## 3 Performance Analysis of NOMA-Enabled HSTRNs
### Ergodic Capacity
#### 3.3.1 DF Protocol  
Let us now derive the ergodic capacity for relay employing DF protocol. The achievable capacity at the \(D_{1}\) can be computed by  
\[ \begin{split}\mathcal{C}_{D_{1}}^{DF}&=\frac{1- \chi}{2}\text{log}_{2}\left(1+\min\left(\Gamma_{R\to x_{1}}^{DF},\Gamma_{D_{1} \to x_{1}}^{DF}\right)\right)\\ &=\frac{1-\chi}{2}\text{log}_{2}\left(1+\min\left(\frac{P_{S}\| \mathbf{h}_{SR}\|_{F}^{2}\operatorname{\[PERSON]}_{1}}{P_{S}\|\mathbf{h}_{SR}\|_{F}^ {2}\operatorname{\[PERSON]}_{1}+N_{0}},\frac{P_{R}\|\mathbf{h}_{1}\|^{2}\operatorname {\[PERSON]}_{1}}{P_{R}\|\mathbf{h}_{1}\|^{2}\operatorname{\[PERSON]}_{2}+N_{0}}\right) \right).\end{split}  \tag{28}\]  
In practice, the amount of harvested energy at the relay is always small, hence the transmit power of relay is much lower than that of the source. Thus, it is reasonable to assume that theSINR at the destinations is lower than the SINR at the relay, i.e., \(\frac{P_{5}\|\mathbf{h}_{\mathrm{S}R}\|_{2}^{2}\[PERSON]_{1}}{P_{5}\|\mathbf{h}_{ \mathrm{S}R}\|_{2}^{2}\[PERSON]_{1}+N_{0}}>\frac{P_{6}\|\mathbf{h}_{1}\|^{2}\[PERSON]_{1}} {P_{6}\|\mathbf{h}_{1}\|^{2}\[PERSON]_{2}+N_{0}}\). Hence, Equation (28) can be rewritten as  
\[ \begin{split}\mathcal{C}_{\mathrm{D}_{1}}^{DF}&= \frac{1-\chi}{2}\text{log}_{2}\left(1+\frac{\phi\tilde{\gamma}_{SR}\|\mathbf{h }_{1}\|^{2}\Xi_{1}}{\phi\tilde{\gamma}_{SR}\|\mathbf{h}_{1}\|^{2}\Xi_{2}+1} \right)\\ &=\frac{1-\chi}{2}\text{log}_{2}\left(\frac{\phi\tilde{\gamma}_{ SR}\|\mathbf{h}_{1}\|^{2}+1}{\phi\tilde{\gamma}_{SR}\|\mathbf{h}_{1}\|^{2}\Xi_{2}+1} \right).\end{split}  \tag{29}\]  
With the instantaneous capacity is derived in Equation (29), the ergodic capacity for user \(D_{1}\) can be obtained as  
\[ \begin{split}\mathcal{C}_{D_{1}}^{DF}=& E\left\{ \frac{1-\chi}{2}\text{log}_{2}\left(\phi\tilde{\gamma}_{SR}\|\mathbf{h}_{1}\|^ {2}+1\right)\right\}\\ &-E\left\{\frac{1-\chi}{2}\text{log}_{2}\left(\phi\tilde{\gamma}_ {SR}\|\mathbf{h}_{1}\|^{2}\Xi_{2}+1\right)\right\}\\ =&\underbrace{\frac{1-\chi}{2\ln 2}\int\limits_{0}^{ \infty}\frac{1-F_{H_{1}}\left(x\right)}{1+x}dx}_{L_{1}}-\underbrace{\frac{1- \chi}{2}\int\limits_{0}^{\infty}\frac{1-F_{H_{2}}\left(x\right)}{1+x}dx}_{L_ {2}},\end{split}  \tag{30}\]  
where \(H_{1}=\phi\tilde{\gamma}_{SR}\|\mathbf{h}_{1}\|^{2}\), \(H_{2}=\phi\tilde{\gamma}_{SR}\|\mathbf{h}_{1}\|^{2}\Xi_{2}\). Based on result from [3.471.9] in [34], Equation (17) and after some algebraic manipulations, \(F_{H_{1}}\left(x\right)\) can be obtained as  
\[ \begin{split} F_{H_{1}}\left(x\right)&=1-\sum_{i_{1 }=0}^{m_{SR}-1}\cdots\sum_{i_{M}=0}^{m_{SR}-1}\Theta\left(M\right)\sum_{p=0}^ {\Delta-1}\frac{(\Delta-1)!2^{-N_{1}m_{1}-p+1}}{p!\Gamma\left(N_{1}m_{1}\right) \left(\beta-\delta\right)^{\Delta}}\\ &\times\left(\sqrt{\frac{4x\left(\beta-\delta\right)}{\rho_{5} \phi\Lambda_{1}}}\right)^{N_{1}m_{1}+p}K_{N_{1}m_{1}-p}\left(\sqrt{\frac{4x \left(\beta-\delta\right)}{\rho_{5}\phi\Lambda_{1}}}\right).\end{split}  \tag{31}\]  
With the help of [Eq. 9.34.3 and Eq. 7.811.5] in [34] and after some variable substitutions and manipulations, \(J_{1}\) and \(J_{2}\) can be written as  
\[L_{1}=\frac{1-\chi}{2\ln 2}\sum_{i_{1}=0}^{m_{SR}-1}\cdots\sum_{i_{M}=0}^{m_{ SR}-1}\Theta\left(M\right)\sum_{p=0}^{\Delta-1}\frac{\Gamma\left(\Delta \right)}{\Gamma\left(N_{1}m_{1}\right)\,p! (\beta-\delta)^{\Delta}}G_{1,3}^{3,1 }\left(\frac{\beta-\delta}{\Lambda_{1}\phi\[PERSON]_{2}\rho_{5}}\left| \begin{array} []{c}0\\ 0,N_{1}m_{1},p\end{array} \right.\right). \tag{32}\]  
Similarly, \(L_{2}\) can be calculated as follows  
\[L_{2}=\frac{1-\chi}{2\ln 2}\sum_{i_{1}=0}^{m_{SR}-1}\cdots\sum_{i_{M}=0}^{m_{ SR}-1}\Theta\left(M\right)\sum_{p=0}^{\Delta-1}\frac{\Gamma\left(\Delta \right)}{\Gamma\left(N_{1}m_{1}\right)\,p! (\beta-\delta)^{\Delta}}G_{1,3}^{3,1 }\left(\frac{\beta-\delta}{\Lambda_{1}\phi\Xi_{2}\rho_{5}}\left| \begin{array} []{c}0\\ 0,N_{1}m_{1},p\end{array} \right.\right). \tag{33}\]  
With the help of Equations (32) and (33), we obtain the achievable capacity of \(D_{1}\).