# Feedback Approach for the Relay Channel with Noisy Feedback and Its Security Analysis
## Appendix B Proof of Theorem 2
### Coding Procedure
Furthermore, Relay does not keep quiet and utilizes the AF strategy to transmit a scaled version of \(Y^{{}^{\prime}}_{r,n-1}\); that is, \(X_{r,n}{}^{{}^{\prime}}=\beta Y^{{}^{\prime}}_{r,n-1}=\beta(\|\mathbf{h}_{ \text{sr}}\|X^{{}^{\prime}}_{n-1}+\eta^{{}^{\prime}}_{r,n-1})=\beta(\|\mathbf{ h}_{\text{sr}}\|\sqrt{P/\alpha_{f,n-2}}\varepsilon_{f,n-2}+\eta^{{}^{\prime}}_{r,n-1})\), and the value of the Relay's AF coefficient is \(\beta=\sqrt{P_{r}/(\|\mathbf{h}_{\text{sr}}\|^{2}P+\sigma_{r}^{2})}\). * Next, the \(n\)-th channel output signal of Destination received is \(Y_{n}=\|\mathbf{h}_{\text{sd}}\|X^{{}^{\prime}}_{n}+|h_{rd}|X_{r,n}{}^{{}^{ \prime}}+\eta_{n}=\|\mathbf{h}_{\text{sd}}\|\sqrt{P/\alpha_{f,n-1}}\varepsilon _{f,n-1}+|h_{rd}|\beta(\|\mathbf{h}_{\text{sr}}\|\sqrt{P/\alpha_{f,n-2}} \varepsilon_{f,n-2}+\eta^{{}^{\prime}}_{r,n-1})+\eta_{n}\). Destination first calculates the auxiliary signal about the received message \(\widehat{Y}_{n}=\|\mathbf{h}_{\text{sr}}\|\cdot|h_{rd}|\beta\sqrt{P/\alpha_{ f,n-2}}\widehat{\varepsilon}_{n-2}\), then forms the innovation \(\mathbf{I}_{n}\) based on the fact that \(\varepsilon_{n-1}=\varepsilon_{n-2}-\widehat{\varepsilon}_{n-2}\). \(\mathbf{I}_{n}\) is denoted by \(\mathbf{I}_{n}\triangleq Y_{n}-\widehat{Y}_{n}=\|\mathbf{h}_{\text{sd}}\|\sqrt{P/ \alpha_{f,n-1}}\varepsilon_{f,n-1}+|h_{rd}|\beta(\|\mathbf{h}_{\text{sr}}\| \sqrt{P/\alpha_{f,n-2}}(\varepsilon_{f,n-1}+\eta^{{}^{\prime}}_{f,n-2}/\gamma_ {n-2})+\eta^{{}^{\prime}}_{r,n-1})+\eta_{n}\). * Then Destination evaluates MMSE of \(\varepsilon_{n-1}\) based on \(\mathbf{I}_{n}\) and updates the estimation of \(\Theta\) by computing \(\widehat{\Theta}_{n}=\widehat{\Theta}_{n-1}-\frac{E[\varepsilon_{n-1}\mathbf{I }_{n}]}{E[\mathbf{I}_{n}^{2}]}\mathbf{I}_{n}=\Theta+\varepsilon_{n}\). The estimation error at time instant \(n(3\leq n\leq N)\) is denoted by \(\varepsilon_{n}\triangleq\widehat{\Theta}_{n}-\Theta=\varepsilon_{n-1}-\frac{ E[\varepsilon_{n-1}\mathbf{I}_{n}]}{E[\mathbf{I}_{n}^{2}]}\mathbf{I}_{2}\), and the variance of \(\varepsilon_{n}\) is \(\alpha_{n}\triangleq\text{Var}(\varepsilon_{n})\). Finally, at time instant \(N\), Destination computes the final estimation \(\widehat{\Theta}_{N}=\Theta+\varepsilon_{N}\). According to the mapping method in Appendix B.2, if the final estimation error satisfies both of the following conditions  
\[\varepsilon_{R,N}\in\left[-\frac{0.5}{2^{\frac{NR(N,\varepsilon)}{2}}},\frac{0. 5}{2^{\frac{NR(N,\varepsilon)}{2}}}\right),\ \ \varepsilon_{I,N}\in\left[-\frac{0.5}{2^{\frac{NR(N, \varepsilon)}{2}}},\frac{0.5}{2^{\frac{NR(N,\varepsilon)}{2}}}\right), \tag{29}\]  
the final estimation \(\widehat{\Theta}_{N}\) is closest to the \(\Theta\), and thus Destination successfully decodes the message \(M\). For a better understanding, the schematic diagram of the scheme proposed above is shown in Figure A1. The coefficient's derivation and the analysis of error probability for Theorem 2 are fundamentally consistent. We omit the similar parts directly and focus on elaborating the differences between them. See details below.