## 3 Performance Analysis
### Preliminary Results
#### 3.1.2 Satellite Channel Model
Furthermore, the PDF of \(\gamma_{SJ}=\gamma_{SJ}\big{|}C_{SJ}^{\max}g_{SJ}\big{|}^{2}\) is given by:  
\[f_{\gamma_{SJ}}\left(x\right)=\frac{\alpha_{SJ}}{\widehat{\gamma}_{SJ}}e^{- \frac{\beta_{SJ}}{\gamma_{SJ}}}{}_{1}F_{1}\left(m_{SJ};1;\frac{\delta_{SJ}}{ \widehat{\gamma}_{SJ}}x\right),x>0, \tag{19}\]  
where \({}_{1}F_{1}\left(a;b;x\right)\) denotes the confluent hypergeometric function defined in [34]. \(\gamma_{SJ}\) is the average SNR between Alice and the \(\xi\)-th R with expression as \(\overline{\gamma}_{SJ}=\frac{\delta Q}{\overline{\theta}_{k}^{2}}\), \(\alpha_{SJ}=\Big{(}\frac{2b_{SJ}m_{SJ}}{2b_{SJ}m_{SJ}+\Omega_{SJ}}\Big{)}^{m_{ SJ}}/2b_{SJ}\), \(\beta_{SJ}=\frac{1}{2b_{SJ}}\), \(\delta_{SJ}=\frac{\Omega_{SJ}}{2b_{SJ}\left(2b_{SJ}m_{SJ}+\Omega_{SJ}\right)}\) with \(\Omega_{SJ}\), \(2b_{SJ}\) and \(m_{SJ}\geq 0\) being the average power of the LOS component, the average power of the multipath component and the fading severity parameter ranging from 0-\(\infty\), respectively. By considering \(m\) being an integer, the PDF of \(\gamma_{SJ}\) is given by:  
\[f_{\gamma_{SJ}}\left(x\right)=\alpha\sum_{k=0}^{m_{SJ}-1}\frac{\left(1-m_{SJ} \right)_{k}\big{(}-\delta_{SJ}\big{)}^{k}}{\left(k!\right)^{2}\left(\gamma_{SJ }\right)^{k+1}}x^{k}\exp\left(-\Delta_{SJ}x\right), \tag{20}\]  
where \(\Delta_{SJ}=\frac{\beta_{SJ}-\delta_{SJ}}{\gamma_{SJ}}\) and \(\left(\cdot\right)_{k}\) is the Pochhammer symbol. Hence, the CDF of \(\gamma_{SJ}\) is given by:  
\[F_{\gamma_{SJ}}\left(x\right)=1-\alpha\sum_{k=0}^{m_{SJ}-1}\sum_{t=0}^{k} \frac{\left(1-m_{SJ}\right)_{k}\big{(}-\delta_{SJ}\big{)}^{k}}{k!\big{(} \gamma_{SJ}\big{)}^{k+1}t!\Delta_{SJ}^{k-t+1}}x^{t}e^{-\Delta_{SJ}x}. \tag{21}\]  
In the manuscript, we consider the worst case (The worst case is for the secondary users. In this paper, it means that all the primary users operated on the same frequency spectrum and worked together; hence, the SNR is the sum of the SNR of all primary users. On this foundation, the left available spectrum is minimal, which is the worst case for the secondary user. ); hence:  
\[\gamma_{SP}{=}\sum\limits_{i=1}^{M}{\gamma_{SP_{i}}}. \tag{22}\]  
From [24], the probability distribution function (PDF) of \(\gamma_{SP}\) is given by:  
\[f_{\gamma_{SP}}\left(x\right)=\sum\limits_{\xi_{1}=0}^{m_{SP}-1}\cdots\sum \limits_{\xi_{M}=0}^{m_{SP}-1}\Xi\left(M\right)x^{\Lambda_{SP}-1}e^{-\Delta_{ SP}x}, \tag{23}\]  
where:  
\[\Xi\left(M\right)\overset{\Delta}{=}\prod\limits_{\tau=1}^{M}{\zeta\left({ \xi_{\tau}}\right)\alpha_{SP}^{M}\prod\limits_{\
u=1}^{M-1}{B\left({\sum \limits_{l=1}^{v}{\xi_{l}+\
u,\xi_{\
u+1}+1}}\right)}},\]  
\[\Lambda_{SP}\overset{\Delta}{=}\sum\limits_{\tau=1}^{M}{\xi_{\tau}+M},\zeta \left({\xi_{\tau}}\right)=\frac{{\left({1-m_{SP}}\right)_{\xi_{\tau}}\left({- \delta_{SP}}\right)^{\xi_{\tau}}}}{{\left({\xi_{\tau}!}\right)^{2}\left({\xi_ {\tau}}{\xi_{\tau}}\right)^{\xi_{\tau+1}}}},\,\Delta_{SP}=\frac{{\beta_{SP}- \delta_{SP}}}{{\gamma_{SP}}},\,\alpha_{SP}\overset{\Delta}{=}\frac{{\left({ \frac{{2{\beta_{SP}}m_{SP}+\Omega_{SP}}}{{2{\beta_{SP}}}}}\right)^{m_{SP}}}}{{ 2{\beta_{SP}}}},\,\beta_{SP}\overset{\Delta}{=}\frac{{1}}{{2{\beta_{SP}}}},\]  
\(\delta_{SP}\overset{\Delta}{=}\frac{{\Omega_{SP}}}{{2{\beta_{SP}}\left({2{ \beta_{SP}}m_{SP}+\Omega_{SP}}\right)}}\) and \(B\left(.,.\right)\) denotes the Beta function [34].