# Physical-Layer Security Analysis over M-Distributed Fading Channels
## 5 Numerical Results
### ESC Results  
Figure 9 shows the ESC versus \(E(\gamma_{\rm B})\) with different \(E(\gamma_{\rm E})\) when \((\alpha_{\rm B},\beta_{\rm B})=(\alpha_{\rm E},\beta_{\rm E})=(2,2)\). In Figure 9, when \(E(\gamma_{\rm B})\) is small, the ESC is almost zero. With the increase of \(E(\gamma_{\rm B})\), the ESC increases accordingly. For a fixed \(E(\gamma_{\rm B})\), the ESC performance degrades with the increase of \(E(\gamma_{\rm E})\). Moreover, for the lower bound of the ESC, the performance gap between theoretical results and simulation results is very small, and thus the derived lower bound of ESC is accurate. Furthermore, when \(E(\gamma_{\rm B})\) is small, the difference between the lower bound of ESC and the exact results of ESC is large. However, with the increase of \(E(\gamma_{\rm B})\), the gap becomes smaller and smaller. Therefore, at high SNR regime, the derived lower bound of ESC can be employed to evaluate the system performance without time-intensive simulations.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(E(\gamma_{\rm B})\) & **10 dB** & **20 dB** & **30 dB** & **40 dB** & **50 dB** \\ \hline \(e_{3}\) & 0.780\% & 0.125\% & 7.777\% & 6.753\% & 2.283\% \\ \hline \hline \end{tabular}
\end{table}
Table 3: Relative errors of the probability of SPSC.  
Figure 8: Probability of SPSC versus \(E(\gamma_{\rm B})\) for different fading channels.  
Figure 10 shows the ESC versus \(E(\gamma_{\rm B})\) with different \((\alpha_{\rm E},\beta_{\rm E})\) when \((\alpha_{\rm B},\beta_{\rm B})=(2,2)\). As seen in Figure 9, the ESC also increases with the increase of \(E(\gamma_{\rm B})\) in Figure 10. Moreover, the gaps among the theoretical results of the lower bound of ESC, the simulation results of the lower bound of ESC and the exact results of ESC are very small for large \(E(U)\). This conclusion is the same as that in Figure 9. Furthermore, with the increase of \((\alpha_{\rm E},\beta_{\rm E})\), the ESC performance degrades.  
Table 4 shows the relative errors of ESC when \((\alpha_{\rm B},\beta_{\rm B})=(2,2)\) and \((\alpha_{\rm E},\beta_{\rm E})=(4,4)\). It can be observed that the error \(e_{2}\) is small for all cases. However, when \(E(\gamma_{\rm B})\) is small, the error \(e_{1}\) is large. Small \(E(\gamma_{\rm B})\) represents the main channel is much worse than the eavesdropping channel, which is quite rare in practical scenarios. Moreover, with the increase of \(E(\gamma_{\rm B})\), \(e_{1}\) decreases rapidly. Therefore, the derived lower bound of ESC is valid in the high SNR regime.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \(E(\gamma_{\rm B})\) & **25 dB** & **30 dB** & **35 dB** & **40 dB** & **45 dB** & **50 dB** \\ \hline \(e_{1}\) & 31.911\% & 11.668\% & 3.879\% & 1.845\% & 0.713\% & 0.195\% \\ \(e_{2}\) & 3.112\% & 0.401\% & 0.419\% & 0.258\% & 0.221\% & 0.450\% \\ \hline \end{tabular}
\end{table}
Table 4: Relative errors of ESC.  
Furthermore, Figure 11 shows the lower bound of ESC versus \(E(\gamma_{\rm B})\) for different fading channels. The lower bounds of ESC over the K, exponential, and Gamma-Gamma fading channels are obtained by (38), (42) and (46), respectively. Moreover, the lower bound of ESC over the M-distributed fading channel (i.e., (25)) with special parameters are also provided as shown in Table 1. Once again, it can be seen that the results over the M-distributed fading channel with special parameters show close agreement with that of the other fading channels.