I Introduction
With the evolution of wireless communication networks, the fifthgeneration (5G) and beyond has sparked a lot of concerns on high data rate, massive connectivity and spectrum utilization. The standards of 5G new radio have been completed currently and researchers are exploring the potential of emerging technologies for the nextgeneration communications [1, 2, 3]. As one of a promising multiple access candidate, nonorthogonal multiple access (NOMA) has the advantages in terms of spectral efficiency and link density. The distinctive feature of NOMA is that multiple users are allowed to occupy the same time/bandwidth resource blocks by utilizing the superposition coding scheme [4, 5]. It has been demonstrated that NOMA has ability to attain the better outage probability and ergodic rate compared to conventional orthogonal multiple access (OMA) [6, 7]. For the emerging sixthgeneration (6G) communication networks, it becomes pivotal to support massive intelligent equipments with different requirements.
By extending NOMA to cooperative communication, cooperative NOMA was proposed in [8], where the nearby user with better channel condition was referred to halfduplex (HD) decodeandforward (DF) relaying to transfer the signals for the distant users. To further enhance spectrum efficiency, the authors of [9, 10] investigated the outage behavior and ergodic rate of fullduplex (FD) cooperative NOMA, where the performance of FD NOMA outperforms HD NOMA in the low signaltonoise radio (SNR). With the emphasis on green communication, the simultaneous wireless power transfer (SWIPT) based NOMA system was studied in [11], where the nearby user is viewed as DF relaying to forward the information. On the other hand, the authors of [12] analyzed the outage performance of a pair of users for amplifyandforward (AF) relaying based NOMA systems. As a further development, the outage probability and ergodic performance of multiple users for AF NOMA systems were surveyed in [13, 14] over Nakagami fading channels. Explicit insights for understanding the impact of FD mode on AF NOMA system, the authors of [15] characterized the outage behaviors of users with an opportunistic power split factor. Apart from the above works, NOMA technique has been widely applied to multiple communication scenarios. Regarding to safety applications, the secrecy outage probability of a pair of users was analyzed in [16] for NOMA networks by invoking stochastic geometry. With the objective of improving terrestrial user connections, the authors of [17] highlighted the trajectory design and power allocation of NOMAbased unmanned aerial vehicle networks. Additionally, the application of NOMA to satellite communications was investigated [18], where the ergodic capacity and energy efficiency are derived analytically.
In view of recent attentions, intelligent reflecting surface (IRS) has been as a prospective technology for 6G wireless communications [19]. More specifically, IRS is a lowcost planar array consisting of a large of passive reflecting elements, which is ability to reconfigure the wireless propagation environment through a programmable controller. In [20], the authors proposed the concept of digital metamaterials, which manipulates the electromagnetic waves by coding ‘0’ and ‘1’ elements with control sequences (i.e., 1bit coding). Recently, several application scenarios of IRSaided were introduced [21] that: 1) Creating the lineofsight (LoS) link between the BS and users via signal reflection; 2) Applying an IRS to enhance the physical layer secrecy; and 3) Deploying an IRS to realize SWIPT for a large amount of devices and so on. Similar to these paradigms, the authors of [22] surveyed the symbol error probability of IRSbased communication networks in a general mathematical framework. By using intelligent reflecting elements, the authors in [23] revealed that IRS has the ability to attain the high rate and energy efficiency comparing to DF relaying. When the eavesdropping channels are stronger than that of legitimate channels, the authors of [24] maximized the secrecy rate of legitimate users by designing the IRS’s reflecting beamforming.
