I Introduction
The demand for nextgeneration (NG) networks having high energy efficiency (EE) has been rapidly increasing [5G_NR]. A variety of sophisticated wireless technologies have been proposed for NG networks, including massive multipleinput multipleoutput (MIMO) and millimeter wave (mmWave) communications [5G_NR_2]. Recently, costefficient reconfigurable intelligent surfaces (RISs) have been proposed for cooperative NG networks [LIS_zhangjiayi_mag, LIS_smart, RIS_mag_basar].
To enhance both the spectrum efficiency (SE) and EE of NG networks, nonorthogonal multiple access (NOMA) has been proposed as a promising technique of opportunistically capitalizing on the users’ specific channel state information (CSI) differences [NOMA_mag_Ding, PairingDING2016, Massive_NOMA_Cellular_IoT]. NOMA networks are capable of serving multiple users at different qualityofservice (QoS) requirements in the same time/frequency/code resource block [NOMA_5G_beyond_Liu, Islam_NOMA_survey, NOMA_large_heter]. Hence, inspired by the aforementioned benefits of NOMA and RIS techniques, we explore the network’s performance enhanced by the intrinsic integration of the powerdomain NOMA^{1}^{1}1In this article, we use NOMA to refer to powerdomain NOMA for simplicity. and RIS techniques in the downlink (DL).
Ia Prior Work
In recent years, RIS based techniques have received considerable attention owing to their beneficial applications [LIS_magazine_multi_scenarios, reconfig_meta_surf_1, reconfig_meta_surf_2]. The RIS aided system comprises an array of intelligent surface units, each of which can independently absorb energy and shift the phase of the incident signal. By appropriately adjusting the reflection angles and amplitude coefficients of RIS elements, the electromagnetic signal can be reconfigured for wireless transmission. The performance of RISaided and relayassisted networks was compared in [LIS_compare_relay], indicating that RISaided networks may have better network performances, provided that the number of RISs is high enough. The associated energy consumption model was proposed in [glob_energy_model, energy_model_LIS], where the EE of the proposed network was optimized. Numerous application scenarios, such as RISs aided physical layer security relying on cooperative jamming techniques have also been considered [Renzo_PHY_security_confe, PLS_LIS_ZhangRui]. The RIS components are capable of blocking the signal of eavesdroppers, hence enhancing the secrecy performance. RIS assisted simultaneous wireless information and power transfer (SWIPT) was proposed in [Swipt_LIS_ZhangRui] for the users located in coverageholes. In the 5G new radio (NR) standard, the coverage area is significantly reduced for carriers beyond 6GHz [Renzo_mmwave_signal_enhancement]. Hence, a sophisticated signal alignment strategy was employed at the RISs for coverage area enhancement in mmWave scenarios [Lv_coverage_enhancement]. However, in most previous research, continuous amplitude coefficients and phase shifts were assumed at the RISs [Zhou_MISO_multi_cluster], whilst in practice the phase shifts of RISs may not be continuous. Thus discrete phase shifts were considered in [ZhangRui_MISO_beams_discrete_2] for a multipleinput singleoutput (MISO) assisted RIS network. The channel capacity of a RISaided MIMO network was maximized, where both analog and digital beamforming, as well as hybrid beamforming were considered [shuowen_RIS]. Furthermore, focusing on the user’s fairness, a fairnessoriented design (FOD) was proposed in a RISaided MIMO network [Hou_RIS_MIMO_global_algrithm].
To further enhance both the SE and EE of the DL, NOMA and RIS techniques were integrated in [DING_RIS_NOMA_letter]. The RISs can be deployed for enhancing the power level of the celledge users, where the cellcenter users treat the reflected signal as interference [DING_RIS_NOMA_letter]. Both continuous and discrete phase shifters were used in a RISaided MISO NOMA network [yuanwei_NOMA_RIS]. Naturally, the BSuser link plays a key role [MISO_with_directlink]. A RISaided NOMA network was also investigated in [NOMA_RIS_Fu], whilst the BSuser link and the BSRIS link, as well as the RISuser link were assumed to experience Rayleigh fading. The associated bit error ratio (BER) was evaluated in the case of Rayleigh fading in [LIS_perform_Anal]. However, both the BS and RISs are part of the infrastructure, and the RISs are typically positioned for exploiting the lineofsight (LoS) path with respect to the fixed BS in NG networks for increasing the received signal power. Hence, the impact of fading environments on RIS networks has also attracted attention [RIS_NOMA_Rice]. A fairnessoriented algorithm was proposed in a RISaided NOMA network [RIS_NOMA_Rice], where Rician fading channels were used for modelling the channel gains. Note that when the Nakagami and Rice fading parameter obey the following constraint , these fading channels are identical [wireless_communication_goldsmith, eq. (3.38)].
