# Energy Efficient Secure Communication Model against Cooperative Eavesdropper
## 2 System Model
### System Model Equations  
The relationship between transmit power of jamming signal and information symbols is given as  
\[ \begin{split} X_{I}=S_{I}P_{I};\ \ \mathbf{X}_{I}=\mathbf{S}_{I}P_{I}; \\ \beta=\frac{T_{I}P_{I}}{P_{I}}.\end{split}  \tag{1}\]  
The unamplified version of \(\mathbf{X}_{I}\) and \(X_{J}\) are denoted as \(S_{I}\) and \(S_{J}\), respectively. \(\mathbf{S}_{I}=[S_{I1}S_{I2}...S_{I\overline{I}_{J}}]^{T}\), where \(S_{I1}\), \(S_{I1}...S_{I\overline{I}_{J}}\) are the unamplified version of modulated signals (\(X_{I1}X_{I2}...X_{I\overline{I}_{J}}\)) transmitted from J. The transmit powers of \(\mathbf{X}_{I}\) and \(\mathbf{X}_{J}\) are denoted as \(P_{I}\) and \(T_{I}P_{I}\), respectively, where \(P_{I}\) is the transmit power of single jamming signal stream and all the jamming signal streams are considered to have equal expected values. The ratio between the total transmit power of the Jamming signal and information signal is denoted as \(\beta\). The relationship between transmit SNR at A and J is given as  
\[ \begin{split}\gamma_{A}=\frac{\mathbb{E}[|X_{I}|^{2}]}{\mathbb{E }[|N_{E1}|^{2}]}=\frac{\sigma_{X,I}^{2}}{\sigma_{N,A}^{2}}=\frac{P_{I}}{ \sigma_{N,A}^{2}};\\ \gamma_{I}=\frac{\mathbb{E}[|\mathbf{X}_{I}^{T}|^{2}]/T_{I}}{ \mathbb{E}[|N_{E1}|^{2}]}\frac{\sigma_{X,I}^{2}/T_{I}}{\sigma_{N,J}^{2}}= \frac{P_{I}}{\sigma_{N,J}^{2}};\\ \frac{\gamma_{I}T_{I}}{\gamma_{A}}=\beta=\frac{T_{I}P_{I}}{P_{I}},\end{split}  \tag{2}\]  
where \(\sigma_{N,A}^{2}=\sigma_{N,J}^{2}\). The received power of \(\mathbf{X}_{I}\) and \(X_{I}\) at E1 and E2 are, respectively, given in the following equations and for the sake of simplicity, only the path loss factor is considered and the channel fading co-efficient is neglected. The received power of \(\mathbf{X}_{J}\) and \(X_{I}\) at E1 are denoted as \(T_{I}P_{I,E1}\) and \(P_{I,E1}\), respectively. The received power of \(\mathbf{X}_{I}\) and \(X_{I}\) at E2 are denoted as \(P_{I,E2}\) and \(P_{I,E2}\), respectively. Then, the relationships between transmit power and received power at E1 and E2 are, respectively, given as  
\[ \begin{split} P_{J,E1}=P_{J}{}^{-D_{J,E1}};\ \ P_{I,E1}=P_{I}{}^{-D_{A,E1}};\ \ \beta_{E1}=\frac{T_{I}P_{I,E1}}{P_{I,E1}},\\ P_{J,E2}=P_{J}{}^{-D_{J,E2}};\ \ P_{I,E2}=P_{I}{}^{-D_{A,E2}};\ \ \beta_{E2}=\frac{T_{I}P_{I,E2}}{P_{I,E2}}, \end{split}  \tag{3}\]  
where \(\beta_{E1}\) and \(\beta_{E2}\) are denoted as the ratio between the total received power of jamming signal and information signal at E1 and E2, respectively. The received signal at B is given as  
\[Y_{R}= \begin{cases}H_{AB}X_{I}+N_{B};&\text{if }\mathcal{L}(\mathcal{V})=0,\\ H_{AB}X_{I}+N_{B}+\mathbf{H}_{JB}X_{J};&\text{otherwise},\end{cases}  \tag{4}\]  
where \(\mathcal{L}(\mathcal{V})=0\) denotes the null space vector of the precoded jamming signal for the respective channel link. The received signal at E1 is  
\[Y_{E1}=H_{AE1}X_{I}+\mathbf{H}_{JE1}\mathbf{X}_{J}+N_{E1}, \tag{5}\]  
where \(\mathbf{H}_{JE1}\mathbf{X}_{J1}\) is considered as a noise term. The channel estimate error at E1 is given as \(\varepsilon_{H,AE1}=f_{MSE}\{\tilde{\mathbf{H}}_{AE1}-\mathbf{H}_{AE1}\}\), where \(f_{MSE}(x)\) represents the expected minimum meansquare error function. In similar way, the channel estimates and the channel estimate errors of other channels are denoted. The SINR of \(Y_{E1}\) are, respectively, given as  
\[ \begin{split}\Gamma_{E1}&=\frac{\mathbb{E}[|H_{AE1}X_{ I}|^{2}]}{\varepsilon_{H,AE1}\mathbb{E}[|X_{I}|^{2}]+\mathbb{E}[|\mathbf{H}_{ JE1}\mathbf{X}_{I}^{T}|\mathbf{|}|^{2}]+\mathbb{E}[|N_{E1}|^{2}]}\\ &=\frac{\sigma_{X,I}^{2}\sigma_{H,AE1}^{2}}{\varepsilon_{H,AE1} \sigma_{X,I}^{2}+\sigma_{X,I}^{2}\sigma_{H,JE1}^{2}+\sigma_{N,E1}^{2}}\\ &=\frac{P_{I,E1}}{(\varepsilon_{H,AE1}+\beta_{E1})P_{I,E1}+ \sigma_{N,E1}^{2}}.\end{split}  \tag{6}\]  
For the sake of simplicity, \(\sigma_{H,AE1}^{2}\) and \(\sigma_{H,JE1}^{2}\) are set to \(1\).