# \(\alpha\)-Fluctuating Nakagami-\(m\) Fading Model for Wireless Communications
## 3 \(\alpha\)-Fluctuating Nakagami Fading Channel Model
### Statistical Model
The MGF (7), defined as  
\[\mathcal{M}_{\gamma}(s)=\mathbb{E}\{e^{\beta\gamma}\} \tag{11}\]can be evaluated with the help of the derived CDF, i.e.,  
\[\mathcal{M}_{\gamma}(-s)=s\int_{0}^{\infty}F_{\gamma}(\gamma)e^{-s\gamma}\mathrm{ d}\gamma. \tag{12}\]  
Treating the integral as the Laplace transform \(\mathcal{L}\{f(\gamma)\}(s)\) of a function \(f(\gamma)\) evaluated at \(s\), applying the identity (see [47], Theorem 2.3, Equation (2.5.16)):  
\[\mathcal{L}\left\{\mathrm{H}_{p,d}^{m,n}\left[\gamma\left| \begin{array}{c}(a _{i},\alpha_{i})_{1,p}\\ (b_{j},\beta_{j})_{1,q}\end{array} \right.\right]\right\}(s)=\frac{1}{t}\mathrm{ H}_{p+1,q}^{m,n+1}\left[\frac{1}{s}\left| \begin{array}{c}(0,1),(a_{i}, \alpha_{i})_{1,p}\\ (b_{j},\beta_{j})_{1,q}\end{array} \right.\right], \tag{13}\]  
and simplifying the expression via the reduction property ([47], Equation (2.1.1)) finalizes the proof of (7). It must be highlighted that (5)-(7) are valid \(\forall m\geq 0.5\) and \(m_{s},\alpha\) that satisfy the condition \(\frac{m_{s}\alpha}{2}>1\) (as to the expression for \(\mathbb{U}\)). It is immediately observed that substitution \(\alpha=2\) in (5) and (6) after some simplifications results in equivalent expressions for the underlying Fluctuating Nakagami model (see [18], Equations (12) and (13)). It is important to observe that the special functions referenced in Theorem 1 and all that follows, as well as the accompanying Corollaries, are easily accessible within modern computer algebra systems (CAS) (e.g., Wolfram Mathematica, MATLAB, Maple, etc.) for both symbolic and numerical computations. Additionally, further numerical simulations indicate that, in almost all scenarios of practical interest, these functions can be computed with high efficiency through univariate numerical integration in the complex domain speeding up the computations, in addition to utilizing the existing CAS methods. For example, in the case of the MGF (7), it can be rewritten in the following form:  
\[\mathcal{M}_{\gamma}(s)=\frac{m_{s}}{\Gamma(m)}\frac{1}{2\pi i}\int_{\mathcal{ L}}\frac{\Gamma(-\sigma)\Gamma\big{(}m+\frac{2\sigma}{\alpha}\big{)}}{\big{(}m_{s}- \frac{2\sigma}{\alpha}\big{)}}\Bigg{(}\frac{s^{-1}}{\lambda^{\frac{2}{2}} \tilde{\gamma}}\Bigg{)}^{-\sigma}\mathrm{d}\sigma, \tag{14}\]  
where \(\mathcal{L}\) is the contour chosen in such a way as to separate the poles \(\Gamma(-\sigma)\) and \(\Gamma\big{(}m+\frac{2\sigma}{\alpha}\big{)}\). For example, it can be chosen as \(\mathcal{L}=\big{(}-\frac{\alpha m}{4}-i\infty;-\frac{\alpha m}{4}+i\infty \big{)}\). From the practical standpoint, it is important to have at hand the channel statistics for infinitely increasing SNR (the so-called high-SNR asymptotics) since it is needed to upper-bound the channel performance. The derived expressions in the high-SNR region can be summarized as follows. **Corollary 1**. : _The high-SNR asymptotics (i.e., \(\tilde{\gamma}\to\infty\)) of PDF, CDF, and MGF for the \(\alpha\)-Fluctuating Nakagami model are given by_  
\[f_{\gamma}(\gamma)\big{|}_{\tilde{\gamma}\to\infty} \backsim\frac{m_{s}\delta^{-m}}{(m_{s}+m)\Gamma(m)\gamma}\bigg{(} \frac{\gamma}{\tilde{\gamma}}\bigg{)}^{\frac{\alpha m}{2}}, \tag{15}\] \[F_{\gamma}(\gamma)\big{|}_{\tilde{\gamma}\to\infty} \backsimeq\frac{m_{s}\delta^{-m}}{(m_{s}+m)\Gamma(m+1)}\bigg{(} \frac{\gamma}{\tilde{\gamma}}\bigg{)}^{\frac{\alpha m}{2}},\] (16) \[\mathcal{M}_{\gamma}(s)\big{|}_{\tilde{\gamma}\to\infty} \backsimeq\frac{am_{s}\Gamma\big{(}\frac{\alpha m}{2}\big{)}}{2(m_ {s}+m)\Gamma(m)}\frac{\lambda^{-m}}{(s\tilde{\gamma})^{\frac{\alpha m}{2}}}. \tag{17}\]Proof. : The expression (15)-(17) can be proved by observing that for a vanishingly small argument (i.e., for \(\bar{\gamma}\to\infty\)), the Fox H-functions can be approximated by the single terms of their power-series expansions, i.e. (see [47], Corollary 1.11.1),  
\[\mathrm{H}_{p,d}^{m,m}\!\!\left(\frac{x}{\bar{\gamma}}\left| \begin{array}{c} \left(a_{i}\,A_{i}\right)_{i=1,p}\\ \left(b_{i},B_{i}\right)_{j=1,d}\end{array}