IB Motivations and Contributions
The abovementioned papers mainly studied the network’s fairness, whilst there is a paucity of investigations on the SE improvement of NOMA networks. To comprehensively analyze the network’s performance enhanced by RISs, a RISaided SISONOMA network is proposed. Motivated by the potential joint benefits of RISs and NOMA networks, whilst relying on analog beamforming [Yi_anlog_beam], in this article we will analyse the performance of a RISaided NOMA DL scenario, where a priorityoriented design (POD) is proposed, which is also capable of enhancing the SE. In the proposed POD, we improve the performance of the user having the best channel gain, where all other users rely on RISaided beamforming. In contrast to previous contributions [Hou_RIS_MIMO_global_algrithm], we will show that the proposed POD outperforms the FOD in terms of its SE.
Against to above background, our contributions can be summarized as follows:

We propose a novel RISaided NOMA network, where a POD is employed for enhancing the SE. The impact of the LoS transmission on the reflected BSRISuser links are exploited. Furthermore, the impact of the proposed design on the attainable performance is characterized in terms of its outage probability (OP), ergodic rate, SE and EE.

Explicitly, we derive closedform expressions of both the OP and of the ergodic rate for the proposed RISaided NOMA network. Both the bestcase and worstcase of the OP and of the ergodic rate are derived. Both accurate and approximate closedform expressions are derived. Furthermore, both the diversity orders and highSNR slopes are obtained based on the OP and ergodic rate. The results confirm that the diversity order can be enhanced by increasing the number of RISs.

The simulation results confirm our analysis, illustrating that: 1) the BSuser link can be ignored when the number of RISs is high enough; 2) the RISaided NOMA network relying on the optimal power allocation factors is capable of outperforming its OMA counterpart; 3) the SE of the proposed POD can be significantly enhanced compared to the FOD, when the number of RISs is high enough.
IC Organization and Notations
The rest of the paper is organized as follows. In Section II, the model of RISaided NOMA networks is discussed. Our analytical results are presented in Section III, while our numerical results are provided in Section IV for verifying our analysis, followed by our conclusions in Section V. The distribution of a circularly symmetric complex Gaussian (CSCG) random variable with mean
and covariance matrix is denoted by ; and stands for “distributed as”. and represent the probability and expectation, respectively.Ii System Model
Let us consider the RISaided NOMA DL, where a BS equipped with a single transmit antenna (TA) is communicating with users, each equipped with a single receive (RA) antenna. We have intelligent surfaces at the appropriate location. By appropriately adjusting the reflection angles and amplitude coefficients of the RIS elements, the electromagnetic signal can be beneficially manipulated. Fig. 1 illustrates the wireless communication model for a single BS.
Iia RISAided SISONOMA Network
We first provide a fundamental model to illustrate the network performance affected by RISs. In order to illustrate the LoS links between the BS and RISs, the smallscale fading vector is defined as
(1) 
where is a
element vector whose elements represent the Nakagami fading channel gains. The probability density function (PDF) of the elements can be expressed as
(2) 
where denotes the fading parameter, and represents the Gamma function. Note that when is an integer.
It is assumed that there are a total of users in the cluster, where the pair of users, user and user with , are superimposed for DL transmission in NOMA. Hence, the smallscale fading vector between the RISs and user is defined as
(3) 
Similarly, the smallscale fading vector between the RISs and user is given by
(4) 
where and are element vectors whose elements represent the Nakagami fading channel gains having fading parameters of and , respectively.
Due to the strong scattering environment, the BSuser link between the BS and user as well as that between the BS and user are modelled by Rayleigh fading, which can be expressed as and , respectively.
In DL transmission, the BS sends the following signal to the paired NOMA users:
(5) 
where and denote the signal intended for user and user , respectively, with and representing the power allocation factors of user and user , respectively. Based on the NOMA protocol, .
