I/Q Imbalance and Imperfect SIC on Two-Way Relay NOMA Systems

: Non-orthogonal multiple access (NOMA) system can meet the demands of ultra-high data rate, ultra-low latency, ultra-high reliability and massive connectivity of user devices (UE). However, the performance of the NOMA system may be deteriorated by the hardware impairments. In this paper, the joint effects of in-phase and quadrature-phase imbalance (IQI) and imperfect successive interference cancellation (ipSIC) on the performance of two-way relay cooperative NOMA (TWR C-NOMA) networks over the Rician fading channels are studied, where two users exchange information via a decode-and-forward (DF) relay. In order to evaluate the performance of the considered network, analytical expressions for the outage probability of the two users, as well as the overall system throughput are derived. To obtain more insights, the asymptotic outage performance in the high signal-to-noise ratio (SNR) region and the diversity order are analysed and discussed. Throughout the paper, Monte Carlo simulations are provided to verify the accuracy of our analysis. The results show that IQI and ipSIC have signiﬁcant deleterious effects on the outage performance. It is also demonstrated that the outage behaviours of the conventional OMA approach are worse than those of NOMA. In addition, it is found that residual interference signals (IS) can result in error ﬂoors for the outage probability and zero diversity orders. Finally, the system throughput can be limited by IQI and ipSIC, and the system throughput converges to a ﬁxed constant in the high SNR region. In Figure 4, we show the outage probabilities of D 1 and D 2 with ipSIC under the different levels of phase mismatch. The outage performances of D 1 and D 2 deteriorate with the increase of the value of phase mismatch. The greater the gap between the value of the phase mismatch and zero degrees, the greater the adverse effect of the phase mismatch on outage performance.


Introduction
The demand for new services and data rate has exploded for wireless communication, due to the growth of data traffic in the mobile Internet. To this end, a higher data rate and massive mobile terminal access are required for the 5 th Generation (5G) systems [1][2][3]. Many new technologies such as massive multiple-input multiple-output (MIMO), non-orthogonal multiple access (NOMA), millimetre wave and device-to-device communication have been proposed to meet these demands [4]. Among these techniques, NOMA can significantly improve the spectral efficiency and reduce the system latency. Different from the conventional orthogonal multiple access (OMA), NOMA provides services for multiple users on the same resource by using superposition coding at the transmitter and successive interference cancellation (SIC) at the receiver [5,6]. Take an example of downlink: signals from the base station (BS) are superimposed and sent to NOMA users. Users with stronger channel gains decode

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Based on the above works, we investigate the effects of IQI and ipSIC on the performance of a TWR C-NOMA network over the Rician fading channels. It is worth noting that this is a valuable problem for practical system design and analysis. As far as the authors know, although a system with IQI or ipSIC has been studied in some papers, the TWR C-NOMA system model with the condition of IQI and ipSIC under Rician fading channels has not been previously studied.
• We derive analytical expressions for outage probabilities of both the far and near users. The results show that IQI and ipSIC have deleterious effects on the outage performance and residual IS. In order to gain better insights into the system performance, we compare the outage performance of NOMA and OMA for both the far and near users, and the results show that the outage performance of NOMA is better than that of OMA. By comparing Rician and Rayleigh fading channel conditions, it is found that the throughput of our considered system with Rician or Rayleigh fading channels is almost the same in ideal conditions, and IQI and ipSIC have worse effects on the system throughput with Rayleigh fading channels than on Rician.
• We carry out the asymptotic analysis in the high SNR region. Furthermore, based on asymptotic outage probability, the diversity order is derived to analyse the diversity gain of the system. It is demonstrated that residual IS can result in error floors for the outage probability and zero diversity orders.

Organization
The rest of this paper is organized as follows. In Section 2, we describe the TWR C-NOMA system model in detail. In Section 3, the analytical and asymptotic been expressions of the outage probabilities, diversity order and the system throughput are obtained. Numerical examples of the derived analytical expressions and Monte Carlo simulations are given in Section 4. Finally, the conclusions are presented in Section 5.

