## 2 System Description
### Second Time Slot  
In the second time slot, \(R\) uses the AF protocol to forward the received data to \(D\) by power \(P_{\text{R}}\). Let \(X_{\text{R}}\) be the signal of \(R\), then the signal \(y_{\text{LD}}\) received by \(D\) is  
Figure 1: System model for a dual-hop wireless/power line hybrid communication system with amplify-and-forward (AF) relay.  
\[y_{\text{LD}}=H_{\text{LD}}\sqrt{P_{\text{R}}}X_{\text{R}}+n_{\text{LD}}, \tag{5}\]  
where \(H_{\text{LD}}\) is the power line fading coefficient that satisfies the following \(\text{LogN}(\mu_{\text{D}},\sigma_{\text{D}}^{2})\) distribution:  
\[F(H_{\text{LD}};\mu_{\text{D}},\sigma_{\text{D}})=\frac{1}{H_{\text{LD}}\sigma_ {\text{D}}\sqrt{2\pi}}\exp(-\frac{\left(\ln H_{\text{LD}}-\mu_{\text{D}}\right) ^{2}}{2}), \tag{6}\]  
where \(\mu_{\text{D}}\) and \(\sigma_{\text{D}}\) are the mean and mean square deviation of \(\ln H_{\text{LD}}\), respectively. Let \(\text{E}\big{(}H_{\text{LD}}\big{|}^{2}\big{)}=\exp(2\mu_{\text{D}}+2\sigma_{ \text{D}}^{2})=1\), that is, \(\mu_{\text{D}}=-\sigma_{\text{D}}^{2}\). The envelope energy of the channel fading can be normalized to ensure that the channel fading does not change the average power of the signal. On the basis of the properties of the LogN distribution, \(\big{|}H_{\text{LD}}\big{|}^{2}\) satisfies the \(\text{LogN}(2\mu_{D},4\sigma_{D}^{2})\) distribution.  
To accurately characterize the PLC channel, the noise is assumed to consist of background and impulsive noise components [19]. These noise types are modeled using the two-term Bernoulli-Gaussian noise model [20]. Therefore, the PDF of the total noise can be simply expressed as  
\[n_{\text{LD}}=n_{g}+n_{i}, \tag{7}\]  
where \(n_{g}\) is considered the Gaussian with zero mean and variance \(\sigma_{g^{\prime}}^{2}\), while the impulsive part \(n_{i}\) is modeled as the Bernoulli-Gaussian random process. As \(n_{\text{LD}}\) is the total noise \(n_{i}=bA\), \(A\) is the white Gaussian noise with mean zero, and \(b\) is the Bernoulli process with the probability mass function.  
\[\begin{split} P_{r}(b=1)&=p\\ P_{r}(b=0)&=1-p\end{split} \tag{8}\]  
Therefore, the PDF of the total noise can be simply expressed as  
\[f_{n_{\text{LD}}}=(1-p)N(0,\sigma_{g}^{2})+pN(0,\sigma_{i}^{2}+\sigma_{g}^{2}), \tag{9}\]  
where \(N(0,\sigma^{2})\) refers to the Gaussian distribution with mean zero and \(\sigma^{2}\) as variance, and \(p\) is the arrival probability of the impulsive component of the Bernoulli-Gaussian noise. The noise power on the PLC link is \(\sigma_{g}^{2}\) in the presence of only the background noise, while the total noise power is \(\sigma_{g}^{2}(1+T)\) in the presence of background and impulsive noises, where \(T=\sigma_{i}^{2}/\sigma_{g}^{2}\) is the impulsive noise index.  
Let \(\Delta_{\text{LD}}=P_{\text{R}}/N_{\text{P}}\), and then the instantaneous SNR of the power line receiver can be expressed as  
\[\gamma_{\text{LD}}=\big{|}H_{\text{LD}}\big{|}^{2}\Delta_{\text{LD}}. \tag{10}\]  
Therefore, the total SNR \(\gamma_{\text{GL}}\) of the dual-hop AF relay protocol communication system is  
\[\gamma_{\text{GL}}=\frac{\gamma_{\text{GR}}\gamma_{\text{LD}}}{\gamma_{\text{ GR}}+\gamma_{\text{LD}}+1}. \tag{11}\]  
When a high SNR (\(P_{\text{S}}/N_{\text{W}}\), \(P_{\text{R}}/N_{\text{P}}\)\(\gg 1\)) exists, Equation (11) can be approximated as  
\[\gamma_{\text{GL}}\approx\frac{\gamma_{\text{GR}}\gamma_{\text{LD}}}{\gamma_{ \text{GR}}+\gamma_{\text{LD}}}=\frac{1}{1/\gamma_{\text{GR}}+1/\gamma_{\text {LD}}}. \tag{12}\]