## 2 System Model
### Mathematical Modeling of the PE Tracking System  
The received electrical SNR with a slow fading channel can be expressed by the following equation [10]:  
\[SNR(h_{FSO})=\frac{2P_{t}^{2}h^{2}R^{2}}{\sigma_{t}^{2}} \tag{7}\]  
where \(h_{FSO}=h_{p}h_{d}\) includes PE \(h_{p}\) and \(h_{d}\), representing the path loss component caused by other atmospheric attenuation. \(h_{p}\) is an independent random variable. In order to simplify the calculation and analysis, a simplified theoretical model is used in this section, which only considers the direct path loss factor and path attenuation factor of the beam and ignores the influence of AT. The path loss component can be determined by [PERSON]'s law as  
\[h_{d}=exp(-\alpha_{atmo}z) \tag{8}\]  
where \(z\) is the propagation distance and \(\alpha_{atmo}\) is the attenuation coefficient caused by AT. In the presence of geometric diffusion and PE, the received power of the detector can be expressed by the following equation [10]:  
\[h_{p}\approx A_{0}exp\bigg{(}\frac{-2r^{2}}{w_{eq}^{2}}\bigg{)} \tag{9}\]  
where \(A_{0}=[erf(v)]^{2}\) is the power collected at radial displacement \(r=0\). \(w_{eq}^{2}\dot{=}w_{z}^{2}\big{(}(\sqrt{\pi}erf(v))\big{)}\big{/}\big{(}2 exp \big{(}-v^{2}\big{)}\) is the equivalent beam width. \(v=\sqrt{\pi}a/\sqrt{2}w_{z}\) is the beam width at distance \(z\). \(a\) is the radius of the detection aperture. The performance of FSO communication under PE can be evaluated based on outage probability.  
The performance of the proposed FSO system is provided by the probability of interruption caused by PE. The channel capacity \(C(h)\), which is a measure of the channel's ability to transmit information, depends on the channel state \(h\). The status is affected byfactors such as AT and PE. In the proposed FSO system, an interruption occurs when \(C(h)\) is insufficient to support the required data rate \(R_{0}\). And it can be expressed by the following equation:  
\[P_{out}(R_{0})=P_{r}(C(SNR_{Rx}(h))<R_{0}) \tag{10}\]  
\[\begin{array}{c}P_{out}(R_{0})=P_{r}\big{(}SNR_{Rx}(h)<C^{-1}(R_{0})\big{)}\\ =P_{r}\Big{(}\frac{2P_{r}^{2}h^{2}R^{2}}{\sigma_{0}^{2}}<C^{-1}(R_{0})\Big{)} \end{array} \tag{11}\]  
The outage probability can be expressed by the following equation:  
\[P_{out}=P_{r}(h<h_{0}) \tag{12}\]  
where \(h_{0}=\left(\left(\left(C^{-1}(R_{0})\sigma_{n}^{2}\right)/\left(2P_{r}^{2}R^{ 2}\right)\right)\right)^{1/2}\). The outage probability is the cumulative density function of \(h\) evaluated as \(h_{0}\). It can be expressed by the following equation:  
\[\begin{array}{c}P_{out}=\int_{0}^{h_{0}}f_{h_{p}}\big{(}h_{p}\big{)}dh_{p}\\ =k^{\prime}\int_{0}^{h_{0}}\left(\ln(\frac{A_{0}}{h_{p}})\right)^{m-1}\big{(}h _{p}\big{)}^{\zeta-1}dh_{p}.\end{array} \tag{13}\]  
On substitution, \(x=ln\big{(}A_{0}/h_{p}\big{)}\)  
\[P_{out}=k^{\prime}A_{0}^{\zeta}\int_{\infty}^{ln(\frac{A_{0}}{h_{0}})}x^{m-1} exp(-x\zeta)dx \tag{14}\]  
where  
\[k^{\prime}=\frac{\xi^{m}}{\Gamma(m)A_{0}^{\zeta^{m}}} \tag{15}\]  
Using the approximation of  
\[\int_{u}^{\infty}x^{v-1}exp(-\mu x)dx=\mu^{-v}\Gamma(v,u\mu) \tag{16}\]  
\[P_{out}=k^{\prime}A_{0}^{\zeta}\zeta^{-m}\Gamma\big{(}m,\zeta ln\big{(}\frac{A_{0}}{h_{0}} \big{)}\big{)} \tag{17}\]