# Feedback Approach for the Relay Channel with Noisy Feedback and Its Security Analysis
## Appendix B Proof of Theorem 2
### Coding Procedure  
The specific coding process is similar to what was described in Section 3.2.3. For the similar parts, we omit them directly, focusing here on elaborating the distinctions from Section 3.2.3. _After channel transformation, at time \(1\),_  
* Source sends \(X^{{}^{\prime}}_{1}=\sqrt{6P}\Theta\), the relay keeps quiet and Destination receives \(Y_{1}=\|\mathbf{h}_{\text{sd}}\|X^{{}^{\prime}}_{1}+\eta_{1}=\|\mathbf{h}_{ \text{sd}}\|\sqrt{6P}\Theta+\eta_{1}\). * Once it receives \(Y_{1}\), Destination first computes the estimation of \(\Theta\) by \(\widehat{\Theta}_{1}=\frac{Y_{1}}{\|\mathbf{h}_{\text{sd}}\|\sqrt{6P}}=\Theta+ \frac{\eta_{1}}{\|\mathbf{h}_{\text{sd}}\|\sqrt{6P}}\triangleq\Theta+\varepsilon _{1}\). The first estimation error is denoted by \(\varepsilon_{1}\triangleq\widehat{\Theta}_{1}-\Theta=\frac{\eta_{1}}{\| \mathbf{h}_{\text{sd}}\|\sqrt{6P}}\), and the variance of \(\varepsilon_{1}\) is \(\alpha_{1}\triangleq\text{Var}(\varepsilon_{1})=\frac{e^{2}}{6P\|\mathbf{h}_{ \text{sd}}\|^{2}}\). * Then, Destination uses the two-dimensional _modulo_-\(\Lambda\) function to encode \(\widehat{\Theta}_{1}\) as \(X_{f,1}=\mathbb{M}_{\Lambda}[\gamma_{1}\widehat{\Theta}_{1}+V_{1}]\) and transmits it back to Source via a noisy feedback channel. _At time \(2\),_
* Once receiving \(Y^{{}^{\prime}}_{f,1}\), Source also uses the two-dimensional _modulo_-\(\Lambda\) function to decode this message and computes the noisy versions of Destination's estimation error, i.e., \(\varepsilon_{f,1}=\frac{1}{\gamma_{1}}\mathbb{M}_{\Lambda}[Y^{{}^{\prime}}_{f,1 }-\gamma_{1}\Theta-V_{1}]=\frac{1}{\gamma_{1}}\mathbb{M}_{\Lambda}[\gamma_{1} \varepsilon_{1}+\eta^{{}^{\prime}}_{f,1}]\). If the modulo-aliasing error does not occur, using Property (2) of Proposition 2, Source can obtain \(\varepsilon_{f,1}=\varepsilon_{1}+\eta^{{}^{\prime}}_{f,1}/\gamma_{1}\). * The Source sends \(X^{{}^{\prime}}_{2}=\sqrt{P/\alpha_{f,1}}\varepsilon_{f,1}\) to Destination, where \(\alpha_{f,1}\triangleq\text{Var}\Big{(}\varepsilon_{f,1}\Big{)}\). At this time instant, Relay also keeps quiet, so Destination receives the output signal \(Y_{2}=\|\mathbf{h}_{\text{sd}}\|_{X^{{}^{\prime}}_{2}}^{{}^{\prime}}+\eta_{2}= \|\mathbf{h}_{\text{sd}}\|\sqrt{P/\alpha_{f,1}}\varepsilon_{f,1}+\eta_{2}\). * Next, Destination evaluates MMSE of \(\varepsilon_{1}\) based on \(Y_{2}\) and updates the estimation of \(\Theta\) by computing \(\widehat{\Theta}_{2}=\widehat{\Theta}_{1}-\frac{E[\varepsilon_{1}Y_{2}^{ \eta}]}{E[Y_{2}Y_{2}^{\eta}]}Y_{2}=\Theta+\varepsilon_{2}\). The second estimation error is denoted by \(\varepsilon_{2}\triangleq\widehat{\Theta}_{2}-\Theta=\varepsilon_{1}-\frac{ E[\varepsilon_{1}Y_{2}^{\eta}]}{E[Y_{2}Y_{2}^{\eta}]}Y_{2}\), and the variance of \(\varepsilon_{2}\) is \(\alpha_{2}\triangleq\text{Var}(\varepsilon_{2})\). Then, Destination encodes \(X_{f,2}=\mathbb{M}_{\Lambda}[\gamma_{2}\widehat{\Theta}_{2}+V_{2}]\) and transmits it back to Source via a noisy feedback channel. _At time \(n\)\((3\leq n\leq N)\)_,_
* Once it receives \(Y^{{}^{\prime}}_{f,n-1}\), Source computes the noisy versions of Destination's estimation error, i.e., \(\varepsilon_{f,n-1}=\frac{1}{\gamma_{n-1}}\mathbb{M}_{\Lambda}[\gamma_{n-1} \varepsilon_{n-1}+\eta^{{}^{\prime}}_{f,n-1}]\). If the modulo-aliasing error does not occur, using Property (2) of Proposition 2, Source can get \(\varepsilon_{f,n-1}=\varepsilon_{n-1}+\eta^{{}^{\prime}}_{f,n-1}/\gamma_{n-1}\). * The Source sends \(X^{{}^{\prime}}_{n}=\sqrt{P/\alpha_{f,n-1}}\varepsilon_{f,n-1}\) to both the Relay and Destination, where \(\alpha_{f,n-1}\triangleq\text{Var}\Big{(}\varepsilon_{f,n-1}\Big{)}\).