Without loss of generality, we focus our attention on user , and the signal received by user from the BS through RISs is given by
(6) 
where denotes the transmit power of the BS, is a diagonal matrix, which accounts for the effective phase shift applied by all intelligent surfaces, represents the amplitude reflection coefficient of RISs, while , and denotes the phase shift introduced by the th intelligent surface. It is assumed that the CSIs are perfectly known at the RIS controller [ZhangRui_MISO_beams_discrete_2]. and denote the distance between the BS and RISs as well as that between the RISs and user , while denotes the distance between the BS and user . Furthermore, and denote the path loss exponent of the BSRISuser links and BSuser links. Finally,
denotes the additive white Gaussian noise (AWGN), which is modeled as a realization of a zeromean complex circularly symmetric Gaussian variable with variance
.Iii RIS Design for the Prioritized User in NOMA Networks
In this section, we first design the phase shifts and reflection amplitude coefficients for the RISs. Our new channel statistics, OPs, ergodic rates, SE and EE are illustrated in the following subsections.
Iiia RIS Design
When the direct BSuser signal and reflected BSRISuser signals are cophased, the channel gain of user is given by
(7) 
It is assumed that there are users in the cluster, and then the achievable channel gains of users are ordered as follows [Hou_naka_order]:
(8) 
We then turn our attention to the RIS design. It is assumed that the RISs mainly focus on providing maximum channel gain to the prioritized user for enhancing the SE. Without loss of generality, we assume that the prioritized user is the one having the best ordered channel gain.
In this article, in order to simultaneously control multiple RISs, the global CSI is assumed to be perfectly available at the RIS controller. Since user is the prioritized user, we aim for maximizing users’ received power by designing the phase shifts and reflection amplitude coefficients of RISs as follows:
(9)  
Thus by utilizing our signal alignment technique, our objective can be achieved by phaseshifting the signals received at the RISs, which is capable of significantly improving the received power.
Thus, we first define a channel vector as follows:
(10) 
Hence, the design of the th RIS can be expressed as
(11) 
where denotes the angle of the element, and denotes the th element of .
We then generate the effective vector of user as follows:
(12) 
Thus, the phase shifts of the RISs can be further transformed into
(13) 
Since the phase shifts are designed for the prioritized user , the effective channel gain for user can be written as , which cannot be evaluated. However, for the effective channel gain we have:
(14) 
We then consider the situation that two users, i.e. user and user having indexes of , are paired to perform NOMA.
IiiB New Channel Statistics
In this subsection, we derive new channel statistics for the proposed RISaided NOMA network, which will be used for evaluating the OPs and ergodic rates in the following subsections.
Lemma 1.
Let us assume that the fading parameters of the elements in and are and , respectively. The elements of the channel vectors are independently and identically distributed (i.i.d.). On the one hand, the worstcase distribution of the effective channel gain of user can be formulated as
(15) 
where
represents the Gamma distribution, and
. On the other hand, the bestcase distribution of the effective channel gain of user can be expressed by(16) 
where
Proof.
Please refer to Appendix A. ∎
IiiC Outage Probability
In this article, the OP of user is defined by
(17)  
where and represent the target rates of user and user , respectively.
We then focus our attention on the SINR analysis of user having the best channel gain. The cellcentre user first decodes the signal of the celledge user with the following SINR:
(18) 
Once the signal of user is decoded successfully, user decodes its own signal at an SINR of:
(19) 
Let us now turn our attention to calculating the OP of user based on Theorems and .
Theorem 1.
Assuming that , the worstcase of the closedform OP expression of user can be expressed as
(20) 
where , , , , , , , and represents the lower incomplete Gamma function.
Proof.
Please refer to Appendix B. ∎
Theorem 2.
Assuming that , the bestcase of the closedform OP expression of user can be expressed as
(21) 
where , and .
Proof.
Similar to Appendix B, Theorem 2 can be readily proved. ∎
It is however quite challenging to directly obtain engineering insights from (20) and (21) due to the th power of the lower incomplete Gamma function. Thus, in order to gain further insights in the highSNR regime, the approximate behaviors are analyzed, when the SNR is sufficiently high, i.e. when the transmit SNR obeys .
Corollary 1.
