# Modeling and Analysis of Two-Way Relay Non-Orthogonal Multiple Access Systems

A two-way relay non-orthogonal multiple access (TWR-NOMA) system is investigated, where two groups of NOMA users exchange messages with the aid of one half-duplex (HD) decode-and-forward (DF) relay. Since the signal-plus-interference-to-noise ratios (SINRs) of NOMA signals mainly depend on effective successive interference cancellation (SIC) schemes, imperfect SIC (ipSIC) and perfect SIC (pSIC) are taken into account. In order to characterize the performance of TWR-NOMA systems, we first derive closed-form expressions for both exact and asymptotic outage probabilities of NOMA users' signals with ipSIC/pSIC. Based on the derived results, the diversity order and throughput of the system are examined. Then we study the ergodic rates of users' signals by providing the asymptotic analysis in high SNR regimes. Lastly, numerical simulations are provided to verify the analytical results and show that: 1) TWR-NOMA is superior to TWR-OMA in terms of outage probability in low SNR regimes; 2) Due to the impact of interference signal (IS) at the relay, error floors and throughput ceilings exist in outage probabilities and ergodic rates for TWR-NOMA, respectively; and 3) In delay-limited transmission mode, TWR-NOMA with ipSIC and pSIC have almost the same energy efficiency. However, in delay-tolerant transmission mode, TWR-NOMA with pSIC is capable of achieving larger energy efficiency compared to TWR-NOMA with ipSIC.

## Authors

• 10 publications
• 56 publications
• 8 publications
• 44 publications
• 26 publications
• ### Outage Performance of Two-Way Relay Non-Orthogonal Multiple Access Systems

This paper investigates a two-way relay nonorthogonal multiple access (T...
01/24/2018 ∙ by Xinwei Yue, et al. ∙ 0

• ### Performance Analysis of Intelligent Reflecting Surface Assisted NOMA Networks

Intelligent reflecting surface (IRS) is a promising technology to enhanc...
02/23/2020 ∙ by Xinwei Yue, et al. ∙ 0

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

In this paper, a novel cooperative non-orthogonal multiple access (NOMA)...
12/18/2018 ∙ by Xinwei Yue, 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

• ### Outage Probability Analysis of Uplink NOMA over Ultra-High-Speed FSO-Backhauled Systems

In this paper, we consider a relay-assisted uplink non-orthogonal multip...
09/08/2018 ∙ by Mohammad Vahid Jamali, 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

• ### Reconfigurable Intelligent Surfaces Aided Multi-Cell NOMA Networks: A Stochastic Geometry Model

By activating blocked users and altering successive interference cancell...
08/17/2020 ∙ by Chao Zhang, et al. ∙ 0

##### This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

## 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 ultra-reliable and low latency communications (URLLC). Apart from crux technologies, such as massive multiple-input multiple-output (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, non-orthogonal 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, point-to-point 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 larger-scale networks for both single-antenna and multiple-antenna 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 (MCR-NOMA) 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 decode-and-forward (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 amplify-and-forward (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 full-duplex (FD) cooperative NOMA was characterized in terms of outage probability [18].

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 [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 multi-user TWR channels for half-duplex (HD) AF relays. By applying physical-layer network coding (PNC) schemes, the performance of two-way 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 two-way full-duplex (FD) DF relay on different multi-user scheduling schemes [23]. In [24], the authors investigated the performance of multi-antenna TWR networks in which both AF and DF protocols are examined, respectively. Taking residual self-interference 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.

### I-a 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 TWR-NOMA 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 TWR-NOMA. Motivated by these, we investigate the performance of TWR-NOMA 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:

1. We derive the closed-form expressions of outage probability for TWR-NOMA 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 delay-limited transmission mode.

2. We show that the outage performance of TWR-NOMA is superior to TWR-OMA in the low signal-to-noise ratio (SNR) regime. We observe that due to the effect of IS at the relay, the outage probabilities for TWR-NOMA 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 TWR-NOMA.

3. We study the ergodic rate of users’ signals for TWR-NOMA 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 TWR-NOMA systems. We demonstrate that the ergodic rates for TWR-NOMA converge to throughput ceilings in high SNR regimes.

