## 3 Performance Analysis  
### Mutual Approximation Algorithm Based on the PDF and MGF Equations  
Determining the PDF of each branch SNR is necessary to obtain the system's outage probability and BER. However, the PDF of the destination node \(D\) of the hybrid fading system is difficult to solve, making it difficult, in turn, to analyze the system performance. At the same time, it is mathematically difficult to deal with because the LogN variable integration does not have a closed expressionin the process of analyzing the performance of the system BER. Therefore, the \(\left|H_{\text{LD}}\right|^{2}\) of LogN(\(2\mu_{\text{D}}\),\(4\alpha_{D}^{2}\)) distribution is approximated to \(G(H_{2,\text{GD}};\alpha_{\text{D}},\beta_{\text{D}})\) based on the similarity between two distributions [21] for further analysis of the system. The approximation accuracy is poor because the PDF approximation is obtained in the same way as the mean and variance [22]. Therefore, we propose the PDF approximation algorithm based on the MGF equations in this work, and the obtained approximation function has high precision. \(\left|H_{\text{LD}}\right|^{2}\) satisfies the LogN(\(2\mu_{\text{D}},4\alpha_{D}^{2}\)) distribution, and the MGF is presented as  
\[M_{\text{LD}}(s;\mu_{D},\sigma_{D})=\sum\limits_{n=1}^{N}\frac{w_{n}}{\sqrt{n}} \exp(-s_{i}\exp(\sqrt{2}2\sigma_{D}a_{n}+2\mu_{D})), \tag{13}\]  
where \(s_{i}\left(i=1,2\right)\) is an adjustable variable. \(N\) is the Hermite integration order, and \(w_{\text{n}}\) and \(a_{\text{n}}\) are the weights of the Gauss-Hermite formula and their abscissas [23], respectively. Similarly, the MGF equation of \(G(H_{2,\text{GD}};\alpha_{\text{D}},\beta_{\text{D}})\) is satisfied with the following equation:  
\[M_{\text{GD}}((s;\alpha_{\text{D}},\beta_{D}))=\left(1+\beta_{D}s_{i}\right)^{ -\alpha_{\text{D}}}. \tag{14}\]  
Then the L2G of \(\left|H_{\text{LD}}\right|^{2}\) using the MGF equations \(M_{\text{GD}}((s;\alpha_{D},\beta_{D})=M_{\text{LD}}(s;\mu_{D},\sigma_{D})\). The approximated parameters \(\alpha_{\text{D}}\) and \(\beta_{\text{D}}\) can be solved by using the solve function in MATLAB. \[\sum\limits_{n=1}^{N}\frac{w_{n}}{\sqrt{n}}\exp(-s_{i}\exp(\sqrt{2}2\sigma_{D} a_{n}+2\mu_{D})=\left(1+\beta_{D}s_{i}\right)^{-\alpha_{\text{D}}}. \tag{15}\]  
The values of the independent variables \(s_{1}\) and \(s_{2}\) in the MGF equations are directly related to the determination of the approximate parameters \(\alpha_{\text{D}}\) and \(\beta_{\text{D}}\) and the approximate accuracy of the PDF. This study proposes a mathematical optimization model based on the \(s\) vectors optimization algorithm with the minimum difference of the PDF curves, the key parameters of the approximate fading PDF are obtained by solving a set of optimal \(s_{1}\) and \(s_{2}\) combinations. For the known power line fading parameters \(\mu_{\text{D}}\) and \(\sigma_{\text{D}}\), we use PDF Equations (4) and (6) to solve the wireless fading parameters \(\alpha_{D}\) and \(\beta_{D}\) with \(s_{1}\) and \(s_{2}\) as variables. Hence, the following mathematical model is established:  
\[\underset{s_{1},s_{2}}{\min}\sum\limits_{i=1}^{N}(F(H_{i};2\mu_{D},2\sigma_{D} )-G(H_{i};\alpha_{D},\beta_{D}))^{2}, \tag{16}\]  
subject to:  
(1) \(\sum\limits_{n=1}^{N}\frac{w_{n}}{\sqrt{n}}\exp(-s_{1}\exp(\sqrt{2}2\sigma_{D} a_{n}+2\mu_{D}))=\left(1+\beta_{D}s_{1}\right)^{-\alpha_{\text{D}}}\); (2) \(\sum\limits_{n=1}^{N}\frac{w_{n}}{\sqrt{n}}\exp(-s_{2}\exp(\sqrt{2}2\sigma_{D} a_{n}+2\mu_{D}))=\left(1+\beta_{D}s_{2}\right)^{-\alpha_{\text{D}}}\); (3) \(H_{i}=0.01+0.05\times(i-1)\); (4) \(i=1,2,\ldots,N\); (5) \(s_{1}>0\); (6) \(s_{2}>0\);  
where \(H_{i}\) denote the sample values of the PDF of SNR with fading, and \(N\) is the total number of sample values of the PDF (\(N=100\) is selected in this study).