# Exploiting Full/Half-Duplex User Relaying in NOMA Systems

In this paper, a novel cooperative non-orthogonal multiple access (NOMA) system is proposed, where one near user is employed as decode-and-forward (DF) relaying switching between full-duplex (FD) and half-duplex (HD) mode to help a far user. Two representative cooperative relaying scenarios are investigated insightfully. The first scenario is that no direct link exists between the base station (BS) and far user. The second scenario is that the direct link exists between the BS and far user. To characterize the performance of potential gains brought by FD NOMA in two considered scenarios, three performance metrics outage probability, ergodic rate and energy efficiency are discussed. More particularly, we derive new closed-form expressions for both exact and asymptotic outage probabilities as well as delay-limited throughput for two NOMA users. Based on the derived results, the diversity orders achieved by users are obtained. We confirm that the use of direct link overcomes zero diversity order of far NOMA user inherent to FD relaying. Additionally, we derive new closed-form expressions for asymptotic ergodic rates. Based on these, the high signal-to-noise radio (SNR) slopes of two users for FD NOMA are obtained. Simulation results demonstrate that: 1) FD NOMA is superior to HD NOMA in terms of outage probability and ergodic sum rate in the low SNR region; and 2) In delay-limited transmission mode, FD NOMA has higher energy efficiency than HD NOMA in the low SNR region; However, in delay-tolerant transmission mode, the system energy efficiency of HD NOMA exceeds FD NOMA in the high SNR region.

## Authors

• 8 publications
• 33 publications
• 7 publications
• 35 publications
• 36 publications
• ### Modeling and Analysis of Two-Way Relay Non-Orthogonal Multiple Access Systems

A two-way relay non-orthogonal multiple access (TWR-NOMA) system is inve...
01/20/2019 ∙ by Xinwei Yue, et al. ∙ 0

• ### Performance Analysis of NOMA with Fixed Gain Relaying over Nakagami-m Fading Channels

This paper studies the application of cooperative techniques for non-ort...
12/18/2018 ∙ by Xinwei Yue, et al. ∙ 0

• ### On the Outage Analysis and Finite SNR Diversity-Multiplexing Tradeoff of Hybrid-Duplex Systems for Aeronautical Communications

A hybrid-duplex aeronautical communication system (HBD-ACS) consisting o...
12/21/2017 ∙ by Tan Zheng Hui Ernest, et al. ∙ 0

• ### Reconfigurable Intelligent Surface Aided NOMA Networks

Reconfigurable intelligent surfaces (RISs) constitute a promising perfor...
12/20/2019 ∙ by Tianwei Hou, et al. ∙ 0

• ### Spatially Random Relay Selection for Full/Half-Duplex Cooperative NOMA Networks

This paper investigates the impact of relay selection (RS) on the perfor...
12/21/2018 ∙ by Xinwei Yue, et al. ∙ 0

• ### Analysis of Outage Probabilities for Cooperative NOMA Users with Imperfect CSI

Non-orthogonal multiple access (NOMA) is a promising spectrally-efficien...
09/25/2018 ∙ by Xuesong Liang, et al. ∙ 0

• ### MIMO Assisted Networks Relying on Large Intelligent Surfaces: A Stochastic Geometry Model

Large intelligent surfaces (LISs) constitute a promising performance enh...
10/02/2019 ∙ by Tianwei Hou, et al. ∙ 0

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## I Introduction

With the rapid increasing demand of wireless networks, the requirements for efficiently exploiting the spectrum is of great significance in new radio (NR) usage scenarios [2]. To achieve higher spectral efficiency of the fifth generation (5G) mobile communication network, non-orthogonal multiple access (NOMA) has received a great deal of attention  [3]. Recently, several NOMA schemes have been researched in detail, such as power domain NOMA (PD-NOMA) [4], sparse code multiple access (SCMA) [5], pattern division multiple access (PDMA) [6], multiuser sharing access (MUSA) [7]

, etc. Generally speaking, NOMA schemes can be classified into two categories, namely power-domain NOMA

111In this paper, we focus on power-domain NOMA and use NOMA to represent PD-NOMA. and code-domain NOMA. Downlink multiuser superposition transmission (DL MUST), the special case of NOMA, has been studied for 3rd generation partnership project (3GPP) in [8]. The pivotal characteristic of NOMA is to allow multiple users to share the same physical resource (i.e., time/fequency/code) via different power levels. At the receiver side, the successive interference cancellation (SIC) is carried out [9].

