## 2 Related Work and System Model
### Related Work-Sco\(-\)AN: Perfect Channel Estimation  
SCO\(-\)AN is proposed in [29]. In this section, SCO\(-\)AN is introduced in detail. The goal of physical layer security is to maximize the secrecy capacity of a communication system. In the wireless wiretap communication model, it is not possible to increase the channel capacity of the legitimate receiver. AN is then proposed to reduce the eavesdropper's channel capacity while the legitimate receiver's channel capacity remains intact. Inspired by AN, the secrecy capacity optimization artificial noise (SCO\(-\)AN) isproposed in [29]. Unlike the traditional AN, SCO\(-\)AN has a slight impact on the legitimate receiver's channel capacity but reduces the capacity of eavesdropping channels much more significantly. Hence, the system's overall secrecy capacity increases. Next, we compute the analytical expression of using SCO\(-\)AN, in a manner parallel to our computations of AN above. [PERSON] adds SCO\(-\)AN to the transmission signal:  
\[\mathbf{x}_{k}=\mathbf{w}_{g}+\mathbf{s}_{k}, \tag{8}\]  
In [29], the transmitted signal is \(\mathbf{s}_{k}=\mathbf{p}_{k}\mathbf{u}_{k}\), where \(\mathbf{u}_{k}\) is the information-bearing signal with variance of \(\sigma_{u}^{2}\) and \(\mathbf{p}_{k}\) obeys the Gaussian distribution. \(\mathbf{p}_{k}\) satisfies the following conditions: (a) \(\mathbf{H}_{k}\mathbf{p}_{k}\
eq 1\); and (b)\(\|\mathbf{p}_{k}\|=\)1. \(\mathbf{w}_{g}\in\mathbb{C}^{N_{A}}\) denotes the SCO\(-\)AN. To facilitate calculations, we assume that \(\mathbf{w}_{g}=\mathbf{Z}_{k}\mathbf{v}_{g}\), where \(\mathbf{Z}_{k}\) is a standard orthonormal basis of \(\mathbf{H}_{k}\) and \(\mathbf{v}_{g}\) is a complex random variables with variance \(\sigma_{g}^{2}\). The signals received by the [PERSON] and [PERSON] are:  
\[\mathbf{z}_{k}=\mathbf{H}_{k}\mathbf{s}_{k}+\mathbf{H}_{k}\mathbf{w}_{g}+ \mathbf{n}_{k}, \tag{9}\]  
\[\mathbf{y}_{k}=\mathbf{G}_{k}\mathbf{s}_{k}+\mathbf{G}_{k}\mathbf{w}_{g}+ \mathbf{e}_{k}, \tag{10}\]  
where \(\mathbf{z}_{k}\) denotes the signal received by [PERSON] and \(\mathbf{y}_{k}\) denotes the signal received by [PERSON]. For the SISO wireless communication system, all the elements in (8)-(10) are complex variables. So the lower bound on secrecy capacity after adding SCO\(-\)AN is:  
\[ \begin{split} C_{\text{sec}}^{g}&=I(Z;S)-I(Y;S)\\ &=\text{log}\bigg{(}1+\frac{|H_{k}p_{k}|^{2}\sigma_{u}^{2}}{E|H_{ k}w_{g}|^{2}+\sigma_{u}^{2}}\bigg{)}-\text{log}\bigg{(}1+\frac{|G_{k}p_{k}|^{2} \sigma_{u}^{2}}{E|G_{k}w_{g}|^{2}+\sigma_{e}^{2}}\bigg{)},\end{split}  \tag{11}\]  
where \(\text{E}\big{|}H_{k}w_{g}\big{|}^{2}=(H_{k}Z_{k}Z_{k}^{\dagger}H_{k}^{\dagger}) \sigma_{g}^{2}\), and \(\text{E}\big{|}G_{k}w_{g}\big{|}^{2}=(G_{k}Z_{k}Z_{k}^{\dagger}G_{k}^{\dagger}) \sigma_{g}^{2}\). \(C_{sec}^{g}\) denotes the secrecy capacity after adding SCO\(-\)AN. In (11), \(C_{\text{sec}}^{g}\) is a non-convex function about \(\sigma_{g}^{2}\). For the MIMO wireless communication system, \(\mathbf{H}_{k}\) and \(\mathbf{G}_{k}\) are gaussian complex martixs, \(\mathbf{x}_{k}\), \(\mathbf{w}_{g}\), \(\mathbf{s}_{k}\), \(\mathbf{n}_{k}\) and \(\mathbf{e}_{k}\) are gaussian vectors. So the lower bound on secrecy capacity after adding SCO\(-\)AN is:  
\[ \begin{split} C_{\text{sec}}^{g}&=I(Z;S)-I(Y;S)\\ &=\text{log}\bigg{(}\frac{|(\mathbf{H}_{k}Z_{k}Z_{k}^{\dagger} \mathbf{H}_{k}^{\dagger})\sigma_{g}^{2}+\mathbf{I}\sigma_{u}^{2}+\mathbf{H}_ {k}Z_{k}Z_{k}^{\dagger}\mathbf{H}_{k}^{\dagger}\sigma_{u}^{2}|}{|(\mathbf{H}_ {k}Z_{k}Z_{k}^{\dagger}\mathbf{H}_{k}^{\dagger})\sigma_{g}^{2}+\mathbf{I} \sigma_{u}^{2}|}\bigg{)}\\ &\quad-\text{log}\bigg{(}\frac{|(\mathbf{G}_{k}Z_{k}Z_{k}^{ \dagger}\mathbf{G}_{k}^{\dagger})\sigma_{g}^{2}+\mathbf{I}\sigma_{e}^{2}+ \mathbf{G}_{k}Z_{k}Z_{k}^{\dagger}\mathbf{G}_{k}^{\dagger}\sigma_{u}^{2}|}{|( \mathbf{G}_{k}Z_{k}Z_{k}^{\dagger}\mathbf{G}_{k}^{\dagger})\sigma_{g}^{2}+ \mathbf{I}\sigma_{e}^{2}|}\bigg{)}\end{split}  \tag{12}\]  
(12) is a function of \(\sigma_{g}^{2}\).