## I Introduction

With the purpose to meet the requirements of future radio access, the design of non-orthogonal multiple access (NOMA) technologies is important to enhance spectral efficiency and user access [1]. The major viewpoint of NOMA is to superpose multiple users by sharing radio resources (i.e., time/frequencey/code) over different power levels [2, 3, 4]. Then the desired signals are detected by exploiting the successive interference cancellation (SIC) [5]. Very recently, the integration of cooperative communication with NOMA has been widely discussed in many treaties [6, 7, 8, 9]. Cooperative NOMA has been proposed in [6], where the user with better channel condition acts as a decode-and-forward (DF) relay to forward information. 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 [7]. Considering the impact of imperfect channel state information (CSI), the authors in [8] investigated the performance of amplify-and-forward (AF) relay for downlink NOMA networks, where the exact and tight bounds of outage probability were derived. To further enhance spectrum efficiency, the performance of full-duplex (FD) cooperative NOMA was characterized in terms of outage behaviors [9], where user relaying was capable of switching operation between FD and HD mode.

Above existing treaties on cooperative NOMA are all based on one-way 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, two-way relay (TWR) technique introduced in [10] 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. In [11], the authors studied the outage behaviors of DF relay with perfect and imperfect CSI conditions. In terms of CSI and system state information (SSI), the system outage behavior was investigated for two-way full-duplex (FD) DF relay on different multi-user scheduling schemes [12].

Motivated by the above two technologies, we focus our attentions on the outage behaviors of TWR-NOMA systems, where two groups of NOMA users exchange messages with the aid of a relay node using DF protocol. Considering both perfect SIC (pSIC) and imperfect SIC (ipSIC), we derive the closed-form expressions of outage probabilities for users’ signals. To provide valuable insights, we further derive the asymptotic outage probabilities of users’ signals and obtain the diversity orders. We show that the outage performance of TWR-NOMA is superior to TWR-OMA in the low signal-to-noise ratio (SNR) regime. We demonstrate that the outage probabilities for TWR-NOMA converge to error floors due to the effect of interference signal (IS) at the relay. We confirm that the use of pSIC is incapable of overcoming the zero diversity order for TWR-NOMA. Additionally, we discuss the system throughput in delay-limited transmission mode.

## Ii System Model

We consider a two-way relay NOMA communication scenario which consists of one relay , two pairs of NOMA users and . Assuming that and are the nearby users in group and , respectively, while and are the distant users in group and , respectively. The exchange of information between user groups and is facilitated via the assistance of a decode-and-forward (DF) relay with two antennas, namely and . User nodes are equipped with single antenna and can transmit the superposed signals [13, 14]. In addition, we assume 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 quasi-static block Rayleigh fading channels and disturbed by additive white Gaussian noise with mean power . We denote that , , and are denoted as the complex channel coefficient of , , and links, respectively. The channel power gains , , and

are assumed to be exponentially distributed random variables (RVs) with the parameters

, , respectively. It is assumed that the perfect CSIs of NOMA users are available at for signal detection.During the first slot, the pair of NOMA users in transmit the signals to just as uplink NOMA. Due to is equipped with two antennas, when the 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 normalized 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 signal-to-interference-plus-noise ratio (SINR) at to detect is given by

(3) |

where denotes the transmit signal-to-noise 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 [15] 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 normalized transmission power at . 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 conditions. 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 is given by

(7) |

From above process, the exchange of information is achieved between the NOMA users for and .

## Iii Outage Probability

In this section, the performance of TWR-NOMA is characterized in terms of outage probability.

#### Iii-1 Outage Probability of

In TWR-NOMA, the outage events of are explained as follow: 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. Hence the outage probability of with ipSIC for TWR-NOMA is expressed as

(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 TWR-NOMA.

###### Theorem 1.

The closed-form expression for the outage probability of for TWR-NOMA 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 TWR-NOMA with pSIC is given by

(10) |

#### Iii-2 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 TWR-NOMA.

###### Theorem 2.

The closed-form 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 TWR-NOMA with pSIC is given by

(13) |

#### Iii-3 Diversity Order Analysis

To obtain deeper insights for TWR-NOMA systems, the asymptotic analysis are presented in high SNR regimes based on the derived outage probabilities. The diversity order is defined as

[16, 17](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.

#### Iii-4 Throughput Analysis

In delay-limited 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 TWR-NOMA with ipSIC/pSIC is calculated as [7]

(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.

Monte Carlo simulations repeated | iterations |
---|---|

Power allocation coefficients of NOMA | |

Targeted data rates | BPCU |

BPCU | |

Pass loss exponent | |

The distance between R and or | m |

The distance between R and or | m |

## Iv Numerical Results

In this section, numerical results are provide to investigate the impact levels of IS on outage probability for TWR-NOMA systems. The simulation parameters used are summarized in Table I, where BPCU is short for bit per channel use. Due to the reciprocity of channels between and , the outage behaviors of and in are presented to illustrate availability of TWR-NOMA. Without loss of generality, the power allocation coefficients of and are set as and , respectively. and are set to be and , respectively.

