## 2 Fisher-Snedecor \(\mathcal{F}\) Fading Channel Model
### Bivariate Fisher-Snedecor \(\mathcal{F}\) Distribution
#### 2.2.1 Joint PDF of Bivariate Fisher-Snedecor \(\mathcal{F}\) Composite Envelope
\begin{array}{c}f_{R_{1},R_{2}}(r_{1},r_{2})=\sum\limits_{l=0}^{\infty}\sum \limits_{l=0}^{\infty}\frac{4(m_{1})_{k}(n_{1})\rho_{G}^{l}\rho_{G}^{l}\rho_{G }}{l!\beta(m_{1}+\rho_{1}+l)!\beta(m_{2}+k,n_{2}+l)}\\ \times\frac{\beta_{1}^{m_{1}+k}\beta_{2}^{m_{2}+k}r_{2}^{2}(m_{1}+k)-1}{(1+\beta _{1}r_{1}^{2})^{\lambda_{1}}(1+\beta_{2}r_{2}^{2})^{\lambda_{2}}}\\ \times F_{2}\left[\lambda_{2};m_{2}-m_{1},n_{2}-n_{1};m_{2}+k,n_{2}+l;\frac{ \rho_{N}\beta_{2}r_{2}^{2}}{\beta_{2}r_{2}^{2}+1},\frac{\rho_{G}}{\beta_{2}r_ {2}^{2}+1}\right],\end{array}  \tag{9}\]  
where \(\beta_{i}=m_{i}(1-\rho_{G})/((n_{i}-1)(1-\rho_{N})\Omega_{i}),\lambda_{i}=m_{i} +k+n_{i}+l,\Theta=(1-\rho_{N})^{m_{2}}(1-\rho_{G})^{n_{2}}\), \((i=1,2)\), \(F_{2}[\cdot]\) is the Appell Hypergeometric function defined in ([43], Equation (9.180.2)). Furthermore, by using the generalized Lauricella series function defined in ([47], Equation A.20), the compact form of (9) can be written as:  
\[ \begin{array}{c}f_{R_{1},R_{2}}(r_{1},r_{2})=\prod\limits_{l=1}^{2}\left( \frac{2\beta_{i}^{n_{l}}r_{l}^{2m_{l}-1}}{(1+\beta_{i}r_{l}^{2})^{m_{l}+n_{l}} \beta_{(m_{l}+n_{l})}}\right)\\ \times F_{2}^{\underline{0}\underline{0}\underline{0}\underline{1}\underline{1} \underline{1}}\begin{bmatrix}(m_{2}+n_{2};1,1,1),(m_{1}+n_{1};1,1,0,0)\\ (m_{2};1,0,1,0),(n_{2};1,0,1)\end{bmatrix}[-][-][-][-][-][-][-][-][-][-][-][-]}\Big{|} \rho_{N}A_{1},\rho_{G}A_{2},\rho_{N}A_{3},\rho_{G}A_{4}\Big{]},\end{array}  \tag{10}\]  
where: \(A_{1}=\prod\limits_{l=1}^{2}\frac{\beta_{i}r_{l}^{2}}{(1+\beta_{i}r_{l}^{2})},A_ {2}=\prod\limits_{i=1}^{2}\frac{1}{(1+\beta_{i}r_{l}^{2})},A_{3}=\beta_{2}r_{2}^{ 2}A_{4},A_{4}=\frac{1}{(1+\beta_{i}r_{2}^{2})},F_{2}^{\underline{0}\underline{0} \underline{0}\underline{1}\underline{1}}\) in (10) is defined as:  
\[ \begin{array}{c}F_{2}^{\underline{0}\underline{0}\underline{0}\underline{0} \underline{1}\underline{1}}\begin{bmatrix}(m_{2}+n_{2};1,1,1),(m_{1}+n_{1};1,1,0 )\\ (m_{2};1,0,1,0),(n_{2};1,0,1)\end{bmatrix}[-][-][-][-][-][-][-][-][-][-][-][-]\\ =\sum\limits_{k=0}^{\infty}\sum\limits_{l=0}^{\infty}\sum\limits_{l=0}^{\infty} \sum\limits_{l=0}^{\infty}\sum\limits_{j=0}^{(m_{2}+n_{2})_{k+l+i+j}(m_{1}+n_{1}) _{k+l}(m_{2}-m_{1})_{j}(n_{2}-n_{1})_{j}}\\ \times\left(\rho_{N}A_{1}\right)^{k}(\rho_{G}A_{2})^{l}(\rho_{N}A_{3})^{l}(\rho_{G }A_{4})^{l}.\end{array}  \tag{11}\]  
Additionally, the joint PDF in (10) can be expressed in terms of the multivariable H-function defined in ([47], Equation (A.31)). For the identical shaping parameters case, i.e., \(m_{1}=m_{2}=m\) and \(n_{1}=n_{2}=n_{,}\)\(F_{2}[\cdot]\) in (9) equals unity with the aid of Lemma 2.1 in [43] and the identity (9.155.4) defined in [43]. Hence, a simplified formula of (9) can be deduced as ([38], Equation (5)):  
\[f_{R_{1},R_{2}}(r_{1},r_{2})=\sum\limits_{k=0}^{\infty}{\sum\limits_{l=0}^{ \infty}{\frac{4\rho_{\text{N}}^{k}\rho_{\text{G}}^{l}\phi\Gamma(\lambda)( \Xi_{1}\Xi_{2})^{(m+k)}}{k!\Pi!\Gamma(m)\Gamma(n)B(k+m,l+n)}}}\prod\limits_{i=1}^ {2}{\frac{r_{i}^{2m+2k-1}}{\left(\Xi_{i}r_{i}^{2}+1\right)^{\lambda}}} \tag{12}\]  
where \(\Xi_{i}=m(1-\rho_{\text{G}})/((n-1)(1-\rho_{N})\Omega_{i})\), \(\lambda=m+k+n+l\), \(\phi=(1-\rho_{N})^{m}(1-\rho_{\text{G}})^{\text{n}}\).