Notations
In this paper, the main notations are shown as follows: (·) * is the operator of the conjugate. The notation f (x) represents the probability density function (PDF) of the independent variable x, while F (x) represents the cumulative density function (CDF) of x. E (·) is the expectation operator. ∑ (·) represents the operation of summation, and I 0 (·) is the zeroth order modified Bessel function of the first kind. x * represents the conjugation of x. x ∼ N 0, σ 2 represents that x follow Gaussian distributions. In addition, the list of acronyms used in this paper is listed in Table 1.

System Model
We consider a TWR C-NOMA network over Rician fading channels as depicted in Figure 1, which consists of one relay R and a near user D 1 and a far user D 2 . It was assumed that there is no direct link between D 1 and D 2 due to severe shadowing or obstacle. In this case, D 1 and D 2 can only communicate via a half-duplex (HD) DF relay. We also assumed that the two users and the relay were equipped with a single antenna. In this study, h RD 1 and h RD 2 are modelled as R to D 1 link and R to D 2 link, respectively.

System Model
We consider a TWR C-NOMA network over Rician fading channels as depicted in Figure 1, which consists of one relay R and a near user D 1 and a far user D 2 . It was assumed that there is no direct link between D 1 and D 2 due to severe shadowing or obstacle. In this case, D 1 and D 2 can only communicate via a half-duplex (HD) DF relay. We also assumed that the two users and the relay were equipped with a single antenna. In this study, h RD 1 and h RD 2 are modelled as R to D 1 link and R to D 2 link, respectively.

I/Q Imbalance Signal Model
Considering IQI at the transmitter (TX) [37], the time domain baseband signal can be obtained as: where s and s * denote the baseband IQI-free signal and the IQI impaired signal, respectively; µ t and ν t are the IQI coefficients; and ν * t is the conjugation of ν t . As in [37], µ t and ν t are given as follows:

I/Q Imbalance Signal Model
Considering IQI at the transmitter (TX) [37], the time domain baseband signal can be obtained as: where s and s * denote the baseband IQI-free signal and the IQI impaired signal, respectively; µ t and ν t are the IQI coefficients; and ν * t is the conjugation of ν t . As in [37], µ t and ν t are given as follows: Electronics 2020, 9, 249 5 of 16 where g t ∼ N 0, σ 2 g t and ϕ t ∼ N 0, σ 2 ϕ t are the TX amplitude and phase mismatch, respectively, which follow the Gaussian distribution. Similarly, the time domain baseband representation of the RX IQI impaired signal can be obtained as:ŷ where y * is referred to as the mirror signal introduced by the IQI. µ t and ν t are the IQI coefficients caused by RX, which can be expressed as: where g r ∼ N 0, σ 2 g r and ϕ r ∼ N 0, σ 2 ϕ r denote the RX amplitude and phase mismatch, respectively, which follow the Gaussian distribution. Therefore, the severity of TX IQI and RX IQI can be expressed as an image rejection ratio, which are defined as follows [38]:

