# Secure Information Transmission with Self Jamming SWIPT
## 2 System Model
Furthermore, let us denote the channel correlation between between \(\mathbf{H}_{E2\xi}\) and \(\mathbf{H}_{E2R}\) as \(\rho_{E2}\). \(\rho_{E1}\) and \(\rho_{E2}\) are expressed in Appendix B. The error due to \(\rho_{E1}\) and \(\rho_{E2}\) is denoted as \(\epsilon_{\rho,E1}\) and \(\epsilon_{\rho,E2}\), respectively. The channel precodings at \(E1\) and \(E2\) from the perspective of \(\xi(b)\), respectively, are given as:  
\[X_{E1}^{\prime}(1+\epsilon_{\rho,E2}) =(X_{E1}^{(k)}+\epsilon_{A,E1}^{(k)})(1+\epsilon_{\rho,E2})=A_{E1} ^{(k)}(H_{E2R}^{(k)}+\epsilon_{E2R}^{(k)})(1+\epsilon_{\rho,E2}), \tag{9}\] \[X_{E2}^{\prime}(1+\epsilon_{\rho,E1}) =(X_{E2}^{(k)}+\epsilon_{A,E2}^{(k)})(1+\epsilon_{\rho,E1})=A_{E2} ^{(k)}(H_{E1R}^{(k)}+\epsilon_{E1R}^{(k)})(1+\epsilon_{\rho,E1}).\]  
From the perspective of \(\xi\), \((1+\epsilon_{\rho,E2})\) and \((1+\epsilon_{\rho,E2})\) are the impacts on \(X_{E1}^{\prime}\) and \(X_{E2}^{\prime}\), respectively, which degrade the precoded EH signals and increase the precoding errors. The received signal at \(\xi(b)\) is given as:  
\[Y_{\xi(b)}^{(k)}= \begin{cases}H_{T\xi}^{(k)}X_{I}^{(k)}+J^{(k)}+ \epsilon_{\xi(b),E1}^{(k)}+\epsilon_{\xi(b),E2}^{(k)};&\text{if }\mathcal{L}( \mathcal{V}^{(k)})=\epsilon_{\xi(b)}^{(k)},\\ H_{T\xi}^{(k)}X_{I}^{(k)}+J^{(k)}+H_{E1R}^{(k)}X_{E1}^{\prime}(1+\epsilon_{\rho,E2})+H_{E2R}^{(k)}X_{E2}^{\prime}(1+\epsilon_{\rho,E1});&\text{otherwise}.\end{cases}  \tag{10}\]where \(\epsilon^{(k)}_{\xi(b),E1}=\frac{H^{(k)}_{\Gamma k}(X^{(k)}_{E1}+\epsilon^{(k)}_{ \epsilon,E1})(1+\epsilon_{\rho,E2})}{X^{(k)}_{E2}}\), \(\epsilon^{(k)}_{\xi(b),E2}=\frac{H^{(k)}_{\Gamma k}(X^{(k)}_{E2}+\epsilon^{(k)}_ {\epsilon,E2})(1+\epsilon_{\rho,E2})}{X^{(k)}_{E1}}\), and \(\epsilon^{(k)}_{\xi(b),E1}\) and \(\epsilon^{(k)}_{\xi(b),E2}\) are the additional noise components to \(Y^{(k)}_{\xi(b)}\). Then, \(\mathcal{L}(H^{(k)}_{E1\xi}X^{(k)}_{E1}+H^{(k)}_{E2\xi}X^{(k)}_{E2})=\epsilon^ {(k)}_{\xi(b)}\), where \(\epsilon^{(k)}_{\xi(b)}=\epsilon^{(k)}_{\xi(b),E1}+\epsilon^{(k)}_{\xi(b),E2}\). The SINR of \(\mathbf{Y}_{\xi(b)}\) for the condition \(\mathcal{L}(\mathcal{V}^{(k)})=\epsilon^{(k)}_{\xi(b)}\) is given as:  
\[\Gamma_{\xi(b)}=\frac{\mathbb{E}[|\mathbf{X}_{I}^{T}\mathbf{H}_{T\xi}|^{2}]}{ \mathbb{E}[(\sum_{k=0}^{K-1}|\epsilon^{(k)}_{\xi(b),E1}|^{2})]+\mathbb{E}[( \sum_{k=0}^{K-1}|\epsilon^{(k)}_{\xi(b),E2}|^{2})]+\mathbb{E}[||\mathbf{J}||^ {2}]}=\frac{\sigma^{2}_{X,I}\sigma^{2}_{H,T\xi}}{\sigma^{2}_{\epsilon,\xi(b), E1}+\sigma^{2}_{\epsilon,\xi(b),E2}+\sigma^{2}_{\gamma}}. \tag{11}\]  
The best scenario for \(\xi\) is \(\xi(c)\), where, the precoding errors due to the channel estimation errors can be avoided at \(\mathcal{T}\). However, there will be a precoding error due to \(H^{(k)}_{E1R}(1+\epsilon_{\rho,E1})\) and \(H^{(k)}_{E2R}(1+\epsilon_{\rho,E2})\). Hence, the precoded EH signals can be written as:  
\[ \begin{split}& X^{(k)}_{E1}(1+\epsilon_{\rho,E2})=A^{(k)}_{E1}H^{(k)} _{E2R}(1+\epsilon_{\rho,E2}),\\ & X^{(k)}_{E2}(1+\epsilon_{\rho,E1})=A^{(k)}_{E2}H^{(k)}_{E1R}(1+ \epsilon_{\rho,E1}).\end{split}  \tag{12}\]  
Then, the received signal at \(\xi(c)\) is given as:  
\[Y^{(k)}_{\xi(c)}= \begin{cases}H^{(k)}_{T\xi}X^{(k)}_{I}+N^{(k)}+\epsilon^{(k) }_{\xi(c),E1}+\epsilon^{(k)}_{\xi(c),E2};&\text{if }\mathcal{L}(\mathcal{V}^{(k)})= \epsilon^{(k)}_{\xi(c)},\\ H^{(k)}_{T\xi}X^{(k)}_{I}+N^{(k)}+H^{(k)}_{E1R}X^{(k)}_{E1}(1+\epsilon_{\rho,E2})+H^{(k)}_{E2R}X^{(k)}_{E2}(1+\epsilon_{\rho,E1});&\text{otherwise}, \end{cases}  \tag{13}\]  
where \(\epsilon^{(k)}_{\xi(c),E1}=H^{(k)}_{E1R}(1+\epsilon_{\rho,E2})\) and \(\epsilon^{(k)}_{\xi(c),E2}=H^{(k)}_{E2R}(1+\epsilon_{\rho,E1})\). Then, \(\mathcal{L}(H^{(k)}_{E1\xi}X^{(k)}_{E1}+H^{(k)}_{E2\xi}X^{(k)}_{E2})=\epsilon^ {(k)}_{\xi(c)}\), where \(\epsilon^{(k)}_{\xi(c)}=\epsilon^{(k)}_{\xi(c),E1}+\epsilon^{(k)}_{\xi(c),E2}\).