# Unified Performance Analysis of MIMO Mixed RF/FSO Relaying System
## 4 Error Performance  
In this section, the closed form expression for a bit error rate of a asymmetric dual-hop MIMO mixed RF/FSO DF relaying system is derived. We use the unified expression of binary modulation schemes BER in [19]  
\[\overline{P_{e}}=\frac{q^{p}}{2\Gamma(p)}\int_{0}^{\infty}e^{-q\gamma_{eq}} \gamma_{eq}{}^{p-1}F_{\gamma_{min}}(\gamma_{eq})d\gamma_{eq}, \tag{15}\]  
where \(p=0.5\), \(q=1\) for BPSK modulation schemes. Other modulation scheme parameters are available in Reference [20]. Substituting Equation (14) into Equation (15), we obtain  
\[\begin{split}\overline{P_{e}}& =\frac{q^{p}}{2\Gamma(p)}\int_{0}^{\infty}e^{-q\gamma_{eq}}{}^{p-1}F_{\gamma_{ RF}}(\gamma_{eq})d\gamma_{eq}\\ &+\frac{q^{p}}{2\Gamma(p)}\int_{0}^{\infty}e^{-q\gamma_{eq}}{}^{ p-1}F_{FSO}(\gamma_{eq})d\gamma_{eq}\\ &-\frac{q^{p}}{2\Gamma(p)}\int_{0}^{\infty}e^{-q\gamma_{eq}}{}^{ p-1}F_{\gamma_{RF}}(\gamma_{eq})F_{FSO}(\gamma_{eq})d\gamma_{eq}\\ &=I_{1}+I_{2}-I_{3}.\end{split} \tag{16}\]  
For easy calculation, we split this formula into three parts, where integrals are defined and solved as follows  
\[\begin{split} I_{1}&=\frac{q^{p}}{2\Gamma(p)}\int_ {0}^{\infty}e^{-q\gamma_{eq}}{}^{p-1}F_{\gamma_{RF}}(\gamma_{eq})d\gamma_{eq }\\ &=\frac{1}{2}-\frac{q^{p}}{2\Gamma(p)}\sum_{u=0}^{N_{tr}N_{tr}-1} \frac{1}{u!}\left(\frac{1}{\hat{\gamma}_{RF}}\right)^{u}\frac{\Gamma(u+p)}{ \left(q+\frac{1}{\hat{\gamma}_{RF}}\right)^{u+p}}.\end{split} \tag{17}\]  
Using ([16] [Equation (07.34.21.0088.01)])  
\[\begin{split} I_{2}&=\frac{q^{p}}{2\Gamma(p)}\int_ {0}^{\infty}e^{-q\pi}x^{p-1}F_{FSO}(x)dx\\ &=\frac{2^{\alpha_{s}+\beta_{s}-3}}{\pi\Gamma(\alpha_{s})\Gamma( \beta_{s})}\Gamma(p)}G_{2,5}^{4,2}\Bigg{[}\frac{(\alpha_{s}\beta_{s})^{2}}{16 q\hat{\gamma}_{FSO}}\bigg{|}\frac{1-p,1}{\Delta 1,0}\Bigg{]},\end{split} \tag{18}\]  
\[\begin{split} I_{3}&=\frac{q^{p}}{2\Gamma(p)}\int_ {0}^{\infty}e^{-q\pi}x^{p-1}F_{\gamma_{RF}}(x)F_{FSO}(x)dx\\ &=I_{2}-\frac{q^{p}}{\Gamma(p)}\frac{2^{\alpha_{s}+\beta_{s}-3}}{ \pi\Gamma(\alpha_{s})\Gamma(\beta_{s})}\sum_{u=0}^{N_{tr}N_{tr}-1}\frac{1}{u! }\left(\frac{1}{\hat{\gamma}_{RF}}\right)^{u}\left(\frac{q\hat{\gamma}_{RF}+1 }{\hat{\gamma}_{RF}}\right)^{u+p}\\ &\times G_{2,6}^{4,2}\Bigg{[}\frac{(\alpha_{s}\beta_{s})^{2}}{16 \left(q+\frac{1}{\hat{\gamma}_{RF}}\right)\gamma_{FSO}}\bigg{|}\frac{1-p-u,1} {\Delta 1,0}\Bigg{]}.\end{split} \tag{19}\]Then, substituting the expression for \(I_{1}\), \(I_{2}\), and \(I_{3}\) from Equations (17)-(19), respectively. After some mathematical simplification, the closed form of BER is available as follows  
\[\begin{split}\overline{P}_{e}&\!\!=\!\!\frac{1}{2}- \frac{q^{p}}{2\Gamma(p)}\sum_{u=0}^{N_{t}N_{r}-1}\frac{1}{u!}\left(\frac{1}{ \tilde{\gamma}_{RF}}\right)^{u}\frac{\Gamma(u+p)}{\left(q+\frac{1}{\tilde{ \gamma}_{RF}}\right)^{u+p}}\\ &\left(1-\frac{2^{\alpha_{s}+\beta_{s}-2}}{\pi\Gamma(\alpha_{s}) \Gamma(\beta_{s})\Gamma(u+p)}\right.\\ &\left.\times G_{2,6}^{4,2}\left[\frac{\left(\alpha_{s}\beta_{s} \right)^{2}}{16\left(q+\frac{1}{\tilde{\gamma}_{RF}}\right)\tilde{\gamma}_{ FSO}}\middle|\begin{matrix}1-p-u,1\\ \Delta 1,0\end{matrix}\right]\right).\end{split} \tag{20}\]