Signal Model of Joint TX/RX Impaired by IQI
The whole communication process consists of two phases: (1) The first time slot (up-link NOMA process): Two users send their signals to the relay. The relay receives the signals of the users with RX IQI, and the baseband received signalŷ R at the relay is given by: where P u denotes the total transmission power of the two users; n r ∼ CN (0, N 0 ) is the additive white Gaussian noise (AWGN); a 1 and a 2 are the power allocation coefficients corresponding to D 1 and D 2 with a 1 + a 2 = 1 and a 1 > a 2 , respectively. The received signal-to-interference plus noise ratio (SINR) for the relay to decode the signal s 1 is given by: where γ i ∆ = h RD i 2 , |A| 2 = |µ r | 2 + |ν r | 2 and ρ u = P u/ N 0 denotes the transmit SNR at users.
It was assumed that the signal from D 1 can be correctly decoded at the relay. After eliminating s 1 by using SIC, the received SINR for the relay to decode the signal s 2 is expressed as: Electronics 2020, 9,249 6 of 16 where ε ∈ [0, 1] is recorded as a parameter of ipSIC elimination and ε = 0 and ε = 0 denote pSIC and ipSIC, respectively.
(2) The second time slot (downlink NOMA process): The relay decodes the received signal and broadcasts the signals to the two users. The baseband received signals at D 1 with RX IQI denoted bŷ y D 1 can be represented as: where P r denotes the transmission power at the relay. In addition, b 1 and b 2 are the responding power allocation coefficients for s 1 and s 2 with b 1 + b 2 = 1 and b 1 > b 2 , respectively. The baseband received signals at D 2 is given as: where n 1 and n 2 are the AWGN at D 1 and D 2 with variance N 0 , respectively. It is assumed that neither D 1 nor D 2 know their own signals. The received SINR for D 1 to decode the signal s 1 is given by: where ρ r = P r /N 0 represents the transmit SNR at the users, and the received SINR for D 1 to decode s 2 is given by: The received SINR for D 2 to decode the signal s 1 is given by: The PDF of the random variable γ i is expressed as [39]: where λ i is the mean value of γ i , i = 1, 2. K is the Rician fading parameter, which represents the ratio of the power of the LoS component to the average power of the non-LoS component. I 0 (·) is the zeroth order modified Bessel function of the first kind.
Applying the result of [39], the PDF of λ i can be rewritten as follows: Based on this, the CDF of γ i is: Electronics 2020, 9, 249 7 of 16

Performance Analysis
In this section, we derive analytical expressions for the outage probability of the system under consideration. The asymptotic outage behaviour and diversity orders are also discussed. Then, the system throughput is obtained.

Outage Probability Analysis
The outage probabilities for the D 1 → R → D 2 link and the D 2 → R → D 1 link are denoted by P out,D 1 and P out,D 2 , respectively.
The outage event of P out,D 1 occurs in the following cases: (1) the information s 1 cannot be decoded by R; (2) the information s 1 cannot be decoded by D 2 . Thus, the outage probability of D 1 can be written as: In addition, the outage event of P out,D 2 occurs in the following cases: (1) s 1 cannot be decoded by R correctly. (2) s 2 cannot be decoded by R after R can first decode s 1 correctly. (3) s 1 cannot be decoded by D 1 correctly. (4) s 2 cannot be decoded by D 1 , while D 1 can first decode s 1 correctly. Thus, the outage probability of D 2 can be written as: where γ th f = 2R 1 − 1 and γ thm = 2R 2 − 1, withR 1 andR 2 denoting the data rate thresholds for users D 1 and D 2 , respectively.
Theorem 1. The analytical expression of the outage probability of D 1 for the TWR C-NOMA network with IQI and ipSIC can be obtained as: where Proof. Please see Appendix A.
Theorem 2. The analytical expression of the outage probability of D 2 for the TWR C-NOMA network with IQI and ipSIC can be obtained as: Proof. Please see Appendix B.

Asymptotic Outage Probability Analysis
To get more insight into the outage behaviour of the TWR C-NOMA system with IQI and ipSIC, the formulas of asymptotic outage probabilities at the high SNR region are derived in this subsection. (22), when ρ u = ςρ r → ∞ and ς > 0, the asymptotic outage probability of D 1 is derived as:

Proposition 1. Based on
where Proposition 2. Based on (23), when ρ u = ςρ r → ∞ and ς > 0, the asymptotic outage probability of D 2 is given by: where

Diversity Orders
In this subsection, the diversity order is analysed, which is defined as [16]: where ρ = ρ u = ςρ r and P ∞ D n denotes the asymptotic outage probability of D n , n = 1, 2.
Electronics 2020, 9, 249 9 of 16 By using the definition in (26), the diversity orders for the IQI and ipSIC conditions of both D 1 and D 2 are respectively derived as follows: The diversity orders of both D 1 and D 2 for the considered system with the ideal condition (I/Q balance and pSIC) are derived as: Remark 1. An important conclusion from the analysis above is that due to the impact of IQI and residual interference with the use of ipSIC, the diversity orders of D 1 and D 2 are both zero due to the fixed outage probabilities at the high SNR region. As can be observed, there are error floors for D 1 and D 2 with IQI and ipSIC.

