# I/Q Imbalance and Imperfect SIC on Two-Way Relay NOMA Systems
## 3 Performance Analysis
### Outage Probability Analysis  
The outage probabilities for the \(D_{1}\to R\to D_{2}\) link and the \(D_{2}\to R\to D_{1}\) link are denoted by \(\mathrm{P}_{out,D_{1}}\) and \(\mathrm{P}_{out,D_{2}}\), respectively. The outage event of \(\mathrm{P}_{out,D_{1}}\) occurs in the following cases: (1) the information \(s_{1}\) cannot be decoded by \(R\); (2) the information \(s_{1}\) cannot be decoded by \(R\); (2) the information \(s_{1}\) cannot be decoded by \(D_{2}\). Thus, the outage probability of \(D_{1}\) can be written as:  
\[\mathrm{P}_{out,D_{1}}=1-\mathrm{Pr}\left(\Gamma_{R\to s_{1}}>\gamma_{thr} \right)\mathrm{Pr}\left(\Gamma_{D_{2}\to s_{1}}>\gamma_{thr}\right). \tag{20}\]  
In addition, the outage event of \(\mathrm{P}_{out,D_{2}}\) 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:  
\[\mathrm{P}_{out,D_{2}}=1-\mathrm{Pr}\left(\Gamma_{R\to s_{1}}>\gamma_{thr}, \Gamma_{R\to s_{2}}>\gamma_{thm}\right)\mathrm{Pr}\left(\Gamma_{D_{1}\to s_{1} }>\gamma_{thm},\Gamma_{D_{1}\to s_{2}}>\gamma_{thm}\right), \tag{21}\]  
where \(\gamma_{thr}=2^{R_{1}}-1\) and \(\gamma_{thm}=2^{R_{2}}-1\), with \(\bar{R}_{1}\) and \(\bar{R}_{2}\) denoting the data rate thresholds for users \(D_{1}\) and \(D_{2}\), respectively. Theorem 3.1: The analytical expression of the outage probability of \(D_{1}\) for the TWR C-NOMA network with IQI and ipSIC can be obtained as:  
\[\mathrm{P}_{out,D_{1}}^{ IQI, ipSIC} =1-\sum\limits_{l_{3}=0}^{\infty}\sum\limits_{m_{3}=0}^{l_{3}} \frac{\Theta_{1}\left(n+j\right)!K^{l_{3}}a_{2}b_{2}^{m_{3}-1}}{l_{3}!m_{3}! \left(\left|A\right|^{2}\bar{b}_{1}\beta_{f}a_{2}\rho_{u}+\bar{b}_{2}\right)^{ \left(n+j+1\right)}}\] \[\times\left(\frac{\gamma_{thr}}{\left|\mu_{r}\right|^{2}\rho_{r}b _{1}\left(\left|\mu_{t}\right|^{2}-\gamma_{thr}\right|v_{t}^{*}|^{2}\right)} \right)^{m_{3}}e^{-\frac{b_{2}\gamma_{thr}}{\left|\mu_{r}\right|^{2}\rho_{r}b _{1}\left(\left|\mu_{t}\right|^{2}-\gamma_{thr}\right|v_{t}^{*}|^{2}\right)}}, \tag{22}\]  
_where \(\Theta_{1}\)= \(\sum\limits_{j=0}^{\infty}\sum\limits_{l=0}^{\infty}\sum\limits_{m=0}^{l}\sum \limits_{n=0}^{m}\left( \begin{array}{c}m\\ n\end{array} \right)\beta_{0}\beta_{f}^{m}\left(\left|A\right|^{2}\right)^{m} a_{2}^{n}\rho_{u}^{n}e^{-\bar{b}_{1}\beta_{f}\left|A\right|^{2}}\) with \(\beta_{0}=\frac{a_{1}a_{2}K^{l+j}b_{1}^{m-1}b_{2}^{j}}{\left(j!\right)^{2}!1m! }\) and \(\beta_{f}=\frac{\gamma_{thr}}{\rho_{u}a_{1}\left(\left|\mu_{r}\right|^{2}- \gamma_{thr}\right|v_{r}|^{2}\right)}\) with \(\left|\mu_{t}\right|^{2}>\gamma_{thr}\left|v_{t}^{*}\right|^{2}\)._  
Proof. : Please see Appendix A.  
Theorem 3.2: The analytical expression of the outage probability of \(D_{2}\) for the TWR C-NOMA network with IQI and ipSIC can be obtained as:  
\[\mathrm{P}_{out,D_{2}}^{ IQI, ipSIC} =1-\sum\limits_{l_{2}=0}^{\infty}\sum\limits_{m_{2}=0}^{l_{2}}\frac{ \Theta_{1}\Theta_{2}\left(n+j\right)!\left(j_{1}+n_{1}\right)!}{l_{2}!m_{2}! \left(\left|A\right|^{2}\bar{b}_{1}\beta_{f}a_{2}\rho_{u}+\bar{b}_{2}\right)^ {\left(n+j+1\right)}}\] \[\times\frac{K^{l_{2}}a_{1}\bar{b}_{1}^{m_{2}-1}\beta_{0}^{m_{2}}e ^{-\bar{b}_{1}\beta_{0}}}{\left(\bar{b}_{2}\beta_{m}a_{1}\rho_{u}\left(\left| \
u_{r}\right|^{2}+\varepsilon|\mu_{r}\right|^{2}\right)+\bar{b}_{1}\right)^{ \left(j_{1}+n_{1}+1\right)}}, \tag{23}\]_where \(\Theta_{2}=\sum\limits_{j_{1}=0}^{\infty}\sum\limits_{l_{1}=0}^{\infty}\sum\limits_{m _{1}=0}^{l_{1}}\sum\limits_{n_{1}=0}^{m_{1}}\left( \begin{array}{c}m_{1}\\ n_{1}\end{array} \right)\frac{a_{1}a_{2}K^{l+j+1}\Theta_{2}}{\left(j_{1}!\right) ^{2}I_{1}!m_{1}!