4. We analyze the energy efficiency of TWR-NOMA with ipSIC/pSIC in both the delay-limited and tolerant transmission modes. We confirm that TWR-NOMA with ipSIC/pSIC in delay-limited transmission mode has almost the same energy efficiency. Furthermore, in delay-tolerant transmission mode, the energy efficiency of system with pSIC is higher than that of system with ipSIC.

### I-B Organization and Notation

The remainder of this paper is organised as follows. In Section II, the system mode for TWR-NOMA is introduced. In Section III, the analytical expressions for outage probability, diversity order and system throughput of TWR-NOMA are derived. Then the ergodic rates of users’ signals for TWR-NOMA 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

### Ii-a System Description

We focus our attentions on a two-way relay NOMA communication scenario which consists of one relay , two pairs of NOMA users and 111The 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 protocol222Note that increasing the number of paired users, i,e,. pairs of users, will not affect the performance of TWR-NOMA 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 decode-and-forward (DF) relay with two antennas, namely and 333For 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 TWR-NOMA, 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 quasi-static 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.

### Ii-B 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

 yRA1=h1√a1Pux1+h2√a2Pux2+ϖ1IRA2+nRA1, (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

 yRA2=h3√a3Pux3+h4√a4Pux4+ϖ1IRA1+nRA2, (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

 γR→xl=ρ|hl|2alρ|ht|2at+ρϖ1(|hk|2ak+|hr|2ar)+1, (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

 γR→xt=ρ|ht|2atερ|g|2+ρϖ1(|hk|2ak+|hr|2ar)+1, (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

 γDk→xt=ρ|hk|2btρ|hk|2bl+ρϖ2|hk|2+1, (5)

where denotes the impact level of IS at the user nodes. Then detects and gives the corresponding SINR as follows:

 γDk→xl=ρ|hk|2blερ|g|2+ρϖ2|hk|2+1. (6)

Furthermore, the received SINR at to detect can be given by

 γDr→xt=ρ|hr|2btρ|hr|2bl+ρϖ2|hr|2+1. (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 TWR-NOMA 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.

#### Iii-1 Outage Probability of xl

In TWR-NOMA 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 TWR-NOMA system can be given by

 PipSICxl= 1−Pr(γR→xl>γthl) ×Pr(γDk→xt>γtht,γDk→xl>γthl), (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

 PipSICxl=1−e−βlΩl3∏i=1λi(Φ1ΩlΩlλ1+βl−Φ2ΩlΩlλ2+βl +Φ3ΩlΩlλ3+βl)(e−θlΩk−ετlρΩIΩk+ερτlΩIe−θl(Ωk+ερτlΩI)ετlρΩIΩk+1ερΩI), (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

 PpSICxl= 1−e−βlΩl−θlΩk3∏i=1λi(Φ1ΩlΩlλ1+βl−Φ2ΩlΩlλ2+βl +Φ3ΩlΩlλ3+βl). (10)

#### Iii-2 Outage Probability of xt

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

 PipSICxt= 1−Pr(γR→xt>γtht,γR→xl>γthl) ×Pr(γDk→xt>γtht)Pr(γDr→xt>γtht), (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

 PipSICxt=1−e−βlΩl−βtφt−ξΩk−ξΩrφtΩt(1+εβtρφtΩI)(λ′2−λ′1)2∏i=1λ′i ×(Ωlβl+βtΩlφt+Ωlλ′1−Ωlβl+βtΩlφt+Ωlλ′2), (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

 PpSICxt=1−e−βlΩl−βtφt−ξΩk−ξΩrφtΩt(λ′2−λ′1)2∏i=1λ′i ×(Ωlβl+βtΩlφt+Ωlλ′1−Ωlβl+βtΩlφt+Ωlλ′2). (13)

#### Iii-3 Diversity Order Analysis

In order to gain 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 [33]

 d=−limρ→∞log(P∞xi(ρ))logρ, (14)

where denotes the asymptotic outage probability of .