So far, point-to-point NOMA has been studied extensively in [10, 11, 12, 13, 14]. To evaluate the performance of uplink NOMA systems, the authors in [10] proposed the uplink NOMA transmission scheme to achieve higher system rate. The expressions of outage probability and achievable sum rates for uplink NOMA were derived with a novel uplink control protocol in [11]. Regarding downlink NOMA scenarios, authors in [12] analyzed the outage behavior and ergodic rates of NOMA networks, where multiple NOMA users are spatial randomly deployed in a disc. In [13], the cognitive radio inspired NOMA concept was proposed, in which the influence of user pairing with the fixed power allocation in NOMA systems was discussed. As the interplay between NOMA and cognitive radio is bidirectional, NOMA was also applied to cognitive radio networks in [14]. More particularly, the analytical expressions of outage probability was derived and diversity orders were characterized. Apart from the above works, a new opportunistic NOMA scheme was proposed in [15] to improve the efficiency of SIC. In [16], the flexible power allocation mode was researched in terms of outage probability for hybrid NOMA systems. The quantum-assisted multiple users transmission mode for NOMA was proposed in [17], which utilizes the minimum bit error ratio criterion to optimize the predefined transmitted information. The author of [18] has studied the linear MUST scheme for NOMA to maximize sum rate of the entire network. With the emphasis on physical layer security, in [19], authors have adopted two effective approaches, namely protected zone and artificial noise for enhancing the secrecy performance of NOMA networks with the aid of stochastic geometry.

Cooperative communication is a particularly effective approach by providing the higher diversity as well as extending the coverage of networks [20]. Current NOMA research contributions in terms of cooperative communication mainly include two aspects. The first aspect is the application of NOMA into cooperative networks [21, 22, 23, 24]. The coordinated two-point system with superposition coding (SC) was researched in the downlink communication in [21]. The authors in [22, 23] investigated outage probability and system capacity of decode-and-forward (DF) relaying for NOMA. In [24], the outage behavior of amplify-and-forward (AF) relaying with NOMA has been discussed over Nakagami- fading channels. The second aspect is cooperative NOMA, which was first proposed in [25]. The key idea of cooperative NOMA is to regard the near NOMA user as a DF user relaying to help far NOMA user. On the standpoint of considering energy efficiency issues, simultaneous wireless information and power transfer (SWIPT) was employed at the near NOMA user, which was regarded as DF relaying in [26].

Although cooperative NOMA is capable of enhancing the performance gains for far user, it results in additional bandwidth costs for the system. To avoid this issue, one promising solution is to adopt the full-duplex (FD) relay technology. FD relay receives and transmits simultaneously in the same frequency band, which is the reason why it has attracted significant interest to realize more spectrally efficient systems [27]. In a general case, due to the imperfect isolation or cancellation process, FD operation may suffer from residual loop self-interference (LI) which is modeled as a fading channel. With the development of signal processing and antenna technologies, relaying with FD operation is feasible [28]. Recently, FD relay technologies have been proposed as a promising technique for 5G networks in [29]. Two main types of FD relay techniques, namely FD AF relaying and FD DF relaying, have been discussed in [30, 31, 32]. The expressions for outage probability of FD AF relaying were provided in [30], which considers the processing delay of relaying in practical scenarios. In [31], the performance of FD AF relaying in terms of outage probability was investigated considering the direct link. The authors in [32] characterized the outage performance of FD DF relaying. It is demonstrated that the optimal duplex mode can be selected according to the outage probability. Furthermore, the operations of randomly switching between FD and HD mode were considered for enhancing spectral efficiency in [33].

