The demand for next-generation (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 multiple-input multiple-output (MIMO) and millimeter wave (mmWave) communications [5G_NR_2]. Recently, cost-efficient 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, non-orthogonal 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 quality-of-service (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 power-domain NOMA111In this article, we use NOMA to refer to power-domain NOMA for simplicity. and RIS techniques in the downlink (DL).
I-a 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 RIS-aided and relay-assisted networks was compared in [LIS_compare_relay], indicating that RIS-aided 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 coverage-holes. 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 multiple-input single-output (MISO) assisted RIS network. The channel capacity of a RIS-aided 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 fairness-oriented design (FOD) was proposed in a RIS-aided 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 cell-edge users, where the cell-center users treat the reflected signal as interference [DING_RIS_NOMA_letter]. Both continuous and discrete phase shifters were used in a RIS-aided MISO NOMA network [yuanwei_NOMA_RIS]. Naturally, the BS-user link plays a key role [MISO_with_directlink]. A RIS-aided NOMA network was also investigated in [NOMA_RIS_Fu], whilst the BS-user link and the BS-RIS link, as well as the RIS-user 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 line-of-sight (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 fairness-oriented algorithm was proposed in a RIS-aided 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)].
I-B Motivations and Contributions
The above-mentioned 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 RIS-aided SISO-NOMA 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 RIS-aided NOMA DL scenario, where a priority-oriented 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 RIS-aided 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 RIS-aided NOMA network, where a POD is employed for enhancing the SE. The impact of the LoS transmission on the reflected BS-RIS-user 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 closed-form expressions of both the OP and of the ergodic rate for the proposed RIS-aided NOMA network. Both the best-case and worst-case of the OP and of the ergodic rate are derived. Both accurate and approximate closed-form expressions are derived. Furthermore, both the diversity orders and high-SNR 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 BS-user link can be ignored when the number of RISs is high enough; 2) the RIS-aided 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.
I-C Organization and Notations
The rest of the paper is organized as follows. In Section II, the model of RIS-aided 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 meanand covariance matrix is denoted by ; and stands for “distributed as”. and represent the probability and expectation, respectively.
Ii System Model
Let us consider the RIS-aided 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.
Ii-a RIS-Aided SISO-NOMA 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 small-scale fading vector is defined as
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
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 small-scale fading vector between the RISs and user is defined as
Similarly, the small-scale fading vector between the RISs and user is given by
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 BS-user 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:
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
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 BS-RIS-user links and BS-user links. Finally,
denotes the additive white Gaussian noise (AWGN), which is modeled as a realization of a zero-mean 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.
Iii-a RIS Design
When the direct BS-user signal and reflected BS-RIS-user signals are co-phased, the channel gain of user is given by
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]:
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:
Thus by utilizing our signal alignment technique, our objective can be achieved by phase-shifting the signals received at the RISs, which is capable of significantly improving the received power.
Thus, we first define a channel vector as follows:
Hence, the design of the -th RIS can be expressed as
where denotes the angle of the element, and denotes the -th element of .
We then generate the effective vector of user as follows:
Thus, the phase shifts of the RISs can be further transformed into
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:
We then consider the situation that two users, i.e. user and user having indexes of , are paired to perform NOMA.
Iii-B New Channel Statistics
In this subsection, we derive new channel statistics for the proposed RIS-aided NOMA network, which will be used for evaluating the OPs and ergodic rates in the following subsections.
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 worst-case distribution of the effective channel gain of user can be formulated as
where represents the Gamma distribution, and
represents the Gamma distribution, and. On the other hand, the best-case distribution of the effective channel gain of user can be expressed by
Please refer to Appendix A. ∎
Iii-C Outage Probability
In this article, the OP of user is defined by
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 cell-centre user first decodes the signal of the cell-edge user with the following SINR:
Once the signal of user is decoded successfully, user decodes its own signal at an SINR of:
Let us now turn our attention to calculating the OP of user based on Theorems and .
Assuming that , the worst-case of the closed-form OP expression of user can be expressed as
where , , , , , , , and represents the lower incomplete Gamma function.
Please refer to Appendix B. ∎
Assuming that , the best-case of the closed-form OP expression of user can be expressed as
where , and .
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 high-SNR regime, the approximate behaviors are analyzed, when the SNR is sufficiently high, i.e. when the transmit SNR obeys .
Assuming that , the worst-case and best-case of the OP can be approximated in closed form by
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.
Based on Corollary 1, the diversity orders can be determined by using the approximate results, explicitly the worst-case and best-case on the diversity order of user supported by the proposed RIS-aided NOMA network are given by
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.
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 best-case and worst-case of the prioritized user can be approximated by .
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.
