# Feedback Approach for the Relay Channel with Noisy Feedback and Its Security Analysis
## Appendix B Proof of Theorem 2  
### Channel Transformation Strategy  
**Pre-coding strategy for the feedforward MISO channels:** At time instant \(n\)\((1\leq n\leq N)\), the output signal received by Destination is given in (6). Let  
\[ \begin{split}\mathbf{X}_{n}=\frac{\mathbf{h}_{\text{sd}}^{\mathcal{H}} }{\|\mathbf{h}_{\text{sd}}\|}X_{n}^{{}^{\prime}}&\Longrightarrow X _{n}^{{}^{\prime}}=\frac{\mathbf{h}_{\text{sd}}}{\|\mathbf{h}_{\text{sd}}\|} \mathbf{X}_{n},\\ X_{r,n}=\frac{h_{rd}{}^{\mathcal{H}}}{|h_{rd}|}{X_{r,n}}^{{}^{ \prime}}&\Longrightarrow{X_{r,n}}^{{}^{\prime}}=\frac{h_{rd}}{|h_ {rd}|}{X_{r,n}},\end{split}  \tag{22}\]  
where \(\mathbf{h}_{\text{sd}}\in\mathbb{C}^{1\times B},h_{rd}\in\mathbb{C}^{1\times 1 },X_{n}^{{}^{\prime}}\in\mathbb{C}^{1\times 1},{X_{r,n}}^{{}^{\prime}}\in \mathbb{C}^{1\times 1}\), and  
\[ \begin{split} E\Big{[}\mathbf{X}_{n}^{\mathcal{H}}\mathbf{X}_{n}\Big{]}& =E\Bigg{[}\frac{X_{n}^{\prime\mathcal{H}}\mathbf{h}_{\text{sd}}}{ \|\mathbf{h}_{\text{sd}}\|}\frac{\mathbf{h}_{\text{sd}}^{\mathcal{H}}X_{n}^{ \prime}}{\|\mathbf{h}_{\text{sd}}\|}\Bigg{]}=E\Big{[}|X_{n}^{{}^{\prime}}|^{2} \Big{]}=P,\\ E\Big{[}{X_{r,n}}^{{}^{\prime}}{X_{r,n}}^{{}^{\prime}}{}^{\prime \mathcal{H}}\Big{]}&=E\Bigg{[}\frac{h_{rd}X_{r,n}}{|h_{rd}|}\cdot \frac{X_{r,n}^{\mathcal{H}}h_{\text{rd}}^{\mathcal{H}}}{|h_{rd}|}\Bigg{]}=E \Big{[}{X_{r,n}}{X_{r,n}^{\mathcal{H}}}\Big{]}=P_{r},\end{split}  \tag{23}\]  
which indicates that the power constraint of \(X_{n}^{{}^{\prime}}\) and \({X_{r,n}}^{{}^{\prime}}\) are exactly the same as that of \(\mathbf{X}_{n},X_{r,n}\), respectively. Therefore, by substituting (22) into (6), the signal received by Destination can be further expressed as  
\[ \begin{split}& Y_{n}=\mathbf{h}_{\text{sd}}\mathbf{X}_{n}+h_{rd}X_{r,n }+\eta_{n},\\ \Rightarrow& Y_{n}=\mathbf{h}_{\text{sd}}\frac{ \mathbf{h}_{\text{sd}}^{\mathcal{H}}}{\|\mathbf{h}_{\text{sd}}\|}{X_{n}^{{}^{ \prime}}+h_{rd}\frac{h_{rd}{}^{\mathcal{H}}}{|h_{rd}|}{X_{r,n}}^{{}^{\prime}}+ \eta_{n},\\ \Rightarrow& Y_{n}=\|\mathbf{h}_{\text{sd}}\|X_{n}^{{} ^{\prime}}+|h_{rd}|{X_{r,n}}^{{}^{\prime}}+\eta_{n}.\end{split}  \tag{24}\]  
Specifically, the link from Source to Relay is also a MISO channel, and the messages sent by Source and Relay are related. So, at time instant \(n\)\((1\leq n\leq N)\), the signal received by Relay can be further expressed as  
\[ \begin{split}& Y_{r,n}=\mathbf{h}_{\text{sr}}\mathbf{X}_{n}+\eta_{r,n }=\mathbf{h}_{\text{sr}}\frac{\mathbf{h}_{\text{sd}}^{\mathcal{H}}}{\| \mathbf{h}_{\text{sd}}\|}X_{n}^{{}^{\prime}}+\eta_{r,n},\\ \Rightarrow&\frac{\mathbf{h}_{\text{sd}}\mathbf{h} _{\text{sr}}^{\mathcal{H}}}{\|\mathbf{h}_{\text{sd}}\|\cdot\|\mathbf{h}_{\text {sr}}\|}Y_{r,n}=\|\mathbf{h}_{\text{sr}}\|X_{n}^{{}^{\prime}}+\frac{\mathbf{h }_{\text{sd}}\mathbf{h}_{\text{sr}}^{\mathcal{H}}}{\|\mathbf{h}_{\text{sd}}\| \cdot\|\mathbf{h}_{\text{sr}}\|}\eta_{r,n},\\ \Rightarrow&{Y_{r,n}}^{{}^{\prime}}=\|\mathbf{h}_{ \text{sr}}\|X_{n}^{{}^{\prime}}+\eta_{r,n}^{{}^{\prime}},\end{split}  \tag{25}\]  
where \(\mathbf{h}_{\text{sr}}\in\mathbb{C}^{1\times B},\mathbf{h}_{\text{sd}}\in \mathbb{C}^{1\times B},X_{n}^{{}^{\prime}}\in\mathbb{C}^{1\times 1}\), and  
\[Y_{r,n}^{{}^{\prime}}=\frac{\mathbf{h}_{\text{sd}}\mathbf{h}_{\text{sr}}^{ \mathcal{H}}}{\|\mathbf{h}_{\text{sd}}\|\cdot\|\mathbf{h}_{\text{sr}}\|}Y_{r,n },\quad\eta_{r,n}^{{}^{\prime}}=\frac{\mathbf{h}_{\text{sd}}\mathbf{h}_{\text{ sr}}^{\mathcal{H}}}{\|\mathbf{h}_{\text{sd}}\|\cdot\|\mathbf{h}_{\text{sr}}\|} \eta_{r,n}, \tag{26}\]  
\(Y_{r,n}^{{}^{\prime}}\in\mathbb{C}^{1\times 1},\eta_{r,n}^{{}^{\prime}}\in\mathbb{C}^{1\times 1 }\sim\mathcal{CN}(0,\sigma_{r}^{2})\). It indicates that the complex-valued feedforward MISO FRC can be transformed into a complex-valued single-input single-output (SISO) FRC case. **Beamforming strategy for the feedback SIMO channel:** At time \(n\)\((1\leq n\leq N-1)\), the signal received by Source (see (6)) can be proceeded as  
\[