### I-a Motivations and Related Works

While the aforementioned research contributions have laid a solid foundation with providing a good understanding of cooperative NOMA and FD relay technology, the treatises for investigating the potential benefits by integrating these two promising technologies are still in their infancy. Some related cooperative NOMA studies have been investigated in [25, 34]. In [25], it is demonstrated that the maximum diversity order can be obtained for all users, but cooperative NOMA with a direct link was only considered with HD operation mode. In [34]

, the authors investigated the performance of FD device-to-device based cooperative NOMA. However, only the outage performance of far user was analyzed. To the best of our knowledge, there is no existing work investigating the impact of the direct link for FD user relaying on the network performance, which motivates us to develop this treatise. Also, there is lack of systematic performance evaluation metrics i.e., considering ergodic rate and energy efficiency in terms of FD/HD NOMA systems. Different from

[25, 34], we present a comprehensive investigation on adopting near user as a FD/HD relaying to improve the reliability of far user. More specifically, we attempt to explore the potential ability of user relaying in NOMA networks with identifying the following key impact factors.

• Will FD NOMA relaying bring performance gains compared to HD NOMA relaying? If yes, what is the condition?

• What is the impact of direct link on the considered system? Will it significantly improve the network performance in terms of outage probability and throughput?

• Will NOMA relaying bring performance gains compared to conventional orthogonal multiple access (OMA) relaying?

• In delay-limited/tolerant transmission modes, what are the relationships between energy efficiency (EE) and HD/FD NOMA systems?

### I-B Contributions

In this paper, we propose a comprehensive NOMA user relaying system, where near user can switch between FD and HD mode according to the channel conditions. We also consider the setting of two scenarios in which the direct link exists or not between the BS and far user. Based on our proposed NOMA user relaying systems, the primary contributions of this paper are summarised as follows:

1. Without direct link: We derive the closed-form expressions of outage probability for the near user and far user, respectively. For obtaining more insights, we further derive the asymptotic outage probability of two users and obtain diversity orders at high SNR. We demonstrate that FD NOMA converges to an error floor and results in a zero diversity order. We show that FD NOMA is superior to HD NOMA in terms of outage probability in the low SNR region rather than in the high SNR region. In addition, we also obtain the diversity orders of two users for HD NOMA. Furthermore, we analyze the system throughput in delay-limited transmission according to the derived outage probability.

2. Without direct link: We study the ergodic rate of two users for FD/HD NOMA. To gain better insights, we derive the asymptotic ergodic rates of two users and obtain the high SNR slopes. We demonstrate that the ergodic rate of far user converges to a throughput ceiling for FD/HD NOMA in the high SNR region. Moreover, we also demonstrate that FD NOMA outperforms HD NOMA in terms of ergodic sum rate in the low SNR region.

3. With direct link: We first derive the closed-form expression in terms of outage probability for far user. In order to get the corresponding diversity order, we also derive the approximated outage probability of far user. We find that the reliability of far user is improved with the help of direct link. We confirm that the use of direct link overcomes the zero diversity order of far user inherent to conventional FD relaying. For the near user, the diversity order is the same as that of FD relaying. Additionally, we conclude that the superiority of FD NOMA is no longer apparent with the values of LI increasing.

4. With direct link: We analyze the ergodic rate of far user for FD/HD NOMA. For this scenario, it is the fact that the ergodic rate of near user is invariant which is not affected by the direct link. Similarly, we also derive the approximated expressions for ergodic rate and obtain the high SNR slopes. We demonstrate that the use of direct link is incapable of assisting far user to obtain additional high SNR slope.

5. Energy efficiency: We derive expressions in terms of energy efficiency for FD/HD NOMA. We conclude that FD NOMA without/with direct link have a higher energy efficiency corresponding to HD NOMA in the low SNR region for delay-limited transmission mode. However, in delay-tolerant transmission mode, the system energy efficiency of HD NOMA exceeds FD NOMA without/with direct link.

