# \(\alpha\)-Fluctuating Nakagami-\(m\) Fading Model for Wireless Communications
## 3 \(\alpha\)-Fluctuating Nakagami Fading Channel Model
### Statistical Model  
Based on the given physical model (4), the statistical description of the \(\alpha\)-Fluctuating Nakagami fading channel can be derived as follows. **Theorem 1**. : _For the \(\alpha\)-Fluctuating Nakagami fading channel model, the first-order statistical characteristics (i.e., PDF \(f_{\gamma}(\gamma)\), CDF \(F_{\gamma}(\gamma)\), and MGF \(\mathcal{M}_{\gamma}(s)\)) of the instantaneous SNR \(\gamma\) are given by:_  
\[f_{\gamma}(\gamma) =\frac{\alpha m_{s}}{\Omega(m)}\gamma^{-1}\mathrm{G}_{1,2}^{1,1} \Bigg{(}\frac{1}{\Omega}\bigg{(}\frac{\gamma}{\gamma}\bigg{)}^{\frac{\alpha}{2 }}\Bigg{|} \begin{array}{c}1-m_{s}\\ m_{s}-m_{s}\end{array} \Bigg{)}, \tag{5}\] \[F_{\gamma}(\gamma) =\frac{m_{s}}{\Gamma(m)}\mathrm{H}_{2,3}^{1,2}\Bigg{(}\frac{ \gamma}{\Omega^{2}\tau}\Bigg{|} \begin{array}{c}(1,1),\big{(}1-m_{s},\frac{ 2}{\alpha}\big{)}\\ \big{(}m,\frac{2}{\alpha}\big{)},\big{(}-m_{s},\frac{2}{\alpha}\big{)}\end{array}  \Bigg{)},\] (6) \[\mathcal{M}_{\gamma}(s) =\frac{m_{s}}{\Gamma(m)}\mathrm{H}_{2,2}^{1,2}\Bigg{(}\frac{s^{-1 }}{\Omega^{2}\tau}\Bigg{|} \begin{array}{c}(1,1),\big{(}1-m_{s},\frac{2}{ \alpha}\big{)}\\ \big{(}m,\frac{2}{\alpha}\big{)},\big{(}-m_{s},\frac{2}{\alpha}\big{)}\end{array}  \Bigg{)}, \tag{7}\]  
_where \(\Omega=\bigg{[}\frac{\Gamma(m)}{m_{s}}\frac{\big{(}m_{s}-\frac{2}{\alpha} \big{)}}{\Gamma\big{(}m+\frac{2}{\alpha}\big{)}}\bigg{]}^{\frac{\alpha}{2}}\); \(\mathrm{G}(\cdot),\mathrm{H}(\cdot)\) are the Meijer \(G\)- and the Fox \(H\)-function [46], where \(\Gamma(\cdot)\) is the Euler gamma-function [46], and \(\bar{\gamma}\) is the average SNR._  
Proof of Theorem 1.: To prove Theorem 1, one starts with the envelope PDF \(f_{R}(r)\) of the underlying Fluctuating Nakagami model (see [18], Equation (7)). Confining oneself to the standard methodology of random variable transformation, first, one derives the PDF of \(W=R^{2}\) (i.e., the power of the envelope) as \(f_{W}(w)=\frac{1}{2}w^{-\frac{1}{2}}f_{R}(w^{\frac{1}{2}})\). Secondly, perform the \(\alpha\)-root transform of \(W\) (that is, \(P^{\alpha}=W\)) with PDF \(f_{P}(\varrho)=\alpha\rho^{\alpha-1}f_{W}(\varrho^{\alpha})\)). Finally, denoting the second raw moment of the \(\alpha\)-transformed power envelope as \(\psi\) (i.e., \(\psi=\mathbb{E}\Big{\{}(P)^{2}\Big{\}}\), and relating it to the instantaneous SNR \(\gamma\) as \(f_{\gamma}(\gamma)=\frac{1}{2}\sqrt{\frac{\psi}{\gamma\gamma}}f_{P}\Big{(} \sqrt{\frac{\psi\gamma}{\gamma}}\Big{)}\), the required instantaneous SNR PDF of the \(\alpha\)-Fluctuating Nakagami-\(m\) model can be related to the envelope PDF of the underlying Fluctuating Nakagami-\(m\) in the following form:  
\[f_{\gamma}(\gamma)=\frac{\alpha}{4}\bigg{(}\frac{\psi}{\bar{\gamma}}\bigg{)}^ {\frac{\alpha}{4}}\gamma^{\frac{\alpha}{4}-1}f_{R}\Bigg{(}\Bigg{[}\frac{\psi }{\bar{\gamma}}\gamma\Bigg{]}^{\frac{\alpha}{4}}\Bigg{)}. \tag{8}\]  
Combining the last result with ([18], Equation (7)) and evaluating \(\mathbb{E}\Big{\{}(P)^{2}\Big{\}}\) yields  
\[\psi=\frac{m_{s}}{\Gamma(m)}\bigg{(}\frac{m_{s}m}{\big{(}m_{s}-1\big{)}\Omega }\bigg{)}^{-\frac{2}{\alpha}}\frac{\Gamma\big{(}m+\frac{2}{\alpha}\big{)}}{ \big{(}m_{s}-\frac{2}{\alpha}\big{)}}, \tag{9}\]  
which is valid if \(\alpha>2\); the PDF (5) is obtained. The CDF (6), defined as \(F_{\gamma}(\gamma)=\int_{0}^{\gamma}f_{\gamma}(x)\mathrm{d}x\), can be obtained by relating the [PERSON] G-function and Fox H-function as  
\[\frac{\alpha}{2}\mathrm{G}_{1,2}^{1,1}\Bigg{(}\frac{1}{\Omega}\bigg{(}\frac{x} {\bar{\gamma}}\bigg{)}^{\frac{2}{2}}\Bigg{|} \begin{array}{c}1-m_{s}\\ m_{s}-m_{s}\end{array} \Bigg{)}=\mathrm{H}_{2,3}^{1,2}\Bigg{(}\frac{x}{\Omega ^{\frac{2}{2}}\bar{\gamma}}\Bigg{|} \begin{array}{c}\big{(}1-m_{s},\frac{2}{ \alpha}\big{)}\\ \big{(}m,\frac{2}{\alpha}\big{)},\big{(}-m_{s},\frac{2}{\alpha}\big{)}\end{array}  \Bigg{)} \tag{10}\]  
and utilizing ([47], Equation (2.8.18)).