# Outage Probability Analysis in Relaying Cooperative Systems with NOMA Considering Power Splitting
## 2 System Model  
We considered a cooperative relaying non-orthogonal multiple access (CR-NOMA) system depicted Figure 1. Particularly, a source node (\(S\)), a relay node, and a destination node (\(D\)) communicate with each other, but \(R\) is used to support the communication between \(S\) and \(D\) because of the far distance. In this model, we assume \(R\) operates in decode-and-forward (DF) half-duplex (HD) relaying mode. It is worth noting that the direct transmission is also possible. In addition, the additive white Gaussian noise (AWGN) has an impact on all nodes' received signals with zero mean \(n_{0}\) and variance \(N_{0}\). In this work, \(d_{SR}\) and \(d_{RD}\) are used to denote the distances between \(S\)-\(R\) and \(R\)-\(D\) links, respectively. In principle, there are two time slots involved in each system communication process. Regarding the first time slot, \(S\) transfers a signal symbol denoted as \(x_{1}\) with the transmit power \(P_{S}\) to \(R\) and \(D\), where \(\mathbb{E}\left(\left|x_{1}\right|^{2}\right)=1\). Interestingly, thanks to the different receiver mechanisms, two signals are split and differentiated with the power allocation at \(S\) as in [21]. Following that, another signal symbol denoted as \(x_{2}\) is transferred to \(D\) with \(P_{S}\) in the second time slot, in which \(\mathbb{E}\left(\left|x_{2}\right|^{2}\right)=1\). It is noted that \(D\) simultaneously receives \(x_{1}\) from \(R\) with the transmit power, \(P_{R}\) in the second time slot. Regarding the system channels, we represent the channel coefficients of \(S\)-\(R\), \(R\)-\(D\), and \(S\)-\(D\) links as \(f_{1}\), \(f_{2}\), and \(f_{3}\), and they are affected by independent and identically distributed (i.i.d) quasi-static Rayleigh fading channel. In addition, the channel power gain is denoted by \(\left|A\right|^{2}\) with \(A\in\left\{f_{1},f_{2},f_{3}\right\}\) following an exponential distribution with the mean value, \(\Omega_{A}\). In order to examine energy harvesting (EH) in this model, we decided to deploy power spitting (PS) protocol showed Figure 2. In particular, the block time, \(T\) is divided for each transmission slot. In principle, the PS ratio, \(\rho P_{S}\) splits a part of the received signal power with \(0\leq\rho\leq 1\) divided for EH while the remaining part, \((1-\rho)P_{S}\) used for information transmission (IT). Figure 1: System model. Therefore, we express the received signal at \(R\) in the first time slot as  
\[\sqrt{\rho}y_{R,1,x_{1}}=\sqrt{\frac{\rho P_{S}}{d_{SR}^{m}}}f_{1}x_{1}+n_{0}, \tag{1}\]  
where the path-loss exponent is \(m\). Following from (1), the harvested energy at \(R\) is calculated as  
\[E_{h}=\frac{T}{2}\eta\rho P_{S}d_{SR}^{-m}|f_{1}|^{2}, \tag{2}\]  
where the energy conversion efficiency is defined as \(0\leq\eta\leq 1\) relying on the EH circuitry and rectification. The transmit power at \(R\) is calculated with a small portion of the energy stored at itself. In particular, \(w\) and \(wP_{S}\) are used to represent the fractional constant and the transmit power supplied by the battery, respectively. We assume when \(w=0\), the transmit power at \(R\) is solely provided by the energy harvested from S. Hence, we can express the transmit power at \(R\) within \(T/2\) as  
\[P_{R}=\frac{2E_{h}}{T}+wP_{S}=\eta\rho P_{S}d_{SR}^{-m}|f_{1}|^{2}\enspace. \tag{3}\]  
However, the signal \(x_{1}\) received at \(R\) is expressed by  
\[\sqrt{1-\rho}y_{R,1,x_{1}}=\sqrt{\frac{(1-\rho)P_{S}}{d_{SR}^{m}}}f_{1}x_{1}+n_ {0}. \tag{4}\]  
In addition, the same applies for \(D\) as  
\[y_{D,1,x_{1}}=\sqrt{\frac{P_{S}}{d_{SD}^{m}}}f_{3}x_{1}+n_{0}, \tag{5}\]  
where \(d_{SD}\) is denoted the distance between \(S\) and \(D\) link. We define the received signal-to-noise ratio (SNR) as \(\gamma=\mathbb{E}\left\{\left|\mathrm{signal}\right|^{2}\right\}\left/\mathbb{ E}\left\{\left|\mathrm{overall\ noise}\right|^{2}\right\}\right\}\).