# \(\alpha\)-Fluctuating Nakagami-\(m\) Fading Model for Wireless Communications
## 3 \(\alpha\)-Fluctuating Nakagami Fading Channel Model
### Statistical Model
\right.\right)\sim h_{j_{0}}\left( \frac{x}{\bar{\gamma}}\right)^{\frac{b_{j_{0}}}{b_{j_{0}}}}, \tag{18}\]  
where for all the assumed cases (i.e., PDF, CDF, and MGF (5)-(7)) \(j_{0}=1\), \(\frac{b_{j_{0}}}{b_{j_{0}}}=\frac{a\bar{m}}{2}\), and \(h_{j_{0}}\) is evaluated in each case according to ([47], Equation (1.8.5)). Simplifying the obtained expression yields (15)-(17). To complete the first-order statistical description, required for further channel performance assessment, the instantaneous SNR raw moments must be derived. Theorem 4.2: _The \(k\)th raw moment of the instantaneous SNR for the \(\alpha\)-Fluctuating Nakagami fading channel model can be expressed as:_  
\[\mathbb{E}\{\gamma^{k}\}=\frac{m_{s}}{\Gamma(m)}\mathrm{\Omega}^{\frac{2k}{2 }}\frac{\Gamma\!\left(m+\frac{2k}{\pi}\right)}{\left(m_{s}-\frac{2k}{\pi} \right)}\bar{\gamma}^{k}. \tag{19}\]  
Proof. : The proof is obtained by combining the definition of the SNR \(k\)th raw moment, i.e., \(\mathbb{E}\{\gamma^{k}\}=\int_{0}^{\infty}\gamma^{k}f_{\gamma}(\gamma) \mathrm{d}\gamma\), with the derived PDF (5) in the following way:  
\[\mathbb{E}\{\gamma^{k}\} =\int_{0}^{\infty}\gamma^{k}f_{\gamma}(\gamma)\mathrm{d}\gamma \tag{20a}\] \[=\frac{am_{s}}{2\Gamma(m)}\int_{0}^{\infty}\gamma^{k-1}\mathrm{G} _{1,2}^{1,1}\!\!\left(\frac{1}{\bar{\gamma}}\left(\frac{\gamma}{\bar{\gamma}} \right)^{\frac{2}{2}}\left| \begin{array}{c}1-m_{s}\\ m_{s}-m_{s}\end{array} \right.\right)\mathrm{d}\gamma=\] (20b) \[=\frac{m_{s}}{\Gamma(m)}\int_{0}^{\infty}\gamma^{k-1}\mathrm{H} _{1,2}^{1,1}\!\!\left(\frac{\gamma}{\bar{\gamma}^{\frac{2}{2}}\bar{\gamma}} \left| \begin{array}{c}\left(1-m_{s},\frac{2}{\pi}\right)\\ \left(m,\frac{2}{\pi}\right),\left(-m_{s},\frac{2}{\pi}\right)\end{array}  \right.\right)\mathrm{d}\gamma=\] (20c) \[=\frac{m_{s}}{\Gamma(m)}\mathrm{\Omega}^{\frac{2k}{2}}\frac{ \Gamma\!\left(m+\frac{2k}{\pi}\right)}{\left(m_{s}-\frac{2k}{\pi}\right)} \bar{\gamma}^{k}, \tag{20d}\]  
where the transition from (20b) to (20d) was performed via ([47], Equation (2.5.7)). It must be pointed out that the expression (19) is valid for \(\frac{m_{s}}{2k}>1\). The derived expression of the SNR moments helps to evaluate such channel characteristic as amount of fading, which is generally used to describe the fading severity and is defined as the normalized variance of the SNR, i.e., \(\mathrm{AoF}=\frac{\mathbb{E}\{\gamma^{2}\}}{\left(\mathbb{E}\{\gamma\} \right)^{2}}-1\). Corollary 2: _The amount of fading for the \(\alpha\)-Fluctuating Nakagami model is given by_  
\[\mathrm{AoF}=\frac{\left(m_{s}-\frac{2}{\pi}\right)^{2}}{m_{s}\left(m_{s}- \frac{4}{\pi}\right)}\frac{\Gamma(m)\Gamma\!\left(m+\frac{4}{\pi}\right)}{ \left(\Gamma\!\left(m+\frac{2}{\pi}\right)\right)^{2}}-1 \tag{21}\]  
The proof of Corollary 2 is omitted since it trivially follows from the definition of AoF and the derived expression for the raw moments (19) with \(k=2\). The provided first-order statistical description of the \(\alpha\)-Fluctuating Nakagami fading channel model facilitates the evaluation of general metrics describing the overall system performance.