## 2 Fisher-Snedecor \(\mathcal{F}\) Fading Channel Model
### Univariate Fisher-Snedecor \(\mathcal{F}\) Distribution  
The PDF of the instantaneous SNR, \(\gamma\), over the Fisher-Snedecor \(\mathcal{F}\) composite fading channel is given by [29]:  
\[f_{\gamma}(\gamma)=\frac{\Lambda^{n}\gamma^{m-1}}{B(m,n)(\Lambda\gamma+1)^{m+ n}}, \tag{1}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Acronym** & **Description** & **Acronym** & **Description** \\ \hline ABEP/ASEP & Average Bit/Symbol Error Probability & n.i.d. & Non-Independent and Identically Distributed \\ AoF & amount of fading & NCBFSK & non-coherent binary frequency shift keying \\ AWGN & additive white Gaussian noise & OP & outage probability \\ BDPSK & binary differential phase-shift keying & PDF & probability density function \\ CDF & cumulative distribution function & PSC & pure selection combining \\ i.i.d. & independent and identically distributed & SEC & switch-and-examine combining \\ i.n.i.d. & independent and non-identically distributed & SECps & SEC with post-examining selection \\ MGF & Moment-generating function & SDC & Switched diversity combining \\ MRC & maximum ratio combining & SNR & output signal-to-noise ratio \\ MQAM & M-ary quadrature amplitude modulation & SSC & switch-and-stay combining \\ n.i.n.i.d. & non-independent and non-identically distributed & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: A list of acronyms used in this paper.  
where \(\Lambda=m/(n-1)\overline{\gamma}\), which is valid for \(n>1\); \(m\) and \(n\) are the multipath fading severity and the shadowing shape parameter, respectively; \(\overline{\gamma}=\mathbb{E}[\gamma]E_{s}/N_{0}\) represents the average SNR, where \(\mathbb{E}[\cdot]\) denotes expectation, \(E_{s}\) is the per symbol energy, and \(N_{0}\) is the single-sided power spectral density of the complex additive white Gaussian noise (AWGN); \(B(\cdot\cdot\cdot)\) denotes the beta function defined in ([43], Equation (8.384.1)). For the Fisher-Snedecor \(\mathcal{F}\) composite fading model, the received signals suffer heavy shadowing when \(n\to 1\). On the contrary, when \(n\to\infty\), this indicates the absence of shadowing in wireless fading environments, and this model reduces to a Nakagami-\(m\) fading channel as a special case. Furthermore, it also comprises Rayleigh (\(n\to\infty,\ m\to 1\)) and one-sided Gaussian (\(n\to\infty,\ m\to 0.5\)) effects. The corresponding CDF of the instantaneous SNR, \(\gamma\) is also given by [29]:  
\[F_{\gamma}(\gamma)=\frac{(\Lambda\gamma)^{m}}{mB(m,n)}2F_{1}[m+n,m;m+1;-\Lambda \gamma], \tag{2}\]  
where \({}_{2}F_{1}[\cdot]\) is the Gauss hypergeometric function defined in ([43], Equation (9.100)).  
The MGF-based approach is very useful in the performance analysis of wireless digital communication systems. It can be applied to simplify the mathematical analysis and evaluate the ABEP/ASEP and the OP over fading channels. By using (1) and the basic definition of the MGF in [1], and with the aid of ([43], Equation (7.811.5)), the MGF of the instantaneous SNR, \(\gamma\) can be derived as:  
\[MGF_{\gamma}(\gamma)=\int_{0}^{\infty}\exp(-s\gamma)f_{\gamma}(\gamma)d\gamma= \frac{G_{12}^{2,1}[s/\Lambda]_{[0,n}^{1-m}]}{\Gamma(m)\Gamma(n)}, \tag{3}\]  
where \(\Gamma\) (\(\cdot\)) is the gamma function defined in ([43], Equation (8.310/1)] and \(G_{p,q}^{m,n}[\cdot\cdot]\) denotes the univariate Meijer G-function defined in ([43], Equation (9.301)). Similarly, the \(q\)th moment of the instantaneous SNR, \(\gamma\) can also be given as:  
\[\mu_{\gamma}(q)=\int_{0}^{\infty}\gamma^{q}f_{\gamma}(\gamma)d\gamma=\frac{B( m+q,n-q)}{B(m,n)\Lambda^{q}}. \tag{4}\]  
It is noted that (4) is valid only when \(n>q\).