# Feedback Approach for the Relay Channel with Noisy Feedback and Its Security Analysis
## 3 An SK-Type Feedback Scheme for the Gaussian Relay Channel with Noisy Feedback
### Our Achievable Scheme
#### 3.2.3 Coding Procedure
At time \(n\)\((3\leq n\leq N)\), once it receives the feedback signals  
\[Y_{f,n-1}=h_{ds}M_{d}\Big{[}\gamma_{n-1}\widehat{\theta}_{n-1}+\
u_{n-1}\Big{]} +\eta_{n-1}, \tag{37}\]  
Source computes the noisy version of the Destination's estimation errors, i.e.,  
\[\varepsilon_{f,n-1} =\frac{1}{\gamma_{n-1}}M_{d}\bigg{[}\frac{1}{h_{ds}}Y_{f,n-1}- \gamma_{n-1}\theta-\
u_{n-1}\bigg{]} \tag{38}\] \[\overset{\text{(a)}}{=}\frac{1}{\gamma_{n-1}}M_{d}\bigg{[}\gamma _{n-1}\varepsilon_{n-1}+\frac{\eta_{f,n-1}}{h_{ds}}\bigg{]}\] \[X_{2}=\sqrt{\frac{P}{\pi_{f,1}}}\varepsilon_{f,1}, \tag{31}\]0  
where \((a)\) follows from (37), Property (1) of Proposition 1 and the fact that \(\varepsilon_{n-1}=\widehat{\theta}_{n-1}-\theta\). \((b)\) follows from \(\gamma_{n-1}\varepsilon_{n-1}+\eta_{f,n-1}/h_{ds}\in[-d/2,d/2)\), which means that modulo-aliasing error does not occur in Source. After decoding \(\varepsilon_{f,n-1}\), Source transmits  
\[X_{2}=\sqrt{\frac{P}{\pi_{f,1}}}\varepsilon_{f,1}, \tag{31}\]1  
where \(\alpha_{f,n-1}\triangleq\text{Var}\Big{(}\varepsilon_{f,n-1}\Big{)}\), and \(E[X_{n}^{2}]=P\), which satisfies the average power constraint of Source. At time instant \(n(3\leq n\leq N)\), Relay does not remain quiet and transmits the received messages from Source to Destination through an AF strategy, that is, it transmits a scaled version of its observation in the previous step, i.e.,  
\[X_{2}=\sqrt{\frac{P}{\pi_{f,1}}}\varepsilon_{f,1}, \tag{31}\]2 \[X_{2}=\sqrt{\frac{P}{\pi_{f,1}}}\varepsilon_{f,1}, \tag{31}\]3  
where the scaling factor is chosen as  
\[X_{2}=\sqrt{\frac{P}{\pi_{f,1}}}\varepsilon_{f,1}, \tag{31}\]4  
and \(E[X_{r,n}^{2}]=\beta^{2}E[Y_{r,n-1}^{2}]=P_{r}\), which satisfies the average power constraint of Relay. So, Destination obtains the \(n\)-th channel output signal  
\[ \begin{split} Y_{n}&=h_{sd}X_{n}+h_{rd}X_{r,n}+\eta_{n} \\ &=h_{sd}\sqrt{\frac{P}{\alpha_{f,n-1}}}\varepsilon_{f,n-1}+\beta h _{sr}h_{rd}\sqrt{\frac{P}{\alpha_{f,n-2}}}\varepsilon_{f,n-2}\\ &+\beta h_{rd}\eta_{r,n-1}+\eta_{n}.\end{split}  \tag{42}\]  
Upon receiving \(Y_{n}\), Destination first computes the auxiliary signal, that is,  
\[\widehat{Y}_{n}=h_{sr}h_{rd}\beta\sqrt{\frac{P}{\alpha_{f,n-2}}}\widehat{ \varepsilon}_{n-2}, \tag{43}\]  
and this auxiliary signal \(\widehat{Y}_{n}\) will be used to form the innovation \(I_{n}\) based on the fact that \(\varepsilon_{n-1}=\varepsilon_{n-2}-\widehat{\varepsilon}_{n-2}\),  
\[ \begin{split} I_{n}&\triangleq Y_{n}-\widehat{Y}_{n }\\ &=h_{sd}\sqrt{\frac{P}{\alpha_{f,n-1}}}\varepsilon_{f,n-1}+h_{sr} h_{rd}\beta\sqrt{\frac{P}{\alpha_{f,n-2}}}\varepsilon_{n-1}\\ &+h_{sr}h_{rd}\beta\sqrt{\frac{P}{\alpha_{f,n-2}}}\frac{\eta_{f, n-2}}{\gamma_{n-2}h_{ds}}+\beta h_{rd}\eta_{r,n-1}+\eta_{n}.\end{split}  \tag{44}\]  
and it carries out the MMSE of \(e_{n-1}\) based on \(I_{n}\), then updates the estimation of \(\theta\) by computing  
\[\widehat{\theta}_{n}=\widehat{\theta}_{n-1}-\frac{E[\varepsilon_{n-1}I_{n}]}{ E[I_{n}^{2}]}I_{n}=\theta+\varepsilon_{n}, \tag{45}\]  
where the estimation error of \(\theta\) is denoted as  
\[\varepsilon_{n}=\varepsilon_{n-1}-\frac{E[\varepsilon_{n-1}I_{n}]}{E[I_{n}^{2 }]}I_{n}, \tag{46}\]  
and the variance of the estimation error is denoted as  
\[\alpha_{n}\triangleq\text{Var}(\varepsilon_{n}). \tag{47}\]  
At time instant \(N\), Destination computes the final estimation  
\[\widehat{\theta}_{N}=\theta+\varepsilon_{N}. \tag{48}\]  
According to the mapping method in Section 3.2.1, if the final estimation error is given by  
\[\varepsilon_{N}\in\left[-\frac{0.5}{2^{NR(N,\varepsilon)}},\frac{0.5}{2^{NR(N,\varepsilon)}}\right), \tag{49}\]  
the final estimation \(\widehat{\theta}_{N}\) is closest to the transmitted point \(\theta\), and thus Destination successfully decodes the transmitted message \(M\). For better understanding, the schematic diagram of the above proposed scheme is shown in Figure 2.  
Proof. : The detailed proof is given in Appendix A.