## 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  
Let \(X_{i}\) (\(i=1,2\)) be the channel fading envelopes of Nakagami-\(m\) processes, with the joint PDF between \(X_{1}\) and \(X_{2}\) given in ([44], Equation (12)) as:  
\[ \begin{array}{c}f_{X_{1},X_{2}}(x_{1},x_{2})=4(1-\rho_{N})^{m_{2}}\sum\limits \limits_{k=0}^{\infty}\frac{(m_{1})_{k}e^{k}_{k}}{\Gamma_{1}}F_{1}[m_{2}-m_{1},m_{2}+k;\frac{\rho_{N}m_{2}x_{2}^{2}}{\gamma_{2}(1-\rho_{N})}]\\ \times\prod\limits_{i=1}^{2}\left[\frac{m_{i}}{Y_{i}(1-\rho_{N})}\right]^{m_ {i}+\frac{x_{i}^{2(m_{i}+k)-1}}{\Gamma(m_{i}+k)}}\exp\left[-\frac{m_{i}x_{i}^ {2}}{(1-\rho_{N})Y_{i}}\right],\end{array}  \tag{5}\]  
where \(m_{2}\geq m_{1}\geq 1/2\) is the Nakagami-\(m\) fading parameter, \(\rho_{N}\) is the power correlation coefficient between \(X_{1}^{2}\), \(X_{2}^{2}\) and \(Y_{i}\) is the mean fading power \(Y_{i}=\mathbb{E}[X_{i}^{2}]\). Furthermore, \({}_{1}F_{1}(\cdot;\cdot;\cdot)\) is the confluent hypergeometric function defined in ([43], Equation (9.210/1)) and \((x)_{p}=\Gamma(x+p)/\Gamma(x)\) is the Pochhammer's symbol with \(p\in\mathbb{N}\) defined in ([43], p.xiiii). When multipath fading is superimposed on shadowing, \(Y_{i}\) slowly varies and its root mean square can be considered as a random variable following the inverse Nakagami-\(m\) distribution in [29]. Based on the revised channel model in [45], we let \(Y_{i}=w_{i}^{2}\Omega_{i}\) (\(i=1,2\)), where \(w_{i}\) is a normalized inverse Nakagami-\(m\) random variable with \(\mathbb{E}[w_{i}]=1,\Omega_{i}=\mathbb{E}\left[R_{i}^{2}\right]\) is the mean power of the composite signal envelope \(R_{i}\), then the PDF in (5) is conditioned on \(w_{i}\). To model the inverse Nakagami-\(m\) distribution, we let the parameter \(w_{i}=a_{i}/r_{i}\), where \(r_{i}\) follows Nakagami-\(m\) distribution, \(a_{i}=\sqrt{(n_{i}-1)/n_{i}}\). Using a standard transformation of random variables, the PDF of the bivariate inverse Nakagami-\(m\) distribution can be obtained as  
\[ \begin{array}{c}f_{W_{1},W_{2}}(w_{1},w_{2})=4(1-\rho_{G})^{n_{2}}\sum\limits_{ l=0}^{\infty}\frac{(n_{1})\rho_{G}^{l}}{l!}{}_{1}F_{1}[n_{2}-n_{1},n_{2}+l; \frac{\rho_{G}(n_{2}-1)w_{2}^{-2}}{1-\rho_{G}}]\\ \times\prod\limits_{l=1}^{2}\left[\frac{n_{1}-l}{1-\rho_{G}}\right]^{n_{l}+l} \frac{w_{l}^{-2(n_{l}+l)-1}}{\Gamma(n_{l}+l)}\exp\left[-\frac{(n_{l}-1)w_{l}^{ -2}}{1-\rho_{G}}\right],\end{array}  \tag{6}\]  
where \(n_{2}\geq n_{1}\geq 1/2\) is the inverse Nakagami-\(m\) shaping parameter and \(\rho_{G}\) is the power correlation coefficient between \(w_{1}^{2}\) and \(w_{2}^{2}\). In [29], the PDF of the Fisher-Snedecor \(\mathcal{F}\) composite envelope is obtained by averaging the conditional PDF of the Nakagami-\(m\) process over the random variation of the root mean square signal power. To this effect, the PDF of the bivariate Fisher-Snedecor \(\mathcal{F}\) composite distribution is written as:  
\[f_{R_{1},R_{2}}(r_{1},r_{2})=\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}f _{Y_{1}|W_{1},Y_{2}|W_{2}}(r_{1}|w_{1},r_{2}|w_{2})f_{W_{1},W_{2}}(w_{1},w_{2}) dw_{1}dw_{2}. \tag{7}\]  
Based on (5), \(f_{Y_{1}|W_{1},Y_{2}|W_{2}}(r_{1}|w_{1},r_{2}|w_{2})\) in (7) can be obtained as:  
\[ \begin{array}{c}f_{R_{1}|W_{1},R_{2}|W_{2}}(r_{1}|w_{1},r_{2}|w_{1})=4(1-\rho _{N})^{m_{2}}\sum\limits_{k=0}^{\infty}\frac{(m_{1})_{k}\rho_{N}^{k}}{k!}F_{1}[ m_{2}-m_{1},m_{2}+k;\frac{\rho_{N}m_{2}r_{2}^{2}}{w_{2}^{2}\Omega_{2}(1-\rho_{N})}] \\ \times\prod\limits_{l=1}^{2}\left[\frac{m_{l}}{w_{l}^{2}\Omega_{1}(1-\rho_{N} )}\right]^{m_{l}+k}\frac{r_{l}^{2(n_{l}+k)-1}}{\Gamma(n_{l}+k)}\exp\left[- \frac{m_{l}^{2}}{(1-\rho_{N})w_{l}^{2}\Omega_{l}}\right].\end{array}  \tag{8}\]  
Substituting (6) and (8) in (7), and using ([46], Equation (2.2)) with some mathematical manipulation, the joint PDF of the bivariate Fisher-Snedecor \(\mathcal{F}\) composite distribution with arbitrary fading parameters can be derived as:  
\[