# Feedback Approach for the Relay Channel with Noisy Feedback and Its Security Analysis
## Appendix A Coefficient's Derivation and Error Probability Analysis of Theorem 1
To derive the achievable rate of this SK-type scheme, we focus on the final estimation error. At time instant 1, the estimation error and its variance are given in (25) and (26). So we consider the second moment and the following moments. At time instant 2, we first need to calculate the MMSE estimation coefficient of Destination based on \(Y_{2}\), that is,  
\[ \begin{split}& E[\varepsilon_{1}Y_{2}]=h_{sd}\sqrt{\frac{P}{\phi} \sqrt{\alpha_{1}}},\\ & E\Big{[}Y_{2}^{2}\Big{]}=h_{sd}^{2}\frac{P}{\phi}+h_{sd}^{2}P \bigg{(}1-\frac{1}{\phi}\bigg{)}+\sigma^{2},\end{split}  \tag{103}\]  
and the estimation error is given in (34). So, the variance of \(\varepsilon_{2}\) is  
\[ \begin{split}&\alpha_{2}\triangleq\text{Var}(\varepsilon_{2})= \alpha_{1}-\frac{\big{(}E[\varepsilon_{1}Y_{2}]\big{)}^{2}}{E\big{[}Y_{2}^{2} \big{]}^{2}}\\ &=\alpha_{1}\left(1+\frac{h_{sd}^{2}\frac{P}{\phi}}{h_{sd}^{2}P \Big{(}1-\frac{1}{\phi}\Big{)}+\sigma^{2}}\right)^{-1},\end{split}  \tag{104}\]  
where \(\alpha_{1}\) is given in (26). At time instant \(n\) (\(3\leq n\leq N\)), similar to the second moment, we also calculate the MMSE estimation coefficient of Destination based on \(I_{n}\) instead of \(Y_{n}\); that is,  
\[ \begin{split}& E[\varepsilon_{n-1}I_{n}]=\sqrt{\alpha_{n-1}} \left(h_{sd}\sqrt{\frac{P}{\phi}}+h_{sr}h_{rd}\beta_{\uparrow}\sqrt{P}\rho_{ \uparrow n-1}\right),\\ & E\Big{[}I_{n}^{2}\Big{]}=\left(h_{sd}\sqrt{\frac{P}{\phi}}+h_{sr }h_{rd}\beta_{\uparrow}\sqrt{P}\rho_{\uparrow n-1}\right)^{2}\\ &+h_{sr}^{2}h_{rd}^{2}\beta_{\uparrow}^{2}P\bigg{(}\left(1-\frac {1}{\phi}\right)-\left(1-\frac{1}{A_{\uparrow n-1}}\right)^{2}\rho_{\uparrow n -1}^{2}\bigg{)}+F_{\uparrow},\end{split}  \tag{105}\]  
where \(F_{\uparrow}=h_{sd}^{2}P\Big{(}1-\frac{1}{\phi}\Big{)}+h_{rd}^{2}\beta_{ \uparrow}^{2}\sigma_{r}^{2}+\sigma^{2}\). The estimation error at time instant \(n\) (\(3\leq n\leq N\)) is given in (46), so the variance of \(\varepsilon_{n}\) is  
\[ \begin{split}&\alpha_{n}\triangleq\text{Var}(\varepsilon_{n})= \alpha_{n-1}-\frac{\big{(}E[\varepsilon_{n-1}I_{n}]\big{)}^{2}}{E[I_{n}^{2}]}= \alpha_{n-1}\times\\ &\left(1+\frac{\Big{(}h_{sd}\sqrt{\frac{P}{\phi}}+h_{sr}h_{rd} \beta_{\uparrow}\sqrt{P}\rho_{\uparrow n-1}\Big{)}^{2}}{h_{sr}^{2}h_{rd}^{2} \beta_{\uparrow}^{2}P\bigg{(}\Big{(}1-\frac{1}{\phi}\Big{)}-\Big{(}1-\frac{1}{ A_{\uparrow n-1}}\Big{)}^{2}\rho_{\uparrow n-1}^{2}\bigg{)}+F_{\uparrow}}\right)^{-1}, \end{split}  \tag{106}\]  
where \(\rho_{\uparrow n-1}\) is the correlation coefficient between \(\varepsilon_{n-1}\) and \(\varepsilon_{f,n-2}\), denoted by  
\[\rho_{\uparrow n-1}\triangleq\frac{E\Big{[}\varepsilon_{n-1}\varepsilon_{f,n -2}\Big{]}}{\sqrt{\alpha_{n-1}}\sqrt{\alpha_{f,n-2}}}=A_{\uparrow n-1}\sqrt{ \frac{\alpha_{n-1}}{\alpha_{f,n-2}}}, \tag{107}\]  
and  
\[A_{\uparrow n-1}=1+\frac{Ph_{sd}\Big{(}h_{sd}\sqrt{\frac{1}{\phi}}+h_{sr}h_{rd }\beta_{\uparrow}\rho_{\uparrow n-2}\Big{)}\Big{(}\sqrt{\frac{1}{\phi}}-\sqrt {\phi}\Big{)}}{h_{sr}^{2}h_{rd}^{2}\beta_{\uparrow}^{2}P\bigg{(}\Big{(}1-\frac {1}{\phi}\Big{)}-\Big{(}1-\frac{1}{A_{\uparrow n-2}}\Big{)}^{2}\rho_{\uparrow n -2}^{2}\bigg{)}+F_{\uparrow}}. \tag{108}\]It is worth noting that the initial value of \(\rho_{\uparrow n-1}\) is given by  
\[\rho_{\uparrow 2}=\frac{\sigma^{2}}{h_{sd}^{2}P\Big{(}1-\frac{1}{\phi}\Big{)}+ \sigma^{2}}\sqrt{\frac{\alpha_{2}}{\alpha_{1}\phi}}. \tag{100}\]  
From (101) and (102), it is apparent that \(\big{|}\rho_{\uparrow n}\big{|}\
eq\big{|}\rho_{\uparrow n-1}\big{|}\). By applying **Cesaro's Mean Theorem** in [20] to the \(\{\rho_{\uparrow n}\}\), it follows that this sequence will converge to a unique fixed point in \([0,1]\). Hence, we obtain the following equation  
\[