In light of the above discussions, researchers have begun to study the coexistence of IRS and NOMA [25, 26, 27]. Given the users’ rate, the authors in [25]
analyzed the IRS reflection with discrete phase shifts for IRSaided NOMA and OMA. On the condition of user ordering, the beamforming vectors and phase shift matrix were jointly optimized to reduce the transmitting power
[26]. Furthermore, the authors of [27] maximized the minimum the achievable rates of users to ensure user’s fairness and improve the rate performance. For multiantenna scenarios, the system sum rate of IRSNOMA was improved by exploiting the fixed reflecting elements in [28]. To overcome the hardware limitations, a simple transmission of IRSNOMA was designed in [29], where the outage probability of single user is derived with onoff control. As a further potential enhancement, the authors of [30] further studied the impact of coherent phase shifting and random phase shifting on outage behaviors for IRSNOMA networks. In [31], the outage probability and ergodic rate of nearby user for IRSNOMA were evaluated by designing the passive beamforming weights of IRS.Ia Motivation and Contributions
While the aforementioned significant literatures have laid a solid foundation for understanding of IRS and NOMA techniques in wireless communication networks, the treatise for enhancing the spectrum and energy efficiency by integrating these two promising technologies is a straightforward and effective approach. As stated in [20], the digital metamaterial has ability to manipulate electromagnetic waves by programming different coding sequences. Consequently, research on using the thought of 1bit coding scheme to analyze the performance of wireless communication systems is still imperative. In addition, there are undesirable factors i.e., error propagation and quantization error in the SIC process for practical scenarios, which will result in decoding errors. It is significant to take the residual interference from SIC procedure into consideration.
Inspired by this treatises, we specifically investigate the performance of IRSassisted NOMA with the aid of the thought of 1bit coding, where an IRS can be regarded as a relay forwarding the information to multiple NOMA users. Moreover, we focus our attention on discussing the impact of residual interference from ipSIC on outage probability, ergodic rate and energy efficiency for IRSNOMA networks. Additionally, the outage probability and ergodic rate of IRSOMA with 1bit coding are surveyed carefully. According to the above explanations, the primary contributions of this paper are summarized as follows:

We derive exact and asymptotic expressions of outage probability for the th user with ipSIC/pSIC in IRSNOMA networks. Based on theoretical analyses, the diversity order for IRSNOMA is obtained. We demonstrate that the diversity orders of the th user with ipSIC/pSIC are in connection with the number of reflecting elements of IRS and channel ordering. We also derive the closedform expression of outage probability for IRSassisted orthogonal multiple access (IRSOMA).

We confirm that the outage behaviors of IRSNOMA are superior to that of IRSOMA, AF relaying and FD/HD relaying. Furthermore, we investigate the impact of IRS’s deployment on the outage behaviors of IRSNOMA networks. We observe that when the IRS is deployed closely to the BS, the enhanced outage performance is achieved. As the IRS departs from BS, the LoS deteriorates and the outage probability increases.

We derive the exact expressions of ergodic rate of the th user for IRSNOMA networks. To obtain further more insights, we derive exact expression of ergodic rate for the th user and obtain the high SNR slopes. We observe that the ergodic rate of the th user converges to a throughput ceiling in the high SNR regime. As the number of reflecting elements increase, the ergodic performance of the th user is becoming higher compared to relaying schemes. We also derive exact expression of ergodic rate for IRSOMA.

We study the throughput and energy efficiency of nonorthogonal users for IRSNOMA networks in both delaylimited and delaytolerant transmission modes. For delaylimited transmission mode, the energy efficiency of nonorthogonal users outperforms that of IRSOMA. In delaytolerant transmission mode, the th user has a larger the energy efficiency than orthogonal user and distant users. However, the energy efficiency of distant users for IRSNOMA converges to the constant value at high SNRs.
IB Organization and Notations
The rest of this paper is organized as follows. In Section II, the network model and transmission formulation are introduced in detail. In Section III, new exact expressions of outage probability for IRSNOMA are derived. Then the ergodic rates of IRSNOMA are investigated in Section IV. At last, the numerical results are presented to verify theoretical analyses in Section VI and followed by concluding remarks in Section VII. The proofs of mathematics are collected in the Appendix.
