## 3 Performance Analysis
### Performance Analysis of MGF Based on Harmonic Mean of the Double Gamma Distribution System
\begin{array}{l}\sum\limits_{n=1}^{N}\frac{w_{n}}{\sqrt{n}}\exp(-s_{i}\exp( \sqrt{2}\sigma_{\rm G}a_{n}+\mu_{\rm G}))=\\ \sum\limits_{n=1}^{N}\frac{w_{n}}{\sqrt{n}}\exp(-s_{i}\exp(\sqrt{2}\sigma_{\rm A }a_{n}+\mu_{\rm A}))\cdot\sum\limits_{n=1}^{N}\frac{w_{n}}{\sqrt{n}}\exp(-s_{i }\exp(\sqrt{2}\sigma_{\rm B}a_{n}+\mu_{\rm B}))\end{array}  \tag{33}\]  
where \(w_{n}\) and \(a_{n}\) refer to literature [22], \(P_{\rm S}\big{|}H_{\rm LR}\big{|}_{2}/N_{\rm W}\)\(\sim\) LogN\((\mu_{\rm A},\sigma_{\rm A}^{2})\), \(P_{\rm R}\big{|}H_{\rm LD}\big{|}_{2}/N_{\rm P}\)\(\sim\) LogN\((\mu_{\rm B},\sigma_{\rm B}^{2})\). The two equations in the simultaneous Equation (33) can solve \(\mu_{\rm G}\) and \(\sigma_{\rm G}\) by using the fsolve function of MATLAB. The Mehta algorithm has high precision, but the parameters of \(\mu_{\rm G}\) and \(\sigma_{\rm G}\) obtained by equation solution lacks analytical expression. Similar to the modeling method of Equation (16), joint parameter optimization based on the principle of minimum difference of the MGF curve is conducted to fit the LogN distribution parameters of variable \(G\) using an objective function  
\[\min_{s1,s2}\!\!\sum\limits_{k=1}^{N}(M_{\rm GR}((s_{k};\alpha_{R},\beta_{R}) )-M_{\rm LR}(s_{k};\mu_{R},\sigma_{R}))^{2}. \tag{34}\]  
If the threshold is \(R_{\rm th}\) and \(\gamma=\exp(2R_{\rm th})\)\(-\)1, then the \(P_{\rm out}^{\rm LL}\) is  
\[P_{\rm out}^{\rm LL}=P_{\rm r}(I<R_{\rm th})\approx P_{\rm r}\Big{(}\gamma_{ \rm LL}<\gamma\Big{)}. \tag{35}\]  
By substituting the cumulative distribution function \(F(\lambda)=Q(-(\ln\gamma+\mu_{\rm G}))/\sigma_{\rm G})\) of \(\gamma_{\rm LL}\) into Equation (35), we obtain  
\[P_{\rm out}^{\rm LL}=\sum\limits_{\rm j=0}^{1}p_{\rm j}Q(-\frac{\ln\gamma+ \mu_{\rm GJ}}{\sigma_{\rm GJ}}), \tag{36}\]  
where \(p_{0}=1-p\), and \(p_{1}=p\), which represent the probability of whether impulse noise exists in the power line channel. \(\mu_{\rm GJ}\) and \(\sigma_{\rm GJ}\) are the mean and variance of the attenuation distribution for the Gauss channel and the channel with impulse noise \((j=0,1)\), respectively. The BER \(P_{\rm BER}^{\rm LL}\) of the system is obtained by using \(F(\lambda)\) in the following equation:  
\[P_{\rm BER}^{\rm LL}=-\sum\limits_{\rm j=0}^{1}p_{\rm j}\int\limits_{0}^{\infty }\frac{dQ(\sqrt{\lambda})}{d\lambda}F(\lambda)d\lambda=\sum\limits_{\rm j=0}^{1 }p_{\rm j}\int\limits_{0}^{\infty}\frac{1}{2\sqrt{2\pi\lambda}}\exp(-\frac{ \lambda}{2})Q(-\frac{\ln\gamma+\mu_{\rm GJ}}{\sigma_{\rm GJ}})d\lambda. \tag{37}\]  
To simplify the calculation, let \(t=(-\mu_{\rm GJ}-\ln\lambda)/\sigma_{\rm GJ}\), the integral transformation of the above equation is defined as follows:\[P_{\text{BER}}^{\text{LL}}=\sum\limits_{\text{j=0}}^{1}p_{\text{j}}\int\limits_{- \infty}^{\infty}\frac{\sigma_{\text{G}}}{2\sqrt{2\pi}}\exp(-\frac{-\mu_{\text{ Gj}}-t\sigma_{\text{Gj}}}{2})\cdot\exp(-\exp(-\mu_{\text{Gj}}-t\sigma_{\text{Gj}} +\ln 0.5))\cdot Q(t)\cdot(-\alpha_{\text{Gj}}\exp(-\mu_{\text{Gj}}-t\sigma_{ \text{Gj}}))dt. \tag{38}\]  
Then, \(\exp(-\exp(x))\) is approximated to the form of Gaussian function weighting, and the BER \(P_{\text{BER}}^{\text{LL}}\) of the system is expressed as follows [11]:  
\[P_{\text{BER}}^{\text{LL}}=-\sum\limits_{\text{j=0}}^{1}\sum\limits_{\text{k= 1}}^{4}p_{\text{j}}\frac{Z_{\text{kj}}}{X_{\text{kj}}}Q(\frac{Y_{\text{kj}}}{ \sqrt{1+X_{\text{kj}}^{2}}}), \tag{39}\]  
which includes  
\[ \begin{gathered} X_{\text{kj}}=\sqrt{2\sigma_{\text{Gj}}^{2}/R_{ \text{kk}}^{2}}\\ Y_{\text{kj}}=\frac{4\alpha_{\text{G}}(\mu_{\text{Gj}}+\ln 0.5-R_{ \text{kk}})-\mu_{\text{Gj}}R_{\text{kk}}^{2}}{2X_{\text{k}}R_{\text{kk}}^{2}} \\ Z_{\text{kj}}=0.5R_{\text{1k}}\sigma_{\text{Gj}}\exp(\frac{-\mu_{ \text{Gj}}+\gamma_{\text{k}}^{2}}{2})\exp[-(\frac{-\mu_{\text{Gj}}+\ln 0.5-R_{ \text{kk}}}{R_{\text{kk}}})^{2}]\end{gathered}  \tag{40}\]