# Feedback Approach for the Relay Channel with Noisy Feedback and Its Security Analysis
## 4 An SK-Type Scheme for the MISO Fading Relay Channel with Noisy Feedback and Its Security Analysis
### Main Result  
**Theorem 2**.: _For given coding blocklength \(N\) and decoding error probability \(\epsilon\), the lower bound on the \((N,\epsilon)\)-capacity \(C_{\rm FRC}^{\rm NP}(N,\epsilon)\) of the FRC with noisy feedback is given by_  
\[\mathcal{C}_{\rm FRC}^{\rm NP}(N,\epsilon)\geq\mathcal{R}_{\rm FRC}^{\rm NP} (N,\epsilon)=\mathcal{R}_{\rm FRC}^{*}(N,\epsilon)-\mathcal{R}_{\rm FRC}^{ \lozenge}(N,\epsilon), \tag{50}\]  
_where_  
\[\mathcal{R}_{\rm FRC}^{*}(N,\epsilon)\] \[= \frac{1}{2}\log\left(1+\frac{\left(\|\mathbf{h}_{\rm sd}\|\sqrt{ \frac{p}{\Phi}}+\|\mathbf{h}_{\rm sr}\|\cdot|h_{rd}|\beta\sqrt{P}\rho^{*} \right)^{2}}{\Gamma}\right), \tag{51}\]  
\[\mathcal{R}_{\rm FRC}^{\lozenge}(N,\epsilon)=\frac{1}{N}(2\mathcal{R}_{\rm FRC }^{*}(N,\epsilon)+\log\lambda), \tag{52}\]  
\[\Phi=\frac{P_{f}\|\mathbf{h}_{\rm ds}\|^{2}}{P_{f}\|\mathbf{h}_{\rm ds}\|^{2}- L\sigma_{f}^{2}}, \tag{53}\]  
\[L=\frac{1}{3}\bigg{(}Q^{-1}\bigg{(}\frac{\epsilon}{8(N-1)}\bigg{)}\bigg{)}^{2}, \tag{54}\]  
Figure 2: The schematic diagram of the proposed SK-Type finite blocklength (FBL) scheme for the GRC with noisy feedback.  
\[\beta=\sqrt{\frac{P_{r}}{\left\|\mathbf{h}_{\text{sf}}\right\|^{2}P+\sigma_{r}^{2}}}, \tag{55}\]  
\[\lambda=\mathcal{Q}^{-1}\Big{(}\frac{\epsilon}{8}\Big{)}\sqrt{\frac{\sigma^{2}} {3P\left\|\mathbf{h}_{\text{sd}}\right\|^{2}}\Bigg{(}1+\frac{\left\|\mathbf{h} _{\text{sd}}\right\|^{2}\frac{P}{\Phi}}{\left\|\mathbf{h}_{\text{sd}}\right\|^ {2}P\Big{(}1-\frac{1}{\Phi}\Big{)}+\sigma^{2}}\Bigg{)}^{-1}}, \tag{56}\]  
\[\Gamma=\left\|\mathbf{h}_{\text{sf}}\right\|^{2}\left|h_{rd}\right|^{2}\beta^ {2}P\Bigg{(}\bigg{(}1-\frac{1}{\Phi}\bigg{)}-\bigg{(}1-\frac{1}{A(\rho^{*})} \bigg{)}^{2}(\rho^{*})^{2}\Bigg{)}+F, \tag{57}\]  
\[F=\left\|\mathbf{h}_{\text{sd}}\right\|^{2}P\bigg{(}1-\frac{1}{\Phi}\bigg{)}+ \left|h_{rd}\right|^{2}\beta^{2}\sigma_{r}^{2}+\sigma^{2}, \tag{58}\]  
_and the correlation coefficient \(\rho^{*}\) is given by the unique solution in [0, 1] of the following equation_  
\[\rho^{2}=\frac{(A(\rho))^{2}}{\Phi}\times \tag{59}\] \[\left(1+\frac{\bigg{(}\left\|\mathbf{h}_{\text{sd}}\right\|\sqrt {\frac{\Phi}{\Phi}}+\left\|\mathbf{h}_{\text{sr}}\right\|\cdot\left|h_{rd} \right|\beta\sqrt{P}\rho\right)^{2}}{\left\|\mathbf{h}_{\text{sr}}\right\|^{2 }\left|h_{rd}\right|^{2}\beta^{2}P\bigg{(}\Big{(}1-\frac{1}{\Phi}\Big{)}- \Big{(}1-\frac{1}{A(\rho)}\Big{)}^{2}\rho^{2}\bigg{)}+F}\right)^{-1},\]  
\[A(\rho)=1+ \tag{60}\]  
**Remark 3**: _When the blocklength \(N\) is sufficiently large, the achievable rate \(\mathcal{R}_{\text{FRC}}^{\text{NFb}}(N,\epsilon)\) approaches \(\mathcal{R}_{\text{FRC}}^{*}(N,\epsilon)\), which is given in (51)._  
**Remark 4**: _For the MISO FRC without feedback, the AF lower bound \(\mathcal{R}_{\text{FRC}}^{\text{AF}}\) is given by [32], i.e.,_  
\[\mathcal{R}_{\text{FRC}}^{\text{AF}}=\log\left(1+\frac{\left\|\mathbf{h}_{ \text{sd}}\right\|^{2}P}{\sigma^{2}}+\frac{\frac{\left\|\mathbf{h}_{\text{sr }}\right\|^{2}P}{\frac{\left\|h_{rd}\right\|^{2}P}{\sigma_{r}^{2}}}{\frac{ \left\|\mathbf{h}_{\text{sr}}\right\|^{2}P}{\sigma^{2}}+\frac{\left\|h_{rd} \right\|^{2}P_{r}}{\sigma^{2}}+1}\right). \tag{61}\]  
**Proof.** The detailed proof is given in Appendix B. \(\square\)