The main notations used in this paper are shown as follows. denotes expectation operation. and
denote the probability density function (PDF) and cumulative distribution function (CDF) of a random variable
, respectively. The superscripts stand for the conjugatetranspose operation. diag represents a diagonal matrix. denotes the Kronecker product. is a identity matrix.Ii Network Model
Iia Network Descriptions
Considering an IRSassisted NOMA communication scenario as illustrated in Fig. 1, in which a base station (BS) sends the signals to terminal users with the assistance of an IRS. More specifically, assuming that the direct links between BS and users are assumed strongly attenuated and the communication can be only established through the IRS. To provide the straightforward analyses, we assume that the BS and users are equipped single antenna, respectively. The IRS is mounted with reconfigurable reflecting elements, which can be controlled by communication oriented software. The complex channel coefficient between the BS and IRS, and between the IRS and users are denoted as and , respectively. All wireless links in IRSNOMA system are modeled as Rayleigh fading channels and perturbed by additive white Gaussian noise with the mean power . Without loss of generality, the effective cascade channel gains from the BS to IRS and then to users are ordered as [32, 7], where is a diagonal matrix, and denote the fixed reflection amplitude coefficient and the phase shift of the th reflecting element of the IRS, respectively [21, 29]. It is noteworthy that the CSI of all wireless channel are perfectly available at the BS. Note that imperfect CSI is more suitable for evaluating practical scenarios, which will be set aside in our future work.
IiB Signal Model
The BS sends the superposed signals to users by the virtue of an IRS. Hence the received signal reflected by IRS at the th user is given by
(1) 
where is assumed to be normalised the unity power signal for the th user, i.e., . The th user’s power allocation factor satisfies the relationship with , which is for the sake of user fairness. The optimal power allocation coefficients between users will further heighten the performance of NOMA networks, but it is beyond the scope of this paper. denotes the normalized transmission power at the BS. , where denotes the complex channel coefficient from the BS to the th reflecting element of IRS. , where denotes the complex channel coefficient from the th reflecting element of IRS to the th user.
On the basis of NOMA principle, the received signaltointerferenceplusnoise ratio (SINR) at the th user to detect the th user’s information is given by
(2) 
where denotes the transmit SNR and . More precisely, and denote the pSIC and ipSIC operations. Assuming that the remaining interference from ipSIC is modeled as the Rayleigh fading and corresponding complex channel coefficient is denoted by .
After striking out the previous users’ signals with SIC, the received SINR at the th user to detect its own information can be given by
(3) 
IiC IRSNOMA with 1bit Coding
From the perspective of practical communication applications, continuously changing the reflection amplitude and phase shift of each IRS’s element is beneficial to enhance the network performance. This alternative implement needs the accurate design and expensive hardware architecture, which will result in the higher cost of IRS. To facilitate implement and analysis, 1bit coding scheme is selected to achieve the discrete amplitude/phase shift levels for IRSassisted NOMA networks [20, 29], where the elements of diagonal matrix are replaced with 0 or 1. It is the scalable and costeffective solution with the number of reflecting elements becomes larger.
Assuming that the number reflecting elements of IRS are equal to , i.e., , where and are integers. Define , where is a column vector of all ones with . The th column of is denoted by with and for . Based on the above explanations, the SINRs of (2) and (3) with 1bit coding are maximized by randomly choosing and can be given by
(4) 
and
(5) 
respectively, where and are the diagonal matrix with its diagonal elements obtained from and . The following network performance of IRSNOMA is discussed with 1bit coding scheme.
IiD IrsOma
In this subsection, the IRSOMA scheme is regarded as one of the benchmarks for comparison purpose, where an IRS is deployed to assist in the transmission from the BS to a user . On the condition of the above assumptions, the maximized SNR of user with 1bit coding scheme for IRSOMA can be given by
(6) 
where , where denotes the complex channel coefficient from the th reflecting element of IRS to user . and is the diagonal matrix with its diagonal elements obtained from .