### I-C Organization and Notation

The rest of the paper is organized as follows. In Section II, the system model of user relaying for FD NOMA is set up. In Section III, the analytical expressions for outage probability, diversity order and throughput of FD/HD user relaying are derived and analyzed. In Section IV, the performance of user relaying for FD/HD NOMA are evaluated in terms of ergodic rate. Section V considers the system energy efficiency for FD/HD NOMA systems. Analytical results and simulations are presented in Section VI. Section VII concludes the paper.

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

; represents “be proportional to”.

## Ii System Model

We consider a FD cooperative NOMA system consisting of one source, i.e, the BS, that intends to communicate with far user via the assistance of near user illustrated in Fig. 1. is regarded as user relaying and DF protocol is employed to decode and forward the information to . To enable FD communication, is equipped with one transmit antenna and one receive antenna, while the BS and are single-antenna nodes. Note that can switch operation between FD and HD mode. All wireless links in network are assumed to be independent non-selective block Rayleigh fading and are disturbed by additive white Gaussian noise with mean power . , , 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. When operates in FD mode, we assume that an imperfect self-interference cancellation scheme222LI refers to the signals that are transmitted by a FD relaying and looped back to the receiver simultaneously. Through radio frequency (RF) cancellation, antenna cancellation and signal process technologies, etc, those LI can be suppressed to a lower level. However, LI still remains in the receiver due to imperfect self-interference cancellation, when decoding the desired signal. is executed at such as in[31, 35]. The LI is modeled as a Rayleigh fading channel with coefficient , and is the corresponding average power. To analyze HD NOMA, we introduce the switching operation factor detailed in the following.

During the -th time slot, according to [12], receives the superposed signal and loop interference signal simultaneously. The observation at is given by

 yD1[k]= h1(√a1Psx1[k]+√a2Psx2[k]) +hLI√ϖPrxLI[k−τ]+nD1[k], (1)

where is the switching operation factor between FD and HD mode. and denote working in FD and HD mode, respectively. Based on the practical application scenarios, we can select the different operation mode. denotes loop interference signal and denotes the processing delay at with an integer . More particularly, we assume that the time satisfies the relationship . and are the normalized transmission powers at the BS and , respectively. and are the signals for and , respectively. and are the corresponding power allocation coefficients. To stipulate better fairness between the users, we assume that with . The SIC333It is assumed that perfect SIC is employed at , our future work will relax this ideal assumption. [36] can be invoked by for first detecting having a larger transmit power, which has less inference signal. Then the signal of can be detected from the superposed signal. Therefore, the received signal-to-interference-plus-noise ratio (SINR) at to detect ’s message is given by

 γD2→D1=|h1|2a2ρ|h1|2a1ρ+ϖ|hLI|2ρ+1, (2)

where is the transmit signal-to-noise radio (SNR). Note that and are supposed to be normalized unity power signals, i.e, .

After SIC, the received SINR at to detect its own message is given by

 γD1=|h1|2a1ρϖ|hLI|2ρ+1. (3)

In the FD mode, the received signal at is written as . However, the observation at for the direct link is written as . Due to the existence of residue interference (RI) from relaying link, the received SINR at to detect for direct link is given by

 γRI1,D2=|h0|2a2ρ|h0|2a1ρ+κ|h2|2ρ+1, (4)

where denotes the impact levels of RI. Since DF relaying protocol is invoked in , we assume that can decode and forward the signal to successfully for relaying link from to . As a consequence, the observation at for relaying link is written as . Similarly, considering the impact of RI from direct link, the received SINR at to for relaying link is given by

 γRI2,D2=|h2|2ρκ|h0|2ρ+1. (5)

As stated in [33, 31], the relaying link corresponding to direct link from BS to has small time delay for any transmitted signals. In other words, there is some temporal separation between the signal from and BS. To derive the theoretical results for practical NOMA systems, we assume that these signals from and BS are fully resolvable by [34]. Hence, we provide the upper bounds of (4) and (5) in the following parts, which are the received SINRs at to detect for direct link and relaying link, i.e.