System Throughput Analysis
In order to further evaluate the performance of the TWR C-NOMA system, the system throughput is obtained for IQI and ipSIC conditions, as well as for that of the ideal condition, respectively, and given by: whereR 1 andR 2 represent the target rates at the receiver to decode the desired signals s 1 and s 2 , respectively.

Numerical Examples and Discussions
In this section, we provide numerical illustration of our analytical results through Monte Carlo simulations. Unless otherwise specified, the main parameter values used in all our evaluations are provided in Table 2.

Monte Carlo Simulations Repeated 10 5 Iterations
Power allocation coefficients of NOMA a 1 = 0.7, a 2 = 0.3, b 1 = 0.8 and b 2 = 0.2 Targeted data ratesR 1 = 0.2BPCU,R 2 = 0.3BPCU The distance between R and D 1 or D 2 d 1 = 0.5, d 2 = 1 Noise power N 0 = 1 The parameters of ipSIC ε = 0.03 Ideal RF front end g t = g r = 1, ϕ t = ϕ r = 0 • The parameters of IQI g t = g r = 0.8, ϕ t = ϕ r = 5 • Figure 2 plots the outage probabilities of two users versus transmit SNR with different conditions. The analytical curves for the outage probabilities of the IQI and ipSIC are plotted using (22) and (23). An excellent agreement can be observed between the analytical results and Monte Carlo simulations. We considered five conditions in this simulation: (1) IQI and ipSIC; (2) ONLY ipSIC; (3) NO-IQI and pSIC; (4) OMA; (5) Rayleigh fading channels. In addition, the asymptotic outage probability curves are plotted in accordance using (24) and (25). It is noticed from Figure 2 that the error floors exist for both D 1 and D 2 . The reason is that the IS result in zero diversity order. The outage performance of the system with IQI is worse than that of the system without IQI, which means that IQI has deleterious effects on the system outage performance. Moreover, the outage performance of D 2 with ipSIC is worse than that of the system with pSIC, due to the residual interference caused by ipSIC. It can be seen that the outage performances of D 1 and D 2 for OMA are lower than NOMA. It is also interesting to note that the outage probability of D 1 with ipSIC or with pSIC has the same curves. This is due to the fact that the SIC method is not used in the decoding of the signal of D 1 , which is proven by (22). Therefore, the outage performance of the Rayleigh condition is better than the Rician in D 1 , and it is worse than the Rician condition in D 2 .
Power allocation coefficients of NOMA a 1 = 0.7, a 2 = 0.3, b 1 = 0.8 and b 2 = 0.2 Targeted data ratesR 1 = 0.2BPCU,R 2 = 0.3BPCU The distance between R and D 1 or D 2 d 1 = 0.5, d 2 = 1 Noise power N 0 = 1 The parameters of ipSIC ε = 0.03 Ideal RF front end g t = g r = 1, ϕ t = ϕ r = 0 • The parameters of IQI g t = g r = 0.8, ϕ t = ϕ r = 5 • Figure 2 plots the outage probabilities of two users versus transmit SNR with different conditions. The analytical curves for the outage probabilities of the IQI and ipSIC are plotted using (22) and (23). An excellent agreement can be observed between the analytical results and Monte Carlo simulations. We considered five conditions in this simulation: (1) IQI and ipSIC; (2) ONLY ipSIC; (3) NO-IQI and pSIC; (4) OMA; (5) Rayleigh fading channels. In addition, the asymptotic outage probability curves are plotted in accordance using (24) and (25). It is noticed from Figure 2 that the error floors exist for both D 1 and D 2 . The reason is that the IS result in zero diversity order. The outage performance of the system with IQI is worse than that of the system without IQI, which means that IQI has deleterious effects on the system outage performance. Moreover, the outage performance of D 2 with ipSIC is worse than that of the system with pSIC, due to the residual interference caused by ipSIC. It can be seen that the outage performances of D 1 and D 2 for OMA are lower than NOMA. It is also interesting to note that the outage probability of D 1 with ipSIC or with pSIC has the same curves. This is due to the fact that the SIC method is not used in the decoding of the signal of D 1 , which is proven by (22). Therefore, the outage performance of the Rayleigh condition is better than the Rician in D 1 , and it is worse than the Rician condition in D 2 .  