Assuming that , the worstcase and bestcase of the OP can be approximated in closed form by
(22) 
and
(23) 
Proof.
Please refer to Appendix C. ∎
The diversity orders of the prioritized user can be obtained for evaluating the slope of the OP in the following Propositions.
Proposition 1.
Based on Corollary 1, the diversity orders can be determined by using the approximate results, explicitly the worstcase and bestcase on the diversity order of user supported by the proposed RISaided NOMA network are given by
(24) 
and
(25) 
Remark 1.
The results of (24) demonstrate that the diversity orders can be approximated by for the prioritized user , when the number of RISs is high enough. It is also demonstrated that increasing the number of RISs and carefully pairing the NOMA users is capable of significantly improving the outage performance.
Remark 2.
Assuming that , which indicates a strong LoS link between the BS as well as the RISs, and provided that the number of RISs is high enough, the diversity orders on both the bestcase and worstcase of the prioritized user can be approximated by .
Remark 3.
Since the users are ordered by their effective channel gain, and based on the results of (24), in order to minimize the OP of the paired NOMA users, it is preferable to pair the users having the best and the second best effective channel gains.
Remark 4.
Assuming that the number of RISs is high enough, and based on the results of (25), the diversity order of the worstcase on the OP can be further approximated by . Again, assuming that , both the worstcase and bestcase on the diversity order of user can be approximated by , which indicates that the diversity order of both the bestcase and worstcase of the OP are identical.
Remark 5.
Assuming that the BSuser link of user is the dominant component, where the path loss exponent as well as , the diversity order of both the bestcase and worstcase are .
Due to the impact of hostile fading environments in NG networks, it is worth mentioning that no BSuser link may be available between the BS and the paired NOMA users, and the approximate result mainly depends on the th ordered element in (22) and (23). Thus, we continue by providing basic numerical insights using the following Corollary.
Corollary 2.
Due to the hostile fading environment between the BS and the users in NG networks, and assuming that , the th ordered elements in terms of the worstcase and bestcase on the approximate OP of user are given by
(26) 
and
(27) 
where , and .
Remark 6.
Assuming that no BSuser links are expected between the BS and the prioritized NOMA user, based on results of (26), the bestcase and worstcase on the diversity orders of user are seen to be and , respectively.
IiiD Ergodic Rate
The ergodic rate is a salient performance metric related to the SE and EE. Therefore, we focus our attention on analyzing the ergodic rate of user . The approximate ergodic rate expressions of user are given in the following Theorems.
Theorem 3.
Assuming that RISs simultaneously serve user , and , the worstcase on the ergodic rate of user can be expressed in closed form as follows:
(28)  
where , and is obtained by rounding to the nearest integer.
Proof.
Please refer to Appendix D. ∎
Similarly, the bestcase on the ergodic rate of user is formulated in the following Theorem.
Theorem 4.
Assuming that RISs simultaneously serve user , and , the bestcase on the ergodic rate of user can be expressed in closed form as follows:
(29)  
where , and .
Proof.
Similar to Appendix D, the results in (29) can be readily obtained. ∎
To gain deep insights into the system’s performance, the highSNR slope, as the key parameter determining the ergodic rate in the highSNR regime, is worth estimating. Therefore, we first express the highSNR slope as
(30) 
Remark 7.
The results of (31) illustrate that the slope of the ergodic rate in the proposed RISaided NOMA network is one, which is not affected by the number of RISs.
Based on the passive beamforming weight design at the RISs, the distribution of NOMA user , having the lower received power, cannot be evaluated. Hence, we only provide the associated SINR analysis for simplicity. By relying on the NOMA protocols, user treats the signal from user as interference, and the SINR is given by
(32) 
Since the elements in are gleaned from random variables, and based on the insights from [Yi_anlog_beam], the SINR of user can be further approximated as:
(33) 
where denotes the normalized Fejèr Kernel function with parameter . Note that has a period of two, hence
is uniformly distributed over
. Thus, the ergodic rate of user can be expressed as follows.Theorem 5.
Assuming that RISs simultaneously serve user , and , the worstcase and bestcase on the ergodic rate of user can be expressed as follows:
(34) 
and
(35) 
where , , , and .
Proof.
Similar to Appendix D, the results can be readily derived. ∎
Remark 8.