### Iv-a Outage Probability

Fig. 1 plots the outage probabilities of and with both ipSIC and pSIC versus SNR for simulation setting with and dB. The solid and dashed curves represent the exact theoretical performance of and for both ipSIC and pSIC, corresponding to the results derived in (1), (1) and (2), (2), respectively. Apparently, the outage probability curves match perfectly with Monte Carlo simulation results. As can be observed from the figure, the outage behaviors of and for TWR-NOMA are superior to TWR-OMA in the low SNR regime. This is due to the fact that the influence of IS is not the dominant factor at low SNR. Furthermore, another observation is that the pSIC is capable of enhancing the performance of NOMA compare to the ipSIC. In addition, the asymptotic curves of and with ipSIC/pSIC are plotted according to (1), (16) and (2), (16), respectively. It can be seen that the outage behaviors of and converge to the error floors in the high SNR regime. The reason can be explained that due to the impact of residual interference by the use of ipSIC, and result in zero diversity orders. Although the pSIC is carried out in TWR-NOMA system, and also obtain zero diversity orders. This is due to the fact that when the relay first detect the strongest signal in the first slot, it will suffer interference from the weaker signal. This observation verifies the conclusion Remark 1 in Section III.

Fig. 2 plots the outage probabilities of and versus SNR with the different impact levels of IS from to . The solid and dashed curves represent the outage behaviors of and with ipSIC/pSIC, respectively. As can be seen that when the impact level of IS is set to be , there is no IS between and at the relay, which can be viewed as a benchmark. Additionally, one can observed that with the impact levels of IS increasing, the outage performance of TWR-NOMA system degrades significantly. Hence it is crucial to hunt for efficient strategies for suppressing the effect of interference between antennas. Fig. 3 plots the outage probability versus SNR with different values of residual IS from dB to dB. It can be seen that the different values of residual IS affects the performance of ipSIC seriously. Similarly, as the values of residual IS increases, the preponderance of ipSIC is inexistent. When dB, the outage probability of and will be in close proximity to one. Therefore, it is important to design effective SIC schemes for TWR-NOMA.

Fig. 4 plots system throughput versus SNR in delay-limited transmission mode for TWR-NOMA with different values of residual IS from dB to dB. The blue solid curves represent throughput for TWR-NOMA with both pSIC and ipSIC, which can be obtained from (III-4). One can observe that TWR-NOMA is capable of achieving a higher throughput compared to TWR-OMA in the low SNR regime, since it has a lower outage probability. Moreover, the figure confirms that TWR-NOMA converges to the throughput ceiling in high SNR regimes. It is worth noting that ipSIC considered for TWR-NOMA will further degrade throughput with the values of residual IS becomes larger in high SNR regimes.

## V Conclusion

This paper has investigated the application of TWR to NOMA systems, in which two pairs of users can exchange their information between each other by the virtue of a relay node. The performance of TWR-NOMA has been characterized in terms of outage probability and ergodic rate for both ipSIC and pSIC. Furthermore, the closed-form expressions of outage probability for the NOMA users’ signals have been derived. Owing to the impact of IS at relay, there were the error floors for TWR-NOMA with ipSIC/pSIC in high SNR regimes and zero diversity orders were obtained. Based on the analytical results, it was shown that the performance of TWR-NOMA with ipSIC/pSIC outperforms TWR-OMA in the low SNR regime.

## Appendix A: Proof of Theorem 1

Substituting (3), (5) and (6) into (III-1), the outage probability of can be further given by

(A.1) |

where .

To calculate the probability in (Appendix A: Proof of Theorem 1), let . We first calculate the PDF of and then give the process derived of . As is known, follows the exponential distribution with the means , . Furthermore, we denote that , and are also independent exponentially distributed random variables (RVs) with means , and , respectively. Based on [18], for the independent non-identical distributed (i.n.d) fading scenario, the PDF of can be given by

(A.2) |

where , and .

According to the above explanations, is calculated as follows:

(A.3) |

Substituting (A.2) into (A.3) and after some algebraic manipulations, is given by

(A.4) |

where .

can be further calculated as follows:

(A.5) |

where with , with . Combining (A.4) and (Appendix A: Proof of Theorem 1), we can obtain (1). The proof is complete.

## Appendix B: Proof of Theorem 2

Substituting (3), (4), (6) and (7) into (III-2), the outage probability of is rewritten as

(B.1) |

where and .

Similar to (A.2), let , the PDF of is given by

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