 γ1,D2=|h0|2a2ρ|h0|2a1ρ+1, (6)

and

 γ2,D2=|h2|2ρ, (7)

respectively. At this moment, the signals from the relaying link and direct link are combined by maximal ratio combining  (MRC) at

. So the received SINR after MRC at is given by

 γMRCD2=|h2|2ρ+|h0|2a2ρ|h0|2a1ρ+1. (8)

## Iii Outage Probability

When the target rate of users is determined by its quality of service (QoS), the outage probability is an important metric for performance evaluation. We will evaluate the outage performance in two representative scenarios in the following.

### Iii-a User Relaying without Direct Link

In this subsection, the first scenario is investigated in terms of outage probability.

#### Iii-A1 Outage Probability of D1

According to NOMA protocol, the complementary events of outage at can be explained as: can detect as well as its own message . From the above description, the outage probability of is expressed as

 PFDD1=1−Pr(γD2→D1>γFDth2,γD1>γFDth1), (9)

where . 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 FD NOMA.

###### Theorem 1.

The closed-form expression for the outage probability of is given by

 PFDD1=1−Ω1Ω1+ρϖθ1ΩLIe−θ1Ω1, (10)

where . , and . Note (10) is derived on the condition of .

###### Proof.

By definition, denotes the complementary event at and is calculated as

 J1= = ∫∞0∫∞(xϖρ+1)θ1f|hLI|2(x)f|h1|2(y)dxdy = Ω1Ω1+ρϖθ1ΩLIe−θ1Ω1. (11)

Substituting (1) into (9), (10) can be obtained and the proof is completed. ∎

###### Corollary 1.

Based on (10), the outage probability of for HD NOMA with is given by

 PHDD1=1−e−θ2Ω1, (12)

where and denote the target SNRs at to detect and with HD mode, respectively. , and with .

#### Iii-A2 Outage Probability of D2

The outage events of can be explained as below. The first is that cannot detect . The second is that cannot detect its own message on the conditions that can detect successfully. Based on these, the outage probability of is expressed as

 PFDD2,nodir= Pr(γD2→D1<γFDth2) +Pr(γ2,D2<γFDth2,γD2→D1>γFDth2), (13)

where .

The following theorem provides the outage probability of for FD NOMA.

###### Theorem 2.

The closed-form expression for the outage probability of without direct link is given by

 PFDD2,nodir=1−Ω1Ω1+ρϖτ1ΩLIe−⎛⎜⎝τ1Ω1+γFDth2ρΩ2⎞⎟⎠, (14)

where .

###### Proof.

By definition, and denote the first and second outage events, respectively. The process calculated is given by

 J2= Pr(|h1|2<τ1(ϖ|hLI|2ρ+1)) = ∫∞0∫τ1(ϖyρ+1)0f|h1|2(x)f|hLI|2(y)dxdy = 1−Ω1Ω1+ρϖτ1ΩLIe−τ1Ω1. (15)

Applying some algebraic manipulations, is given by:

 J3=Ω1Ω1+ρϖτ1ΩLIe−τ1Ω1⎛⎜⎝1−e−γFDth2ρΩ2⎞⎟⎠. (16)

Combining (2) and (16), (14) can be obtained and the proof is completed. ∎

###### Corollary 2.

Based on (14), the outage probability of without direct link for HD NOMA with is given by

 PHDD2,nodir=1−e−τ2Ω1−γHDth2ρΩ2. (17)

#### Iii-A3 Diversity Analysis

To get more insights, the asymptotic diversity analysis is provided in terms of outage probability investigated in high SNR region. The diversity order is defined as

 d=−limρ→∞log(P∞D(ρ))logρ. (18)
##### D1 for FD NOMA case

Based on analytical result in (10), when , the asymptotic outage probability of for FD NOMA with is given by

 PFD,∞D1=1−Ω1Ω1+ρθ1ΩLI. (19)

Substituting (19) into (18), we can obtain .

