# 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
### Main Result
\[\mathcal{R}_{\text{GRC}}^{*}(N,\epsilon)=\frac{1}{2}\log\left(1+\frac{\left(h_{sd} \sqrt{\frac{P}{\phi}}+h_{sr}h_{rd}\beta_{\uparrow}\sqrt{P}\rho_{\uparrow}^{*} \right)^{2}}{\Gamma_{\uparrow}}\right), \tag{9}\]  
\[\mathcal{R}_{\text{GRC}}^{\lozenge}(N,\epsilon)=\frac{1}{N}(2\mathcal{R}_{ \text{GRC}}^{*}(N,\epsilon)+\log\mu), \tag{10}\]  
\[\mathcal{R}_{\mathrm{GRC}}^{\mathrm{NRp}}(N,\epsilon)\geq\mathcal{R}_{\mathrm{ GRC}}^{\mathrm{NRp}}(N,\epsilon)\geq\mathcal{R}_{\mathrm{GRC}}^{*}(N,\epsilon)- \mathcal{R}_{\mathrm{GRC}}^{\Diamond}(N,\epsilon). \tag{9}\]0  
\[\mathcal{R}_{\mathrm{GRC}}^{\mathrm{NRp}}(N,\epsilon)\geq\mathcal{R}_{\mathrm{ GRC}}^{\mathrm{NRp}}(N,\epsilon)\geq\mathcal{R}_{\mathrm{GRC}}^{*}(N,\epsilon)- \mathcal{R}_{\mathrm{GRC}}^{\Diamond}(N,\epsilon). \tag{9}\]1  
\[\mathcal{R}_{\mathrm{GRC}}^{\mathrm{NRp}}(N,\epsilon)\geq\mathcal{R}_{\mathrm{ GRC}}^{\mathrm{NRp}}(N,\epsilon)\geq\mathcal{R}_{\mathrm{GRC}}^{*}(N,\epsilon)- \mathcal{R}_{\mathrm{GRC}}^{\Diamond}(N,\epsilon). \tag{9}\]2  
\[\mathcal{R}_{\mathrm{GRC}}^{\mathrm{NRp}}(N,\epsilon)\geq\mathcal{R}_{\mathrm{ GRC}}^{\mathrm{NRp}}(N,\epsilon)\geq\mathcal{R}_{\mathrm{GRC}}^{*}(N,\epsilon)- \mathcal{R}_{\mathrm{GRC}}^{\Diamond}(N,\epsilon). \tag{9}\]3  
\[\mathcal{R}_{\mathrm{GRC}}^{\mathrm{NRp}}(N,\epsilon)\geq\mathcal{R}_{\mathrm{ GRC}}^{\mathrm{NRp}}(N,\epsilon)\geq\mathcal{R}_{\mathrm{GRC}}^{*}(N,\epsilon)- \mathcal{R}_{\mathrm{GRC}}^{\Diamond}(N,\epsilon). \tag{9}\]4  
\[\mathcal{R}_{\mathrm{GRC}}^{\mathrm{NRp}}(N,\epsilon)\geq\mathcal{R}_{\mathrm{ GRC}}^{\mathrm{NRp}}(N,\epsilon)\geq\mathcal{R}_{\mathrm{GRC}}^{*}(N,\epsilon)- \mathcal{R}_{\mathrm{GRC}}^{\Diamond}(N,\epsilon). \tag{9}\]5  
_and the correlation coefficient \(\rho_{\uparrow}^{*}\) is given by the unique solution in [0, 1] of the following equation:_  
\[\mathcal{R}_{\mathrm{GRC}}^{\mathrm{NRp}}(N,\epsilon)\geq\mathcal{R}_{\mathrm{ GRC}}^{\mathrm{NRp}}(N,\epsilon)\geq\mathcal{R}_{\mathrm{GRC}}^{*}(N,\epsilon)- \mathcal{R}_{\mathrm{GRC}}^{\Diamond}(N,\epsilon). \tag{9}\]6  
\[\mathcal{R}_{\mathrm{GRC}}^{\mathrm{NRp}}(N,\epsilon)\geq\mathcal{R}_{\mathrm{ GRC}}^{\mathrm{NRp}}(N,\epsilon)\geq\mathcal{R}_{\mathrm{GRC}}^{*}(N,\epsilon)- \mathcal{R}_{\mathrm{GRC}}^{\Diamond}(N,\epsilon). \tag{9}\]7  
\[\mathcal{R}_{\mathrm{GRC}}^{\mathrm{NRp}}(N,\epsilon)\geq\mathcal{R}_{\mathrm{ GRC}}^{\mathrm{NRp}}(N,\epsilon)\geq\mathcal{R}_{\mathrm{GRC}}^{*}(N,\epsilon)- \mathcal{R}_{\mathrm{GRC}}^{\Diamond}(N,\epsilon). \tag{9}\]8  
\[\mathcal{R}_{\mathrm{GRC}}^{\mathrm{NRp}}(N,\epsilon)\geq\mathcal{R}_{\mathrm{ GRC}}^{\mathrm{NRp}}(N,\epsilon)\geq\mathcal{R}_{\mathrm{GRC}}^{*}(N,\epsilon)- \mathcal{R}_{\mathrm{GRC}}^{\Diamond}(N,\epsilon). \tag{9}\]9  
**Remark 1**. : _When the blocklength \(N\) is sufficiently large, the achievable rate \(\mathcal{R}_{\text{GRC}}^{\text{NRP}}(N,\epsilon)\) approaches \(\mathcal{R}_{\text{GRC}}^{*}(N,\epsilon)\), which is given in (9)._  
**Remark 2**. : _The upper bound \(\mathcal{R}_{\text{GRC}}^{\text{FPb}}\) of the GRC with noisy feedback is given in [23], where_  
\[\mathcal{C}_{\mathrm{GRC}}^{\mathrm{NRp}}(N,\epsilon)\geq\mathcal{R}_{\mathrm{ GRC}}^{\mathrm{NRp}}(N,\epsilon)=\mathcal{R}_{\mathrm{GRC}}^{*}(N,\epsilon)- \mathcal{R}_{\mathrm{GRC}}^{\Diamond}(N,\epsilon). \tag{10}\]0\(\rho_{\uparrow}^{*}\) _is the unique solution in \([0,1]\) of the following equation_  
\[\mathcal{C}_{\mathrm{GRC}}^{\mathrm{NRp}}(N,\epsilon)\geq\mathcal{R}_{\mathrm{ GRC}}^{\mathrm{NRp}}(N,\epsilon)=\mathcal{R}_{\mathrm{GRC}}^{*}(N,\epsilon)- \mathcal{R}_{\mathrm{GRC}}^{\Diamond}(N,\epsilon). \tag{10}\]1  
_For the GRC with noisy feedback, letting the power of the feedback channel \(P_{f}\) be sufficiently large, from (11), (15) and (18), we conclude that_  
\[\mathcal{C}_{\mathrm{GRC}}^{\mathrm{NRp}}(N,\epsilon)\geq\mathcal{R}_{\mathrm{ GRC}}^{\mathrm{NRp}}(N,\epsilon)=\mathcal{R}_{\mathrm{GRC}}^{*}(N,\epsilon)- \mathcal{R}_{\mathrm{GRC}}^{\Diamond}(N,\epsilon). \tag{10}\]2  
_Substituting (21) into (17), we conclude that (17) can be rewritten as (20).