# Achievable Rate of NOMA-Based Cooperative Spectrum-Sharing CRN over Nakagami-\(m\) Channels
## 4 Achievable Rate Analysis
### Ergodic Achievable Rates
\biggr{]}\] \[\times H_{0,1}^{1,0}\left[\frac{m_{0}\gamma_{p}}{\overline{ \gamma}_{p}}\biggr{|} \begin{array}{c}\\ (m_{0}-1,1)\end{array} \right]d\gamma_{p}. \tag{24}\]  
The integral in (24) can be evaluated using ([25], Equation (2.6.8)) and further simplified using identities ([25], Equations (1.2.4) and (1.7.1)) to yield \[I_{2} =\frac{1}{\ln(2)\Gamma(m_{0})}\sum_{n=1}^{\infty}\frac{\left(-1 \right)^{n+1}}{n\,\Gamma(n)}(C)^{n}\] \[\times G_{1,2}^{2,1}\Bigg{(}\frac{m_{0}}{\overline{\gamma}_{p}} \bigg{|} \begin{array}{c}1\\ m_{0},n\end{array} \Bigg{)}. \tag{25}\]  
Substituting the values of \(I_{1}\) and \(I_{2}\) in (19), we obtain  
\[I =\frac{1}{\ln(2)\Gamma(m_{0})}G_{3,2}^{1,3}\Bigg{(}\frac{ \overline{\gamma}_{p}}{m_{0}}\bigg{|} \begin{array}{c}1,\;1,\;1-m_{0}\\ 1,\;0\end{array} \Bigg{)}\] \[+\frac{1}{\ln(2)\Gamma(m_{0})}\sum_{n=1}^{\infty}\frac{\left(-1 \right)^{n+1}}{n\,\Gamma(n)}(C)^{n}\] \[\times G_{1,2}^{2,1}\Bigg{(}\frac{m_{0}}{\overline{\gamma}_{p}} \bigg{|} \begin{array}{c}1\\ m_{0},n\end{array} \Bigg{)}. \tag{26}\]  
It can be observed from (26) that our analytical approach has led us to an easily computable expression consisting of a summation and two [PERSON]'s G-functions. Now, substituting the value of \(I\) in (17), we have  
\[\bar{R}_{p}^{\text{NOMA}} =\frac{1}{2\ln(2)\Gamma(m_{0})}G_{3,2}^{1,3}\Bigg{(}\frac{ \overline{\gamma}_{p}}{m_{0}}\bigg{|} \begin{array}{c}1,\;1,\;1-m_{0}\\ 1,\;0\end{array} \Bigg{)}\] \[\times\int_{0}^{\infty}f_{\gamma_{s}}(\gamma_{s})d\gamma_{s}\] \[+\frac{1}{2\ln(2)\Gamma(m_{0})}\sum_{n=1}^{\infty}\frac{\left(-1 \right)^{n+1}}{n\,\Gamma(n)}\left(\frac{\alpha_{0}}{\alpha}\right)^{n}\] \[\times G_{1,2}^{2,1}\Bigg{(}\frac{m_{0}}{\overline{\gamma}_{p}} \bigg{|} \begin{array}{c}1\\ m_{0},n\end{array} \Bigg{)}\] \[\times\underbrace{\int_{0}^{\infty}\bigg{(}\frac{\alpha\,\gamma_{s }}{\alpha\,\gamma_{s}+1}\bigg{)}^{n}f_{\gamma_{s}}(\gamma_{s})d\gamma_{s}}_{ \text{$\text{$\text{$\text{$\text{$\text{$\text{$\text{\text{$\text{ $\text{ $\text{ $\text{ $\text{ $\text{$\text{$\text{$}$\text{$}$\text{$}${\text{$}${\text{$$}${${}}}}}}}$}}}}} \,. \tag{27}\]  
It is a well-known fact in statistics that \(\int_{0}^{\infty}f_{\gamma_{s}}(\gamma_{s})d\gamma_{s}=1\). Furthermore, the expression \(\left(\frac{\alpha\,\gamma_{s}}{\alpha\,\gamma_{s}+1}\right)^{n}\) in \(I_{3}\) can be written in terms of [PERSON]'s H-function using ([25], Equations (1.7.3) and (1.2.4)) as  
\[\left(\frac{\alpha\,\gamma_{s}}{\alpha\,\gamma_{s}+1}\right)^{n}=\frac{1}{ \Gamma(n)}H_{1,1}^{1,1}\bigg{[}\alpha\,\gamma_{s}\bigg{|} \begin{array}{c} (1,1)\\ (n,1)\end{array} \bigg{]}. \tag{28}\]  
We now focus on evaluating \(I_{3}\) by substituting (13) and (28) in it to obtain  
\[I_{3} =\frac{\left(\frac{m_{2}}{\overline{\gamma}_{s}}\right)}{\Gamma( n)\Gamma(m_{2})}\int_{0}^{\infty}H_{1,1}^{1,1}\bigg{[}\alpha\,\gamma_{s} \bigg{|} \begin{array}{c}(1,1)\\ (n,1)\end{array} \bigg{]}\] \[\times H_{0,1}^{1,0}\bigg{[}\frac{m_{2}\gamma_{s}}{\overline{ \gamma}_{s}}\bigg{|} \begin{array}{c}\overline{(m_{2}-1,1)}\end{array}  \bigg{]}\,d\gamma_{s}, \tag{29}\]  
where \(m_{2}\) is the fading severity parameter for the ST-PR link. Substituting \(\alpha\,\gamma_{s}=x\), \(I_{3}\) simplifies into \[I_{3} =\frac{\left(\frac{m_{2}}{\kappa^{\frac{3}{7}}\gamma_{s}}\right)}{ \Gamma(n)\Gamma(m_{2})}\int_{0}^{\infty}H_{1,1}^{1,1}\bigg{[}x\bigg{|}(1,1) \bigg{]}\] \[\times H_{0,1}^{1,0}\bigg{[}\frac{m_{2}\,x}{\alpha\,\overline{ \gamma}_{s}}\bigg{|} \begin{array}{c}\\ (m_{2}-1,1)\end{array} \bigg{]}\,dx. \tag{30}\]  
Integral \(I_{3}\) can be evaluated using ([25], Equation (2.6.8)); then, using relation ([25], Equation (1.2.4)), we obtain  
\[I_{3}=\frac{1}{\Gamma(n)\Gamma(m_{2})}H_{1,2}^{2,1}\bigg{[}\frac{m_{2}}{\alpha \,\overline{\gamma}_{s}}\bigg{|} \begin{array}{c}\\ (m_{2},1),(0,1)\end{array} \bigg{]}. \tag{31}\]  
A simplified expression of \(I_{3}\) is obtained in terms of Meijer's G-function using ([25], Equation (1.7.1)) as  
\[I_{3}=\frac{1}{\Gamma(n)\Gamma(m_{2})}G_{1,2}^{2,1}\bigg{(}\frac{m_{2}}{\alpha \,\overline{\gamma}_{s}}\bigg{|}