I Introduction
With the purpose to improve system throughput and spectrum efficiency, the fifth generation (5G) mobile communication networks are receiving a great deal of attention. The requirements of 5G networks mainly contain key performance indicator (KPI) improvement and support for new radio (NR) scenarios [2], including enhanced mobile broadband (eMBB), massive machine type communications (mMTC), and ultrareliable and low latency communications (URLLC). Apart from crux technologies, such as massive multipleinput multipleoutput (MIMO), millimeter wave and heterogeneous networks, the design of novel multiple access (MA) techniques is significant to make the contributions for 5G networks. Driven by these, nonorthogonal multiple access (NOMA) has been viewed as one of promising technologies to increase system capacity and user access [3]. The basic concept of NOMA is to superpose multiple users by sharing radio resources (i.e., time/frequencey/code) over different power levels [4, 5]. Then the desired signals are detected by exploiting the successive interference cancellation (SIC) [6]. More specifically, downlink multiuser superposition (MUST) transmission [7], which is one of special case for NOMA has been researched for Long Term Evolution (LTE) in 3rd generation partnership project (3GPP) and approved as work item (WI) in radio access network (RAN) meeting.
Until now, pointtopoint NOMA has been discussed extensively in many research contributions [8, 9, 10, 11]. In [8], the authors have investigated the outage performance and ergodic rate of downlink NOMA with randomly deployed users by invoking stochastic geometry. Considering the secrecy issues of NOMA against external eavesdroppers, the authors in [9] investigated secrecy outage behaviors of NOMA in largerscale networks for both singleantenna and multipleantenna transmission scenarios. Explicit insights for understanding the asynchronous NOMA, a novel interference cancellation scheme was proposed in [10], where the bit error rate and throughput performance were analyzed. By the virtue of available CSI, the performance of NOMA based multicast cognitive radio scheme (MCRNOMA) was evaluated [11], in which outage probability and diversity order are obtained for both secondary and primary networks. Very recently, the application of cooperative communication [12] to NOMA is an efficient way to offer enhanced spectrum efficiency and spatial diversity. Hence the integration of cooperative communication with NOMA has been widely discussed in many treaties [13, 14, 15, 16]. Cooperative NOMA has been proposed in [13], where the user with better channel condition acts as a decodeandforward (DF) relay to forward information. Furthermore, in [14], the authors studied the ergodic rate of DF relay for a NOMA system. With the objective of improving energy efficiency, the application of simultaneous wireless information and power transfer (SWIPT) to the nearby user was investigated where the locations of NOMA users were modeled by stochastic geometry [15]. Considering the impact of imperfect channel state information (CSI), the authors in [16] investigated the performance of amplifyandforward (AF) relay for downlink NOMA networks, where the exact and tight bounds of outage probability were derived. Moreover, in [17], the outage behavior and ergodic sum rate of NOMA for AF relay was analyzed under Nakagami fading channels. To further enhance spectrum efficiency, the performance of fullduplex (FD) cooperative NOMA was characterized in terms of outage probability [18].
Above existing treaties on cooperative NOMA are all based on oneway relay scheme, where the messages are delivered in only one direction, (i.e., from the BS to the relay or user destinations). As a further advance, twoway relay (TWR) technique introduced in [19] has attracted remarkable interest as it is capable of boosting spectral efficiency. The basic idea of TWR systems is to exchange information between two nodes with the help of a relay, where AF or DF protocol can be employed. With the emphasis on user selection, in [20], the authors analyzed the performance of multiuser TWR channels for halfduplex (HD) AF relays. By applying physicallayer network coding (PNC) schemes, the performance of twoway AF relay systems was investigated in terms of outage probability and sum rate [21]. It was shown that two time slots PNC scheme achieves a higher sum rate compared to four time slot transmission mode. In [22], the authors studied the outage behaviors of DF relay with perfect and imperfect CSI conditions, where a new relay selection scheme was proposed to reduce the complexity of TWR systems. In terms of CSI and system state information, the system outage behavior was investigated for twoway fullduplex (FD) DF relay on different multiuser scheduling schemes [23]. In [24], the authors investigated the performance of multiantenna TWR networks in which both AF and DF protocols are examined, respectively. Taking residual selfinterference into account, the tradeoffs between the outage probability and ergodic rate were analyzed in [25] for FD TWR systems. In addition, the authors in [26] studied the performance of cooperative spectrum sharing by utilizing TWR over general fading channels. It was worth mentioning that the effective spectrum sharing is achieved by restraining additional cooperative diversity order.