###### Remark 1.

The diversity order of is zero, which is the same as the conventional FD relaying.

##### D1 for HD NOMA case

Based on analytical result in (12), the asymptotic outage probability of for HD NOMA is given by

 PHD,∞D1=θ2Ω1∝1ρ. (20)

Substituting (20) into (18), we can obtain .

##### D2 for FD NOMA case

Based on (14), the asymptotic outage probability of for FD NOMA is given by

 PFD,∞D2,nodir=1−Ω1Ω2ρ−Ω1γFDth2−τ1ρΩ2Ω2ρ(Ω1+τ1ρΩLI). (21)

Substituting (21) into (18), we can obtain .

###### Remark 2.

The diversity order of is zero, which is the same as in FD NOMA.

##### D2 for HD NOMA case

Based on (17), the asymptotic outage probability of for HD NOMA is given by

 PHD,∞D2,nodir=γHDth2ρΩ2+τ2Ω1∝1ρ. (22)

Substituting (22) into (18), we can obtain .

###### Remark 3.

As can be observed that and are a constant independent of , respectively. Substituting (19) and (21) into (18), we see that there are the error floors for outage probability of two users.

#### Iii-A4 Throughput Analysis

In this subsection, the delay-limited transmission mode [26, 37] is considered for FD/HD NOMA.

##### FD NOMA case

In this mode, the BS transmits information at a constant rate , which is subject to the effect of outage probability due to wireless fading channels. The system throughput of FD NOMA without direct link is given by

 RFDl_nodir=(1−PFDD1)R1+(1−PFDD2,nodir)R2, (23)

where and are given in (10) and (14), respectively.

##### HD NOMA case

Similar to (23), the system throughput of HD NOMA without direct link is given by

 (24)

where and are given in (12) and (17), respectively.

### Iii-B User Relaying with Direct Link

In this subsection, we explore a more challenging scenario, where the direct link between the BS and is used to convey information and system reliability can be improved. However, the outage probability of will not be affected by the direct link. As such, we only show outage probability of in the following.

#### Iii-B1 Outage Probability of D2

For the second scenario, the outage events of for FD NOMA is described as below. One of the events is when can be detected at , but the received SINR after MRC at in one slot is less than its target SNR. Another event is that neither nor can detect . Therefore, the outage probability of is expressed as

 PFD,RID2,dir= Pr(γRI1,D2+γRI2,D2<γFDth2,γD2→D1>γFDth2) +Pr(γD2→D1<γFDth2,γRI1,D2<γFDth2). (25)

Unfortunately, the closed-form expression of (III-B1) for can not be derived successfully. However, it can be evaluated by using numerical simulations. To further obtain a theoretical result for , exploiting the upper bounds of received SINRs derived in (6) and (7), the outage probability of is expressed as

 PFDD2,dir= Pr(γMRCD2<γFDth2,γD2→D1>γFDth2) +Pr(γD2→D1<γFDth2,γ1,D2<γFDth2), (26)

where .

The following theorem provides the outage probability of for FD NOMA.

###### Theorem 3.

The closed-form expression for the outage probability of with direct link is given by

 PFDD2,dir={1−e−τ1Ω0−∞∑n=0(−1)neφn!ϕ2n+1[(−1)2n+1ϕ1n+1(n+1)! ×(Ei(ψ)−Ei(ϕ1))+n∑k=0(1+a1ρτ1)n+1eψψk−eϕ1ϕ1k(n+1)n⋯(n+1−k)]} (27)

where , , , and . is the exponential integral function [38, Eq. (8.211.1)].

###### Proof.

See Appendix A. ∎

#### Iii-B2 Diversity Analysis

In this subsection, the diversity order of with direct link for FD NOMA is analyzed in the following.