Assuming that the number of RISs is high enough, and based on the results of (25), the diversity order of the worst-case on the OP can be further approximated by . Again, assuming that , both the worst-case and best-case on the diversity order of user can be approximated by , which indicates that the diversity order of both the best-case and worst-case of the OP are identical.
Assuming that the BS-user link of user is the dominant component, where the path loss exponent as well as , the diversity order of both the best-case and worst-case are .
Due to the impact of hostile fading environments in NG networks, it is worth mentioning that no BS-user 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.
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 worst-case and best-case on the approximate OP of user are given by
where , and .
Assuming that no BS-user links are expected between the BS and the prioritized NOMA user, based on results of (26), the best-case and worst-case on the diversity orders of user are seen to be and , respectively.
Iii-D 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.
Assuming that RISs simultaneously serve user , and , the worst-case on the ergodic rate of user can be expressed in closed form as follows:
where , and is obtained by rounding to the nearest integer.
Please refer to Appendix D. ∎
Similarly, the best-case on the ergodic rate of user is formulated in the following Theorem.
Assuming that RISs simultaneously serve user , and , the best-case on the ergodic rate of user can be expressed in closed form as follows:
where , and .
Similar to Appendix D, the results in (29) can be readily obtained. ∎
To gain deep insights into the system’s performance, the high-SNR slope, as the key parameter determining the ergodic rate in the high-SNR regime, is worth estimating. Therefore, we first express the high-SNR slope as
The results of (31) illustrate that the slope of the ergodic rate in the proposed RIS-aided 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
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:
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.
Assuming that RISs simultaneously serve user , and , the worst-case and best-case on the ergodic rate of user can be expressed as follows:
where , , , and .
Similar to Appendix D, the results can be readily derived. ∎
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 best-case and worst-case on the ergodic rate of user may approach in the high-SNR regime.
In order to provide further insights for RIS-aided 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
where . Similarly, the channel capacity of user can be expressed as
Iii-E 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.
The SE of the proposed RIS-aided NOMA network is given by
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 RIS-aided NOMA network as
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 BS-RIS link is set to m. The length of the RIS-user links are set to m and m, and these of the BS-user links are set to m. The path loss exponents of the reflected BS-RIS-user and the direct BS-user 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 RIS-aided NOMA network. The solid curves and dashed curves represent the worst-case and best-case 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 non-RIS-aided 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 BS-RIS as well as RIS-user 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 RIS-user 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 high-SNR 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 BS-RIS as well as of the RIS-user links increase the ergodic rate of user , where the ergodic rate approaches the best-case for the case of . 2) The triangles are between the best-case and worst-case, which verify the accuracy of our results. 3) As seen from the figure, the high-SNR 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 non-prioritized user . Observe that in the high-SNR 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 high-SNR slope for the case that is large enough, respectively. It is worth noting that the diversity order of the non-prioritized user is the optimized result, which can only be obtained by setting .
DIVERSITY ORDER AND HIGH-SNR SLOPE
5) Comparing the RIS-aided NOMA to an OMA Network: In Fig. 6, we then evaluate the SE of our RIS-aided NOMA network, as well as that of its OMA counterpart . The results of the RIS-aided NOMA and OMA networks are derived by and , respectively. We can see that the RIS-aided 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 RIS-aided 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 cell-edge users.
7) Comparing Half-duplex and Full-duplex 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 full-duplex (FD) and half-duplex (HD) cooperative networks are evaluated. We consider the classic relaying protocols, where the transmissions of HD-relaying are divided into two identical phases. By contrast, the FD-relay is suffering from self-interference. 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 FD-relay has to decode the signal of the paired NOMA users. Based on the insights from Remark 6 for simplicity, it is assumed that no BS-user link exists. Hence, the FD-relay first decodes the signal of user , achieving the following expectation:
where denotes the channel gain between the BS and the FD-relay, denotes the self-interference coefficient of the FD-relay itself, and denotes the transmit power of the relay. Then the FD-relay can decode the signal of user as follows:
Similarly, assuming that SIC can be also invoked successfully by the paired NOMA users, and thus the non-prioritized user treats the signal of user as interference, and the expected data rate can be given by
where denotes the channel gain between the FD-relay and user . On the other hand, by utilizing SIC technique, the transmission rate of user is given by
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 FD-relay in the network, based on (41) to (44), the expected rate of the paired NOMA users in the FD-relay network can be given by
We then consider the HD-relay network, where the expected data rate of the paired NOMA users at the HD-relay can be given by
By applying the classic SIC technique, the expected data rate of paired NOMA users can be written as
Thus, the expected rate of the paired NOMA users in the HD-relay network is given by
In Fig. 8, we evaluate the network throughput of our RIS-aided NOMA network, as well as of the HD-relay and FD-relay aided networks. The results of the FD-relay and HD-relay are given by and