Ia Motivations and Contributions
While the aforementioned theoretical researches have laid a solid foundation for the understanding of NOMA and TWR techniques in wireless networks, the TWRNOMA systems are far from being well understood. Obviously, the application of TWR to NOMA is a possible approach to improve the spectral efficiency of systems. To the best of our knowledge, there is no contributions to investigate the performance of TWR for NOMA systems. Moreover, the above contributions for NOMA have been comprehensively studied under the assumption of perfect SIC (pSIC). In practical scenarios, there still exist several potential implementation issues with the use of SIC (i.e., complexity scaling and error propagation). More precisely, these unfavorable factors will lead to errors in decoding. Once an error occurs for carrying out SIC at the nearby user, the NOMA systems will suffer from the residual interference signal (IS). Hence it is significant to examine the detrimental impacts of imperfect SIC (ipSIC) for TWRNOMA. Motivated by these, we investigate the performance of TWRNOMA with ipSIC/pSIC in terms of outage probability, ergodic rate and energy efficiency, where two groups of NOMA users exchange messages with the aid of a relay node using DF protocol.
The essential contributions of our paper are summarized as follows:

We derive the closedform expressions of outage probability for TWRNOMA with ipSIC/pSIC. Based on the analytical results, we further derive the corresponding asymptotic outage probabilities and obtain the diversity orders. Additionally, we discuss the system throughput in delaylimited transmission mode.

We show that the outage performance of TWRNOMA is superior to TWROMA in the low signaltonoise ratio (SNR) regime. We observe that due to the effect of IS at the relay, the outage probabilities for TWRNOMA converge to error floors in the high SNR regime. We confirm that the use of pSIC is incapable of overcoming the zero diversity order for TWRNOMA.

We study the ergodic rate of users’ signals for TWRNOMA with ipSIC/pSIC. To gain more insights, we discuss one special case that when there is no IS between a pair of antennas at the relay. On the basis of results derived, we obtain the zero high SNR slopes for TWRNOMA systems. We demonstrate that the ergodic rates for TWRNOMA converge to throughput ceilings in high SNR regimes.

We analyze the energy efficiency of TWRNOMA with ipSIC/pSIC in both the delaylimited and tolerant transmission modes. We confirm that TWRNOMA with ipSIC/pSIC in delaylimited transmission mode has almost the same energy efficiency. Furthermore, in delaytolerant transmission mode, the energy efficiency of system with pSIC is higher than that of system with ipSIC.
IB Organization and Notation
The remainder of this paper is organised as follows. In Section II, the system mode for TWRNOMA is introduced. In Section III, the analytical expressions for outage probability, diversity order and system throughput of TWRNOMA are derived. Then the ergodic rates of users’ signals for TWRNOMA are investigated in Section IV. The system energy efficiency is evaluated in Section V. Analytical results and numerical simulations are presented in Section VI, which is followed by our conclusions in Section VII.
The main notations of this paper is shown as follows: denotes expectation operation; and
denote the probability density function (PDF) and the cumulative distribution function (CDF) of a random variable
.Ii System Model
Iia System Description
We focus our attentions on a twoway relay NOMA communication scenario which consists of one relay , two pairs of NOMA users and ^{1}^{1}1The geographical dimensions of clusters and are to ensure that there is a certain distance difference from distant user and nearby user to .. To reduce the complexity of systems, many research contributions on NOMA have been proposed to pair two users for the application of NOMA protocol^{2}^{2}2Note that increasing the number of paired users, i,e,. pairs of users, will not affect the performance of TWRNOMA system. It is worth pointing that within each group, superposition coding and SIC are employed, and across the groups, transmissions are orthogonal. [27, 28]. As shown in Fig. 1, we assume that and are the nearby users in groups and , respectively, while and are the distant users in groups and , respectively. It is worth noting that the nearby user and distant user are distinguished based on the distance from the users to [29]. For example, and are near to , while and are far away from . The exchange of information between user groups and is facilitated via the assistance of a decodeandforward (DF) relay with two antennas, namely and ^{3}^{3}3For the practical scenario, we can assume that the relay is located on a mountain, where the user nodes on both sides of the mountain are capable of exchanging the information between each other.. User nodes are equipped with single antenna. In practical communication process, the complexity of DF protocol is too high to implement. To facilitate analysis, we focus our attention on a idealized DF protocol, where is capable of decoding the users’ information correctly. Relaxing this idealized assumption can make system mode