###### Proposition 1.

Based on the analytical results in (1) and (1), when , the asymptotic outage probabilities of for ipSIC/pSIC with are given by

 PipSICxl,∞=1−3∏i=1λi(Φ1ΩlΩlλ1+βl−Φ2ΩlΩlλ2+βl+Φ3ΩlΩlλ3+βl) ×[1−θlΩk−ετlρΩIΩk+ερτlΩI(1−θl(Ωk+ετlρΩI)ερτlΩIΩk)], (15)

and

 PpSICxl,∞=1−3∏i=1λi(Φ1ΩlΩlλ1+βl−Φ2ΩlΩlλ2+βl+Φ3ΩlΩlλ3+βl), (16)

respectively. Substituting (1) and (16) into (14), the diversity orders of with ipSIC/pSIC are equal to zeros.

###### 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.

Similar to the resolving process of , the asymptotic outage probabilities of with ipSIC/pSIC in high SNR regimes are given by

 PipSICxt,∞=1−λ′1λ′2φtΩt(1+ερβtφtΩI)(λ′2−λ′1) ×(Ωlβl+βtΩ1φt+Ωlλ′1−Ωlβl+βtΩ1φt+Ωlλ′2), (17)

and

 PpSICxt,∞=1−λ′1λ′2φtΩt(λ′2−λ′1) ×(Ωlβl+βtΩ1φt+Ωlλ′1−Ωlβl+βtΩlφt+Ωlλ′2), (18)

respectively. Substituting (2) and (2) into (14), the diversity orders of for both ipSIC and pSIC are zeros.

###### 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 [15, 34]

 Rψdl= (1−Pψx1)Rx1+(1−Pψx2)Rx2 +(1−Pψx3)Rx3+(1−Pψx4)Rx4, (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 TWR-NOMA is investigated for considering the influence of signal’s channel fading to target rate.

#### Iv-1 Ergodic Rate of xl

Since can be detected at the relay as well as at successfully. By the virtue of (3) and (6), the achievable rate of for TWR-NOMA 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. ∎

Substituting (20), the corresponding ergodic rate of is given by

 Rergxl=12ln2∫∞01−FX(x)1+xdx, (21)

where and . Unfortunately, it is difficult to obtain the closed-form 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 TWR-NOMA 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 closed-form expression of ergodic rate for with ipSIC for TWR-NOMA is given by

 RipSICxl,erg= −12ln2⎡⎢⎣AeΨEi(−Ψ)+BeΨΛ1Λ1Ei(−ΨΛ1) +CeΨΛ2Λ2Ei(−ΨΛ2)⎤⎥⎦, (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

 RpSICxl,erg=−12ln2⎡⎢⎣AeΨEi(−Ψ)+CeΨΛ2Λ2Ei(−ΨΛ2)⎤⎥⎦. (23)

#### Iv-2 Ergodic Rate of xt

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

 Rergxt=12ln2∫∞01−FY(y)1+ydy, (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

 RipSICxt,erg=12ln2∫btbl0e−xρatΩt−xρ(bt−xbl)Ωk−xρ(bt−xbl)Ωr(1+x)(1+xΛ3)dx, (25)

and

 RpSICxt,erg=12ln2∫btbl0e−xρatΩt−xρ(bt−xbl)Ωk−xρ(bt−xbl)Ωr1+xdx, (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

 RipSICxt,∞=12(1−Λ3)ln2[ln(1+btbl)−ln(1+btΛ3bl)]. (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

 RpSICxt,∞=12ln2e1ρatΩt[Ei(−1ρatblΩt)−Ei(−1ρatΩt)]. (28)

#### Iv-3 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

 S=limρ→∞R∞xi(ρ)log(ρ), (29)

where denotes the asymptotic ergodic rate of .

##### xl for ipSIC/pSIC case
###### Proposition 3.

Based on the above analytical results in (3) and (23), when , by using  [35, Eq. (8.212.1)] and , where is the Euler constant, the asymptotic ergodic rates of with ipSIC/pSIC in the high regime are given by

 RipSICxl,∞=−12ln2[A(1+Ψ)(ln(Ψ)+Ec)+BΛ1(1+ΨΛ1) (30)

and

 RpSICxl,∞=