# Energy Efficient Secure Communication Model against Cooperative Eavesdropper
## 2 System Model
### Decoding Information by Using Jamming Signal Estimate  
The information signal can be decoded at E1 by following the next steps. 1. The information signal estimate and estimate error are given as \(\tilde{\mathbf{X}}_{I}\) and \(\varepsilon_{X,I}\), respectively. Estimate the jamming signal at E2, by using \(\varepsilon_{X,I}\), \(\tilde{\mathbf{H}}_{AE2}\) and \(\tilde{\mathbf{H}}_{JE2}\) in (11). 2. The jamming signal estimate and estimate error are given as \(\tilde{\mathbf{X}}_{J}\) and \(\varepsilon_{X,J}\), respectively. Estimate the information signal at E1, by using \(\varepsilon_{X,J}\), \(\tilde{\mathbf{H}}_{AE1}\) and \(\tilde{\mathbf{H}}_{JE1}\) in (11). The SINR of jamming signal at E2 can be improved by using \(\tilde{\mathbf{H}}_{AE2}\) and (8) can be written as  
\[ \begin{split}\Gamma_{E2}&=\frac{\mathbb{E}[|| \mathbf{H}_{JE2}\mathbf{X}_{I}^{T}||^{2}]/T_{I}}{\varepsilon_{H,JE2}\mathbb{E}[ ||\mathbf{X}_{J}^{T}||^{2}]+\epsilon_{H,AE2}\mathbb{E}[||X_{I}||^{2}]+ \mathbb{E}[|N_{E1}|^{2}]}\\ &=\frac{P_{J,E2}}{T_{J}P_{J,E2}(\varepsilon_{H,JE2}+\frac{ \varepsilon_{H,AE2}+\varepsilon_{X,J}}{\varepsilon_{E2}})+\sigma_{N,E2}^{2}}. \end{split}  \tag{13}\]  
The SINR of the information signal at E1 can be improved by using \(\tilde{\mathbf{H}}_{AE1}\), and (6) can be written as  
\[ \begin{split}\Gamma_{E1}&=\frac{\mathbb{E}[|H_{AE1} X_{I}|^{2}]}{\varepsilon_{H,AE1}\mathbb{E}[||X_{I}||^{2}]+(\varepsilon_{H, JE1}+\varepsilon_{X,J})\mathbb{E}[||\mathbf{X}_{J}^{T}||^{2}]+\mathbb{E}[|N_{E1}|^{2}]} \\ &=\frac{P_{L,E1}}{P_{L,E1}(\varepsilon_{H,AE1}+(\varepsilon_{H, JE1}+\varepsilon_{X,J})\beta_{E1})+\sigma_{N,E1}^{2}}\\ &=\frac{T_{J}P_{J,E1}}{T_{J}P_{J,E1}(\varepsilon_{H,AE1}+( \varepsilon_{H,JE1}+\varepsilon_{X,J})\beta_{E1})+\sigma_{N,E1}^{2}}.\end{split}  \tag{14}\]  
From the legitimate users' perspective for the given scenario, as in Figure 1, where E1 is located in such a way that reduces the interference of \(\mathbf{X}_{J}\) in the best possible way at E1 and E2 is located in such a way that reduces the interference of \(X_{I}\) in the best possible way at E2. If [PERSON] reduces the transmit power of jamming signal to reduce the SINR at E2, then this will reduce the interference of \(\mathbf{X}_{J}\) at E1 but, at the same time, the quality of jamming signal estimate will reduce at E2. Thus, [PERSON] needs to find an optimum transmit power for the jamming signal. The jammer can take counter measures against the active cooperative eavesdropper by adjusting \(\beta\). The desired value of \(\beta_{E1}\) and \(\beta_{E2}\) for the legitimate users is \(\beta_{E1}\geq 1\) and \(\beta_{E2}\leq 1\), respectively, but this desired condition is not feasible, since received power will change for eavesdroppers at different locations. If \(\beta_{E2}=1\) or \(\beta_{E2}<1\), then from (8) and (13), we can understand that the SINR will be below the threshold to estimate \(\mathbf{X}_{J}\). Even though \(\beta_{E2}=1\) or \(\beta_{E2}<1\) is the desired condition for countering E2 from estimating \(\mathbf{X}_{J}\), and this condition will eventually allow E1 to estimate \(X_{I}\), even without estimating \(\mathbf{X}_{J}\), since the interference of \(\mathbf{X}_{J}\) is minimal. Therefore, J should keep \(\beta_{E2}>1\) and \(\beta_{E2}<1\) and should also maintain the best possible balanced \(\mathbf{X}_{J}\) interference at E1 and \(X_{I}\) at E2. One of the main advantages of J is that two jamming signals combined to form the artificial noise interference for eavesdroppers, but E2 should estimate both the jamming signals individually, and this is evident from (8) and (13). From (2), it is evident that [PERSON] can use two jamming signals with approximately equal power to create artificial noise. Contrary to J, this adversely affects E2 estimation of \(\mathbf{X}_{J}\), the SNR of \(X_{J1}\) and \(X_{J2}\) as an independent signal stream is half the SNR of \(\mathbf{X}_{J}\).