## 3 Performance Analysis
### Mutual Approximation Algorithm Based on the PDF and MGF Equations
The mathematical model considers the matching degree of PDF curves of the LogN and Gamma distributions as the optimization objective and performs the weighting calculation on the square of the difference of the PDF values to the corresponding fading sampling values \(H_{i}\). Specifically,\(\sum\limits_{i=1}^{N}\left(F(H_{i},2\mu_{D},2\sigma_{D})-G(H_{i},\alpha_{D},\beta_ {D})\right)^{2}\), which is called the objective function in this study. When the selected \(s\) value minimizes the objective function, the simultaneous equations can obtain the PDF key parameters of approximated Gamma distribution by the known LogN distribution. The optimal \(s\) value can be solved by an intelligent optimization algorithm, such as the genetic algorithm. Therefore, the numerical calculation and analysis are conducted in this study under channel fading normalization (\(\text{E}\big{(}\big{|}H_{\text{LD}}\big{|}^{2}\big{)}=1\)). Figure 2 presents the PDF curves comparison of the LogN distribution in the second time slot of the system with the approximated Gamma distribution for the different combinations of parameters \(s_{1},s_{2}\) and power line fading parameter \(\sigma_{D}\). 1. (The choice of \(s\) value that will affect the prediction of the PDF curve coincidence is verified. 2. Although the MGF equation approximation lacks analytical expressions, it can be optimized through \(s_{1}\) and \(s_{2}\) with increased accuracy. For the sake of clarity, Figure 3 presents the PDF curve comparison of the Gamma distribution in the first time slot with the approximated LogN distribution under normalized conditions (\(\text{E}\big{(}\big{|}H_{\text{GR}}\big{|}^{2}\big{)}=1\)) for different combinations of the parameters \(s_{1},s_{2}\),and \(m_{\text{R}}\). Using the above method, Table 1 presents the approximate optimization table of the MGF equation for reference in practical applications, where \(s_{1}\) and \(s_{2}\) are the optimal values of different \(\sigma_{D}\), and \(m_{\text{D}}\)  
Figure 3: PDF comparison of \(\big{|}H_{\text{LR}}\big{|}^{2}\) with the Gamma and approximated LogN distributions. Figure 2: Probability density function (PDF) comparison of \(\big{|}H_{\text{GD}}\big{|}^{2}\) with the Log-normal (LogN) and approximated Gamma distributions. and \(\beta_{\text{D}}\) are the PDF parameters that are optimal approximated to the Gamma distribution by the usual \(\sigma_{\text{D}}\), \(s_{1}\) and \(s_{2}\). Similarly, Table 2 shows the optimization results of the LogN distribution approximated by the parameters of the Gamma distribution of the first time slot in the system. As the fading parameters of the channel after the approximation no longer satisfy the normalization condition, it would be cumbersome to use above numerical experimental method to determine the appropriate \(s_{1}\) and \(s_{2}\) values. In addition, the accuracy of \(s_{1}\) and \(s_{2}\) determined by the experiments in a large numerical range will not be guaranteed because of the lack of help from mathematical theory tools. Thus, this study first approximates the normalized PDF of channel fading and then calculates the equivalent parameters of the instantaneous SNR based on the average SNR of the first hop. According to \(\gamma_{\text{LD}}=H_{\text{LD}}^{2}\Delta_{\text{LD}}\) and its properties, the key parameters of \(\gamma_{\text{LD}}\) approximating into \(G\big{(}\gamma_{\text{GD}};\overline{\alpha_{\text{D}}},\overline{\beta_{ \text{D}}}\big{)}\) distribution are  
\[\overline{\alpha_{\text{D}}}=\alpha_{\text{D}}, \tag{17}\]  
\[\overline{\beta_{\text{D}}}=\beta_{\text{D}}\cdot\Delta_{\text{LD}}. \tag{18}\]  
Figure 4 shows a comparison of PDF for L2G based on the joint optimization algorithm of MGF equations and \(s\) vectors when \(\Delta=2\).