Iii Outage probability
As mentioned in conventional NOMA, the SIC scheme is carried out at the th user by decoding and striking out the th user’s information before it detects its own signal. If the th user cannot successfully detect the th user’s information, an outage occurs and is denoted by
(7) 
where with being the target rate at the th user to detect . As a consequence, the outage probability of th user with 1bit coding for IRSNOMA networks can be expressed as
(8) 
It is worth pointing out that the first user (i.e., ) with the worse channel condition does not execute the SIC procedure.
Theorem 1.
Under Rayleigh fading channels, the closedform expression for outage probability of the th user with ipSIC in IRSNOMA networks is given by
(9) 
where , and with . with being the target rate at the th user to detect . . and are the weight and abscissas for the GaussLaguerre integration, respectively. More specifically, is the th zero of Laguerre polynomial and the corresponding the th weight is given by . The parameter is to ensure a complexityaccuracy tradeoff. is the modified Bessel function of the second kind with order . denotes the gamma function [33, Eq. (8.310.1)].
Proof.
See Appendix A. ∎
Corollary 1.
For the special case with substituting into (Appendix A: Proof of Theorem 1), the closedform expression for outage probability of the th user with pSIC in IRSNOMA networks is given by
(10) 
For IRSOMA, an outage is defined as the probability that the instantaneous SNR falls bellow a threshold SNR . Hence the outage probability of user with 1bit coding which can be expressed as
(11) 
where with being the target rate of user to detect . Referring to (1) and removing the order operation, we can derive the outage probability for IRSOMA in the following corollary.
Corollary 2.
The closedform expression of outage probability for IRSOMA networks with 1bit coding is given by
(12) 
Iiia Diversity Analysis
In order to gain better insights, the diversity order is usually selected to evaluate the outage behaviors for communication systems, which is able to describe how fast the outage probability decreases with the transmitting SNR. Hence the diversity order can be expressed as
(13) 
where denotes the asymptotic outage probability in the high SNR regime.
Corollary 3.
Based on analytical result in (1), when , the asymptotic outage probability of the th user with ipSIC for IRSNOMA networks is given by
(14) 
where and with .
Remark 1.
Corollary 4.
For the cases and , the asymptotic outage probability of the th user with pSIC at high SNRs are given by
(15) 
and
(16) 
respectively.
Proof.
To facilitate the calculation, we employ the series representation of Bessel functions to obtain the high SNR approximation. When and , can be approximated as
(17) 
and
(18) 
respectively. Upon substituting (17) and (18) into (1), and then taking the first term of summation term, we can obtain (4) and (4), respectively. The proof is completed. ∎
Remark 2.
Corollary 5.
IiiB Delaylimited transmission
In the delaylimited transmission mode, the BS sends the information at a constant rate, which is subject to outage according to the random fading of wireless channels [34, 9]. Hence the throughput of the th user with ipSIC/pSIC for IRSNOMA networks in the delaylimited transmission mode can be given by
(21) 
Iv Ergodic Rate
In this section, the ergodic rate of the th user with ipSIC/pSIC for IRSNOMA networks is discussed in detail, where the target rates of users are determined by the channel conditions. The th user detects the th user’s information successfully, since it holds . Under this situation, the achievable rate of the th user can be written as . Based on (4) and (5), the ergodic rates of the th user and the th user with ipSIC for IRSNOMA networks can be given by
(22) 
and
(23) 
respectively, where and . One can be seen from the equations, there are no the closedform solutions. However, these expressions can be evaluated numerically by using the standard softwares such as Matlab or Mathematica. By employing pSIC, the ergodic rates of the th user and the th user are presented in the following part.
Theorem 2.
For the special case with substituting into (IV), the closedform expression of ergodic rate for the th user with pSIC in IRSNOMA networks is given by
(24) 
where , , and is a parameter to ensure a complexityaccuracy tradeoff.
