# Secure Information Transmission with Self Jamming SWIPT
## 2 System Model
Therefore, due to precoding, the SINR of \(\mathbf{Y}_{R(a)}\) at \(R(a)\) does not experience any interference from EH signals, which is given as:  
\[\Gamma_{R(a)}=\frac{\mathbb{E}[|\mathbf{X}_{I}^{T}\mathbf{H}_{TR}|^{2}]}{ \mathbb{E}[||\mathbf{N}||^{2}]}=\frac{\sigma_{X,I}^{2}\sigma_{H,TR}^{2}}{ \sigma_{N}^{2}}, \tag{4}\]  
where \(\mathbb{E}[|\mathbf{X}_{I}|^{2}]=\sigma_{X,I}^{2}\). Thus, under the perfect channel condition, the EH signal does not affect the performance of the legitimate receiver \(R\). Irrespective of its location, \(\xi\) receives the signal with low SINR due to the EH signal interference. The received signal at \(\xi(a)\) is given as:  
\[Y_{\xi}^{(k)}(a)=H_{T\xi}^{(k)}X_{I}^{(k)}+H_{E1\xi}^{(k)}X_{E1}^{(k)}+H_{E2 \xi}^{(k)}X_{E2}^{(k)}+J^{(k)}, \tag{5}\]  
where \(Y_{\xi}^{(k)}(a)\) is the PLNC signal, \(H_{T\xi}^{(k)}X_{I}^{(k)}\) is the useful information component, and the EH signal components (\(H_{E1\xi}^{(k)}X_{  
Note that the performance of the legitimate user is dependent on the perfect channel condition, which allows \(E1\) and \(E2\) to perform signal precoding using CSI. With the imperfect channel condition, the performance of \(R\) is dependent on a robust channel estimation technique as in [2,4]. The channel estimates of \(\mathbf{H}_{E1R}\) and \(\mathbf{H}_{E2R}\) are obtained by using training symbols from \(R\) to \(E1\) and \(E2\), respectively. The channel estimates of \(H_{E1R}^{(k)}\) and \(H_{E2R}^{(k)}\), respectively, are given as \(\hat{H}_{E1R}^{(k)}=H_{E1R}^{(k)}+\epsilon_{E1R}^{(k)}\) and \(\hat{H}_{E2R}^{(k)}=H_{E2R}^{(k)}+\epsilon_{E1R}^{(k)}\). Here, \(\epsilon_{E1R}^{(k)}\) and \(\epsilon_{E2R}^{(k)}\) are the channel estimation errors of \(H_{E1R}^{(k)}\) and \(H_{E2R}^{(k)}\), respectively. Then, the new precoded EH signals denoted as \(X_{E1}^{\prime}\) and \(X_{E2}^{\prime}\) are expressed in Appendix A.2. \(\epsilon_{A,E1}^{(k)}\) and \(\epsilon_{A,E2}^{(k)}\) are the precoding errors due to channel estimation error at \(\epsilon_{E2R}^{(k)}A_{E1}^{(k)}\) and \(\epsilon_{E1R}^{(k)}A_{E2}^{(k)}\), respectively. Similarly, the received signal at \(R(b)\) based on the impact of \(\hat{H}_{E1R}^{(k)}\) and \(\hat{H}_{E2R}^{(k)}\) is given as:  
\[Y_{R(b)}^{(k)}= \begin{cases}H_{TR}^{(k)}X_{I}^{(k)}+N^{(k)}+ \epsilon_{R(b),E1}^{(k)}+\epsilon_{R(b),E2}^{(k)};&\text{if }\mathcal{L}( \mathcal{V}^{(k)})=\epsilon_{R(b)}^{(k)},\\ H_{TR}^{(k)}X_{I}^{(k)}+N^{(k)}+H_{E1R}^{(k)}X_{E1}^{\prime}+H_{E2R}^{(k)}X_{ E2}^{\prime};&\text{otherwise},\end{cases}  \tag{7}\]  
where \(\epsilon_{R(b),E1}^{(k)}=H_{E1R}^{(k)}\epsilon_{A,E2}^{(k)}\), \(\epsilon_{R(b),E2}^{(k)}=H_{E2R}^{(k)}\epsilon_{A,E1}^{(k)}\), and \(\epsilon_{R(b),E1}^{(k)}\) and \(\epsilon_{R(b),E2}^{(k)}\) are the additional noise components of \(Y_{R(b)}^{(k)}\) due to precoding error. Then, \(\mathcal{L}(H_{E1R}^{(k)}X_{E1}^{(k)}+H_{E2R}^{(k)}X_{E2}^{(k)})=\epsilon_{R(b )}^{(k)}\), where \(\epsilon_{R(b)}^{(k)}=\epsilon_{R(b),E1}^{(k)}+\epsilon_{R(b),E2}^{(k)}\). The SINR of \(\mathbf{Y}_{R(b)}\) for the condition \(\mathcal{L}(\mathcal{V}^{(k)})=\epsilon_{R(b)}^{(k)}\) is given as:  
\[\Gamma_{R(b)}=\frac{\mathbb{E}[|\mathbf{X}_{I}^{T}\mathbf{H}_{TR}|^{2}]}{ \mathbb{E}[(\sum_{k=0}^{K-1}|\epsilon_{R(b),E1}^{(k)}|)^{2}]+\mathbb{E}[(\sum _{k=0}^{K-1}|\epsilon_{R(b),E2}^{(k)}|)^{2}]+\mathbb{E}[||\mathbf{N}||^{2}]} =\frac{\sigma_{X,I}^{2}\sigma_{H,TR}^{2}}{\sigma_{\epsilon,R(b),E1}^{2}+ \sigma_{\epsilon,R(b),E2}^{2}+\sigma_{N}^{2}}. \tag{8}\]  
Note that, since \(R\) does not need to equalize \(H_{E1R}^{(k)}\) and \(H_{E2R}^{(k)}\), their respective channel estimation errors \(\epsilon_{E1R}^{(k)}\) and \(\epsilon_{E2R}^{(k)}\) are neglected in Equation (8). The performance of \(\xi(b)\) is better than \(\xi(a)\) due to high channel correlations between \(\mathbf{H}_{E1\xi}\) and \(\mathbf{H}_{E1R}\) and also between \(\mathbf{H}_{E2\xi}\) and \(\mathbf{H}_{E2R}\). Let us denote the channel correlation between \(\mathbf{H}_{E1\xi}\) and \(\mathbf{H}_{E1R}\) as \(\rho_{E1}\).