## 3 Performance Analysis
### Performance Analysis of MGF Based on Harmonic Mean of the Double Gamma Distribution System
The outage probability can also be calculated by substituting Equation (22) into the following equation [25] after some necessary alterations:  
\[P_{\text{out}}^{\text{GG}}=P(R_{\text{th}};A,N,Q)=\frac{2-Q\epsilon^{A/2}}{R_ {\text{th}}}\sum_{q=0}^{Q}\left( \begin{array}{c}Q\\ q\end{array} \right)\cdot\sum_{n=0}^{N+q}\frac{(-1)^{n}}{\beta_{\text{th}}} \text{Re}\left(\frac{M_{\text{Z}}\left(\frac{A_{1}+2n}{2R_{\text{th}}}\right) \alpha_{R},\beta_{R},\beta_{D}\right)}{\frac{A_{1}+2n}{2R_{\text{th}}}}\right) +E(A,N,Q) \tag{30}\]  
where \(\beta_{0}=2\) and \(\beta_{\text{th}}=1(n\) is a positive integer), and the values of \(A\), \(N\) and \(Q\) determine the accuracy of the numerical calculation, which is estimated by the overall truncation error term \(E(A,N,Q)\). **Lemma 2**: _If \(X_{l}(I=R\),\(D)\) satisfies \(G(X_{l};\alpha_{l},\beta_{l})\), the diversity order of the AF relay system is min(\(\alpha_{R}\), \(\alpha_{D}\)). See Appendix A for the specific certification process._  
Performance Analysis of the Double LogN Distribution System Based on the Approximation of the LogN Variable Sum  
In this section, we will similarly operate G2L in the first time slot of the system to analyze the performance. In particular, the system performance analysis under the hybrid fading condition is converted into the double LogN distribution condition. Accordingly, the best approximation scheme can be further selected by comparing the accuracy, reliability, and complexity of the two approximation algorithms with those in Sub-Section 3.2. In addition, the method directly calculates the BER and outage probability of the system by using the SNR of the system terminal \(D\), which is beyond the limitation of the MGF and can easily analyze various performance indexes of the system. We assume that the process of approximating the Nakagami-m distribution to the LogN distribution in the first slot has been completed because of the similarity of the approximation process (Section 3.1). Figure 3 shows the comparison of the PDF curves of the approximate \(\left|H_{\text{LR}}\right|^{2}\). At this time, \(H_{\text{LR}}\) and \(H_{\text{LD}}\) are the respective power line fading coefficients of the first and second time slots of the system model, which satisfy the LogN\((\mu_{\text{I}},\sigma_{1}^{2})\) distribution, (I = R,D). The instantaneous SNR of the link is \(\gamma_{\text{LI}}\) = \(|H_{\text{LI}}|^{2}\Delta_{\text{LL}}\), where \(\Delta_{\text{LR}}\) = \(P_{\text{S}}/N_{\text{W}}\) and \(\Delta_{\text{LD}}\) = \(P_{\text{R}}/N_{\text{P}}\) are the average SNRs of the channel. Then, the total SNR \(\gamma_{\text{LL}}\) of the AF relay protocol communication system can be directly written as  
\[\gamma_{\text{LL}}=\frac{1}{\frac{1}{P_{\text{S}}[H_{\text{LR}}]^{2}/N_{\text{ W}}}+\frac{1}{P_{\text{R}}[H_{\text{LD}}]^{2}/N_{\text{P}}}}. \tag{31}\]The instantaneous mutual information \(I\) of the system is  
\[I=\frac{1}{2}\log(1+\gamma_{\rm LL}). \tag{32}\]  
According to the properties of the LogN distribution, \(P_{\rm S}|H_{\rm LR}|^{2}/N_{\rm W}\) and \(P_{\rm R}|H_{\rm LD}|^{2}/N_{\rm P}\) satisfy the LogN distribution when \(P_{\rm S}/N_{\rm W}\) and \(P_{\rm R}/N_{\rm P}\) are constants. As the reciprocal of the LogN variable still satisfies the LogN distribution, \(1/(P_{\rm S}|H_{\rm LR}|^{2}/N_{\rm W})\) and \(1/(P_{\rm R}|H_{\rm LD}|^{2}/N_{\rm P})\) also satisfy the LogN distribution. Hence, the reciprocal of the sum of the LogN distribution variables, namely, the SNR \(\gamma_{\rm LL}\) of the system, also satisfies the LogN distribution. If \(G=1/(P_{\rm S}|H_{\rm LR}|^{2}/N_{\rm W})+1/(P_{\rm R}|H_{\rm LD}|^{2}/N_{\rm P})\), then \(\gamma_{\rm LL}\sim\log(\mu_{\rm G},\sigma_{\rm G}^{2})\). When the known parameters \(\mu_{\rm P}\), \(\sigma_{\rm I}^{2}\) and the Mehta algorithm are used, the distribution parameters \(\mu_{\rm G}\) and \(\sigma_{\rm G}\) of variable \(G\) satisfy the following relations:  
\[