Let us assume that , indicating that the paired NOMA users share an identical channel vector, the Fejèr Kernel function can be considered as one. Hence, based on the insights in [Hou_Single_UAV], the bestcase and worstcase on the ergodic rate of user may approach in the highSNR regime.
In order to provide further insights for RISaided NOMA networks, the ergodic rate of the paired users is also analysed in the OMA scenario using TDMA. The OMA benchmark adopted in this article relies on supporting user and user in a pair of identical time slots. In each time slot, the RISs provide access only for one of the users. Thus, the channel capacity of user in the OMA scenario can be expressed as
(36) 
where . Similarly, the channel capacity of user can be expressed as
(37) 
where .
IiiE Spectrum Efficiency and Energy Efficiency
Based on the analysis of the previous two subsections, a tractable SE expression can be formulated in the following Proposition.
Proposition 3.
The SE of the proposed RISaided NOMA network is given by
(38) 
In NG networks, EE is an important performance metric. Thus, based on insights gleaned from [EE_model_massive_MIMO], we first model the total power dissipation of the proposed RISaided NOMA network as
(39) 
where is the static hardware power consumption of the BS, denotes the efficiency of the power amplifier at the BS, is the power consumption of each user, and represents the power consumption of each RIS controller. Thus, the EE of the proposed network is given by the following Proposition.
Iv Numerical Studies
In this section, numerical results are provided for the performance evaluation of the proposed network. Monte Carlo simulations are conducted for verifying the accuracy of our analytical results. The bandwidth of the DL is set to MHz, and the power of the AWGN is set to dBm. The power attenuation at the reference distance is set to 30 dB, and the reference distance is set to 1 meter. Note that the LoS and NLoS links are indicated by the Nakagami fading parameter, where and are for NLoS and for LoS links, respectively. The target rates are and bits per channel use (BPCU). The power allocation factors of the paired NOMA users are set to and . The number of users is set to , and . The fading environments are set to . The length of the BSRIS link is set to m. The length of the RISuser links are set to m and m, and these of the BSuser links are set to m. The path loss exponents of the reflected BSRISuser and the direct BSuser links are set to as well as , respectively, unless otherwise stated.
1) Impact of the Number of RISs: In Fig. 2, we focus our attention on the OP of the RISaided NOMA network. The solid curves and dashed curves represent the worstcase and bestcase of the analytical results, respectively. We can see that as the number of RISs serving user increases, the OP decreases. This is due to the fact that, as more RISs are employed, the received signal power can be significantly increased as a benefit of the increased diversity order. Observe that the slope of the curves increases with the number of RISs, which validates our Remark 1. Let us assume that and , then the minimum diversity order that can be obtained is for the case of and , which is identical to that of the nonRISaided networks. Observe that as expected the simulation results are located between the best and worst cases, which verifies Remark 4.
2) Impact of Fading Environments: In Fig. 3, we evaluate the OP of the prioritized user in different fading environments. As expected, with the transmit power increases, the OP decreases. Observe that both the BSRIS as well as RISuser links have an impact on the OP, which is in contrast to the FOD of [Hou_RIS_MIMO_global_algrithm], where the fading environment of the RISuser link has almost no effect on the OP.
3) Impact of the Number of Users: Let us now study the impact of the number of users in Fig. 4. Observe that it is preferable to pair the users having the best effective channel gains for minimizing the OP. Based on the results in the highSNR regime, the diversity order is seen to be significantly enhanced by increasing the number of users, because they experience independent fading channels. It is also worth noting that the diversity order is , which verified by the insights gleaned from Remark 2. This is because when the path loss exponent is , the power received from links reflected by the RISs can be nearly ignored.
4) Ergodic Rate: Fig. 5 compares the ergodic rates of paired NOMA users versus the SNR parameterized by the fading parameters and by the number of RISs. Several observations can be drawn as follows: 1) Based on the curves in Fig. 5(a), we can observe that the LoS links of both the BSRIS as well as of the RISuser links increase the ergodic rate of user , where the ergodic rate approaches the bestcase for the case of . 2) The triangles are between the bestcase and worstcase, which verify the accuracy of our results. 3) As seen from the figure, the highSNR slope of user is one, which also verifies Remark 8. 4) The ergodic rate can be significantly increased by employing more RISs, which is because the spatial diversity gain can be significantly increased upon increasing the number of RISs. 5) The ergodic rate of conventional NOMA dispensing with RISs is provided as the benchmark schemes, which can be calculated by setting the number of RISs to . 6) Fig. 5(b) evaluates the ergodic rate of the nonprioritized user . Observe that in the highSNR regime, the slope of user approaches zero in Fig. 5(b), which indicates that the number of RISs has no significant impact on the ergodic rate of user . In TABLE I, we use “D” and “S” to represent the diversity order and highSNR slope for the case that is large enough, respectively. It is worth noting that the diversity order of the nonprioritized user is the optimized result, which can only be obtained by setting .
