# 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 instant 1, Source transmits  
\[X_{1}=\sqrt{12P}\theta, \tag{22}\]  
where \(E[X_{1}^{2}]=P\), which satisfies the average power constraint of Source. Meanwhile, Relay remains quiet, and Destination obtains the first output signal  
\[Y_{1}=h_{sd}X_{1}+\eta_{1}=h_{sd}\sqrt{12P}\theta+\eta_{1}, \tag{23}\]  
then computes the first estimation of \(\theta\) by  
\[\widehat{\theta}_{1}=\frac{Y_{1}}{h_{sd}\sqrt{12P}}=\theta+\frac{\eta_{1}}{h_{ sd}\sqrt{12P}}\triangleq\theta+\varepsilon_{1}, \tag{24}\]  
where \(\varepsilon_{1}\) is the first estimation error, denoted by  
\[\varepsilon_{1}\triangleq\widehat{\theta}_{1}-\theta=\frac{\eta_{1}}{h_{sd} \sqrt{12P}}, \tag{25}\]  
and its variance is  
\[\alpha_{1}\triangleq\text{Var}(\varepsilon_{1})=\frac{\sigma^{2}}{12 Ph_{sd}^ {2}}. \tag{26}\]Then, Destination encodes  
\[X_{f,1}=M_{d}\Big{[}\gamma_{1}\widehat{\theta}_{1}+\
u_{1}\Big{]}, \tag{27}\]  
and sends \(X_{f,1}\) back to Source. Here from Property (1) of Proposition 1 and \(d=\sqrt{12P_{f}}\), the power constraint of \(X_{f,1}\) satisfies \(P_{f}\). Then Source receives the feedback signal  
\[Y_{f,1}=h_{ds}X_{f,1}+\eta_{f,1}=h_{ds}M_{d}\Big{[}\gamma_{1}\widehat{\theta}_ {1}+\
u_{1}\Big{]}+\eta_{f,1}. \tag{28}\]  
At time 2, by using the feedback signals \(Y_{f,1}\) (see (28)), Source firstly computes the noisy versions of the Destination's estimation errors, i.e.,  
\[ \begin{split}\varepsilon_{f,1}&=\frac{1}{\gamma_{1 }}M_{d}\bigg{[}\frac{1}{h_{ds}}Y_{f,1}-\gamma_{1}\theta-\
u_{1}\bigg{]}\\ &\stackrel{{\text{(a)}}}{{=}}\frac{1}{\gamma_{1}}M_{ d}\bigg{[}M_{d}\Big{[}\gamma_{1}\widehat{\theta}_{1}+\
u_{1}\Big{]}+\frac{ \eta_{f,1}}{h_{\text{ds}}}-\gamma_{1}\theta-\
u_{1}\bigg{]}\\ &\stackrel{{\text{(b)}}}{{=}}\frac{1}{\gamma_{1}}M_{ d}\bigg{[}\gamma_{1}\varepsilon_{1}+\frac{\eta_{f,1}}{h_{ds}}\bigg{]},\end{split}  \tag{29}\]  
where \((a)\) follows from (28), \((b)\) follows from Property (1) of Proposition 1 and the fact that \(\varepsilon_{1}=\widehat{\theta}_{1}-\theta\). In accordance with (4), we know that if \(\gamma_{1}\varepsilon_{1}+\eta_{f,1}/h_{ds}\in[-d/2,d/2)\), (29) can be rewritten by  
\[\varepsilon_{f,1}=\frac{1}{\gamma_{1}}M_{d}\bigg{[}\gamma_{1}\varepsilon_{1}+ \frac{\eta_{f,1}}{h_{ds}}\bigg{]}=\varepsilon_{1}+\frac{\eta_{f,1}}{\gamma_{1 }h_{ds}}, \tag{30}\]  
otherwise, it implies that a modulo-aliasing error does occur in Source. Hence understanding how to choose the modulation coefficient \(\gamma_{1}\) is crucial to avoid the modulo-aliasing error. After decoding \(\varepsilon_{f,1}\), Source sends  
\[X_{2}=\sqrt{\frac{P}{\pi_{f,1}}}\varepsilon_{f,1}, \tag{31}\]  
where \(\alpha_{f,1}\triangleq\text{Var}\Big{(}\varepsilon_{f,1}\Big{)}\), and \(E[X_{2}^{2}]=P\), which satisfies the average power constraint of Source. At this time instant, Relay also keeps quiet. And once it receives  
\[Y_{2}=h_{sd}X_{2}+\eta_{2}=h_{sd}\sqrt{\frac{P}{\pi_{f,1}}}\varepsilon_{f,1}+ \eta_{2}, \tag{32}\]  
Destination carries out the MMSE of \(\varepsilon_{1}\) based on \(Y_{2}\) and updates the estimation of \(\theta\) by computing  
\[\widehat{\theta}_{2}=\widehat{\theta}_{1}-\frac{E[\varepsilon_{1}Y_{2}]}{E \big{[}Y_{2}^{2}\big{]}}Y_{2}=\theta+\varepsilon_{1}-\frac{E[\varepsilon_{1}Y_ {2}]}{E\big{[}Y_{2}^{2}\big{]}}Y_{2}=\theta+\varepsilon_{2}, \tag{33}\]  
where the second estimation error \(\varepsilon_{2}\) is denoted as  
\[\varepsilon_{2}=\varepsilon_{1}-\frac{E[\varepsilon_{1}Y_{2}]}{E\big{[}Y_{2}^ {2}\big{]}}Y_{2}, \tag{34}\]and the variance of the estimation error is denoted as  
\[\alpha_{2}\triangleq\text{Var}(\varepsilon_{2}). \tag{35}\]  
_Iteration:_  
At time instant \(n-1\)\((3\leq n\leq N)\), Destination sends  
\[X_{f,n-1}=M_{d}\Big{[}\gamma_{n-1}\widehat{\theta}_{n-1}+\
u_{n-1}\Big{]} \tag{36}\]  
back to Source.