## 3 Performance Analysis of NOMA-Enabled HSTRNs
### Ergodic Capacity
#### 3.3.2 AF Protocol  
The achievable capacity at the \(D_{i}\) in case of AF relaying can be calculated by  
\[\mathcal{C}_{D_{i}}^{AF}=\frac{1-\chi}{2}\text{log}_{2}\left(1+\Gamma_{D_{i}\to x _{i}}^{AF}\right). \tag{36}\]  
Moreover, the ergodic capacity of user \(D_{1}\) can be formulated as  
\[\mathcal{C}_{D_{1}}^{AF}=\frac{1-\chi}{2\ln 2}\int\limits_{0}^{\Xi_{2}/\Xi_{1}} \frac{1-F_{\Gamma_{D_{1}\to x_{1}}}\left(x_{1}\right)}{1+x_{1}}dx_{1}. \tag{37}\]  
In the above expression, the CDF of \(\Gamma_{D_{1}\to x_{1}}\) can be expressed as  
\[\begin{split} F_{\Gamma_{D_{1}\to x_{1}}}\left(x_{1}\right)& =1-2\sum\limits_{i_{1}=0}^{m_{SR}-1}\cdots\sum\limits_{i_{M}=0}^{m_{SR}-1} \Theta\left(M\right)\sum\limits_{n_{1}=0}^{N_{1}m_{1}-1}\sum\limits_{q=0}^{ \Delta-1}\left(\begin{array}{c}\Delta-1\\ q\end{array}\right)\frac{\left(\Lambda_{1}\phi\right)^{q}}{n_{1}!(\beta- \delta)^{\Delta}}\\ &\times\left(\frac{x_{1}\left(\beta-\delta\right)}{\left(\[PERSON]_{1}-\[PERSON]_{2}x_{1 }\right)\Lambda_{1}\phi\rho_{S}}\right)^{\frac{\Delta+q+n_{1}}{2}}e^{-\frac{x _{1}\left(\beta-\delta\right)}{\left(\[PERSON]_{1}-\[PERSON]_{2}x_{1}\right)\rho_{S}}}K_{ \Delta-q-n_{1}}\left(\frac{x_{1}\left(\beta-\delta\right)}{\left(\[PERSON]_{1}-\[PERSON]_{ 2}x_{1}\right)\Lambda_{1}\phi\rho_{S}}\right).\end{split} \tag{38}\]  
Now, using Gaussian-Chebyshev with \(\phi_{n}=\cos\left(\frac{2n-1}{2N}\pi\right)\). After solving the integral, we can obtain \(\mathcal{C}_{D_{1}}^{AF}\) as  
\[\begin{split}\mathcal{C}_{D_{1}}^{AF}&\approx\frac{ 1-\chi}{\ln 2}\sum\limits_{i_{1}=0}^{m_{SR}-1}\cdots\sum\limits_{i_{M}=0}^{m_{SR}-1} \Theta\left(M\right)\sum\limits  
With the help of \(\left(1+x_{2}\right)^{-1}=G_{1,1}^{1,1}\left(x_{2}\left|\begin{array}{c}0\\ 0\end{array}\right.\right)\) and [9.34.3] in [34], we can rewrite \(\mathcal{C}_{D_{2}}^{AF}\) as  
\[\begin{split}\mathcal{C}_{D_{2}}^{AF}&=\frac{1-\chi}{2\ln 2} \sum\limits_{i_{1}=0}^{m_{SR}-1}\cdots\sum\limits_{i_{M}=0}^{m_{SR}-1}\Theta \left(M\right)\sum\limits_{n_{1}=0}^{N_{1}m_{1}-1}\sum\limits_{q=0}^{\Delta-1 }\left(\begin{array}{c}\Delta-1\\ q\end{array}\right)\frac{\left(\Lambda_{1}\phi\right)^{q}}{n_{1}!(\beta-\delta )^{\Delta}}\\ &\times\int\limits_{0}^{\infty}e^{-\frac{\left(\beta-\delta \right)}{\Xi_{2}\rho_{S}}x_{2}}G_{1,1}^{1,1}\left(x_{2}\left|\begin{array}{c }0\\ 0\end{array}\right.\right)G_{0,2}^{2,0}\left(\frac{\left(\beta-\delta \right)}{\Xi_{2}\Lambda_{1}\phi\rho_{S}}x_{2}\left|\begin{array}{c}-\\ \Delta,q+n_{1}\end{array}\right.\right)dx_{2}.\end{split} \tag{43}\]  
Based on [Eq 2.6.2] in [35], the closed-form expression for ergodic capacity of \(D_{2}\) is given by  
\[\begin{split}\mathcal{C}_{D_{2}}^{AF}&=\frac{1-\chi}{2\ln 2 }\sum\limits_{i_{1}=0}^{m_{SR}-1}\cdots\sum\limits_{i_{M}=0}^{m_{SR}-1}\Theta \left(M\right)\sum\limits_{n_{1}=0}^{N_{1}m_{1}-1}\sum\limits_{q=0}^{\Delta-1 }\left(\begin{array}{c}\Delta-1\\ q\end{array}\right.\right)\frac{\left(\Lambda_{1}\phi\right)^{q}}{n_{1}!( \beta-\delta)^{\Delta}}\\ &\times G_{1,[1;0],0,[1;2]}^{1,1,0,1,2}\left(\begin{array}{c}1\\ \frac{\Xi_{2}\rho_{S}}{(\beta-\delta)}\\ \frac{1}{\Lambda_{1}\varphi}\end{array}\right.\left|\begin{array}{c}0;-\\ -\\ 0;\Delta,q+n_{1}\end{array}\right.\right).\end{split} \tag{44}\]