# I/Q Imbalance and Imperfect SIC on Two-Way Relay NOMA Systems
## Appendix B Proof of Theorem 2  
Proof: According to (21), the outage probability for \(D_{2}\) with ipSIC under IQI condition is given by:  
\[\mathbb{P}_{out,D_{2}}^{IQI_{2}jSIC} =1-\Pr\left(\mathbb{\Gamma}_{R\to s_{1}}>\gamma_{thj},\mathbb{ \Gamma}_{R\to s_{2}}>\gamma_{thm}\right)\Pr\left(\mathbb{\Gamma}_{D_{1}\to s_{1} }>\gamma_{thm},\mathbb{\Gamma}_{D_{1}\to s_{2}}>\gamma_{thm}\right)\] \[=1-\underbrace{\Pr\left(\frac{|\mu_{t}|^{2}|\rho_{\
u}\gamma_{1}a_ {1}}{|\
u_{t}|^{2}\rho_{\
u}\gamma_{1}a_{1}+|A|^{2}\rho_{\
u}\gamma_{2}a_{2}+|A |^{2}}>\gamma_{thj},\frac{|\mu_{t}|^{2}\rho_{\
u}\gamma_{2}a_{2}}{\left(|\
u_{t }|^{2}+\varepsilon|\mu_{t}|^{2}\right)\rho_{\
u}\gamma_{1}a_{1}+|\
u_{t}|^{2} \rho_{\
u}\gamma_{2}a_{2}+|A|^{2}}>\gamma_{thm}\right)}_{f_{4}}\] \[\times\underbrace{\Pr\left(\frac{|\mu_{t}|^{2}|\mu_{t}|^{2}\rho_ {\
u}\gamma_{1}b_{1}}{\gamma_{1}\left(|\
u_{t}^{*2}|\mu_{t}|^{2}\rho_{\
u}+|\mu _{t}|^{2}|\mu_{t}|^{2}\rho_{\
u}b_{2}\right)+1}>\gamma_{thj},\frac{|\mu_{t}|^{2 }|\mu_{t}|^{2}\rho_{\
u}\gamma_{1}b_{2}}{\gamma_{1}\left(|\mu_{t}|^{2}|\mu_{t} |^{2}\rho_{\
u}b_{1}+|\
u_{t}^{*2}|^{2}|\mu_{t}|^{2}\rho_{\
u}\right)+1}> \gamma_{thm}\right)}_{f_{4}}.\] (A5)  
Therein, \(f_{3}\) and \(f_{4}\) can be rewritten as follows:  
\[f_{3} =\Pr\left(\gamma_{1}>\beta_{f}\left(a_{2}|A|^{2}\rho_{u}\gamma_{ 2}+|A|^{2}\right),\gamma_{2}>\beta_{m}\left(a_{1}\rho_{u}\gamma_{1}\left(|\
u_ {t}|^{2}+\varepsilon|\mu_{t}|^{2}\right)+|A|^{2}\right)\right)\] \[\qquad=\frac{\Theta_{1}\Theta_{2}\left(n+j\right)!\left(j_{1}+n_{ 1}\right)!}{\left(|A|^{2}\tilde{b}_{1}\beta_{f}a_{2}\rho_{u}+\tilde{b}_{2} \right)^{\left(n+j+1\right)}\left(\tilde{b}_{2}\beta_{m}a_{1}\rho_{u}\left(| \
u_{r}|^{2}+\varepsilon|\mu_{r}|^{2}\right)+\tilde{b}_{1}\right)^{\left(j_{1 }+n_{1}+1\right)}},\] (A6)  
