## 2 Related Work and System Model
### Sco\(-\)AN: Imperfect Channel Estimation  
The Gaussian white noise causes the error of channel estimation. The effect of the imperfect channel estimation should be considered.  
For the SISO communication system, \(H_{\omega o}\) denotes channel estimation error. The channel state information received by [PERSON] is \(\widetilde{H}\):  
\[H_{k}=H_{\omega}+\widetilde{H}, \tag{14}\]  
The signal received by [PERSON] after adding SCO\(-\)AN is:  
\[z_{k}^{*}=(H_{\omega}+\widetilde{H})s_{k}+(H_{\omega}+\widetilde{H})w_{g}+n_{ k}, \tag{15}\]  
For MIMO communication system, \(\mathbf{H}_{\omega}\) denotes channel estimation error. The channel state information received by [PERSON] is \(\widetilde{\mathbf{H}}\) :  
\[\mathbf{H}_{k}=\mathbf{H}_{\omega o}+\widetilde{\mathbf{H}}, \tag{16}\]  
The signal received by [PERSON] after adding SCO\(-\)AN is:  
\[\mathbf{z}_{k}^{*}=(\mathbf{H}_{\omega o}+\widetilde{\mathbf{H}})\mathbf{s}_{ k}+(\mathbf{H}_{\omega o}+\widetilde{\mathbf{H}})\mathbf{w}_{g}+\mathbf{n}_{k}, \tag{17}\]  
We assume that the channel estimation of \(\mathbf{G}\) is perfect.  
For the SISO communication system, \(H_{\omega o}\), \(\widetilde{H}\), and \(Z_{k}\) are independent. Therefore, \(|H_{\omega o}Z_{k}|^{2}=|H_{\omega o}|^{2}|Z_{k}|^{2}\), \(\left|\widetilde{H}Z_{k}\right|^{2}=\left|\widetilde{H}\right|^{2}\left|Z_{k}\right| ^{2}\). The lower bound on secrecy capacity after adding SCO\(-\)AN under imperfect channel estimation is:  
Figure 2: Secrecy capacity comparison of the AN and SCO\(-\)AN versus different SNR.  
\[\begin{split} C^{S}_{\text{sec},\rho o}&=I(Z;S)-I(Y;S)\\ &=\text{log}\bigg{(}1+\frac{\left|\dot{H}_{p_{k}}\right|^{2}\sigma _{n}^{2}}{\sigma_{n}^{2}+\left|H_{o}p_{k}\right|^{2}\sigma_{n}^{2}+\left|\dot{H }_{n}\right|^{2}+\left|\dot{H}_{o}w_{g}\right|^{2}}\bigg{)}-\text{log}\bigg{(} 1+\frac{\left|\dot{G}_{k}p_{k}\right|^{2}\sigma_{n}^{2}}{\left|\dot{G}_{k} \right|^{2}\sigma_{n}^{2}+\sigma_{n}^{2}}\bigg{)}\\ &=\text{log}\bigg{(}1+\frac{\left|\dot{H}_{p_{k}}\right|^{2} \sigma_{n}^{2}}{\sigma_{n}^{2}+\left|H_{o}p_{k}\right|^{2}\sigma_{n}^{2}+\left| \dot{H}_{2}\right|^{2}\sigma_{n}^{2}+\left|H_{o}^{2}\chi_{k}\right|^{2}\sigma _{n}^{2}}\bigg{)}-\text{log}\bigg{(}1+\frac{\left|\dot{G}_{k}p_{k}\right|^{2} \sigma_{n}^{2}}{\left|\dot{G}_{k}\right|^{2}\sigma_{n}^{2}+\sigma_{n}^{2}} \bigg{)}\\ &=\text{log}\bigg{(}1+\frac{\left|\dot{H}_{p_{k}}\right|^{2} \sigma_{n}^{2}}{\sigma_{n}^{2}+\left|H_{o}p_{k}\right|^{2}\sigma_{n}^{2}+ \left|\dot{H}\right|^{2}\left|\dot{\chi}_{k}\right|^{2}\sigma_{n}^{2}+\left|H _{o}\right|^{2}\left|\dot{\chi}_{k}\right|^{2}\sigma_{n}^{2}}\bigg{)}-\text{ log}\bigg{(}1+\frac{\left|\dot{G}_{k}p_{k}\right|^{2}\sigma_{n}^{2}}{\left| \dot{G}_{k}\right|^{2}\left|\dot{\chi}_{k}\right|^{2}\sigma_{n}^{2}+\sigma_{n }^{2}}\bigg{)}\end{split} \tag{18}\]  
In (18), we see that the channel estimation error will affect the channel capacity of the legitimate channel. Meanwhile, the secrecy capacity of the wireless communication system is reduced.  
For the MIMO system, the lower bound on secrecy capacity after adding SCO\(-\)AN under imperfect channel estimation is:  
\[\begin{split} C^{S}_{\text{sec},\rho o}&=I(Z;S)-I(Y; S)\\ &=\text{log}\Bigg{(}\frac{\left|\mathbf{K}_{H}+(\mathbf{H}_{eo}+ \tilde{\mathbf{H}})\mathbf{Z}_{k}\mathbf{Z}_{k}^{\dagger}(\mathbf{H}_{eo}+ \tilde{\mathbf{H}})^{\dagger}\sigma_{u}^{2}\right|}{\left|\mathbf{K}_{H} \right|}\Bigg{)}-\text{log}\Bigg{(}\frac{\left|\mathbf{K}_{G}+\mathbf{G}_{k} \mathbf{Z}_{k}\mathbf{Z}_{k}^{\dagger}\mathbf{G}_{k}^{\dagger}\sigma_{u}^{2} \right|}{\left|\mathbf{K}_{G}\right|}\Bigg{)}\\ &\qquad\qquad\text{In (\ref{eq:C_H_eo}), }\mathbf{K}_{H}=((\mathbf{H}_{eo}+\tilde{\mathbf{H}})\mathbf{Z}_{k}\mathbf{Z}_ {k}^{\dagger}(\mathbf{H}_{eo}+\tilde{\mathbf{H}})^{\dagger})\sigma_{g}^{2}+ \mathbf{I}\sigma_{n}^{2}\text{ and }\mathbf{K}_{G}=(\mathbf{G}_{k}\mathbf{Z}_{k}\mathbf{Z}_{k}^{ \dagger}\mathbf{G}_{k}^{\dagger})\sigma_{g}^{2}+\mathbf{I}\sigma_{e}^{2}.\end{split} \tag{19}\]