# Unified Performance Analysis of MIMO Mixed RF/FSO Relaying System
## 2 System and Channel Model
\(A_{0}=[erf(v)]^{2}\) denotes the fraction of the collected power at \(r=0\), \(v=\sqrt{\pi}R_{a}/\sqrt{2}\)\(w_{z}\), \(w_{z}\) is the beam waist, \(R_{a}\) denotes the aperture radius of the receiver. \(\omega_{zeq}\)=\(\omega_{z}^{2}\sqrt{\pi}erf(v)/2v\exp(-v^{2})\) represents the equivalent beam waist. Reference [13] proposed a new method to approximate the distribution for the sum of L IT fading with Rayleigh pointing error random variables, the PDF is shown as following:  
\[f_{y}(y) \approx\sum_{n=0}^{\infty}\sum_{i=0}^{n}\Lambda(v,n,i)\frac{2^{lg -\alpha_{s}-\beta_{s}+1}}{\Gamma(\alpha_{s})\Gamma(\beta_{s})}L^{L}g^{L}\left( \frac{\alpha_{s}\beta_{s}}{A_{0}\omega}\right)^{\frac{lg}{2}}y^{\frac{lg}{2}-1} \tag{5}\] \[\times G_{L,L+1}^{L+10}\left(2\sqrt{\frac{\alpha_{s}\beta_{s}}{A_{ 0}\omega}}y\right|_{2\beta_{s}+i-lg,0,\cdots,0}^{1,1,\cdots,1}\right)\]  
where \(\gamma_{FSO}\) is the instantaneous SNR of the FSO link and \(\bar{\gamma}_{FSO}\) is the average SNR of the FSO link. \(\alpha_{s}=MN\alpha+\varepsilon\)\(\beta_{s}=MN\beta\), \(g=2\phi^{2}\) denotes the parameter related to misalignment. The parameters \(\alpha\) and \(\beta\) are given by [14]  
\[\alpha\] \[\beta \tag{6}\]  
where \(\sigma_{R}^{2}\) is the Rytov variance, which is given by \(\sigma_{R}^{2}=1.23C_{n}^{2}(\frac{2\pi}{\lambda_{FSO}})^{\frac{\gamma}{2}} \hat{\epsilon}\frac{11}{\hat{\epsilon}}\). \(C_{n}^{2}\) is the refractive index. \(\varepsilon\) is adjustment parameter, which is given by \(\varepsilon=(MN-1)\times(-0.127-0.95\alpha-0.0058\beta)/(1+0.00124\alpha+0.9 8\beta)\). \(\Gamma(\cdot)\) is the Gamma function, \(G_{\gamma}^{\cdot}[\cdot]\) is [PERSON]'s G-function, and, \(L=MN\). The coefficient \(\Lambda(v,n,i)=\frac{(-1)^{i}\sqrt{\pi}\Gamma(2v)\Gamma\left(\frac{1}{2}+n-v \right)L(n,i)}{2^{-i}\Gamma\left(\frac{1}{2}-v\right)\Gamma\left(\frac{1}{2} +n+v\right)nt}\), \(v=\alpha_{s}-\beta_{s}\) is the order of the modified Bessel function of the second kind. \(L(n,i)\) is Lah numbers [15] and it is defined as \(L(n,i)= \begin{pmatrix}n-1\\ i-1\end{pmatrix} \frac{nt}{i! }\), for n,i>0. The correcting factor \(F\) is defined as \(F=\frac{(2+\varepsilon L)^{L}}{L^{L}g^{L-1}(2+\delta)}\). According to the distribution theorem of the random variable function, the PDF of the \(\gamma_{FSO}\) can be expressed as:  
\[f_{FSO}^{p}(\gamma_{FSO})=\left|\frac{d}{d\gamma_{FSO}}\left(\frac{MN}{F}\sqrt {\frac{\gamma_{FSO}}{\bar{\gamma}_{FSO}}}\right)\right|f_{y}\left(\frac{MN}{ F}\sqrt{\frac{\gamma_{FSO}}{\bar{\gamma}_{FSO}}}\right) \tag{7}\]  
Then, substitute Equation (7) into Equation (5), the following expression can be obtained:  
\[f_{FSO}^{p}(\gamma) =\sum_{n=0}^{\infty}\sum_{i=0}^{n}\Lambda(v,n,i)\frac{2^{lg-\alpha _{s}-\beta_{s}+1}}{\Gamma(\alpha_{s})\Gamma(\beta_{s})}L^{L}g^{L}\left(\frac{ \alpha_{s}\beta_{s}}{A_{0}\omega}\cdot\frac{MN}{F}\right)^{\frac{lg}{2}}\left( \frac{1}{\bar{\gamma}}\right)^{\frac{lg}{4}}(\gamma)^{\frac{lg}{4}-1} \tag{8}\] \[\times G_{L,L+1}^{L+10}\left(2\sqrt{\frac{\alpha_{s}\beta_{s}}{A_{ 0}\omega}}\frac{MN}{F}\sqrt{\frac{\gamma}{\bar{\gamma}}}\right)_{2n+i-lg,0, \cdots,0}^{1,1,\cdots,1}\]Using ([16] [Equation (07.34.21.0084.01)]), we can get:  
\[ \begin{split}& F_{FSO}^{p}(\gamma_{th}){=}\int_{0}^{\gamma_{th}}f_{ \gamma}(\gamma){=}\sum_{n=0}^{\infty}\sum_{i=0}^{n}\Lambda(v,n,i)\frac{2^{3 \beta_{s}+2i-Lg-2L-a_{s}-\frac{1}{2}}}{\pi^{3/2}\Gamma(a_{s})\Gamma(\beta_{s}) }L^{L}g^{L}\bigg{(}\frac{\alpha_{s}\beta_{s}}{A_{0}\omega}\frac{MN}{F}\bigg{)}^ {\frac{Lg}{2}}\bigg{(}\frac{\gamma_{th}}{\hat{\gamma}}\bigg{)}^{\frac{Lg}{4}}\\ & G_{4L+1,4L+5}^{4L+4}\left(\begin{array}{c}\frac{1}{16}\left( \frac{\alpha_{s}\beta_{s}}{A_{0}\omega}\frac{MN}{F}\right)^{2}\frac{\gamma_{th }}{\hat{\gamma}}\bigg{|}_{\frac{2\beta_{s}+i-Lg}{4},\frac{2\beta_{s}+i-Lg+2}{4},\frac{2\beta_{s}+i-Lg+3}{4},(0,\frac{1}{2},\frac{2}{3})^{\otimes L},-\frac{Lg }{4}}^{\otimes L}\\ \end{array}\right)\\ &\text{where symbol }(\ast)^{\otimes L}\text{ denotes the L times repetition of a given tuple.}\end{split}