# Outage Probability Analysis in Relaying Cooperative Systems with NOMA Considering Power Splitting
## 3 Performance Analysis
### Exact Outage Performance  
In principle, the \(OP\) denoted as \(OP\) represents the probability in which the instantaneous SNR, \(\gamma\) is set below its own threshold defined as \(\gamma_{0}\). In this paper, we define \(OP\) as \(OP=\Pr\left(\gamma<\gamma_{0}\right)=F_{\gamma}\left(\gamma_{0}\right)\).  
Let us first investigate the CDF in Proposition 1.  
**Proposition 1**.: _OPs for \(x_{1}\) and \(x_{2}\) presented as \(OP_{x_{1}}\) and \(OP_{x_{2}}\) can be computed as_  
\[OP_{x_{1}}=(1-e^{-v_{1}})\times\left(1-\frac{1}{\Omega_{f_{3}}}\int_{x=0}^{ \infty}e^{-\frac{x}{\Omega_{f_{3}}}-v_{2}}\mu_{1}K_{1}\left(\mu_{1}\right)dx \right), \tag{12}\]  
_and_  
\[OP_{x_{2}}=1-\frac{1}{\Omega_{f_{3}}}e^{-v_{1}}\int_{x=0}^{\infty}e^{-\frac{x} {\Omega_{f_{3}}}}\mu_{1}K_{1}\left(\mu_{1}\right)dx, \tag{13}\]  
_where \(v_{1}=\frac{d_{SD}^{m}\gamma_{0}}{\beta D\Omega_{f_{3}}}\), \(v_{2}=\frac{d_{SR}^{m}\gamma_{0}}{\beta(1-\rho)\Omega_{f_{1}}}\) and \(\mu_{1}=\sqrt{\frac{4\overline{d_{SR}^{m}\delta_{RD}^{m}\gamma_{0}}}{\beta D \Omega_{f_{1}}\Omega_{f_{2}}}\left(\frac{\beta}{d_{SD}^{s}}x+1\right)}\)._  
Proof.: Thanks to the CDF for \(\gamma_{R,1,x_{1}}\) and \(\gamma_{D,1,x_{1}}\), we achieve  
\[F_{\gamma_{R,1,x_{1}}}\left(\gamma_{0}\right)=1-e^{-\frac{d_{SR}^{m}\gamma_{0 }}{\beta(1-\rho)\Omega_{f_{1}}}},\] (14a) and \[F_{\gamma_{D,1,x_{1}}}\left(\gamma_{0}\right)=1-e^{-\frac{d_{SD}^{m}\gamma_{0 }}{\beta D\Omega_{f_{3}}}}. \tag{14b}\]  
To obtain the \(OP\) for \(x_{1}\) at \(D\), the CDF of \(\gamma_{D,2,x_{1}}\) must be achieved first. Thus, we condition \(F_{\gamma_{D,2,x_{1}}}\left(\gamma_{0}\right)\) on \(|f_{3}|^{2}\) as follows  
\[F_{\gamma_{D,2,x_{1}}}\left(\gamma_{0}\right) =\Pr\left(|f_{1}|^{2}\leq\frac{\gamma_{0}d_{SR}^{m}\delta_{RD}^{m} \left(\frac{\left|\beta/\gamma\right|^{2}+1}{\delta_{SD}^{m}}\right)}{\beta \rho\beta|^{2}}\right)\] \[=\frac{1}{\Omega_{f_{2}}}\int_{x=0}^{\infty}\left(1-e^{-\frac{1}{ \gamma}\left(\frac{\gamma_{0}d_{SR}^{m}\delta_{RD}^{m}\left(\frac{\left|\beta/ \gamma\right|^{2}+1}{\delta_{SD}^{m}}\right)}{\beta\rho\beta\gamma_{f_{1}}} \right)}\right)e^{-\frac{x}{\Omega_{f_{2}}}}dx \tag{15}\]  
By taking advantage of ((28), Equation (3.324.1)), we have  
\[\int_{0}^{\infty}e^{-\frac{\beta}{24}-\eta\rho x}dx=\sqrt{\frac{\beta}{\eta \rho}}K_{1}\left(\sqrt{\beta\eta\rho}\right). \tag{16}\]  
Thus, the CDF of \(\gamma_{D,2,x_{1}}\) over \(|f_{3}|^{2}\) can be expressed by  
\[F_{\gamma_{D,2,x_{1}}}\left(\gamma_{0}\right)=1-\frac{1}{\Omega_{f_{3}}}\int_ {0}^{\infty}\left(e^{-\frac{x}{\Omega_{f_{3}}}}\mu_{1}K_{1}\left(\mu_{1} \right)\right)dx. \tag{17}\]  
If \(x_{1}\) is not decoded, outage events may occur. Therefore, applying (14a) and (17), the \(OP\) at the end-to-end SNR at \(D\) is written by  
\[\begin{split} OP_{2\ell,x_{1}}&\overset{\Delta}{=}F_ {\gamma_{R,1,x_{1}}}\left(\gamma_{0}\right)+F_{\gamma_{D,2,x_{1}}}\left(\gamma_ {0}\right)-F_{\gamma_{R,1,x_{1}}}\left(\gamma_{0}\right)F_{\gamma_{D,2,x_{1}}} \left(\gamma_{0}\right)\\ &\overset{\Delta}{=}1-\frac{1}{\Omega_{f_{3}}}\int_{x=0}^{\infty} \left(e^{-\frac{x}{\Omega_{f_{3}}}-\frac{d_{SR}^{m}\gamma_{0}}{\beta(1-\rho) \Omega_{f_{1}}}}\mu_{1}K_{1}\left(\mu_{1}\right)\right)dx.\end{split} \tag{18}\]  
Thanks to selection combining technique, the total value of \(OP\), \(x_{1}\) can be computed as  
\[OP_{x_{1}}=F_{\gamma_{D,1,x_{1}}}\left(\gamma_{0}\right)\times OP_{2\ell,x_{1}}. \tag{19}\]  
Substituting (14b) and (18) into (19), (12) was derived. In addition, the \(OP\) of \(x_{2}\) is expressed as\[OP_{x_{2}}=1-\left(1-F_{\gamma_{D,2,x_{1}}}\left(\gamma_{0}\right)\right)\times \left(1-F_{\gamma_{D,2,x_{2}}}\left(\gamma_{0}\right)\right), \tag{20}\]  
where \(F_{\gamma_{D,2,x_{2}}}\) is presented in (14b).  
This ends the proof for Proposition 1.  
**Remark 1**.: _OP varies as a function of PS ratio, \(\rho\) ranging from 0 to 1. It is noted that the more \(\rho\) is, the more transmit power is available at R. As a result, if there is more transmit power at R, fewer outage events are likely to occur._