## 3 Performance Analysis
### Performance Analysis of MGF Based on Harmonic Mean of the Double Gamma Distribution System
Then, by using the PDF of \(\gamma_{\text{GL}}\), which is denoted as \(f_{\gamma_{\text{GL}}}(x)\), we can obtain the PDF of \(1/\gamma_{\text{GL}}\), \(Y=1/\gamma_{\text{GR}}+1/\gamma_{\text{GD}}\) and \(Z=1/Y\), which are presented as follows:  
\[f_{1/\gamma_{\text{GL}}}(x)=\frac{1}{x^{2}}f_{\gamma_{\text{GL}}}\left(\frac{1 }{x}\right)\cdot U(x),\] (20-a)  
\[f_{Y}(y)=\int_{0}^{y}f_{1/\gamma_{\text{GR}}}(y-x)f_{1/\gamma_{\text{GD}}}(x) dx\cdot U(y),\] (20-b)  
\[f_{Z}(z)=\frac{1}{z^{2}}\int_{0}^{\frac{1}{z}}f_{1/\gamma_{\text{GR}}}\left( \frac{1}{z}-x\right)f_{1/\gamma_{\text{GD}}}(x)dx\cdot U(z)\] (20-c)  
where \(U(z)=1\) for \(z\geq 0\) and \(U(z)=0\) for \(z<0\). Then, by substituting the PDF of \(\gamma_{\text{GL}}\), which is \(G(X_{\text{I}},\alpha_{\text{I}},\beta_{\text{I}})\) into Equation (20-c), we derive the following:  
\[f_{Z}(z)=\int_{0}^{1}\frac{z^{\alpha_{R}+\alpha_{D}-1}\exp\left(-2\left(\frac{ 1}{\beta_{R}t}+\frac{1}{\beta_{D}(1-t)}\right)\right)}{\Gamma(\alpha_{R}) \Gamma(\alpha_{D})\beta_{R}^{\alpha_{R}}\beta_{D}^{\alpha_{D}}t^{\alpha_{R}+1 }(1-t)^{\alpha_{D}+1}}dt\cdot U(z). \tag{21}\]  
According to the MGF definition \(M_{Z}(s)=\int_{0}^{\infty}e^{-sz}f_{Z}(z)dz\), the following expression is obtained:  
\[M_{Z}(s;\alpha_{R},\alpha_{D},\beta_{R},\beta_{D})=\int_{0}^{1} \frac{1}{\Gamma(\alpha_{R})\Gamma(\alpha_{D})\beta_{R}^{\alpha_{R}}\beta_{D}^ {\alpha_{D}}t^{\alpha_{R}+1}(1-t)^{\alpha_{D}+1}}\times \tag{22}\] \[\left(\int_{0}^{\infty}z^{\alpha_{R}+\alpha_{D}-1}\exp\left(- \left(\frac{1}{\beta_{R}t}+\frac{1}{\beta_{D}(1-t)}+s\right)/2 dz\right)dt\right.\]  
When \(\int_{0}^{\infty}x^{\
u-1}\exp(-ux)dx=\Gamma(v)/u^{\
u}\) (\(u>0\), \(v>0\)) is used [23], the MGF of Equation (22) can be rewritten as  
\[M_{Z}(s;\alpha_{R},\alpha_{D},\beta_{R},\beta_{D})=\int_{0}^{1}\frac{\beta_{ R}^{\alpha_{D}}\cdot\beta_{D}^{\alpha_{R}}\cdot t^{\alpha_{D}-1}(1-t)^{\alpha_{R}-1}}{ \text{B}(\alpha_{R},\alpha_{D})\cdot\left(\beta_{D}+2\Delta t-\beta_{R}\beta _{D}st^{2}\right)^{\alpha_{R}+\alpha_{D}}}dt=\int_{0}^{1}\Omega(s,t)dt, \tag{23}\]  
where B(\(\alpha_{R}\), \(\alpha_{D}\)) is the beta function. Finally, according to the definition of the Appell function F1 [23], a concise form of the MGF is obtained as  
\[M_{Z}(s;\alpha_{R},\alpha_{D},\beta_{R},\beta_{D})=\left(\frac{\beta_{D}}{ \beta_{R}}\right)^{\alpha_{D}}\times F_{1}(\alpha_{D};\alpha_{R}+\alpha_{D}, \alpha_{R}+\alpha_{D};\alpha_{R}+\alpha_{D};k_{1},k_{2}). \tag{24}\]  
Hence, the final MGF only involves a single integral, which can be solved by quad and other integral functions in MATLAB or Appell F1 function in Mathematics. In particular, when the PDF parameters of \(X_{\text{R}}\) and \(X_{\text{D}}\)  
where \({}_{2}F_{1}(\cdot)\) represents the hypergeometric function [29]. The function \({}_{2}F_{1}(\cdot)\) can also be expressed as the following simple integral:  
Figure 5:0  
For the coherently \(M\)-PSK modulation, the system BER is defined as  
Figure 5:1  
With the help of the MGF derived in Equation (22), we can then compute the BER according to [27] or use the double integral function in MATLAB directly, where \(g_{\text{PSK}}=\sin^{2}(\pi/M)\). If \(\Theta=\frac{(M-1)\pi}{M}\), Equation (28) can be further simplified to  
\[ \begin{array}{l}P_{\text{BER}}^{\text{GG}}\approx\left(\frac{\Theta}{2\pi}- \frac{1}{6}\right)M_{\text{Z}}(g_{\text{PSK}};\alpha_{R},\alpha_{D},\beta_{R}, \beta_{D})+\frac{1}{4}M_{\text{Z}}\left(\frac{4}{3}g_{\text{PSK}};\alpha_{R},\alpha_{D},\beta_{R},\beta_{D}\right)\\ +\left(\frac{\Theta}{2\pi}-\frac{1}{4}\right)M_{\text{Z}}\left(\frac{g_{\text {PSK}}}{\sin^{2}\Theta};\alpha_{R},\alpha_{D},\beta_{R},\beta_{D}\right)\end{array}  \tag{29}\]  
Similarly, the outage probability \(P_{\text{out}}\) is defined as the probability value when the instantaneous total SNR is lower than the fixed threshold (\(R_{\text{th}}\)).