Figure 3 plots the outage probabilities of the two users versus the parameter of ipSIC for the system with IQI and ideal conditions. It is clearly seen that the outage performance of D 2 becomes worse with the increase of ipSIC parameter. This is because the residual interference caused by ipSIC is harmful to the outage performance of D 2 . In addition, the parameter of ipSIC has no effect on the outage probability of D 1 , which is proven by (22). Finally, the outage performances for both D 1 and D 2 with IQI are worse than that obtained without IQI condition. All in all, IQI reduces the system outage performance. Outage probability versus the transmit SNR of the proposed TWR C-NOMA system for D 1 and D 2 with data rate thresholdsR 1 = 0.2BPCU andR 2 = 0.3BPCU. Figure 3 plots the outage probabilities of the two users versus the parameter of ipSIC for the system with IQI and ideal conditions. It is clearly seen that the outage performance of D 2 becomes worse with the increase of ipSIC parameter. This is because the residual interference caused by ipSIC is harmful to the outage performance of D 2 . In addition, the parameter of ipSIC has no effect on the outage probability of D 1 , which is proven by (22). Finally, the outage performances for both D 1 and D 2 with IQI are worse than that obtained without IQI condition. All in all, IQI reduces the system outage performance.     Figure 5 plots the outage probabilities of the two users with respect to the amplitude mismatch with ipSIC and pSIC conditions. It is clearly seen that the closer the amplitude mismatch parameter is to one, the better the outage performance of D 1 and D 2 becomes. The greater the gap between the value of the amplitude mismatch and one, the greater the harmful effect of the amplitude mismatch on    Figure 5 plots the outage probabilities of the two users with respect to the amplitude mismatch with ipSIC and pSIC conditions. It is clearly seen that the closer the amplitude mismatch parameter is to one, the better the outage performance of D 1 and D 2 becomes. The greater the gap between the value of the amplitude mismatch and one, the greater the harmful effect of the amplitude mismatch on system performance.  Figure 5 plots the outage probabilities of the two users with respect to the amplitude mismatch with ipSIC and pSIC conditions. It is clearly seen that the closer the amplitude mismatch parameter is to one, the better the outage performance of D 1 and D 2 becomes. The greater the gap between the value of the amplitude mismatch and one, the greater the harmful effect of the amplitude mismatch on system performance.  The effects of the three different conditions for the system throughput are evaluated in Figure 6, in which we fix the parameters of IQI as g t = g r = 0.8, ϕ t = ϕ r = 5 • and the ipSIC parameters as ε = 0.03 for the IQI and ipSIC conditions. For only the ipSIC condition, we fix the parameters of IQI as g t = g r = 1, ϕ t = ϕ r = 0 • and ipSIC parameters as ε = 0.03. For the ideal condition, we use the parameters of IQI as g t = g r = 1, ϕ t = ϕ r = 0 • and ipSIC parameters as ε = 0. In the SNR range from -10dB to 45dB, the analysis curves of the system throughput with IQI and/or ipSIC condition is always lower than that of the ideal condition. It should be noted that the system throughput enhances as the SNR increase. Moreover, both system throughputs gradually tend to be stable in high SNR region. It can be observed from Figure 6 that the system throughput under two channels is the same in the low SNR region, and at high SNRs, the Rayleigh fading is slightly better than the throughput of the Rician fading channels in the ideal condition. However, in other non-ideal conditions, it is obviously worse than Rician fading channels. Therefore, a conclusion can be obtained that there is almost no difference of the considered system throughput with the ideal condition in the two channel cases, and IQI and ipSIC have worse effects on the system throughput with Rayleigh fading channels than on Rician.  