results. In these figures, we denote ''Simulation'' for the simulated curves relying on Monte-carlo and ''Analytical'' for analytical ones obtained with the expressions derived in the previous sections.

FIGURE 2.
Outage probability of D 1 versus η in two schemes with different target rates ν 1 . Fig. 2 compares the outage probability versus the transmit SNR at the base station for two schemes by considering target rate as ν 1 = 0.5 and ν 1 = 1. By increasing transmit SNR η, one can achieve better outage performance, especially such outage behavior can be improved significantly at very high SNR regime, i.e. when η = 50 (dB), the outage probability of the first user in NOMA-RIS system is around 10 −3 . It can be clearly seen that SINRs to detect signal for both users depend on the thresholds, while such threshold is decided by ν 1 = 0.5. As a result, higher requirement of target rate leads to worse outage performance. Obviously, by achieving advances of RIS, NOMA-RIS exhibits better performance compare with NOMA scheme using relay. Performance gap can be observed clearly when we compare performance of NOMA with and without RIS once η belongs to the range from 20 (dB) to 45 (dB). Besides, the Monte-Carlo based simulation results are matched very well with the analytical results, which confirms the exactness of our derived expressions.
We make another comparison of the outage probability performance when adopting the considered NOMA-RIS with the varying number of RIS reflecting elements N , as shown in Fig. 3. The higher value of N leads to significant improvement in term of outage probability performance at high SNR region. For example, the outage performance in the NOMA-RIS system can be reduced significantly for N = 20. When the number of reflecting elements N increases, this advantage of RIS to the NOMA-RIS system is further  enlarged. In addition, with the increase of N from 20 to 40, the advantages of the NOMA-RIS system are more significant.
Similarly, the outage probability of the second user D 2 also reduces at high SNR as presented in Fig. 4. For instance, at an average transmit SNR of 50 dB, the outage probability for the case of ν 2 = 0.5 equals to 10 −3 . When we compare two cases of target rate ν 2 , the performance gap at NOMA-RIS is smaller than that of NOMA relying on relay. Although decoding procedures and conditions for computing outage probability at user D 2 is different compared to user D 1 , but the similar trends of such outage behavior can be seen in Fig. 2,  Fig. 3 and Fig. 4. This is because the expressions of outage probability depend mainly on the transmit SNR η, and hence same trends are concluded for both users.
As can be seen from Fig. 5, the number of reflecting meta-surface N contributes to improvement of outage probability for the second user, similar case as seen in Fig. 3. Since main equations of outage probability contain SINRs represented in (2), (4) and (6) which then contain  channel coefficients improved by increasing number of meta-surface N , and hence the outage probability would be better at higher value of N .
To compare outage performance of two users under the impact of the number of meta-surface N , Fig. 6 demonstrates the outage probability versus the transmit SNR for the NOMA-RIS and orthogonal multiple access (OMA) enabled RIS systems. It is seen clearly that RIS system relying NOMA scheme shows better outage probability compared with RIS using OMA. As can be seen from Fig. 6, the second user D 2 exhibits its superiority compared with that of user D 1 in terms of outage performance. It can be explained that different power coefficients P 1 , P 2 allocated two users make influence to expressions of SINRs, and then the corresponding outage probability. In addition, conditions of decoding signals at two users are different, and hence values of outage probability of two users are dissimilar. In cases of η = 40dB and N = 10, OP I D1 = 0.0002, OP I D2 = 0.0012 and OMA equal 0.004. When we compare our study with OMA, the improvement of OP I D1 and OP I D2 are respectively 0.38% and 0.28%.  In Fig. 7, by increasing the number of metasurfaces N from 2 to 14, the outage behavior can be improved significantly, especially for two cases, η = 30dB, and η = 40dB. The performance gap among two users is very small for case of η = 40dB regardless of any values of N . It can be explained by intelligent adjustment of phase shifts by enabling RIS, so that the SINRs at destinations can be maximized, and hence the outage probability can be improved. Therefore, the role of RIS design is crucial to the main task of enhancing performance at destinations.
To extend our consideration on outage performance, we compare three schemes in Fig. 8. The main finding in Fig. 8    99.93%, 67.48% and 98.65%. Similarly, we can easy obtain the improvement of D 2 in Scheme I. In Fig. 9, we consider how imperfect phase shift make influence to RIS-aided systems at two destinations D 1 and D 2 . It can be intuitively seen that the performance loss might occur since the ideal and practical phase shifts are emphasized as factor affecting the outage performance. From [44], the practical phase shifts lead to the performance loss by considering n (ϕ n ) = (1 − ω min ) sin(ϕ n −κ)+1 2 ς +ω min , where ω min represents for the minimum amplitude, κ is denoted as the horizontal distance between π 2 and ω min , ς is the steepness of function curve. In this circumstance, our setup parameters are ω min = 0.8, κ = 0.43π and ς = 1.6. It can be concluded that a certain performance loss will be raised when we compare the considered system relying on the practical and ideal phase shifts.
In view of the average capacity performance in Fig. 10, the superiority of the first user D 1 over the second user D 2 becomes more significant as η increases. This is because of the reason that (5) mainly depends on η, and hence the average capacity. Similar trends as outage probability presented in Fig. 2-Fig. 6, the average capacity has main affection from the number of meta-surface N . In three cases of N , the average capacity values of user D 2 are same when we consider them at the point η is greater than 45 (dB). We can explain that average rates depend on SINRs, while (5), (6) are SINRs which indicate difference for detecting signals x 1 , x 2 , respectively. In (5), SINR is linear increase of C 1 which relies on SNR η, while SINR in (6) contains C 2 in both numerator and denominator, and hence average rate is limited at high η region. In the contrast, considering performance among three cases of N , the average capacity values of user D 1 show larger gaps.
It is worth noting that NOMA-RIS shows its advantage once we compare it with NOMA using traditional relay, shown in Fig. 11. The superiority of the NOMA-RIS can be concluded for two users, which shows the impact of architecture of RIS to improve the average capacity at both users. As a result, the proposed protocol outperforms the NOMA with relaying in terms of outage probability and average capacity. It is worth noting that the NOMA-RIS shows significant improvement when transmit SNR η is greater 40 (dB).