Proof.
See Appendix B. ∎
Theorem 3.
For the special case with substituting into (23), the exact expression of ergodic rate for the th user with pSIC in IRSNOMA networks is given by
(25) 
Proof.
See Appendix C. ∎
For IRSOMA, based on (6), the ergodic rate of user with 1bit coding can be expressed as
(26) 
Corollary 6.
Similar to the derivation process in (3), the exact expression of ergodic rate for IRSOMA with 1bit coding is given by
(27) 
Iva Slope Analysis
Similar to the diversity order, the high SNR slope aims to capture the diversification of ergodic rate with the transmitting SNRs, which can be defined as
(28) 
where denotes the asymptotic ergodic rate in the high SNR regime.
According to (Appendix B: Proof of Theorem 2), when , the asymptotic ergodic rate of the th user with pSIC is given by
(29) 
Remark 4.
As can be seen from (3) that the exact derivation of the approximation at high SNRs appears mathematically intractable. Furthermore, we focus our attention on evaluating the slope of ergodic rate for the th user via the assistance of its upper bound. By noticing that is a concave function for , we invoke the Jensen s inequality to derive an upper bound as
(30) 
We performe the derivative operation on (Appendix C: Proof of Theorem 3) in Appendix C and some manipulates, the upper bound is given by
(31) 
where is constant with increasing the SNRs. Upon substituting (31) into (28) and further applying L’hospital rule, we can obtain the high SNR slope of ergodic rate for the th user in the following remark.
IvB DelayTolerate Transmission
In the delaytolerant mode, the BS sends the information at any fixed rate upper bounded by the ergodic capacity. Hence the throughput of the th user and the th user with pSIC for IRSNOMA networks is given by
(32) 
To facilitate comparison, the diversity orders and high SNR slopes of the th user with ipSIC/pSIC for IRSNOMA are summarized in TABLE I, where we use “D” and “S” to represent the diversity order and high SNR slope, respectively.
Mode  SIC  User  D  S 
IRSOMA  ——  User  
User  
IRSNOMA  ipSIC  User  
pSIC  User  
User  

V Energy Efficiency
In wireless communication systems, the energy efficiency (EE) can be interpreted as the user’s data rate divided by the energy consumption, which can be expressed as
(33) 
More specifically, based on above throughput analyses, the energy efficiency of IRSNOMA networks is given by
(34) 
where and denotes the entire communication process time.
Monte Carlo simulations repeated  iterations 

Pass loss exponent  
The power allocation factors for users  
The targeted data rates for users  BPCU 
BPCU  
BPCU  
The distance from BS to IRS  m 
The distance from IRS to users  m 
m  
m 
Vi Numerical Results
In this section, the numerical results are presented to confirm the rationality of the derived theoretical expressions for IRSNOMA networks. We show the impact of the reflecting elements on the performance of the IRSassisted NOMA communication network. Monte Carlo simulation parameters used are summarized in TABLE II, where BPCU denotes the short for bit per channel use. Assume that three users
are taken into consideration and the distance from the BS to IRS, and then to the terminal users are normalized to unity. As a further development, the variances of complex channel coefficients are set to be
, , and , respectively. The complexityvsaccuracy tradeoff parameter is set to be and simulation results are denoted by . Without loss of the generality, the conventional OMA, AF relaying and FD/HD DF relaying are selected as the benchmarks for the purpose of comparison. The target rate of the orthogonal user is equal to .Via Outage Probability
Fig. 2 plots the outage probability of three users versus SNR for a simulation setting with , , , , , and BPCU. The theoretical analysis curves of outage probability for users with pSIC are plotted according to (1). It is obvious that the Monte Carlo simulation outage probability curves excellently agree with analytical results across the entire average SNR range. The asymptotic outage probability converges to the analytical expressions given in (4), which proves the effectiveness of our theoretical derivation. As can be seen from the figure that the outage performance of the nearest user () is higher than that of the distant users ( and ). This is due to the fact that the nearby user attains the higher diversity order, which verifies the insights in Remark 2. Another observation is that the outage behaviors of IRSNOMA with pSIC are superior than that of IRSOMA (12), AF relaying [35] and FD/HD relaying [36, Eq. (7) and Eq. (8)]. The reasons are that: 1) IRSNOMA can realize much better user fairness than IRSOMA for multiple users; 2) FD DF relay suffers from loop interference due to signal leakage and needs the advanced loop interference cancellation technologies, which will lead to the higher cost; and 3) IRSNOMA operates in the FD mode provides the more spectrum efficient than HD DF relaying.
