## 2 System and Channel Model
### Channel Model  
In satellite-ground laser communication, EW distribution can be applied to all turbulence conditions and give the experimental data an excellent fitting effect when the aperture smoothing effect occurs. Therefore, the EW probability distribution model is established in the ground-satellite uplink, and its probability density function (PDF) is as follows [24]:  
\[f_{I}(I)=\frac{\alpha\beta}{\eta}\bigg{(}\frac{I}{\eta}\bigg{)}^{\beta-1}exp \left[-\bigg{(}\frac{I}{\eta}\bigg{)}^{\beta}\right]\times\bigg{\{}1-exp \left[-\bigg{(}\frac{I}{\eta}\bigg{)}^{\beta}\right]\bigg{\}}^{\alpha-1} \tag{1}\]  
In the uplink system of ground-to-satellite laser communication, \(\alpha\) is the shape parameter related to the receiving aperture, \(\beta\) is the shape parameter of the scintillation index, and \(\eta\) is the proportional parameter associated with the irradiation intensity. The expressions for these parameters are given as [25]:  
\[\alpha=3.931\bigg{(}\frac{D_{R}}{\rho_{0}}\bigg{)}^{-0.519}\]  
\[\beta=\bigg{(}\alpha\sigma_{I}^{2}\bigg{)}^{\frac{-6}{\eta}}\]  
\[\eta=\frac{1}{\alpha\Gamma\Big{(}1+\frac{1}{\beta}\Big{)}g(\alpha,\beta)} \tag{2}\]  
\[\rho_{0}=\bigg{(}1.46C_{n}^{2}(2\pi/\lambda)^{2}L\bigg{)}^{-3/5} \tag{3}\]  
\[g(\alpha,\beta)=\sum_{j=0}^{\infty}\frac{(-1)^{j}(j+1)^{-(1+\beta)/\beta}}{j! \Gamma(\alpha-j)} \tag{4}\]  
where \(\lambda\) is the wavelength of the Gaussian beam, \(L\) is the length of the FSO link, \(C_{n}^{2}\) is the refractive index structure constant, \(D_{R}\) is the receiving aperture, and \(\sigma_{I}^{2}\) is the scintillation index of the uplink non-Kolmogorov turbulence, which is defined as follows [26]:  
\[\begin{split}\sigma_{I}^{2}&=4\pi^{2}k^{2}A(a)\beta _{0}u^{2-a}\int\limits_{l_{0}}^{H}C_{n}^{2}(h)\sum\limits_{n=1}^{\infty}\frac {\left[\Lambda\
u_{p}\left(1-\frac{h-h_{0}}{H-h_{0}}\right)\right]^{2n}}{ \pi^{4}}\left(\frac{k_{2}}{\lambda}\right)^{2n+2-a}\times\\ &\Bigg{\{}\frac{\Gamma\left(\frac{\delta}{\eta}-n-1\right)}{ \Gamma\left(\frac{\delta}{\eta}\right)}+\frac{\Gamma\left(n+1-\frac{1}{\delta }\right)}{\Gamma\left(n+1\right)}\Bigg{\{}\bigg{(}\frac{k_{2}}{\lambda}\bigg{)} ^{2}\bigg{[}\frac{\mu^{2}}{k_{2}^{2}}+\frac{\Lambda\left(H-h_{0}\right)\left(1 -\frac{h-h_{0}}{H-h_{0}}\right)^{2}}{k}\bigg{]}\bigg{\}}^{\frac{\delta}{\eta} -n-1}\Bigg{\}}dh+\\ &\pi^{2}k^{2}A(a)\ \beta_{0}u^{2-a}\int\limits_{l_{0}}^{H}C_{n}^{2}(h) \Big{(}\frac{k_{2}}{\lambda}\Big{)}^{2-a}\Gamma\big{(}1-\frac{\delta}{2} \big{)}\times Re\Bigg{\{}\begin{array}{l}\bigg{\{}\bigg{(}\frac{k_{2}}{ \lambda}\bigg{)}^{2}\bigg{[}\frac{k_{2}^{2}}{k^{2}}+\frac{\Lambda\left(H-h_{0 }\right)\left(1-\frac{h-h_{0}}{H-h_{0}}\right)^{2}}{k}\bigg{]}\bigg{\}}^{ \frac{\delta}{\eta  
In summary, considering the influence of light intensity scintillation, beam drift, and the angle of arrival fluctuation and using \(f_{\gamma}(\gamma)=\frac{1}{2\sqrt{\gamma\gamma}}f_{w}\Big{(}\sqrt{\gamma/\gamma \overline{\gamma}}\Big{)}\) transformation [28], the closed expression of the \(\gamma\) probability density function of the EW distribution under the combined action of the three is:  
\[f_{\gamma}(\gamma)=\frac{1}{2\sqrt{\gamma\overline{\gamma}}}\sum_{l=1}^{\infty }w_{i}\frac{\alpha\beta}{\eta}\bigg{(}\frac{\sqrt{\gamma}}{\eta\sqrt{\gamma}} \bigg{)}^{\beta-1}exp\left[-\bigg{(}\frac{\sqrt{\gamma}}{<I>\eta\sqrt{\gamma} }\bigg{)}^{\beta}\right]\times\Bigg{\{}1-exp\left[-\bigg{(}\frac{\sqrt{\gamma }}{<I>\eta\sqrt{\gamma}}\bigg{)}^{\beta}\right]\Bigg{\}}^{\alpha-1} \tag{7}\]  
where \(w_{i}\) is the i'th Gaussian coefficient of the angle of arrival fluctuations, and \(<I>\) is the average received light intensity.