## 5 Conclusions  
In this study, a novel type of dual-hop wireless power line hybrid communication system based on the AF relay protocol is analyzed. The proposed system uses the hybrid channel fading based on the Nakagami-m and LogN distributions, and the Bernoulli-Gaussian noise model is used for the mathematical modeling and theoretical deduction and analysis of the system. The difficulty in analyzing the system performance increases because the PDF at the end of the hybrid fading system is difficult to solve. Therefore, the MGF equation based on PDF is used to transform the performance analysis of the hybrid fading system into the performance analysis of the double Gamma fading system and further derive the precise closed-form expressions of the outage probability and BER of the system. At the same time, we execute the process of G2L, and the performance indicators of the double LogN distribution system are analyzed by using the approximation of the LogN variable sum, integral change, Mehta, and other algorithms. The numerical results show that the accuracy of the L2G algorithm is higher than that of the G2L algorithm. However, the L2G algorithm is also sensitive to parameter selection because of its high complexity. The G2L algorithm is beyond the limits of the MGF  
Figure 9: Comparison of the system analytical BER and the fading parameters \(m_{\text{R}}\) and \(\sigma_{\text{D}}\) under various modulation schemes.  
and can be used to analyze many performance indicators. The fading and noise parameters of each link can affect the system performance to varying degrees. Hence, the LogN distribution link is crucial in the entire system, and the combination of environmental variables of the link can be adjusted to optimize the system performance.  
Conceptualization, Z.C.; formal analysis, C.Y.; investigation, J.Y.; resources, D.H.  
The research is supported by the National Natural Science Foundation of China (Nos. 61601182 and 61771195), Natural Science Foundation of Hebei Province (F2017502059 and F2018502047), and Fundamental Research Funds for the Central Universities (No. 2019 MS088).  
The authors declare no conflict of interest.  
Note that \(M_{Z}(s;\alpha_{R},\alpha_{D},\beta_{R},\beta_{D})=\int_{0}^{\infty}\Omega(s,t )dt-\int_{1}^{\infty}\Omega(s,t)dt\). The right integral \(\int_{1}^{\infty}\Omega(s,t)dt\) is transformed first using \(t=1+x/s\). The right integral is then presented as  
\[\int_{0}^{\infty}\frac{\beta_{R}^{\alpha_{D}}\beta_{D}^{\alpha_{R}}(1+\frac{x} {s})^{\alpha_{D}-1}(-\frac{x}{s})^{\alpha_{R}-1}\frac{x}{2}}{B(\alpha_{R}, \alpha_{D})\left\langle\beta_{R}+\left(\frac{x}{s}-\beta_{D}}{s}\beta_{D} \right)\right\rangle_{-\frac{x}{s}\beta_{D}}\frac{x}{s}\frac{x}{s}^{\alpha_{ R}+\alpha_{D}}dx}\stackrel{{ s\rightarrow\infty}}{{\sim}}\int_{0}^{\infty}\frac{\beta_{R}^{\alpha_{D}} \beta_{D}^{\alpha_{R}}\frac{x}{s}\frac{(-x)^{\alpha_{R}-1}}{2}}{B(\alpha_{R}, \alpha_{D})\left\langle\beta_{R}-\beta_{R}\beta_{D}\right\rangle^{\alpha_{R} +\alpha_{D}}}dx, \tag{10}\]  
with the aid of [30], Equation (10) can be simplified as  
\[\int_{1}^{\infty}\Omega(s,t)dt\stackrel{{ s\rightarrow\infty}}{{\approx}}-s^{- \alpha_{R}}\beta_{R}^{-\alpha_{R}}. \tag{11}\]  
Similarly, by using \(t=x/s\), the left integral \(\int_{0}^{\infty}\Omega(s,t)dt\) in the MGF can be transformed into  
\[\int_{0}^{\infty}\Omega(s,t)dt\stackrel{{ s\rightarrow\infty}}{{ \approx}}s^{-\alpha_{D}}\beta_{D}^{-\alpha_{D}}. \tag{12}\]  
As \(s{\rightarrow}\infty\), \(|M_{y}(s;\alpha_{R},\alpha_{D},\beta_{R},\beta_{D})|=b|s|^{-d}+o(|s|^{-d})\), then the diversity order for the relaying system is \(d\)[31]. Based on Equations (11) and (12), we have \(M_{Z}(s;\alpha_{R},\alpha_{D},\beta_{R},\beta_{D})\approx s^{-\alpha_{D}}\beta _{D}^{-\alpha_{D}}+s^{-\alpha_{R}}\beta_{R}^{-\alpha_{R}}\) (as \(s{\rightarrow}\infty\)), and the diversity order is \(\min(\alpha_{R},\alpha_{D})\) for the system model in this study.