Fig. 3 plots the outage probability of three users versus SNR for a simulation setting with , , , , , and BPCU. The exact and approximate analyses curves of outage probability for users with ipSIC are plotted by (1) and (3), respectively. The simulation results matches closely with the theoretical analysis. The important observation is that the outage probability of distant users with ipSIC converge to an error floor in the high SNR regime and thus obtain a zero diversity order. The reason is that there is the residual interference from ipSIC for IRSNOMA. This phenomenon is also confirmed by the conclusions in Remark 1. Additionally, it is worth noting that the farthest user () does not carry out the SIC operation, since it has the worst channel conditions. Compared to the benchmarks in Fig. 2, we observe that IRSNOMA with ipSIC is also capable of achieving the lower outage behaviors. Certainly, with the value of residual interference increasing, the achieved outage probability of IRSNOMA converges to the worst error floors. As a result, it is important to consider the influence of ipSIC on the network performance for IRSNOMA in the practical scenario.
Fig. 4 plots the outage probability versus SNR for a simulation system with different reflecting elements of IRS and dB. One can observe that the setting of the reflecting elements for IRSNOMA is significant to provide the network performance. With increasing the number of reflecting elements
, the lower outage probabilities are attained for multiple users. This behaviors are caused by the fact that the application of IRS to NOMA networks provides a new degree of freedom to enhance the wireless link performance. This phenomenon is also certificate the completion of
Remark 2, where both the number of reflecting elements and channel ordering determine the slope of outage probability for IRSNOMA. Another observation is that all outage probability curves of each user have the same slopes, which manifests that the diversity orders of users are the same. This appearance demonstrates the insight we derived from the analytical results given by (4).Fig. 5 plots the outage probability versus SNR for a simulation setting with , , , , , BPCU and dB. The approximated outage probability curves of users are plotted corresponding to (4), which match precisely with the simulation results. As can be observed from the figure that as the number of reflecting elements increases, the outage probability of users is becoming much smaller. The main reason behind this is that IRSNOMA with 1bit coding provides much more diversity orders given by Remark 2. It is worth mentioning that the outage probability curves of each user has a different diversity order, which confirms the analytical result derived in (4). Fig. 6 plots the outage probability versus SNR with the different target rate for , , and . One can observe that adjusting the target rate of users largely affect the outage performance. With the values of target rate increasing, the outage behaviors of users for IRSNOMA networks are becoming much worse, which is in line with the conventional NOMA networks [7].
To illustrate the impact of IRS’s deployment on the performance, Fig. 7 plots the outage probability as a function of the normalized distance between the BS and users, with , , , , BPCU, BPCU. We can observe that when the IRS is deployed closely to BS, the outage performance of nonorthogonal users is becoming much better. This phenomenon can be explained that the IRS can receive the clear LoS signals from the BS for the purpose of maximizing its received signal power. As the IRS departs from BS, the LoS deteriorates and outage probability of users increases seriously. When the IRS is in the middle of the BS and users, the worst outage behaviors of users are attained in the IRSNOMA networks. This is due to the fact that the IRS is neither closed to the BS nor to users. After this point, the performance begins to improve again. This is because that the IRS is close to NOMA users and enhance the reflecting signals received by users. Such an outage behavior can be useful to establish an optimal deployment of IRS in NOMA networks. As a consequence, the deployment scenarios of IRS should take into account some practical constraints.
