# \(\alpha\)-Fluctuating Nakagami-\(m\) Fading Model for Wireless Communications
## 4 Performance Analysis of the \(\alpha\)-Fluctuating Nakagami Model
### Capacity Analysis of the \(\alpha\)-Fluctuating Nakagami Model  
The channel capacity of the wireless communication link in the presence of multipath fading effects is commonly defined as  
\[\overline{\mathrm{C}}=\int_{0}^{\infty}\log_{2}(1+\gamma)f_{\gamma}(\gamma) \mathrm{d}\gamma. \tag{31}\]  
The obtained statistical description of the \(\alpha\)-Fluctuating Nakagami channel model presented in Section 2 helps to derive the exact and asymptotic expressions for \(\overline{\mathrm{C}}\).  
**Theorem 4**: _The average channel capacity of the \(\alpha\)-Fluctuating Nakagami model can be expressed as_  
\[\overline{\mathrm{C}}=\frac{m_{s}}{\Gamma(m)\ln 2}\mathrm{H}_{3,4}^{3,2}\left( \frac{\{\Omega^{-\frac{2}{s}}\}}{\gamma}\left|\begin{array}{c}(0,1),\big{(}1- m_{s},\frac{2}{\alpha}\big{)},(1,1)\\ \big{(}m,\frac{2}{\alpha}\big{)},(0,1),(0,1),\big{(}-m_{s},\frac{2}{\alpha} \big{)}\end{array}\right.\right). \tag{32}\]  
To prove Theorem 4, one starts with applying the contour integral representation of \(\mathrm{G}_{1,2}^{1,1}(\cdot)\) in (5) and rearranging the integration order:  
\[\overline{\mathrm{C}}=\frac{m_{s}\log_{2}e}{\Gamma(m)}\frac{1}{2\pi i}\int_{ \mathcal{H}}\frac{\Gamma\big{(}m+\frac{2s}{\alpha}\big{)}\Gamma\big{(}m_{s}- \frac{2s}{\alpha}\big{)}}{\Gamma\big{(}m_{s}+1-\frac{2s}{\alpha}\big{)}}\left( \int_{0}^{\infty}\frac{\ln(1+\gamma)}{\gamma^{s+1}}\mathrm{d}\gamma\right) \left(\frac{1}{\Omega^{\frac{2}{s}}\gamma}\right)^{-s}\mathrm{d}s. \tag{33}\]  
The inner integral over \(\gamma\) via ([49], Equation (2.6.9.21)) can be expressed as \(\frac{\pi\csc(\pi s)}{s}\) (for \(0<\Re\{s\}<1\)). Using the reflection and recurrence property of the gamma-function (see [46], Equation (5.5.3) and [46], Equation (5.5.1), respectively), which relate it to the cosecant function, helps to represent the inner integral solely in terms of the gamma-functions, i.e., \(\frac{\pi\csc(\pi s)}{s}=\frac{\Gamma^{2}(s)\Gamma(1-s)}{\Gamma(1+s)}\). Lastly, one can shift the integration contour \(\mathcal{H}\) in such a way that \(0<\Re\{s\}<\frac{m_{s}\delta}{2}\) and note that the obtained integral and its integration contour is a valid Mellin-Barnes representation of the Fox H-function in (32), which finalizes the proof.  
The high-SNR approximation of the average capacity for the \(\alpha\)-Fluctuating Nakagami model is given by  
\[\overline{\mathrm{C}}\big{|}_{\gamma\to\infty}\approx\log_{2}\left(\lozenge ^{\frac{2}{s}}\gamma\right)+\frac{2}{\alpha m_{s}\ln 2}\left(1+m_{s}\psi^{(0)}(m) \right), \tag{34}\]  
where \(\psi^{(0)}(\cdot)\) is the digamma-function [46]. The asymptotic channel capacity is obtained by noting that (see [17], Equations (6) and (7)):  
\[\overline{\mathrm{C}}\big{|}_{\gamma\to\infty}\approx\log_{2}\gamma+\frac{1}{ \ln 2}\frac{\mathrm{d}}{\mathrm{d}k}\left(\frac{\mathbb{E}\{\gamma^{k}\}}{( \mathbb{E}\{\gamma\})^{k}}\right)\Bigg{|}_{k=0}. \tag{35}\]  
Applying Theorem 4, performing differentiation and limiting operation with \(k\to 0\), after some simplifications, (34) follows.  
To the best of the author's knowledge, the \(\alpha\)-Fluctuating Nakagami fading model has not been reported in the technical literature yet, and the derived results of its statistical description are novel.