VII. CONCLUSION
In this paper, the benefits of joint NOMA and RIS techniques have been investigated. The base station enables fixed power allocation scheme to send superimposed signals to serve a dedicated group of two users. It shows significant benefit when we adopt RIS and hence the improvement can be shown by comparing NOMA system with and without RIS. Especially, we consider various practical circumstances, such as, direct link, system without RIS, imperfect phase shifts. For such setup, we derived the analytical closed-form formulas of the outage probability and average capacity, which are useful to understand how the system/channel parameters affecting to the system's performance. The main results showed that the NOMA RIS outperforms OMA RIS, which explicitly confirms advances of RIS in terms of outage performance and average capacity. Our analytical and simulation results reveal that the higher number of the reflective elements at the RIS can be intelligently controlled to boost the outage probability and average capacity significantly at high SNR regime.

APPENDIX
To compute the closed-form expression of average capacity, we first calculate the CDF of γ x 1 D 1 , and it can be obtained as Substituting (49) into (19), we can have Using [47, Eq. 5], we can write Next, (49) can be calculated as can be obtained as where he is also a Senior Researcher at the Instituto de Telecomunicações. He has participated in several national and European projects. He has led several research projects in broadband wireless communications at the national level. His interests include multicarrier-based systems, cooperative networks, precoding, multiuser detection, and massive MIMO and millimeter wave communications. Within these research topics, he has published more than 150 technical articles in international journals and conference proceedings. He served as a member of the TPC at several international conferences. He is currently an Associate Editor of IEEE ACCESS and IET Signal Processing. He has numerous publications in peer-reviewed journals and conferences. His current research interests include wireless relaying techniques, cooperative communications, cognitive relaying networks, device-to-device communications, reconfigurable intelligent surfaces, signal processing, physical layer security, and MIMO systems. He also served as a TPC member, the session chair, the program co-chairs, and a reviewer for various national and international conferences. He is serving as a Reviewer in a number of international journals, including the IEEE TRANSACTIONS ON