ViB Ergodic Rate
Fig. 8 plots the ergodic rates versus SNR, with , , and . The exact curves of ergodic rate for the th user and th user with pSIC are plotted based on (2) and (3), respectively. One can observe that the ergodic rates of distant users for IRSNOMA outperform that of AF relaying and FD/HD relaying in the low SNR regime, which are consistent to FD/HD NOMA systems[9, 10]. As the SNR value increases, the ergodic rate of distant users converges to a throughput ceiling, which is also confirmed in Remark 4. This is due to the fact that the distant user will suffer from the interference from the nearby users’ signals when it decodes their own signals. Another observation is that the ergodic rate of nearest user is much greater than that of nonorthogonal users, IRSOMA (6), AF relaying and FD/HD relaying. The origin for this behavior is that it is closest to the IRS and has the best channel conditions. In addition, Fig. 9 plots the ergodic rates versus SNR with different reflecting elements. As can be observed from this figure that with the increasing reflecting elements of IRS, the ergodic performance of the nearest user with pSIC is enhanced and has the same slopes, which confirms the insights in Remark 5. The distant users’ performance has no obvious variety due to the effects of interference signals. We conclude that IRSNOMA cannot circumvent the problem of slope for the distant users.
ViC Energy Efficiency
Fig. 10 plots the energy efficiency for IRSNOMA networks in the delaylimited transmission mode, with 5 W and S. The energy efficiency curve of nonorthogonal users for IRSNOMA with ipSIC/pSIC is plotted according to (34). It can be observed that the energy efficiency of nonorthogonal users outperforms that of IRSOMA. This is because that nonorthogonal users is capable of achieving the larger data rate in the delaylimited transmission mode. This phenomenon indicates that IRSNOMA networks have the ability to supply the higher energy efficiency. As a further advance, Fig. 11 plots the energy efficiency for IRSNOMA in delaytolerant transmission mode, with 15 W and S. We observe that the energy efficiency of nearest user is much larger than that of orthogonal user and distant users. This is due to that the nearest user is closed to the IRS and has a greater throughput. Another observation is that the energy efficiency of distant users for IRSNOMA converges to the constant value at high SNRs. The reason is that the distant users suffer from the nearby user’s interference, when it detects its own information.
Vii Conclusion
In this paper, an IRS has been invoked in downlink NOMA networks for enhancing the performance of multiple users, where 1bit coding scheme is taken into account. More specifically, we have derived the exact expressions of outage probability and ergodic rate for users with ipSIC/pSIC in IRSassisted NOMA networks. Based on the approximated analyses, the diversity order of the th user is related to the number of reflecting elements and channel ordering. With increasing the number of reflecting elements, the outage probability of users with ipSIC/pSIC for IRSNOMA networks decreases. Due to the interference from the nearby users’ signals, a of high SNR slope for ergodic rate is obtained by the distant users. Simulation results have shown that the outage behaviors of IRSNOMA are superior to that of IRSOMA, AF relaying and FD/HD DF relaying. The nearest user has a larger ergodic rate than orthogonal user and distant users. Finally, the throughput and energy efficiency of nonorthogonal users for IRSNOMA were discussed both in delaylimited and delaytolerant transmission modes. The application of IRS to NOMA provided a new degree of freedom to enhance the wireless link performance.
Appendix A: Proof of Theorem 1
The proof starts by assuming , and then
(A.1) 
Hence the outage probability of the th user with ipSIC need to further calculate . Applying the complementary set and some algebraic manipulations, it can be calculated as follows:
(A.2) 
where , , with and . Based on the previous assumption, the cascade channel gains from the BS to IRS and then to users with 1bit coding scheme are also sorted as . As a further advance, we focus our attention the CDF in the following part.
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