# Performance Analysis of Hard-Switching Based Hybrid FSO/RF System over Turbulence Channels
## 2 System Model
### Outage Probability  
It is assumed that the FSO link follows Gamma-Gamma fading distribution so the corresponding probability density function (PDF) for the SNR is given as [14; 15]:  
\[f_{FSO}=\frac{2(\alpha\beta)^{(\alpha+\beta)/2}I^{(\alpha+\beta-2)/2}}{\Gamma( \alpha)\Gamma(\beta)}K_{\alpha-\beta}\Big{(}2\sqrt{\alpha\beta l}\Big{)}, \tag{4}\]  
where \(\Gamma(. )\) is the gamma function, \(K_{v}(. )\) is the modified Bessel function of second kind and order \(v\), while \(\alpha,\beta\) are the parameters of the distribution and can be estimated directly from the link's parameters and the atmospheric turbulence strength [15]. For FSO applications the specific parameters are given as [16]:  
\[\alpha\cong exp\Bigg{[}\frac{0.49\ \sigma_{R}^{2}}{\left(1+1.11\sigma_{R}^{1/ 5}\right)^{7/6}}\Bigg{]}-1 \tag{5}\]  
and  
\[\beta\cong exp\Bigg{[}\frac{0.51\ \sigma_{R}^{2}}{\left(1+0.69\ \sigma_{R}^{1/ 5}\right)^{7/6}}\Bigg{]}-1 \tag{6}\]  
These parameters are related to Rytov variance \(\sigma_{R}^{2}\) and here, the plane wave propagation without the inner scale is assumed. Thus, the Rytov variance is given as, [12]:  
\[\sigma_{R}^{2}=1.23\Big{(}\frac{2\pi}{\lambda}\Big{)}^{7/6}C_{n}^{2}v^{11/6} \tag{7}\]  
where \(\lambda\) is the operational wavelength of the FSO link, \(v\) is the link's length and \(C_{n}^{2}\) is the refractive-index structure parameter which actually represents the atmospheric turbulence strength. Generally, fluctuations in refractive index for wave propagation through turbulence atmosphere are considered isotropic in nature. But near the earth's surface, i.e., 2-4 m above the surface, the propagation is anisotropic due to moving atmosphere and stationary ground interaction [17]. Mutual coherence functions (MCF) for one ccc-dimensional vertical and horizontal directions are measurably different in anisotropic turbulence regimes. For our proposed model, it is assumed that communication established at the rooftop is present in a zone where the turbulence can be taken into account as isotrop and much lower than near ground turbulence. For these cases, the value of \(C_{n}^{2}\) could be around \(10^{-17}m^{-2/3}\), [18]. Next, the PDF of \(\gamma_{FSO}\) can be estimated from (4) using a power transformation of random variables as [6; 13]:  
\[f_{FSO}(\gamma_{FSO})=\frac{(\alpha\beta/\sqrt{\overline{\gamma}_{FSO}})^{\frac{( \alpha+\beta)}{2}}\gamma_{FSO}\frac{\alpha+\beta-4}{4}}{\Gamma(\alpha)\Gamma( \beta)}K_{\alpha-\beta}\Bigg{[}2\sqrt{\alpha\beta\sqrt{\frac{\gamma_{FSO}}{ \overline{\gamma}_{FSO}}}}\Bigg{]} \tag{8}\]  
where \(\gamma_{FSO}\) and \(\overline{\gamma}_{FSO}=\frac{E_{s}\mu^{2}R^{2}P_{FSO}^{2}G_{FSO}^{2}}{\sigma_ {FSO}^{2}}E[I]^{2}\) stand for the instantaneous and the expected electrical SNR, respectively, while the symbol \(E[. ]\) stands for the expected value and here, taking the value \(E[I]=1\), [13]. On the other hand, the RF link experiences Rayleigh fading as it is subjected to a multi-path effect. Accordingly, the signal to noise ratio corresponds to following exponential PDF [9]:  
\[f_{RF}(\gamma_{RF})=\ \frac{1}{\overline{\gamma_{RF}}}e^{-\gamma_{RF}/\overline{ \gamma_{RF}}} \tag{9}\]  
where \(\overline{\gamma_{RF}}\) is the time expected SNR of the received signal just before the modulated envelope is detected. To compute outage probability for a fixed transmission rate \(R_{0}(\mathrm{bps/Hz})=\log_{2}(1+\gamma_{th})\) where, \(\gamma_{th}=2^{R_{0}}-1\) is the receiver's SNR threshold. The probability for which end-to-end SNR of the system is less than specific threshold \(\gamma_{th}\) is called outage probability, [6; 19]. Thus, the outage probability either for the FSO link or the alternative RF connection estimated as [20]:  
\[\mathrm{P}_{outage}=Pr(\mathrm{SNR}<\gamma_{th-x})\, \tag{10}\]  
where the subscript \(th\)-\(x\), represents either \(th\)-\(FSO\) or \(th\)-\(RF\) depending on the SNR threshold value of each specific communication link, i.e., FSO or RF link, respectively.