# Feedback Approach for the Relay Channel with Noisy Feedback and Its Security Analysis
## 5 Numerical Results  
The aim of these numerical results is to evaluate the performance of our proposed schemes versus different parameters. Here, note that the simulation results for the fading scenario are based on an average of 1000 independent channel realizations. Figure 4 shows the performance analysis of the SK-Type scheme for the GRC with noisy feedback. Here, the achievable rate \(\mathcal{R}_{\text{GRC}}^{\text{NRp}}(N,e)\) of this proposed scheme is given in (8). In Figure 4a, we show the relationship between the achievable rate \(\mathcal{R}_{\text{GRC}}^{\text{NRp}}(N,e)\), the coding blocklength \(N\) and the decoding error probability \(e\). From Figure 4a, we see that when the coding blocklength \(N\) and the decoding error probability \(e\) increase, the achievable rate \(\mathcal{R}_{\text{GRC}}^{\text{NRp}}(N,e)\) also increases. Figure 4b depicts the relationship between the achievable rate \(\mathcal{R}_{\text{GRC}}^{\text{NRp}}(N,e)\) and its asymptotic value \(\mathcal{R}_{\text{GRC}}^{*}(N,e)\). Figure 4b shows that as blocklength \(N\) increases, there exists an asymptotic value \(\mathcal{R}_{\text{GRC}}^{*}(N,e)\) for the achievable rate \(\mathcal{R}_{\text{GRC}}^{\text{NRp}}(N,e)\). And as blocklength \(N\) and the power of the feedback channel \(P_{f}\) increase, it will approach its asymptotic value more closely. This simulation result is the same as the theoretical result in Remark 1. Figure 4c illustrates the relationship between the relative rate \(\Delta\mathcal{R}\) and the relative power \(\Delta P\), where the relative rate \(\Delta\mathcal{R}\) and the relative power \(\Delta P\) are, respectively, defined as  
\[\Delta\mathcal{R}=\frac{\mathcal{R}_{\text{GRC}}^{\text{NRp}}(N,e)}{\mathcal{ R}_{\text{GRC}}^{\text{PRp}}},\Delta P=\frac{P_{f}}{P} \tag{68}\]  
where \(\mathcal{R}_{\text{GRC}}^{\text{NRp}}(N,e)\) is given in (8) and \(\mathcal{R}_{\text{GRC}}^{\text{PRp}}\) is given in (19). From Figure 4c, it shows that when the power of the Destination-Source feedback channel \(P_{f}\) is sufficiently large, the achievable rate \(\mathcal{R}_{\text{GRC}}^{\text{NRp}}(N,e)\) of the noisy feedback scheme in Theorem 1 approaches the rate \(\mathcal{R}_{\text{GRC}}^{\text{PRp}}\) of the noiseless feedback scheme in [23]. Figure 5 shows the performance analysis of the SK-Type scheme for the MISO FRC with noisy feedback. Here, the achievable rate \(\mathcal{R}_{\text{GRC}}^{\text{NRp}}(N,e)\) of this proposed scheme is given  
in (50). The conclusions drawn in Figure 4a,b are consistent with the results in the Gaussian case; i.e, when the coding blocklength \(N\) and the decoding error probability \(\epsilon\) increase, the achievable rate \(\mathcal{R}_{\text{FRC}}^{\text{NRp}}(N,\epsilon)\) also increases. And as blocklength \(N\) increases, there exists an asymptotic value \(\mathcal{R}_{\text{FRC}}^{*}(N,\epsilon)\) for the achievable rate \(\mathcal{R}_{\text{FRC}}^{\text{NRp}}(N,\epsilon)\). Figure 5c illustrates the influence of the number of the Source's antennas \(B\) on the achievable rate \(\mathcal{R}_{\text{FRC}}^{\text{NRp}}(N,\epsilon)\) in the proposed SK-Type coding scheme. From Figure 5c, we conclude that as the number of the Source's antennas \(B\) increases, the achievable rate \(\mathcal{R}_{\text{FRC}}^{\text{NRp}}(N,\epsilon)\) also increases, which indicates that the number of the Source's antennas \(B\) can bring a gain to the achievable rate \(\mathcal{R}_{\text{FRC}}^{\text{NRp}}(N,\epsilon)\) of our proposed SK-Type coding scheme. Figure 5d compares the achievable rates of the SK-Type coding scheme for the MISO FRC with noisy feedback and the scheme without feedback [32], where both Relays use the AF strategy. Figure 5d shows that our proposed scheme can significantly enhance the achievable rate even in the presence of noise in the feedback channel.