## 5 Numerical Results and Discussion
### Results of the Channel Model  
For atmospheric and coronal turbulence, values of their Rytov variances are shown in Appendix A.1. We use \(\sigma_{ht}^{2}=0.4\) and \(\sigma_{hc}^{2}=0.6\) for weak turbulence, and \(\sigma_{ht}^{2}=2\) and \(\sigma_{hc}^{2}=3\) for strong turbulence. Consider that the radius of light spot have been normalized at the receiver, we use \(\sigma_{p}=0.1\) and \(\sigma_{p}=0.3\) to represent small and large pointing errors,  
\begin{table}
\begin{tabular}{l l l} \hline
**Parameter** & **Symbol** & **Value** \\ \hline Optical wavelength & \(\lambda\) & \(1550\) nm \\ PPM symbol size & \(Q\) & \(16\) \\ Slot width & \(T_{s}\) & \(2\times 10^{-8}\) s \\ Optical filter bandwidth & \(B_{o}\) & \(12.5\times 10^{9}\) Hz \\ APD quantum efficiency & \(\eta_{a}\) & \(0.9\) \\ APD Bandwidth & \(B_{e}\) & \(2.5\times 10^{7}\) Hz \\ Dark current & \(I_{D}\) & \(1\times 10^{-9}\) A \\ Load resistance & \(R_{L}\) & \(146,650\)\(\Omega\) \\ Ionization ratio & \(k_{a}\) & \(0.007\) \\ Field of view & \(\Omega_{F}\) & \(150\) urad \\ Spectral albedo of the Earth & \(R_{f}\) & \(0.25\) \\ Optical receiving aperture diameter & \(d_{r}\) & \(0.1\) m \\ Background radiation spectral radiance & \(N_{b}(\lambda)\) & \(6.4\times 10^{-20}\) W/Hz \\ \hline \end{tabular}
\end{table}
Table 1: Simulation parameters.  
respectively. For simulation of AOA fluctuations, we use \(\sigma_{\theta}=20\)\(\mu\)rad for small fluctuations and use \(\sigma_{\theta}=100\)\(\mu\)rad for large fluctuations.  
Our derived closed-form expression for the uplink channel model is shown as (17). The analytical results and the Monte Carlo simulation results under different \(\sigma_{ht}^{2}\), \(\sigma_{hs}^{2}\), \(\sigma_{p}\), and \(\sigma_{\theta}\) are shown in Figure 4. It can be seen that the analytical results agree well with Monte Carlo simulation results, which confirms the accuracy of our derived closed-form expression of the uplink model. Moreover, it is clear that the average channel fading coefficient, i.e., mean of \(h\), decreases with the increase in \(\sigma_{ht}^{2}\), \(\sigma_{hc}^{2}\), \(\sigma_{p}\), and \(\sigma_{\theta}\).  
In addition, as shown in Figure 4a,b, respectively, the distribution shapes of the PDF curves under various values of \(\sigma_{ht}^{2}\), \(\sigma_{hs}^{2}\), and \(\sigma_{p}\) are different. However, in Figure 4c, the distribution shapes of the PDF curves are similar for different AOA fluctuations. Specifically, curvilinear trends are the same, and only absolute PDF values are different when \(\sigma_{\theta}\) is \(20\)\(\mu\)rad and \(100\)\(\mu\)rad. The reasons for these cases are that various atmospheric turbulence, coronal turbulence, and pointing errors will cause different fading in the channel. However, in the model of AOA fluctuations, \(h_{a}\) is reasonably assumed to be two values, \(1\) and \(0\), which correspond to the optical beam being captured successfully and unsuccessfully, respectively. That is, \(h_{a}\) does not influence the channel distribution when \(h_{a}=1\), and the probability of \(h_{a}=0\) increases with increasing \(\sigma_{\theta}\) when FOV is given. Hence, in addition to the results shown in Figure 4c, it is reasonable to infer that the probability of \(h_{a}=0\) of the blue curve is bigger than that of the red curve.