The effects of the three different conditions for the system throughput are evaluated in Figure 6, in which we fix the parameters of IQI as g t = g r = 0.8, ϕ t = ϕ r = 5 • and the ipSIC parameters as ε = 0.03 for the IQI and ipSIC conditions. For only the ipSIC condition, we fix the parameters of IQI as g t = g r = 1, ϕ t = ϕ r = 0 • and ipSIC parameters as ε = 0.03. For the ideal condition, we use the parameters of IQI as g t = g r = 1, ϕ t = ϕ r = 0 • and ipSIC parameters as ε = 0. In the SNR range from −10dB to 45dB, the analysis curves of the system throughput with IQI and/or ipSIC condition is always lower than that of the ideal condition. It should be noted that the system throughput enhances as the SNR increase. Moreover, both system throughputs gradually tend to be stable in high SNR region. It can be observed from Figure 6 that the system throughput under two channels is the same in the low SNR region, and at high SNRs, the Rayleigh fading is slightly better than the throughput of the Rician fading channels in the ideal condition. However, in other non-ideal conditions, it is obviously worse than Rician fading channels. Therefore, a conclusion can be obtained that there is almost no difference of the considered system throughput with the ideal condition in the two channel cases, and IQI and ipSIC have worse effects on the system throughput with Rayleigh fading channels than on Rician.  The effects of the three different conditions for the system throughput are evaluated in Figure 6, in which we fix the parameters of IQI as g t = g r = 0.8, ϕ t = ϕ r = 5 • and the ipSIC parameters as ε = 0.03 for the IQI and ipSIC conditions. For only the ipSIC condition, we fix the parameters of IQI as g t = g r = 1, ϕ t = ϕ r = 0 • and ipSIC parameters as ε = 0.03. For the ideal condition, we use the parameters of IQI as g t = g r = 1, ϕ t = ϕ r = 0 • and ipSIC parameters as ε = 0. In the SNR range from -10dB to 45dB, the analysis curves of the system throughput with IQI and/or ipSIC condition is always lower than that of the ideal condition. It should be noted that the system throughput enhances as the SNR increase. Moreover, both system throughputs gradually tend to be stable in high SNR region. It can be observed from Figure 6 that the system throughput under two channels is the same in the low SNR region, and at high SNRs, the Rayleigh fading is slightly better than the throughput of the Rician fading channels in the ideal condition. However, in other non-ideal conditions, it is obviously worse than Rician fading channels. Therefore, a conclusion can be obtained that there is almost no difference of the considered system throughput with the ideal condition in the two channel cases, and IQI and ipSIC have worse effects on the system throughput with Rayleigh fading channels than on Rician.  Figure 6. System throughput versus the transmit SNR of the proposed TWR C-NOMA system for D 1 and D 2 with data rate thresholdsR 1 = 0.2BPCU andR 2 = 0.3BPCU. Figure 6. System throughput versus the transmit SNR of the proposed TWR C-NOMA system for D 1 and D 2 with data rate thresholdsR 1 = 0.2BPCU andR 2 = 0.3BPCU.

Conclusions
In this paper, the performance of a TWR C-NOMA system with IQI and ipSIC over the Rician fading channels was investigated. The analytical expressions of these performance metrics were obtained. In addition, the asymptotic outage behaviour and the diversity order were discussed in the high SNR region. Based on the analytical results, it was shown that the outage performance of the NOMA based system was better than that of OMA. The results revealed that the system outage performance could significantly deteriorate by the IQI and ipSIC. It was also presented that the IS resulted in a zero diversity order in the system. Moreover, the system throughput was found to improve with the increase of SNR, in which it converged to a fixed constant in the high SNR region.