# Closed-Form Performance Analysis of the Inverse Power Lomax Fading Channel Model
## 3 IPL Fading Channel Model Application
### Error Rate Analysis  
The average bit error rate is defined as the instantaneous error rate averaged with the respect to the PDF of the instantaneous SNR:  
\[\bar{\mathrm{P}}_{e}\ =\ \int_{0}^{\infty}\mathrm{P}_{e}(\gamma)f_{\gamma}( \gamma)\mathrm{d}\gamma=\delta_{1}\sum_{j=1}^{\delta_{3}}\int_{0}^{\infty}Q \Big{(}\sqrt{2\delta_{2,j}\gamma}\Big{)}f_{\gamma}(\gamma)\mathrm{d}\gamma, \tag{23}\]  
where \(Q(x)\) is the Gaussian Q-function [46] and the coefficients \(\left\{\delta_{1},\delta_{2,j},\delta_{3}\right\}\) depend on the choice of the modulation format. For M-QAM or M-PSK, they are given in Table 1.  
**Theorem 4**. : _The average error rate for the IPL channel model \(\overline{\mathrm{P}}_{\mathrm{er}}\) and its high-SNR asymptotics are given by_  
\[ \begin{cases}\overline{\mathrm{P}}_{\mathrm{er}}=\frac{\delta_{1}}{2\sqrt{\pi \Gamma}(\alpha)}\sum_{j=1}^{\delta_{2}}\mathrm{H}_{2,1}^{1,2}\bigg{(}\frac{ \tilde{\Omega}_{\alpha,\beta}}{(\delta_{2,j}\tilde{\gamma})^{\beta}}\bigg{|} \begin{array}{c}(1,1),\Big{(}\frac{1}{2},\beta)\\ (\alpha,1)\end{array}\bigg{)},\\ \overline{\mathrm{P}}_{\mathrm{er}}\bigm{|}_{\gamma\to\infty} \simeq\frac{\delta_{1}\tilde{\Omega}_{\alpha,\beta}^{\alpha}}{2\sqrt{\pi}\tilde {\gamma}^{\alpha\beta}}\Gamma\bigg{(}\alpha\beta+\frac{1}{2}\bigg{)}\sum_{j=1} ^{\delta_{3}}\delta_{2,j}^{-\alpha\beta}.\end{cases}  \tag{24}\]  
Proof of Theorem 4.: To prove Theorem 4, one can make use of the connection between the Gaussian Q-function and the complementary error function \(\mathrm{erfc}(\cdot)\)[46]:  
\[\overline{\mathrm{P}}_{\mathrm{er}}=\frac{\delta_{1}}{2}\sum_{j=1}^{\delta_{3 }}\int_{0}^{\infty}\mathrm{erfc}\bigg{(}\sqrt{\delta_{2,j}\gamma}\bigg{)}f_{ \gamma}(\gamma)\mathrm{d}\gamma. \tag{25}\]  
Denoting the integral over \(\gamma\) as \(\mathrm{J}_{\mathrm{Q}}\), it can be further expressed as  
\[\mathrm{J}_{\mathrm{Q}} =\int_{0}^{\infty}\mathrm{erfc}\Big{(}\sqrt{\delta_{2,j}\gamma} \Big{)}f_{\gamma}(\gamma)\mathrm{d}\gamma=\sqrt{\frac{\delta_{2,j}}{\pi}}\int _{0}^{\infty}\frac{F_{\gamma}(\gamma)}{\sqrt{\gamma}}e^{-\delta_{2,j}\gamma} \mathrm{d}\gamma, \tag{26a}\] \[=\sqrt{\frac{\delta_{2,j}}{\pi}}\int_{0}^{\infty}\frac{\left(1+ \frac{1}{\Omega_{\alpha,\beta}}\left(\frac{\gamma}{\gamma}\right)^{\beta} \right)^{-\alpha}}{\sqrt{\gamma}}e^{-\delta_{2,j}\gamma}\mathrm{d}\gamma,\] (26b) \[= \tag{26c}\]  
where (26a) is obtained with the help of integration by parts and (26c) via the change in the variable \(t=\left(\frac{\gamma}{\gamma}\right)^{\beta}\). Making use of (11c) and (12), \(\mathrm{J}_{\mathrm{Q}}\) can be represented as  
\[\mathrm{J}_{\mathrm{Q}} =\frac{1}{\sqrt{\pi}(\delta_{2,j}\tilde{\gamma})^{\beta}\Gamma( \alpha)}\int_{0}^{\infty}\mathrm{H}_{1,1}^{1,1}\bigg{(}\frac{t}{\tilde{\Omega }_{\alpha,\beta}}\biggm{|} \begin{array}{c}(1-\alpha,1)\\ (0,1)\end{array} \bigg{)}\mathrm{H}_{1,0}^{0,1}\bigg{(}\frac{t}{(\delta_{2,j} \tilde{\gamma})^{\beta}}\biggm{|}\begin{array}{c}(\frac{1}{2}-\beta,\beta) \\ \mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf\mathbf{}}}}}}}}}}}}}}}} \end\Big{\Big{\rm{\rm{d}}}} \tag{27a}\] \[=\frac{\tilde{\Omega}_{\alpha,\beta}}{\sqrt{\pi}(\delta_{2,j} \tilde{\gamma})^{\beta}\Gamma(\alpha)}\mathrm{H}_{2,1}^{1,2}\bigg{(}\frac{ \tilde{\Omega}_{\alpha,\beta}}{(\delta_{2,j}\tilde{\gamma})^{\beta}}\biggm{|} \begin{array}{c}(0,1),\Big{(}\frac{1}{2}-\beta,\beta)\\ (\alpha-1,1)\end{array} \bigg{)}, \tag{27b}\]  
where (27b) is evaluated with equality (2.25.1.1) from [52]. Finally, combining (27b) with (25) and making use of the power-transformation property of the Fox H-function (see (1.59) in [51]), the closed-form expression for the average error rate can be equated as  
\[\overline{\mathrm{P}}_{\mathrm{er}}=\frac{\delta_{1}}{2\sqrt{\pi\Gamma}(\alpha )}\sum_{j=1}^{\delta_{3}}\mathrm{H}_{2,1}^{1,2}\bigg{(}\frac{\tilde{\Omega}_{ \alpha,\beta}}{(\delta_{2,j}\tilde{\gamma})^{\beta}}\biggm{|} \begin{array}{c}( 1,1),\Big{(}\frac{1}{2},\beta)\\ (\alpha,1)\end{array} \bigg{)}.