# Physical-Layer Security Analysis over M-Distributed Fading Channels
## 5 Numerical Results
### SOP Results  
Figure 3 shows the SOP versus \(E(\gamma_{\rm B})\) with different \(E(\gamma_{\rm E})\) when \(\gamma_{\rm th}=0\) dB and \((\alpha_{\rm B},\beta_{\rm B})=(\alpha_{\rm E},\beta_{\rm E})=(2,2)\). It can be observed that the SOP performance improves with the increase of \(E(\gamma_{\rm B})\). This indicates that the larger the average strength of the main channel is, the better the system performance becomes. Moreover, the theoretical results of the lower bound of SOP match the simulation results very well. This shows the accuracy of the lower bound of SOP. Furthermore, when \(E(\gamma_{\rm E})=1\) dB, which does not satisfy the condition \(E(\gamma_{\rm E})\gg 1\), and thus the performance gap between the lower bound of SOP and the exact result of the SOP is observable. With the increase of \(E(\gamma_{\rm E})\), the gap becomes smaller and smaller. This indicates that when \(E(\gamma_{\rm E})\gg 1\), the derived lower bound of SOP can be used to evaluate the system performance, without reliable on time-intensive simulations.  
Figure 4 shows the SOP versus \(E(\gamma_{\rm B})\) with different \(\gamma_{\rm th}\) when \(E(\gamma_{\rm W})=18\) dB and \((\alpha_{\rm B},\beta_{\rm B})=(\alpha_{\rm E},\beta_{\rm E})=(2,2)\). As can be observed, the SOP decreases with the increase of \(E(\gamma_{\rm B})\), which coincides with the conclusion in Figure 3. Moreover, for the lower bound of SOP, the gap between theoretical results and simulation results is very small, which verifies the accuracy of the derived lower bound. In this simulation, the lower bound of SOP and the exact result of the SOP almost coincide for larger \(E(\gamma_{\rm B})\) and \(E(\gamma_{\rm E})\), which shows the tightness of the lower bound of the SOP. Furthermore, it can be seen that the value of the SOP increases with the increase of \(\gamma_{\rm th}\). This indicates that large SNR threshold will degrade the system performance.  
Table 2 shows the accuracy of the SOP for different \(E(\gamma_{\rm B})\) when \(E(\gamma_{\rm E})=18\) dB, \((\alpha_{\rm B},\beta_{\rm B})=(\alpha_{\rm E},\beta_{\rm E})=(2,2)\) and \(\gamma_{\rm th}=-4\) dB. As can be seen, all the relative errors \(e_{1}\) and \(e_{2}\) are quite small. This indicates that the derived lower bound of SOP in (16) is very accurate to evaluate the system performance.  
In this subsection, the lower bounds of the SOP over different fading channels are also provided, as shown in Figure 5. In this figure, the lower bounds of the SOP over the K, exponential, and Gamma-Gamma fading channels are derived in (36), (40) and (44), respectively. Moreover, the lower bounds of the SOP over the M-distributed fading channel (i.e., (16)) with special parameters are also provided. As can be seen in Table 1, when \(\Omega=0\) and \(\rho=0\), the M-distributed channel reduces to the K channel; when \(\Omega=0\), \(\rho=0\) and \(\alpha\rightarrow\infty\), the M-distributed channel reduces to the exponential channel; when \(\rho=1\) and \(\Omega^{\prime}=1\), the M-distributed channel reduces to the Gamma-Gamma channel. As can be observed, the results of (16) with \(\Omega=0\) and \(\rho=0\) match with that of (36) very well. The results of (16) with \(\Omega=0\), \(\rho=0\) and \(\alpha\rightarrow\infty\) and the results of (40) coincide with each other. Similarly, the difference between (16) with \(\rho=1\) and \(\Omega^{\prime}=1\) and (44) is so small and it can be ignored. These conclusions indicate that the derived expression of SOP over the M-distributed fading channel is a general expression, which can be accurately extended to other well-known fading channels.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \(E(\gamma_{\rm B})\) & **10 dB** & **20 dB** & **30 dB** & **40 dB** & **50 dB** \\ \hline \(e_{1}\) & 3.699\% & 3.205\% & 1.880\% & 1.936\% & 0.293\% \\ \(e_{2}\) & 0.436\% & 0.041\% & 0.795\% & 0.651\% & 0.156\% \\ \hline \end{tabular}
\end{table}
Table 2: Relative errors of SOP.