## 4 Performance Analysis
### Average Bep/SEP
#### 4.2.2 ABEP/ASEP in the Erfc Function Form  
A. Single-Link System  
By inserting \(Pe(\varepsilon|\gamma)\) in the erfc function form and (7) into (37), the ABEP/ASEP of the single-link system over \(\alpha\)-\(\mu\)/IGA fading channels can be given as  
\[ \begin{array}{ll}P_{e}(\varepsilon)&=\frac{\alpha C}{2\Gamma(\mu)\Gamma(n)} \underbrace{\int_{0}^{\infty}\gamma^{-1}erfc(\sqrt{E\gamma})H_{1,1}^{1,1}[ \lambda\gamma^{0.5\alpha}|_{(\mu,1)}^{(1-n,0.5\alpha)}]d\gamma}_{l_{3}}\\ &-\frac{\alpha D}{2\Gamma(\mu)\Gamma(n)}\underbrace{\int_{0}^{\infty}\gamma^{ -1}erfc^{2}(\sqrt{E\gamma})H_{1,1}^{1,1}[\lambda\gamma^{0.5\alpha}|_{(\mu,1)} ^{(1-n,0.5\alpha)}]d\gamma}_{l_{4}}.\end{array} . \tag{38}\]  
In order to solve the integrals (\(l_{3}\) and \(l_{4}\)) in (38), we employ the identities [45] (Equations (8.4.14.2) and (8.3.2.21)) to represent the erfc function in terms of [PERSON]'s H function as  
\[erfc(\sqrt{E\gamma})=\pi^{-0.5}H_{1,2}^{2,0}[E\gamma]_{(0,1),(0.5,1)}^{(1,1)}]. \tag{39}\]  
By inserting (39) into \(l_{3}\) and \(l_{4}\), with the help of [46] (Equation (2.8.4) and [47] (Equation (2.3)) and performing some algebraic operations, \(l_{3}\) and \(l_{4}\) can be obtained, respectively, as  
\[ \begin{array}{ll}I_{3}=\pi^{-0.5}H_{3,2}^{1,3}[\lambda E^{-0.5\alpha}|_{( \mu,1),(0.5\alpha)}^{(1-n,0.5\alpha),(1,0.5\alpha),(0.5\alpha)}],\\ I_{4}=\pi^{-1}H_{2,1;1;2;1,1}^{0.22;0.1,1}[1,\lambda E^{-0.5\alpha}|_{(0,1,0.5 \alpha)}^{(1,1,0.5\alpha),(0.5\alpha)}|_{(0,1),(0,1)}^{(1-n,0.5\alpha)}].\end{array}  \tag{40}\]  
Consequently, if we substitute (40) into (38), the exact ABEP/ASEP expression of the single-link system over \(\alpha\)-\(\mu\)/IGA fading channels can be obtained. Following the similar procedure used in (40) and utilizing (10), the exact approximated ABEP/ASEP expressions of the single-link system by using the mixed \(\alpha\)-\(\mu\) model can be deduced as  
\[ \begin{array}{ll}P_{e}(\varepsilon)&=\sum\limits_{i=1}^{N}\frac{Car^{-0.5 }\varepsilon_{i}}{2\Gamma(\mu)}H_{2,2}^{1,2}[(\psi_{i}E)^{-0.5\alpha}|_{(\mu, 1),(0,0.5\alpha)}^{(1,0.5\alpha),(0.5\alpha)}]\\ &-\sum\limits_{i=1}^{N}\frac{Dar^{-1}\varepsilon_{i}}{2\Gamma(\mu)}H_{2,1;1,2,0,1}^{0.22;0.1,0}[1,(\psi_{i}E)^{-0.5\alpha}|_{(0,1;0.5\alpha)}^{(1;1,0.5 \alpha),(0.5\alpha)}|_{(0,1),(0.5,1)}^{(1,1)}|_{(\mu,1)}^{-}]\end{array}  \tag{41}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Communication Systems** & **ABEP (P\({}_{\mathbf{e}}\))** \\ \hline Single-link system by using (9) & \(\frac{\Delta a}{2\Gamma(\mu)\Gamma(n)}H_{2,1}^{1,2}[\frac{\lambda}{2\Gamma( \mu)}|_{(\mu,1)}^{(1-n,0.5\alpha),(1,0.5\alpha)}]\) \\ Single-link system by using (12) & \(\sum\limits_{i=1}^{N}\frac{\Delta c_{i}}{2\Gamma(\mu)}H_{1,1}^{1,1}[(B\psi_{i} )^{-0.5\alpha}|_{(\mu,1)}^{(1,0.5\alpha)}]\) \\ L-branch MRC system & \(\sum\limits_{i=1}^{N}A_{i}H_{2,1}^{1,2}[\frac{\lambda_{i}}{2\Gamma(\mu)}|_{( \mu,1)}^{(1-n,0.5\alpha),(1,0.5\alpha)}]\) \\ L-branch MRC system & \(\sum\limits_{i=1}^{N_{1}}\cdots\sum\limits_{i=1}^{N_{k}}A_{i}\prod\limits_{i=1 }^{N}\frac{\Delta\Gamma\left(\frac{\lambda_{i}\varepsilon_{i}}{2\Gamma(\mu)}H_{ 1,1}^{1,1}[(B\psi_{i})^{-0.5\alpha}|_{(\mu,1)}^{(1,0.5\alpha)}]\right)}{\left( \mu,1\right)}\) \\ Two-branch MRC system by using (32) & \(\sum\limits_{k=0}^{\infty}\sum\limits_{l=0}^{\infty}\frac{0.25\Delta 4\psi_{l}^{0.5\alpha}\psi_{l}^{0.5\alpha}}{2\Gamma(\mu)\Gamma(\mu+\mu)\Gamma (l+n)}\prod\limits_{i=1}^{\Gamma}H_{2,1}^{1,2}[\frac{\theta_{i}}{2\Gamma(\mu)} |_{(\mu+\mu,1)}^{(1-n-l,0.5\alpha),(1,0.5\alpha)}]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: ABEP expressions of DPSK for various systems. and caused a new case in which some associated coefficients \(\alpha_{j}^{(i)}\) and \(\beta_{j}^{(i)}\) defined in [49] (Equation (34)) are negative real numbers (e.g., \(-1\)), not positive real numbers as defined in [38] (Equation (A.5)).