## 4 Case II: Active Eavesdropper  
### Active Eavesdropper System Model  
The wiretap channel system model with the use of FD self-jamming and an active eavesdropper is shown in Figure 13.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\(\rho_{SI}\)} & \multirow{2}{*}{\(SNR_{E,max}\)} & \multicolumn{2}{c}{**Without Feedback at User A**} & \multicolumn{2}{c}{**Blind Feedback at User A**} \\ \cline{3-6}  & & \(SNR_{A,min}\) & \(S_{g}\) & \(SNR_{A,min}\) & \(S_{g}\) \\ \hline
0 & 4.1 & 23.8 & 19.7 & 16.5 & 12.4 \\
5 & 5.8 & 24.5 & 18.7 & 16.8 & 10.8 \\
10 & 8.1 & 25.1 & 17 & 17.2 & 9.1 \\
15 & 10.3 & 25.6 & 15.3 & 17.6 & 7.3 \\
20 & 12.4 & 26.7 & 14.3 & 17.9 & 5.5 \\
25 & 14.3 & 27.5 & 13.2 & 18.1 & 3.8 \\
30 & 16.6 & 29 & 12.4 & 18.3 & 1.7 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Security gap \(S_{g}\) in case of passive eavesdropper.  
Figure 12: \(S_{g}\) versus \(\rho_{SI}\) in the case of a passive eavesdropper.  
In this case, both user A and user E operate in the FD transmission mode to simultaneously receive the intended information message from user B and transmit the self-jamming or jamming signal to other users. In particular, user B wants to send his encoded message \(\mathbf{x}_{B}\) to the legitimate receiver user A by the main channel, while the eavesdropper user E not only tries to listen to and decode user B's message by the wiretap channel but also simultaneously broadcasts their jamming signal to user A. Consequently, the received signals in the digital domain at user A and user E are given by:  
\[y_{A}[n] =y_{BA}[n]+y_{AA}[n]+y_{EA}[n]+w_{A}[n] \tag{6}\] \[=(\sqrt{p_{B}}\mathbf{x}_{B}*\mathbf{h}_{BA})[n]+(\sqrt{p_{A}} \mathbf{x}_{A}*\mathbf{h}_{AA})[n]+(\sqrt{p_{E}}\mathbf{x}_{E}*\mathbf{h}_{EA}) [n]+w_{A}[n];\]  
and  
\[y_{E}[n] =y_{BE}[n]+y_{EE}[n]+y_{AE}[n]+w_{E}[n] \tag{7}\] \[=(\sqrt{p_{B}}\mathbf{x}_{B}*\mathbf{h}_{BE})[n]+(\sqrt{p_{E}} \mathbf{x}_{E}*\mathbf{h}_{EE})[n]+(\sqrt{p_{A}}\mathbf{x}_{A}*\mathbf{h}_{AE} )[n]+w_{E}[n],\]  
respectively.  
It can be seen that the signal-to-noise ratio at user A is reduced due to the impact of the jamming signal from user E, which leads to an increase in noise at the receiver of user A. Therefore, in addition to the proposed blind feedback scheme, the joint iterative SI channel estimation and equalization processes with the semi-blind algorithm, which have been studied in [28], should be used at user A in order to eliminate the SI component and estimate the intended signal \(\mathbf{k}_{S0\,l}\) because the proposed semi-blind algorithm shows its robustness in the low region of the SNR, compared to the blind algorithm. Indeed, the principle of this algorithm is to use at least four pilot symbols between the transceivers (which is a sufficient number of pilot symbols, as shown in [28]) to perform the channel estimation processes as well as the feedback loop. At the receiver side of user E, in order to distinguish the decoding behavior of the legitimate receiver (user A) and the eavesdropper (user E) and to also show the robustness of two proposed feedback schemes over the conventional schemes without feedback, user E will only use the blind scheme without feedback and the semi-blind scheme without feedback. In case of the semi-blind scheme without feedback, it is also assumed that four pilot symbols are observed by user E.  
Next, we will briefly mention and summarize the joint iterative semi-blind channel estimation and equalization processes and name it the semi-blind feedback scheme.