# \(\alpha\)-Fluctuating Nakagami-\(m\) Fading Model for Wireless Communications
## 4 Performance Analysis of the \(\alpha\)-Fluctuating Nakagami Model
### ABER Performance of the \(\alpha\)-Fluctuating Nakagami Model  
For a broad variety of coherent modulations widely used in wireless communications [48] (e.g., BPSK, GMSK (for high \(\gamma\)), M-PSK, square M-QAM, etc. ), ABER can be efficiently approximated as a combination of Gauss Q-functions \(Q(\cdot)\), i.e.,  
\[\overline{\mathrm{P}}_{er}^{c}=\delta_{1}\sum_{j=1}^{\delta_{3}}\int_{0}^{ \infty}Q(\sqrt{2\delta_{2j}\gamma})f_{\gamma}(\gamma)\mathrm{d}\gamma, \tag{22}\]  
where the combination of coefficients \(\{\delta_{1},\delta_{2j},\delta_{3}\}\) are properly defined for the particular modulation. For example, in the case of M-QAM: \(\left\{\frac{4(1-1/\sqrt{M})}{\log_{2}M},\frac{3(2j-1)^{2}}{2(M-1)},\frac{ \sqrt{M}}{2}\right\}\) and for M-PSK: \(\left\{\frac{1}{\max(2\log_{2}M)},2\log_{2}M\sin^{2}\left(\frac{(2j-1)\pi}{M} \right),\max\left(1,\frac{M}{4}\right)\right\}\). For several non-coherent modulations, ABER can be efficiently approximated by [48]:  
\[\overline{\mathrm{P}}_{er}^{nc}=\sum_{j=1}^{\epsilon_{3}}\epsilon_{1,j}\int_{0 }^{\infty}e^{-\epsilon_{2,j}\gamma}f_{\gamma}(\gamma)\mathrm{d}\gamma, \tag{23}\]  
where, for example, \(\{\epsilon_{1,j},\epsilon_{2,j},\epsilon_{3}\}\) are equal to \(\left\{\frac{(-1)^{j+1}}{j+1}(\frac{M-1}{j}),\frac{j}{j+1},M-1\right\}\) for M-PSK and \(\left\{\frac{1}{2},1,1\right\}\) for DBPSK. Applying the results of the derived statistical representation, the ABER can be expressed in closed form as follows. **Theorem 3**: _For the \(\alpha\)-Fluctuating Nakagami fading channel model, the average bit error rate can be expressed as:_  
* _For the coherent modulations (e.g.,_ \(M\)_-QAM and M-PSK):_ \[\overline{\mathrm{P}}_{er}^{c}=\frac{\delta_{1}m_{s}}{2\sqrt{\pi\Gamma}(m)} \sum_{j=1}^{\epsilon_{3}}\mathrm{H}_{3,3}^{1.3}\left(\frac{\delta_{2j}^{-1}}{ \delta_{2}^{1.2}\gamma}\left| \begin{array}{c}\left(\frac{1}{2},1\right), \left(1,1\right),\left(1-m_{s},\frac{2}{\alpha}\right)\\ \left(m,\frac{2}{\alpha}\right),\left(-m_{s},\frac{2}{\alpha}\right),\left(0,1 \right)\end{array} \right.\right),\] (24)
* _For the non-coherent modulations (e.g.,_ \(D\)_PSK, BFSK, and M-FSK):_ \[\overline{\mathrm{P}}_{er}^{nc}=\frac{m_{s}}{\Gamma(m)}\sum_{j=1}^{\epsilon_ {3}}\epsilon_{1,j}\mathrm{H}_{3,3}^{1.3}\left(\frac{\delta_{2j}^{-\frac{2}{ \alpha}}}{\epsilon_{2,j}\gamma}\left| \begin{array}{c}\left(0,1\right),\left( 1,1\right),\left(1-m_{s},\frac{2}{\alpha}\right)\\ \left(m,\frac{2}{\alpha}\right),\left(-m_{s},\frac{2}{\alpha}\right),\left(0,1 \right)\end{array} \right.\right),\] (25)  
To prove Theorem 3, one can note that, via integration-by-parts, (22) is expressed as:  
\[\int_{0}^{\infty}Q(\sqrt{2\delta_{2j}\gamma})f_{\gamma}(\gamma)\mathrm{d} \gamma=\sqrt{\frac{\delta_{2j}}{4\pi}}\int_{0}^{\infty}\frac{e^{-\delta_{2j} \gamma}}{\sqrt{\gamma}}F_{\gamma}(\gamma)\mathrm{d}\gamma. \tag{26}\]The latter integral can be regarded as the Laplace transform \(\mathcal{L}\{\cdot,s\}\) evaluated at \(s=\delta_{2,j}\):  
\[\overline{\mathrm{P}}_{\sigma r}^{c}=\frac{\delta_{1}}{2\sqrt{\pi}}\sum_{j=1}^{ \delta_{3}}\delta_{2,j}^{-\frac{1}{2}}\mathcal{L}\Big{\{}\frac{F_{\gamma}( \gamma)}{\sqrt{\gamma}};\delta_{2,j}\Big{\}}. \tag{27}\]  
Applying ([47], Corollary 2.3.1, Equation (2.5.25)), after some simplifications, (26) follows. For non-coherent modulations, using the same strategy as in (26) yields:  
\[\overline{\mathrm{P}}_{\sigma r}^{nc}=\sum_{j=1}^{\epsilon_{2,j}}\epsilon_{1, j}\epsilon_{2,j}\mathcal{L}\big{\{}F_{\gamma}(\gamma);\epsilon_{2,j}\big{\}}. \tag{28}\]  
Noting that the Laplace transform of the CDF can be related to the MGF (i.e., \(\mathcal{L}\big{\{}F_{\gamma}(\gamma);\epsilon_{2,j}\big{\}}=\epsilon_{2,j}^ {-1}\mathcal{M}_{\gamma}(\epsilon_{2,j})\)), after applying (7) and some straightforward modifications, (27) is obtained, finalizing the proof.