## 4 Numerical Results  
Numerical Monte Carlo (MC) simulation results are presented to show the correctness of our analytical results. With no loss of generality, we assumed \(\delta_{R}^{2}=\delta_{D}^{2}=1\), and through the whole figures we instructed \(\overline{\gamma}_{SR}=\overline{\gamma}_{SP}=\overline{\gamma}_{RD}= \overline{\gamma}_{RP}=\overline{\gamma}\), \(k_{1}=k_{2}=k_{SP}=k_{RP}=k\), \(T_{SR}=T_{RD}\)=\(T_{SP}=T_{RP}=L\) and \(R_{s}=10\)bit/s/Hz, \(U\in\{SR,SP\}\). The system and channel fading parameters are presented in Table 1[22] and Table 2[35], respectively. Figures 2 and 3 illustrate the OP of the system versus different \(\overline{\gamma}\) with \(L=10\) and \(\gamma_{0}=3\) dB in FHS. As presented in both figures, the analytical results were in accurate agreement with MC simulations; besides, the asymptotic plots matched very well with the exact plots at high SNRs, which verifies the correctness of our derivation. They also showed that HIs and CEEs affect the system performance to a much higher degree at higher SNRs than that of lower SNRs. Furthermore, it was found that at high SNRs, HIs played the main role, while at low SNRs, the CEEs were the dominant factor. In addition, when the system suffered from HIs, the OP at high SNRs would have a lower bound. The larger the HIs level was, the larger the bound was. Besides, the OP would be decreased with the decreasing of \(M\) and \(\sigma\); this is because with the decrease of \(M\) and \(\sigma\), the power of the secondary user would be enhanced, which would lead to a lower OP. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Value** \\ \hline Satellite Orbit & GEO \\ Frequency Band & \(f=2\) GHz \\
3-dB Angle & \(\overline{\theta}_{k}=0.8^{\circ}\) \\ Maximal Beam Gain & \(G_{max}=48\) dB \\ Antenna Gain & \(G_{ES}=4\) dB \\ \hline \hline \end{tabular}
\end{table} 
Table 1: System parameters. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Shadowing** & \(m_{U}\) & \(b_{U}\) & \(\Omega_{U}\) \\ \hline Frequent heavy shadowing (FHS) & 1 & 0.063 & 0.0007 \\ Average shadowing (AS) & 5 & 0.251 & 0.279 \\ Infrequent light shadowing (ILS) & 10 & 0.158 & 1.29 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Channel parameters. In Figures 4 and 5, OP is plotted against different \(\gamma_{0}\) with \(\overline{\gamma}\) = 30 dB and \(L=10\) in FHS. We found that there was an SNDER ceiling for the OP plots, which means that there existed a special average SNR value that corresponded to the largest SNDER of the HI system. In other words, there was a fixed SNDER threshold beyond which the OP value always remained one, which has been proven in Equation (30). Moreover, we observe that different \(M\) systems will have the same ceilings and that different \(\sigma\) systems have the same ceilings as well. The aggregate level of HIs affects the ceiling effect all on its own. Figure 6 plots the OP of the system versus different \(\overline{\gamma}\) with \(M=3\), \(\sigma=0.8\), \(\gamma_{0}=3\) dB and \(L=10\). We can obtain that the OP will be larger when the channel was under heavy fading. Moreover, we also found that the OP would be higher when the system was under HIs. Figure 7 examines the OP of the system versus the training symbol length \(L\) with \(M=3\), \(\gamma_{0}=3\) dB and \(\sigma=0.8\). We can observe that the OP would be lower with the length of pilot symbol increasing as a result of that when \(L\) was larger, the CSI of the channel would be more accurate. The more accurate CSI would lead to the better system performance. Figure 8 illustrates the throughput of the system versus different \(\overline{\gamma}\) with \(M=3\), \(L=10\) and \(\gamma_{0}=3\) dB in the FHS condition. From the figure, we observe that the throughput was lower than that of the target rate \(R_{s}\) for the reason that the system suffered HIs.