# Feedback Approach for the Relay Channel with Noisy Feedback and Its Security Analysis
## Appendix B Proof of Theorem 2
### Coefficient's Derivation and Error Probability Analysis  
The error events of the SK-type FBL coding scheme for the MISO FRC with noisy feedback described in Appendix B.4 consist of  
1. A modulo-aliasing error occurs in Source at time instant \(n(1\leq n\leq N-1)\). The event is defined as \[C_{n}\triangleq\Big{\{}\gamma_{n}\varepsilon_{n}+\eta^{{}^{\prime}}_{f,n} \
otin\Lambda\Big{\}},\] (103) where \(\varepsilon_{n}\) and \(\eta^{{}^{\prime}}_{f,n}\) are both complex numbers, so (103) can be further expressed as \[ \begin{split} C_{n}\triangleq&\bigg{\{}\gamma_{n} \varepsilon_{R,n}+\eta^{{}^{\prime}}_{fR,n}\
otin\left[-\frac{d}{2},\frac{d}{2 }\right)\bigg{\}}\bigcup\\ &\bigg{\{}\gamma_{n}\varepsilon_{I,n}+\eta^{{}^{\prime}}_{fI,n} \
otin\left[-\frac{d}{2},\frac{d}{2}\right)\bigg{\}},\end{split} \] (104) where \(d=\sqrt{6P_{f}}\) and the subscripts \(R\) and \(I\) represent the real and imaginary parts of the original complex elements, respectively. 2. A decoding error occurs in Destination at time instant \(N\). The event is defined as \[ \begin{split} C_{N}\triangleq&\Bigg{\{} \varepsilon_{R,N}\
otin\left[-\frac{0.5}{2\frac{\overline{N(N,\varepsilon)}} {2}},\frac{\overline{N(N,\varepsilon)}}{2}\right)\bigg{\}}\bigcup\\ &\Bigg{\{}\varepsilon_{I,N}\
otin\left[-\frac{0.5}{2\frac{ \overline{N(N,\varepsilon)}}{2}},\frac{\overline{N(N,\varepsilon)}}{2}\right) \bigg{\}}.\end{split} \] (105)  
Hence, the error probability is bounded by (104). And for a given decoding error probability \(\varepsilon\), from (103)-(105), let  
\[\Pr\Big{(}\widetilde{C}_{N}\Big{)}=\frac{\varepsilon}{2},\Pr\Big{(} \widetilde{C}_{n}\Big{)}=\frac{\varepsilon}{2(N-1)}. \tag{106}\]  
Since no modulo-aliasing error occurs at time instant \(n\)\((1\leq n\leq N-1)\), from (103) and (104) and the fact that \(\eta^{{}^{\prime}}_{f,n}\) is a circularly symmetric Gaussian distribution, we conclude that \(\gamma_{n}\varepsilon_{R,n}+\eta^{{}^{\prime}}_{fR,n}\sim\mathcal{N}(0,(\gamma_{n}^ {2}\kappa_{n}+\sigma_{f}^{2}/\|\mathbf{h}_{\text{ds}}\|^{2})/2)\), \(\gamma_{n}\varepsilon_{I,n}+\eta^{{}^{\prime}}_{fI,n}\sim\mathcal{N}(0,(\gamma_{ n}^{2}\alpha_{n}+\sigma_{f}^{2}/\|\mathbf{h}_{\text{ds}}\|^{2})/2)\). So we have  
\[ \begin{split}\text{Pr}\Big{(}\widetilde{C}_{n}\Big{)}& =2\text{Pr}\bigg{\{}\gamma_{n}\varepsilon_{R,n}+\eta^{{}^{\prime} }_{fR,n}\
ot\in\left[-\frac{d}{2},\frac{d}{2}\right)\bigg{\}}\\ &=4\text{Q}\left(\sqrt{\frac{3P_{f}}{\gamma_{n}^{2}\alpha_{n}+ \frac{\sigma_{f}^{2}}{\|\mathbf{h}_{\text{ds}}\|^{2}}}}\right),\end{split}  \tag{100}\]  
where \(d=\sqrt{6P_{f}}\). Similar to the GRC scenario, we also introduce a looseness parameter \(L\), which is denoted by  
\[L\triangleq\frac{P_{f}}{\gamma_{n}^{2}\alpha_{n}+\frac{\sigma_{f}^{2}}{\| \mathbf{h}_{\text{ds}}\|^{2}}}=\frac{1}{3}\bigg{(}Q^{-1}\bigg{(}\frac{ \epsilon}{8(N-1)}\bigg{)}\bigg{)}^{2}. \tag{101}\]  
Then, we determine the appropriate selection of coefficient \(\gamma_{n}\),  
\[\gamma_{n}=\sqrt{\Bigg{(}\frac{P_{f}}{L}-\frac{\sigma_{f}^{2}}{\|\mathbf{h}_{ \text{ds}}\|^{2}}\Bigg{)}\frac{1}{\alpha_{n}}}, \tag{102}\]  
where \(\gamma_{n}\) is the real-value number. According to \(\alpha_{f,n}\triangleq\text{Var}(\varepsilon_{f,n})\), we obtain  
\[\alpha_{f,n}=\alpha_{n}\frac{P_{f}\|\mathbf{h}_{\text{ds}}\|^{2}}{P_{f}\| \mathbf{h}_{\text{ds}}\|^{2}-L\sigma_{f}^{2}}\overset{\text{(a)}}{=}\alpha_{ n}\Phi. \tag{103}\]  
where \((a)\) follows from \(\Phi\triangleq P_{f}\|\mathbf{h}_{\text{ds}}\|^{2}/(P_{f}\|\mathbf{h}_{\text{ ds}}\|^{2}-L\sigma_{f}^{2})\). The computation process of estimation errors at each time instant is analogous to that described in Appendix A. Here, we will omit the detailed procedures and provide the results directly. At time instant 1, \(\varepsilon_{1}=\frac{\eta_{1}}{\sqrt{6P}\|\mathbf{h}_{\text{ds}}\|}\) and \(\alpha_{1}=\frac{\epsilon^{2}}{6P\|\mathbf{h}_{\text{ds}}\|^{2}}\). At time instant 2, \(\varepsilon_{2}=\varepsilon_{1}-\frac{E\left[\varepsilon_{1},\gamma_{2}^{R} \right]}{E\left[\gamma_{2}\gamma_{2}^{R}\right]}\gamma_{2}\), where  
\[