## 3 Performance Analysis
### Performance Analysis of MGF Based on Harmonic Mean of the Double Gamma Distribution System  
The performance analysis under hybrid fading is transformed into the performance analysis under the double Gamma distribution after L2G in the second time slot \(R\)-\(D\) link. Many studies have investigated the BER and outage performance of the AF relaying system under Nakagami-m fading. However, these studies have the following limitations: (1) The use of the approximate function for end-to-end SNR, such as \(\gamma_{\rm GR}\gamma_{\rm GGD}/(\gamma_{\rm GR}+\gamma_{\rm GD}+1)\approx\min( \gamma_{\rm GR},\gamma_{\rm GD})\)[24], leads to imprecise results. (2) Numerous performance formulas contain multiple series [25], Whittaker functions [26], or Appell F4 functions [27], thereby causing high computation complexity. (3) Some methods can only evaluate the system outage probability [28]. (4) Some precise conclusions are only suitable for a special fading parameter m (integer [24; 25] or integer plus one-half [25]). To reduce the computational complexity of the harmonic mean, some studies use the least function approximation calculation. Some basic frame works are also proposed but with increased complexity, where in two Appell functions need to be calculated [16]. Motivated by all of the above, we present an effective performance evaluation approach using the basic theory of probability. The MGF for terminal-to-terminal instantaneous SNR is derived first in the form of the Appell function F1, which involves a single integral that can be computed by the general software MATLAB. Suitable for the arbitrary Nakagami-m fading parameter m, the MGF is then used to derive the system performance of the diversity order, BER, and outage probability. The channel coefficients \(H_{\rm GR}\) and \(H_{\rm GD}\) of the \(S\)-\(R\) and \(R\)-\(D\) links are subject to Nakagami-m fading with the shape parameters \(m_{\rm R}\) and \(m_{\rm D}\) and satisfy \(\rm E[|\)\(H_{\rm GR}|^{2}]=\Omega_{R}=1\) and \(\rm E[|\)\(H_{\rm GD}|^{2}]=\Omega_{D}\). Then, the instantaneous SNR of the links are \(\gamma_{\rm GR}=|\)\(H_{\rm GR}|^{2}\Delta_{\rm GR}\) and \(\gamma_{\rm GD}=|\)\(H_{\rm GD}|^{2}\Delta_{\rm GD}\), where \(\Delta_{\rm GR}\) and \(\Delta_{\rm GD}\) are the average SNRs. Consequently, each \(\gamma_{\rm GI}(\rm I=R,D)\) satisfies the Gamma distribution denoted as \(\gamma_{\rm GI}\)-\(G(\gamma_{\rm GI};\alpha_{\rm I},\beta_{\rm I})\), where \(\alpha_{\rm I}=m_{\rm I}\) and \(\beta_{\rm I}=\Delta_{\rm GI}\Omega_{\rm I}/m_{\rm I}\). The equivalent end-to-end SNR \(\gamma_{\rm GG}\) at terminal \(D\) is similar to that of Equation (11). \(F_{1}(\cdot)\) represents the first type of Appell hypergeometric function, and the MGF of the harmonic mean of the two variables with the Gamma distribution satisfies the following lemma. **Lemma 1**. : _If \(X_{\rm I}(I=R,D)\) satisfies \(G(X_{\rm I},\alpha_{I},\beta_{I})\), then the MGF of \(X_{\rm I}\)'s harmonic mean \(Z=X_{R}\times X_{D}/(X_{R}+X_{D})\) has the following expression:_  
\[M_{Z}(s;\alpha_{R},\alpha_{D},\beta_{R},\beta_{D})=\left(\frac{\beta_{D}}{ \beta_{R}}\right)^{\alpha_{D}}\times F_{1}(\alpha_{D};\alpha_{R}+\alpha_{D}, \alpha_{R}+\alpha_{D};\alpha_{R}+\alpha_{D};k_{1},k_{2}), \tag{19}\]  
Figure 5: Comparison of PDF for instantaneous SNR with the Gamma and approximated LogN distributions using the best \(s\) value optimization method. _where \(k_{1}=\beta_{R}\beta_{D}s/\left(\rho-\sqrt{\beta_{R}\beta_{D}^{2}s+\rho^{2}}\right)\), \(k_{2}=\beta_{R}\beta_{D}s/\left(\rho+\sqrt{\beta_{R}\beta_{D}^{2}s+\rho^{2}}\right)\) and \(\rho=\left(\beta_{R}\beta_{D}s+\beta_{R}-\beta_{D}\right)/2\)._  
**Proof. ** We mainly derive the PDF and MGF for \(\gamma_{\text{GI}}\).