# I/Q Imbalance and Imperfect SIC on Two-Way Relay NOMA Systems
## 2 System Model
### 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 signal \(\hat{g}_{R}\) at the relay is given by:  
\[\hat{g}_{R}=\mu_{r}\left(h_{RD_{1}}\sqrt{a_{1}P_{u}}s_{1}+h_{RD_{2}}\sqrt{a_{2 }P_{u}}s_{2}\right)+\
u_{r}\left(h_{RD_{1}}^{*}\sqrt{a_{1}P_{u}}s_{1}^{*}+h_{ RD_{2}}^{*}\sqrt{a_{2}P_{u}}s_{2}^{*}\right)^{*}+\mu_{r}n_{r}+\
u_{r}n_{r}^{*}, \tag{9}\]  
where \(P_{u}\) denotes the total transmission power of the two users; \(n_{r}\sim CN\left(0,N_{0}\right)\) 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:  
\[\Gamma_{R\to s_{1}}=\frac{\left|\mu_{r}\right|^{2}\!\rho_{u}\gamma_{1}a_{1}}{ \left|\
u_{r}\right|^{2}\!\rho_{u}\gamma_{1}a_{1}+\left|A\right|^{2}\!\rho_{u }\gamma_{2}a_{2}+\left|A\right|^{2}}, \tag{10}\]  
where \(\gamma_{i}\overset{\Delta}{=}\left|h_{RD_{i}}\right|^{2}\), \(\left|A\right|^{2}=\left|\mu_{r}\right|^{2}+\left|\
u_{r}\right|^{2}\) and \(\rho_{u}=P_{u}/N_{0}\) 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:  
\[\Gamma_{R\to s_{2}}=\frac{\left|\mu_{r}\right|^{2}\!\rho_{u}\gamma_{2}a_{2}}{ \left|\
u_{r}\right|^{2}\!\rho_{u}\gamma_{1}a_{1}\!+\!\left|\
u_{r}\right|^{2} \!\rho_{u}\gamma_{2}a_{2}\!+\!\left|\mu_{r}\right|^{2}\!\rho_{u}\gamma_{1}a_{1 }+\left|A\right|^{2}}, \tag{11}\]where \(\varepsilon\in[0,1]\) is recorded as a parameter of ipSIC elimination and \(\varepsilon=0\) and \(\varepsilon\
eq 0\) 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 by \(\hat{y}_{D_{1}}\) can be represented as:  
\[\hat{y}_{D_{1}}=\mu_{t}h_{RD_{1}}\left(\mu_{r}\sqrt{b_{1}P_{r}}s_{1}+\mu_{r} \sqrt{b_{2}P_{r}}s_{2}\right)+\
u_{t}^{*}h_{RD_{1}}\left(\mu_{r}\sqrt{b_{1}P_{ r}}s_{1}+\mu_{r}\sqrt{b_{2}P_{r}}s_{2}\right)^{*}+n_{1}, \tag{12}\]  
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:  
\[\hat{y}_{D_{2}}=\mu_{t}h_{RD_{2}}\left(\mu_{r}\sqrt{b_{1}P_{r}}s_{1}+\mu_{r} \sqrt{b_{2}P_{r}}s_{2}\right)+\
u_{t}^{*}h_{RD_{2}}\left(\mu_{r}\sqrt{b_{1}P_{ r}}s_{1}+\mu_{r}\sqrt{b_{2}P_{r}}s_{2}\right)^{*}+n_{2}, \tag{13}\]  
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.