##### D2 for FD NOMA case

For with direct link, it is challenging to obtain diversity order from (3). We can use Gaussian-Chebyshev quadrature to find an approximation from (III-B1) and the approximated expression of outage probability for at high SNR is given by

 PFD,approD2,dir=⎡⎢ ⎢⎣τ1Ω0−⎛⎜ ⎜⎝1−Ω2τ1+2Ω0τ1(a2−a1γFDth2)2Ω0Ω2⎞⎟ ⎟⎠ ×τ1π2NΩ0N∑n=1(1+(sn+1)τ1a2Ω2((sn+1)τ1a1ρ+2)−snτ12Ω0) ×√1−s2n]Ω1(Ω1+τ1ρΩLI)+(1−Ω1(Ω1+τ1ρΩLI))τ1Ω0, (28)

where is a parameter to ensure a complexity-accuracy tradeoff, . Substituting (III-B2) into (18), we can obtain .

###### Remark 4.

From above explanation, the observation is that the direct link to convey information is an effective way to overcome the problem of zero diversity order for .

##### D2 for HD NOMA case

The outage performance of for HD NOMA has been investigated in [25] and we can obtain .

#### Iii-B3 Throughput Analysis

Based on the derived results of outage probability above, we obtain the throughput expressions for FD/HD NOMA in delay-limited transmission mode as below.

##### FD NOMA case

As suggested in Section III-A4, the system throughput of FD NOMA with direct link is given by

 RFDl_dir=(1−PFDD1)R1+(1−PFDD2,dir)R2, (29)

where and can be obtained from (10) and (3), respectively.

##### HD NOMA case

Similar to (29), the system throughput of HD NOMA with direct link is given by

 RHDl_dir=(1−PHDD1)R1+(1−PHDD2,dir)R2, (30)

where and can be obtained from (12) and [25, Eq. (11)].

## Iv Ergodic rate

When user’s rates are determined by their channel conditions, the ergodic sum rate is an important metric for performance evaluation. Hence the performance of FD/HD user relaying are characterized in terms of ergodic sum rates in the following.

### Iv-a User Relaying without Direct Link

#### Iv-A1 Ergodic Rate of D1

On the condition that can detect , the achievable rate of can be written as . The ergodic rate of for FD NOMA can be obtained in the following theorem.

###### Theorem 4.

The closed-form expression of ergodic rate for without direct link for FD NOMA is given by

 RFDD1= −e1ρΩLIEi(−1ρΩLI)]. (31)
###### Proof.

See Appendix B. ∎

As such, we can derive the ergodic rate of for HD NOMA in the following corollary.

###### Corollary 3.

The ergodic rate of for HD NOMA is given by

 RHDD1=−e1a1ρΩ12ln2Ei(−1a1ρΩ1). (32)

#### Iv-A2 Ergodic Rate of D2

Since should be detected at as well as at for SIC, the achievable rate of without direct link for FD NOMA is written as . The corresponding ergodic rate is given by

 RFDD2,nodir=1ln2∫∞01−FX1(x1)1+x1dx1, (33)

where with . Obviously, it is difficult to obtain the CDF of . However, in order to derive an accurate closed-form expression for the ergodic rate applicable to high SNR region, the following theorem provides the high SNR approximation.

###### Theorem 5.

The asymptotic expression for ergodic rate of without direct link for FD NOMA in the high SNR region is given by

 RFD,∞D2,nodir ×(Ω1a2Ω1−ξ)−ea2Ω1ρΩ2ξξ[Ei(−a2ξ+a1a2Ω1ρa1ξΩ2) −Ei(−a2Ω1ρΩ2ξ)](a1a2Ω21+a2Ω1ξa2Ω1−ξ)}, (34)

where .

