# Achievable Rate of NOMA-Based Cooperative Spectrum-Sharing CRN over Nakagami-\(m\) Channels
## 4 Achievable Rate Analysis
### Ergodic Achievable Rates
\begin{array}{c}1-n\\ m_{2},0\end{array} \bigg{)}. \tag{32}\]  
The ergodic achievable rate expression at the PR after substituting the value of \(I_{3}\) in (27) can be expressed as  
\[\bar{R}_{p}^{\rm NOMA} =\frac{1}{2\ln(2)\Gamma(m_{0})}G_{3,2}^{1,3}\bigg{(}\frac{ \overline{\gamma}_{p}}{m_{0}}\bigg{|} \begin{array}{c}1,\;1,\;1-m_{0}\\ 1,\;0\end{array} \bigg{)}\] \[+\frac{1}{2\ln(2)\Gamma(m_{0})\Gamma(m_{2})}\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n\,\Gamma(n)\Gamma(n)}\bigg{(}\frac{\alpha_{0}}{\alpha} \bigg{)}^{n}\] \[\times G_{1,2}^{2,1}\bigg{(}\frac{m_{0}}{\overline{\gamma}_{p}} \bigg{|} \begin{array}{c}1\\ m_{0},n\end{array} \bigg{)}\,G_{1,2}^{2,1}\bigg{(}\frac{m_{2}}{\alpha\, \overline{\gamma}_{s}}\bigg{|} \begin{array}{c}1-n\\ m_{2},0\end{array} \bigg{)}. \tag{33}\]  
Expression (33) contains two [PERSON]'s G-functions inside the summation, which may cause slower convergence of the above expression. Therefore, it is more useful to replace [PERSON]'s G-functions in terms of Tricomi confluent hypergeometric functions ([23], Equation (9.211.4)) using identity [27] for rapid convergence of the summation. Finally, we obtain a closed-form expression of the ergodic achievable rate at the PR as  
\[\bar{R}_{p}^{\rm NOMA}=\frac{1}{2\ln(2)\Gamma(m_{0})}G_{3,2}^{1,3 }\bigg{(}\frac{\overline{\gamma}_{p}}{m_{0}}\bigg{|} \begin{array}{c}1,\;1, \;1-m_{0}\\ 1,\;0\end{array} \bigg{)}\] \[+\frac{\left(\frac{m_{0}}{\overline{\gamma}_{p}}\right)^{m_{0}} \left(\frac{m_{2}}{\alpha\,\overline{\gamma}_{s}}\right)^{m_{2}}}{2\ln(2) \Gamma(m_{2})}\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}\Big{(}\frac{\alpha_{0}} {\alpha}\Big{)}^{n}\Gamma(m_{2}+n)\] \[\times\Psi\left(m_{0},1+m_{0}-n,\frac{m_{0}}{\overline{\gamma}_{ p}}\right)\Psi\left(m_{2}+n,1+m_{2},\frac{m_{2}}{\alpha\,\overline{\gamma}_{s}} \right). \tag{34}\]  
It is important to mention here that only two or three terms are sufficient for the convergence of the infinite series in (34). **Ergodic Achievable Rate at the SR**: Using (7), the ergodic achievable data rate at the \(k^{\rm th}\) SR can be determined using (13) and (15) in (14) as \[\bar{R}_{s,k}^{\text{NOMA}}=\mathbb{E}\{R_{s,k}^{\text{NOMA}}\}\] \[\quad=\frac{\left(\frac{m_{22}}{\overline{\gamma}_{sk}}\right)}{2 \text{ln}(2)\Gamma(m_{22})}\int_{0}^{\infty}H_{2,2}^{1,2}\bigg{[}\frac{\gamma_ {sk}\,\alpha_{k}}{\gamma_{sk}\,\alpha+1}\bigg{|} \begin{array}{c}(1,1),(1,1) \\ (1,1),(0,1)\end{array} \bigg{]}\] \[\quad\times H_{0,1}^{1,0}\bigg{[}\frac{m_{22}\gamma_{sk}}{ \overline{\gamma}_{sk}}\bigg{|} \begin{array}{c}(m_{22}-1,1)\end{array}  \bigg{]}\,d\gamma_{sk}, \tag{35}\]  
where \(m_{22}\) is the fading severity parameter of the link between ST and the \(k^{\text{th}}\) SR link and where \(\gamma_{sk}=\rho_{s}|g_{s,k}|^{2}\), \(\bar{\gamma}_{sk}=\mathbb{E}\{\gamma_{sk}\}\), and \(\bar{\pi}\)=\(\sum_{i=k+1}^{k}\alpha_{i}\). Now, using the definition of [PERSON]'s H-function ([25], Equation (1.1.1)), expression (35) becomes  
\[\bar{R}_{s,k}^{\text{NOMA}}=\frac{\left(\frac{m_{22}}{\overline{ \gamma}_{sk}}\right)}{2\text{ln}(2)\Gamma(m_{22})}\frac{1}{\left(2\pi\iota \right)^{2}}\int_{\mathfrak{L}_{1}}\int_{\mathfrak{L}_{2}}\frac{\Gamma(1+s) \Gamma(-s)^{2}}{\Gamma(1-s)}\] \[\quad\times\Gamma(m_{22}-1+t)(\alpha_{k})^{-s}\bigg{(}\frac{m_{2 2}}{\overline{\gamma}_{sk}}\bigg{)}^{-t}\underbrace{\int_{0}^{\infty}\gamma_{ sk}^{-s-t}(\gamma_{sk}\bar{\pi}+1)^{s}d\gamma_{sk}}_{l_{4}}\] \[\quad ds\,dt \tag{36}\]  
where \(\mathfrak{L}_{1}\) and \(\mathfrak{L}_{2}\) are contours of the \(s\)- and \(t\)-complex planes, respectively. Now, integral \(I_{4}\) can be evaluated using ([23], Equations (3.194.3) and (8.384.1)), yielding  
\[I_{4}=(\bar{\pi})^{-1+s+t}\frac{\Gamma(1-s-t)\Gamma(-1+t)}{\Gamma(-s)}, \tag{37}\]  
where \(Re\{t\}\)\(>\)\(1\)\(>\)\(Re\{s+t\}\).