# Closed-Form Performance Analysis of the Inverse Power Lomax Fading Channel Model
## 3 IPL Fading Channel Model Application
### Error Rate Analysis
\tag{28}\] \begin{table}
\begin{tabular}{c c c} \hline \hline  & \(\delta_{\mathbf{1}}\) & \(\delta_{\mathbf{2,j}}\) & \(\delta_{\mathbf{3}}\) \\ \hline M-QAM & \(\frac{4}{\log_{2}M}\Big{(}1-\frac{1}{\sqrt{M}}\Big{)}\) & \(\frac{3(2j-1)^{2}}{2(M-1)}\) & \(\frac{\sqrt{M}}{2}\) \\ \hline M-PSK & \(\frac{1}{\max(\frac{2}{2},\log_{2}M)}\) & \(2\sin^{2}\Big{(}\frac{(2j-1)\pi}{M}\Big{)}\) & \(\max\Big{(}1,\frac{M}{4}\Big{)}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Modulation-specific coefficients for M-QAM and M-PSK. To derive the high-SNR asymptotics of (28), one can make use of Fox H-function expansion in the case of an infinitely small argument (see expression (3) and (4) in [53]):  
\[\mathrm{H}_{2,1}^{1,2}\!\left(\frac{\dot{\Omega}_{\alpha,\beta}}{(\delta_{2,j} \mathcal{T})^{\beta}}\right)^{\!\!\beta}\left| \begin{array}{c}(1,1),\left( \frac{1}{2},\beta\right)\\ (\alpha,1)\end{array} \right.\right)\bigg{|}_{\mathcal{T}\to\infty}\simeq\left( \frac{\dot{\Omega}_{\alpha,\beta}}{(\delta_{2,j}\mathcal{T})^{\beta}}\right)^{ \omega}h_{j}, \tag{29}\]  
where \(\omega\) and \(h_{j}\) are defined in [53] (see expression (3) and (4), respectively). For the IPL model, \(\omega=\alpha\) and \(h_{j}=\Gamma(\alpha)\Gamma\!\left(\alpha\beta+\frac{1}{2}\right)\), which finally helps to derive the high-SNR asymptotics for the average error rate of the IPL model as  
\[\overline{\mathrm{P}}_{\mathrm{er}}\bigm{|}_{\mathcal{T}\to\infty}\simeq\frac {\delta_{1}\dot{\Omega}_{\alpha,\beta}^{\alpha}}{2\sqrt{\pi}\dot{\gamma}^{ \alpha\beta}}\Gamma\!\left(\alpha\beta+\frac{1}{2}\right)\sum_{j=1}^{\delta_{ 3}}\delta_{2,j}^{-\alpha\beta}. \tag{30}\]  
\(\square\)  
Similar to the approach used for the MGF expression, calculations for (28) can be efficiently performed through direct numerical integration. This method can significantly speed up calculations for certain parameter values by utilizing the integral representation of the Fox H-function as follows:  
\[\overline{\mathrm{P}}_{\mathrm{er}}=\frac{\delta_{1}}{2\sqrt{\pi}\Gamma(\alpha )}\sum_{j=1}^{\delta_{3}}\frac{1}{2\pi i}\int_{-\frac{\alpha}{2}-i\infty}^{- \frac{\alpha}{2}+i\infty}\Gamma(-\sigma)\Gamma\!\left(\frac{1}{2}-\beta\sigma \right)\Gamma(\alpha+\sigma)\!\left(\frac{\dot{\Omega}_{\alpha,\beta}}{( \delta_{2,j}\mathcal{T})^{\beta}}\right)^{-\sigma}\!\mathrm{d}\sigma. \tag{31}\]  
Analogous to the outage probability asymptotic representation, the average bit error rate can be represented as (see [55]) \(\overline{\mathrm{P}}_{\mathrm{er}}\bigm{|}_{\mathcal{T}\to\infty}=(\mathcal{O }_{c}\mathcal{\bar{\gamma}})^{-\mathcal{O}_{d}}\), where \(\mathcal{O}_{d}\) and \(\mathcal{O}_{c}\) are the diversity and coding gains derived from the ABER. **Corollary 2**. : _The ABER diversity and coding gains of the IPL channel model are expressed in terms of the channel parameters as follows: \(\mathcal{O}_{d}=\alpha\beta,\mathcal{O}_{c}=\frac{\delta_{1}\dot{\Omega}_{ \alpha,\beta}^{\alpha}}{2\sqrt{\pi}}\Gamma\!\left(\alpha\beta+\frac{1}{2} \right)\sum_{j=1}^{\delta_{3}}\delta_{2,j}^{-\alpha\beta}\)._  
It can be noted that the diversity gains derived from the OP and ABER are equal, and the ABER coding gain accounts for modulation-specific parameters.