Access Mode  Rx  D  S 

RISaided NOMA  1  
0  
Conventional NOMA  1  
0  
OMA  0.5  
0.5 
DIVERSITY ORDER AND HIGHSNR SLOPE
5) Comparing the RISaided NOMA to an OMA Network: In Fig. 6, we then evaluate the SE of our RISaided NOMA network, as well as that of its OMA counterpart . The results of the RISaided NOMA and OMA networks are derived by and , respectively. We can see that the RISaided NOMA network is capable of outperforming its OMA counterpart in terms of its SE by appropriately setting the power allocation factors. Observe that the SE gap between the RISaided NOMA network and its OMA counterpart becomes higher, when the number of RISs is increased, which indicates that it is preferable to employ more RISs for enhancing the SE.
6) Comparing the POD to the FOD: In Fig. 7, we evaluate the SE of the proposed POD. The SE of the FOD in [Hou_RIS_MIMO_global_algrithm] is provided as the benchmark schemes. Observe from the figure that the SE of the POD is higher than the FOD of [Hou_RIS_MIMO_global_algrithm], which indicates that the proposed POD becomes more competitive compared to the FOD. This is due to the fact that the proposed POD is conceived for attainting the maximum network throughput for the prioritized user, which is capable of providing higher SE. By contrast, the FOD is mainly focused on the fairness, which can provide higher throughput for the celledge users.
7) Comparing Halfduplex and Fullduplex Relay Networks:
In order to provide further engineering insights, combined with the insights inferred from [yuanwei_cooperative, DF_relaying_outage, cooperative_Yue], the network throughputs of alternative fullduplex (FD) and halfduplex (HD) cooperative networks are evaluated. We consider the classic relaying protocols, where the transmissions of HDrelaying are divided into two identical phases. By contrast, the FDrelay is suffering from selfinterference. It is assumed that both the BS and relay, as well as the users are equipped by a single antenna. Similar to (7), we first rank the entire set of users according to their effective channel gains. We then evaluate the SE of the FD network, where the FDrelay has to decode the signal of the paired NOMA users. Based on the insights from Remark 6 for simplicity, it is assumed that no BSuser link exists. Hence, the FDrelay first decodes the signal of user , achieving the following expectation:
(41) 
where denotes the channel gain between the BS and the FDrelay, denotes the selfinterference coefficient of the FDrelay itself, and denotes the transmit power of the relay. Then the FDrelay can decode the signal of user as follows:
(42) 
Similarly, assuming that SIC can be also invoked successfully by the paired NOMA users, and thus the nonprioritized user treats the signal of user as interference, and the expected data rate can be given by
(43) 
where denotes the channel gain between the FDrelay and user . On the other hand, by utilizing SIC technique, the transmission rate of user is given by
(44) 
More specifically, the size of data rate for user and user depend on four kinds of data rates, such as 1) the data rate for the relay to detect user ; 2) the data rate for the relay to detect user ; 2) The data rate for user ; and 3) the data rate for user . Among the FDrelay in the network, based on (41) to (44), the expected rate of the paired NOMA users in the FDrelay network can be given by
(45) 
and
(46) 
We then consider the HDrelay network, where the expected data rate of the paired NOMA users at the HDrelay can be given by
(47) 
and
(48) 
By applying the classic SIC technique, the expected data rate of paired NOMA users can be written as
(49) 
and
(50) 
Thus, the expected rate of the paired NOMA users in the HDrelay network is given by
(51) 
and
(52) 
In Fig. 8, we evaluate the network throughput of our RISaided NOMA network, as well as of the HDrelay and FDrelay aided networks. The results of the FDrelay and HDrelay are given by and
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