where \(\Theta_{2}=\sum\limits_{j_{1}=0}^{\infty}\sum\limits_{l_{1}=0}^{\infty}\sum \limits_{m_{1}=0}^{l_{1}}\sum\limits_{n_{1}=0}^{m_{1}}\left( \begin{array}{c} m_{1}\\ n_{1}\end{array} \right)\frac{\Omega_{1,2}k_{1}^{2}+\Omega_{1}\Theta_{1}}{(j_{1})^{2}l _{1}!m_{1}! }e^{-\tilde{b}_{2}\beta_{m}|A|^{2}}\); \(\beta_{m}=\frac{\gamma_{thm}}{\rho_{u}a_{2}\left(|\mu_{r}|^{2}-\gamma_{thm}| \
u_{t}|^{2}\right)}\), with \(|\mu_{r}|^{2}>\gamma_{thm}|\
u_{t}|^{2}\); \(\Theta_{3}=\mathbb{Z}_{2}^{m_{1}-1}\mathbb{I}_{1}^{j_{1}}\mathbb{\mu}_{m}^{m_{ 1}}a_{1}^{n_{1}}\mathbb{\rho}_{u}^{\left(|\
u_{r}|^{2}+\varepsilon|\mu_{r}|^{2} \right)^{n_{1}}}\left(|A|^{2}\right)^{m_{1}-n_{1}}\). \[f_{4} =\Pr\left(\frac{|\mu_{t}|^{2}|\mu_{t}|^{2}\rho_{r}\gamma_{1}b_{1 }}{\gamma_{1}\left(|\
u_{t}|^{2}|\mu_{t}|^{2}\rho_{r}+|\mu_{t}|^{2}|\mu_{t}|^{ 2}\rho_{t}b_{2}\right)+1}>\gamma_{thr},\frac{|\mu_{t}|^{2}|\
u_{t}|^{2}\rho_{r} \gamma_{1}b_{2}}{\gamma_{1}\left(|\mu_{t}|^{2}|\mu_{t}|^{2}\rho_{r}b_{1}+|\
u _{t}^{*2}|^{2}|\mu_{r}|^{2}\rho_{r}\right)+1}>\gamma_{thm}\right)\] \[=\Pr\left(\gamma_{1}>\beta_{0}=\max\left(\beta_{1},\beta_{2} \right)\right)\] \[=\frac{\tilde{a}_{1}}{b_{1}}\sum\limits_{j_{2}=0}^{\infty}\sum \limits_{n_{2}=0}^{l_{2}}\frac{K_{1}^{2}\mathbb{I}_{1}^{m_{2}}}{l_{2}!m_{2}!} \mathbb{\rho}_{0}^{\alpha_{2}}e^{-\tilde{b}_{1}\beta_{0}},\] (A7)  
where \(\beta_{0}=\max\left(\beta_{1},\beta_{2}\right);\beta_{2}=\frac{\gamma_{thm} }{|\mu_{t}|^{2}\rho_{r}\left(|\mu_{t}|^{2}b_{2}-\gamma_{thm}\left(|\epsilon| \mu_{t}|^{2}b_{1}+|\
u_{t}^{*2}|^{2}\right)\right)}\), with \(\varepsilon=[0,1]\), \(b_{2}>\frac{\gamma_{thm}\left(|\epsilon|\mu_{t}|^{2}b_{1}+|\
u_{t}^{*2}|^{2} \right)}{|\mu_{t}|^{2}}\) and \(\beta_{1}=\frac{\gamma_{thj}}{|\mu_{t}|^{2}\rho_{r}\left(|\mu_{t}|^{2}b_{1}- \gamma_{thm}\left(|\
u_{t}^{*}|^{2}+|\mu_{t}|^{2}b_{2}\right)\right)}\), with \(b_{1}>\frac{\gamma_{thj}\left(|\
u_{t}^{*2}|^{2}+|\mu_{t}|^{2}b_{2}\right)}{| \mu_{t}|^{2}}\). Substituting (A6* [PERSON] et al. (2019) [PERSON]; [PERSON]; [PERSON] Performance Analysis of Impaired SWIPT NOMA Relaying Networks Over Imperfect Weibull Channels. _IEEE Syst. J._**2019**, 1-4. [CrossRef]
* [PERSON] et al.