close to the practical scenario, but this is beyond the scope of this treatise. Additionally, to evaluate the impact of error propagation on TWRNOMA, ipSIC operation is employed at relay and nearby users. It is assumed that the direct links between two pairs of users are inexistent due to the effect of strong shadowing. Without loss of generality, all the wireless channels are modeled to be independent quasistatic block Rayleigh fading channels and disturbed by additive white Gaussian noise with mean power . Furthermore, , , and are denoted as the complex channel coefficient of , , and links, respectively. We assume that the channels from user nodes to and the channels from to user nodes are reciprocal. In other words, the channels from user nodes to have the same fading impact as the channels from to the user nodes [30, 25, 31]. The channel power gains , , and
are assumed to be exponentially distributed random variables (RVs) with the parameters
, , respectively. Note that the perfect CSIs of NOMA users are available at for signal detection.IiB Signal Model
During the first slot, the pair of NOMA users in transmit the signals to just as uplink NOMA. Since is equipped with two antennas, when receives the signals from the pair of users in , it will suffer from interference signals from the pair of users in . More precisely, the observation at for is given by
(1) 
where denotes IS from with . denotes the impact levels of IS at . is the transmission power at user nodes.
, and , are the signals of , and , , respectively, i.e, . , and , are the corresponding power allocation coefficients. Note that the efficient uplink power control is capable of enhancing the performance of the systems considered, which is beyond the scope of this paper. denotes the Gaussian noise at for , .
Similarly, when receives the signals from the pair of users in , it will suffer from interference signals from the pair of users in as well and then the observation at is given by
(2) 
where denotes the interference signals from with .
Applying the NOMA protocol, first decodes ’s information by the virtue of treating as IS. Hence the received signaltointerferenceplusnoise ratio (SINR) at to detect is given by
(3) 
where denotes the transmit signaltonoise ratio (SNR), , .
After SIC is carried out at for detecting , the received SINR at to detect is given by
(4) 
where and denote the pSIC and ipSIC employed at , respectively. Due to the impact of ipSIC, the residual IS is modeled as Rayleigh fading channels [32] denoted as
with zero mean and variance
.In the second slot, the information is exchanged between and by the virtue of . Therefore, just like the downlink NOMA, transmits the superposed signals and to and by and , respectively. and denote the power allocation coefficients of and , while and are the corresponding power allocation coefficients of and , respectively. is the transmission power at and we assume . In particular, to ensure the fairness between users in and , a higher power should be allocated to the distant user who has the worse channel condition. Hence we assume that with and with . Note that the fixed power allocation coefficients for two groups’ NOMA users are considered. Relaxing this assumption will further improve the performance of systems and should be concluded in our future work.
According to NOMA protocol, SIC is employed and the received SINR at to detect is given by
(5) 
where denotes the impact level of IS at the user nodes. Then detects and gives the corresponding SINR as follows:
(6) 
Furthermore, the received SINR at to detect can be given by
(7) 
From above process, the exchange of information is achieved between the NOMA users for and . More specifically, the signal of is exchanged with the signal of . Furthermore, the signal of is exchanged with the signal of .
Iii Outage Probability
In this section, the performance of TWRNOMA is characterized in terms of outage probability. Due to the channel’s reciprocity, the outage probability of and are provided in detail in the following part.
Iii1 Outage Probability of
In TWRNOMA system, the outage events of are explained as: i) cannot decode correctly; ii) The information cannot be detected by ; and iii) cannot detect , while can first decode successfully. To simplify the analysis, the complementary events of are employed to express its outage probability. As a consequence, the outage probability of with ipSIC for TWRNOMA system can be given by
(8) 
where , and . with being the target rate at to detect and with being the target rate at to detect .
The following theorem provides the outage probability of for TWRNOMA.
Theorem 1.
The closedform expression for the outage probability of for TWRNOMA with ipSIC is given by
(9) 
where . , and . . , and . . with and with .
Proof.
See Appendix A. ∎
Corollary 1.
Based on (1), for the special case , the outage probability of for TWRNOMA with pSIC is given by
(10) 
Iii2 Outage Probability of
Based on NOMA principle, the complementary events of outage for have the following cases. One of the cases is that can first decode the information and then detect . Another case is that either of and can detect successfully. Hence the outage probability of can be expressed as
(11) 
where , and .
