## 4 Asymptotic SER  
In this section, we present an accurate approximation for the average SER, which provides a reasonable insight into the performance of the system under consideration in the high SNR regime. Let us assume that when SNR takes a very large value, then \(m_{1}\) is approximated by its closest integer [6; 21]; therefore, by using the Binomial expansion in Equation (11), we have:  
\[I_{1}=\sum\limits_{l=0}^{m_{1}-1}\binom{m_{1}-1}{l}\frac{\left(\alpha_{1} \left(1+z\frac{E_{l}}{\sigma_{1}^{2}}\right)y+\alpha_{1}C\right)\left(s\frac{E _{l}}{\sigma_{1}^{2}}+\beta_{1}\left(1+z\frac{E_{l}}{\sigma_{1}^{2}}\right) \right)^{l}y^{l}\left(C\beta_{1}\right)^{m_{1}-l-1}}{\left(\left(s\frac{E_{l} }{\sigma_{1}^{2}}+\left(\beta_{1}-\delta_{1}\right)\left(1+z\frac{E_{l}}{ \sigma_{1}^{2}}\right)\right)y+C\left(\beta_{1}-\delta_{1}\right)\right)^{m_{1 }}} \tag{29}\]  
By substituting Equation (4) in Equation (12), we get:  
\[I_{2}=\int\limits_{0}^{\infty}I_{1}\lambda_{2}y^{m_{2}-1}e^{-\epsilon_{2}y}dy \tag{30}\]  
By substituting Equation (29) in Equation (30) and after interchanging summation and integration, we get:  
\[I_{2}=\lambda_{2}\sum\limits_{l=0}^{m_{1}-1}\frac{\left(\frac{\alpha_{1}^{m_{ 1}-1}}{\left(s\frac{E_{l}}{\sigma_{1}^{2}}+\beta_{1}\left(1+z\frac{E_{l}}{ \sigma_{1}^{2}}\right)\right)^{l}\left(C\beta_{1}\right)^{m_{1}-l-1}}}{\left(s \frac{E_{l}}{\sigma_{1}^{2}}+\left(\beta_{1}-\delta_{1}\right)\left(1+z\frac{ E_{l}}{\sigma_{1}^{2}}\right)\right)^{m_{1}}}\int\limits_{0}^{\infty}\frac{\left(a_{1} \left(1+z\frac{E_{l}}{\sigma_{1}^{2}}\right)y+a_{1}C\right)y^{m_{2}+l-1}}{ \left(y+\frac{E_{l}-\delta_{1}}{s\frac{E_{l}}{\sigma_{1}^{2}}+\left(\beta_{1}- \delta_{1}\right)\left(1+z\frac{E_{l}}{\sigma_{1}^{2}}\right)}\right)^{m_{1}} \epsilon^{2}y}dy \tag{31}\]After multiplication and the simplification of terms inside the integral in Equation (31), we obtain:  
\[I_{2} = \lambda_{2}\sum\limits_{l=0}^{m_{1}-1}\binom{m_{1}-1}{l}\frac{ \left(s\frac{\frac{E_{l}}{e_{l}^{2}}+\beta_{1}\left(1+z\frac{E_{l}}{e_{l}^{2}} \right)}\right)^{l}(C\beta_{1})^{m_{1}-l-1}}{\left(s\frac{E_{l}}{e_{l}^{2}}+( \beta_{1}-\delta_{1})\left(1+z\frac{E_{l}}{e_{l}^{2}}\right)\right)^{m_{1}}}\] \[\times\] \[+\]  
By using [28] (Equation (2.3.6.9)), we can solve the integrals in Equation (32) and get:  
\[I_{2} = \lambda_{2}\sum\limits_{l=0}^{m_{1}-1}\binom{m_{1}-1}{l}\frac{ \left(s\frac{E_{l}}{e_{l}^{2}}+\beta_{1}\left(1+z\frac{E_{l}}{e_{l}^{2}} \right)\right)^{l}(C\beta_{1})^{m_{1}-l-1}}{\left(s\frac{E_{l}}{e_{l}^{2}}+( \beta_{1}-\delta_{1})\left(1+z\frac{E_{l}}{e_{l}^{2}}\right)\right)^{m_{1}}}\] \[\times\] \[\times\] \[+\] \[I_{2}=\int\limits_{0}^{\infty}I_{1}\lambda_{2}y^{m_{2}-1}e^{-\epsilon_{2}y}dy \tag{30}\]0  
where \(\Psi(a,b;z)\) is defined in [23] (Equation (9.210.2)). By substituting Equations (5) and (33) in Equation (20), using the fact that \(\overline{\gamma}\) takes very large value and employing [23] (Equation (9.211.4)), we finally get the MGF for the high SNR case as:  
\[I_{2}=\int\limits_{0}^{\infty}I_{1}\lambda_{2}y^{m_{2}-1}e^{-\epsilon_{2}y}dy \tag{30}\]1  
where:  
\[ \begin{split}& T_{1}=\alpha_{1}\lambda_{2}\lambda_{3}C^{m_{1}-1} \sum\limits_{l=0}^{m_{1}-1}\binom{m_{1}-1}{l}\beta_{1}^{m-l-1}\left(\beta_{1}- \delta_{1}\right)^{m_{2}+l+1-m_{1}}\\ &\times\left(\int\limits_{0}^{\infty}\left(1+z\frac{E_{l}}{e_{l} ^{2}}\right)\left(s\frac{E_{l}}{e_{l}^{2}}+\beta_{1}\left(1+z\frac{E_{l}}{e_{l }^{2}}\right)\right)^{l}\left(s\frac{E_{l}}{e_{l}^{2}}+(\beta_{1}-\delta_{1}) \left(1+z\frac{E_{l}}{e_{l}^{2}}\right)\right)^{-(m_{2}+l+1)}z^{m_{2}-l}e^{-e \circ z}dz\\ &\times\int\limits_{0}^{\infty}\frac{e_{l}e_{l}^{2}+(\beta_{1}- \delta_{1})\left(1+z\frac{E_{l}}{e_{l}^{2}}\right)}{s\frac{E_{l}}{e_{l}^{2}}+( \beta_{1}-\delta_{1})\left(1+z\frac{E_{l}}{e_{l}^{2}}\right)}\,t^{m_{2}+l} \left(1+t\right)^{-m_{1}}dt\right)\end{split}  \tag{35}\]and:  
\[