###### Proof.

See Appendix C. ∎

For , the ergodic rate of without direct link for HD NOMA is given by

 RHDD2,nodir=12ln2∫a2a10e−yρ(a2−ya1)Ω1−yρΩ21+ydy. (35)

As can be seen from the above expression, (35) does not have a closed-form solution. Corollary 4 gives the high SNR approximation.

###### Corollary 4.

The asymptotic expression for ergodic rate of without direct link for HD NOMA in the high SNR region is given by

 RHD,∞D2,nodir=e1ρΩ22ln2[Ei(−1ρa1Ω2)−Ei(−1ρΩ2)]. (36)
###### Proof.

See Appendix D. ∎

#### Iv-A3 Slope Analysis

In this subsection, the high SNR slope is evaluated, which is the key parameter determining ergodic rate in high SNR region. The high SNR slope is defined as

 (37)
##### D1 for FD NOMA case

Based on (4), when , by using  [38, Eq. (8.212.1)] and , where is the Euler constant, the asymptotic ergodic rate of for FD NOMA is given by

 RFD,∞D1= a1Ω1ln2(ΩLI−a1Ω1)[(1+1a1ρΩ1)(ln(1a1ρΩ1) +C)−(1+1ρΩLI)(ln(1ρΩLI)+C)]. (38)

Substituting (IV-A3) into (37), we can obtain .

##### D1 for HD NOMA case

Based on (32), the asymptotic ergodic rate of for HD NOMA in the high SNR region is given by

 RHD,∞D1=−12ln2(1+1a1ρΩ1)[ln(1a1ρΩ1)+C]. (39)

Substituting (39) into (37), we can obtain .

##### D2 for FD NOMA case

Based on above analysis, substituting (5) into (37), we can obtain .

##### D2 for HD NOMA case

Such as (IV-A3), substituting (36) into (37), we can obtain .

###### Remark 5.

Based on above analysis, the ergodic rate of converges to a throughput ceiling in the high SNR region for FD/HD NOMA without direct link.

Combing (5) and (IV-A3), the asymptotic expression for ergodic sum rate of FD NOMA without direct link is expressed as

 RFD,∞sum,nodir=a1Ω1ln2(ΩLI−a1Ω1)[(1+1a1ρΩ1) +1ln2{e1ρΩ2[Ei(−1ρa1Ω2)−Ei(−1ρΩ2)](Ω1a2Ω1−ξ) +ea2Ω1ρΩ2ξξ[Ei(−a2Ω1ρΩ2ξ)−Ei(−a2ξ−a1a2Ω1ρa1ξΩ2)] ×(a1a2Ω21+a2Ω1ξa2Ω1−ξ)}. (40)

Similarly, combing (36) and (39), the asymptotic expression for ergodic sum rate of HD NOMA without direct link is expressed as

 RHD,∞sum,nodir= −12ln2(1+1a1ρΩ1)[ln(1a1ρΩ1)+C] +e1ρΩ22ln2[Ei(−1ρa1Ω2)−Ei(−1ρΩ2)]. (41)

#### Iv-A4 Throughput Analysis

In this subsection, the throughput in delay-tolerant transmission for FD/HD NOMA are presented, respectively.

##### FD NOMA case

In this mode, the throughput is determined by evaluating the ergodic rate. Using (4) and (33), the system throughput of FD NOMA without direct link is given by

 RFDt_nodir=RFDD1+RFDD2,nodir. (42)
##### HD NOMA case

Similar to (42), using (32) and (35), the system throughput of HD NOMA without direct link is given by

 RHDt_nodir=RHDD1+RHDD2,nodir. (43)

### Iv-B User Relaying with Direct Link

In this subsection, we investigate the ergodic rate of for FD/HD NOMA with direct link.

#### Iv-B1 Ergodic Rate of D2

Assume that the signal from relaying and direct link can be detected at as well as at for SIC. Moreover, considering the effect of RI between these two links, the achievable rate of is written as . For the sake of simplicity, the achievable rate for can be further written as . Hence, the ergodic rate of for FD NOMA is given by

 RFDD2,dir=1ln2∫∞01