The following theorem provides the outage probability of for TWRNOMA.
Theorem 2.
The closedform expression for the outage probability of with ipSIC is given by
(12) 
where . and . , .
Proof.
See Appendix B. ∎
Corollary 2.
For the special case, substituting into (2), the outage probability of for TWRNOMA with pSIC is given by
(13) 
Iii3 Diversity Order Analysis
In order to gain deeper insights for TWRNOMA systems, the asymptotic analysis are presented in high SNR regimes based on the derived outage probabilities. The diversity order is defined as [33]
(14) 
where denotes the asymptotic outage probability of .
Proposition 1.
Remark 1.
An important conclusion from above analysis is that due to impact of residual interference, the diversity order of with the use of ipSIC is zero. Additionally, the communication process of the first slot similar to uplink NOMA, even though under the condition of pSIC, diversity order is equal to zero as well for . As can be observed that there are error floors for with ipSIC/pSIC.
Proposition 2.
Remark 2.
Based on above analytical results of , the diversity orders of with ipSIC/pSIC are also equal to zeros. This is because residual interference is existent in the total communication process.
Iii4 Throughput Analysis
In delaylimited transmission scenario, the BS transmits message to users at a fixed rate, where system throughput will be subject to wireless fading channels. Hence the corresponding throughput of TWRNOMA with ipSIC/pSIC is calculated as [15, 34]
(19) 
where . and with ipSIC/pSIC can be obtained from (1) and (1), respectively, while and with ipSIC/pSIC can be obtained from (2) and (2), respectively.
Iv Ergodic rate
In this section, the ergodic rate of TWRNOMA is investigated for considering the influence of signal’s channel fading to target rate.
Iv1 Ergodic Rate of
Since can be detected at the relay as well as at successfully. By the virtue of (3) and (6), the achievable rate of for TWRNOMA is written as . In order to further calculate the ergodic rate of , using , the corresponding CDF is presented in the following lemma.
Lemma 1.
The CDF for is given by (20) at the top of the next page, where and , , . and .
Proof.
See Appendix C. ∎
(20) 
Substituting (20), the corresponding ergodic rate of is given by
(21) 
where and . Unfortunately, it is difficult to obtain the closedform expression from (21). However, it can be evaluated by applying numerical approaches. To further obtain analytical results, we consider the special cases of with ipSIC/pSIC for TWRNOMA where there is no IS between the pair of antennas at the relay in the following part.
Based on the above analysis, for the special case that substituting into (21), the ergodic rate of with ipSIC can be obtained in the following theorem.
Theorem 3.
The closedform expression of ergodic rate for with ipSIC for TWRNOMA is given by
(22) 
where , and ; , and . is the exponential integral function [35, Eq. (8.211.1)].
Proof.
See Appendix D. ∎
Corollary 3.
Based on (3), the ergodic rate of for pSIC with can be expressed in the closed form as
(23) 
Iv2 Ergodic Rate of
On the condition that the relay and are capable of detecting , can be also detected by successfully. As a consequence, combining (4), (5) and (7), the achievable rate of is written as . The corresponding ergodic rate of can be expressed as
(24) 
where with and . To the best of authors’ knowledge, (24) does not have a closed form solution. We also consider the special cases of by the virtue of ignoring IS between the pair of antennas at the relay.
For the special case that substituting into (24) and after some manipulations, the ergodic rates of with ipSIC/pSIC is given by
(25) 
and
(26) 
respectively, where with .
As can be seen from the above expressions, the exact analysis of ergodic rates require the computation of some complicated integrals. To facilitate these analysis and provide the simpler expression for the ergodic rate of with ipSIC/pSIC, the following theorem and corollary provide the high SNR approximations to evaluate the performance.
Theorem 4.
The approximation expression for ergodic rate of with ipSIC at high SNR is given by
(27) 
Proof.
See Appendix E. ∎
Corollary 4.
For the special case with , the ergodic rate of for pSIC can be approximated at high SNR as
(28) 
Iv3 Slope Analysis
In this subsection, by the virtue of asymptotic results, we characterize the high SNR slope which is capable of capturing the influence of channel parameters on the ergodic rate. The high SNR slope is defined as
(29) 
where denotes the asymptotic ergodic rate of .
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