## 2 Statistical Characteristics of \(\alpha\)-\(\mu\)/IGA Distribution
### Bivariate Case  
In the performance evaluations of wireless digital communication systems, the bivariate distributions are often used to analyze the performance of diversity receivers over correlated fading environments, such as [1; 29]. So far, the bivariate \(\alpha\)-\(\mu\)/IGA composited distribution has not been considered in the previous works. In this context, we will derive the analytical expressions of the statistical properties of the bivariate \(\alpha\)-\(\mu\)/IGA distribution. Let \(R_{i}\) (\(i=1\), \(2\)) represent the channel fading envelope of \(\alpha\)-\(\mu\) processes, then, the joint conditional PDF between \(R_{1}\) and \(R_{2}\) can be expressed from [36] (Equation (28)) as  
\[ \begin{array}{ll}f_{R_{1}|W_{1},R_{2}|W_{2}}\big{(}r_{1}|w_{1},r_{2}|w_{1}\big{)} &=\frac{\pi_{1}\pi_{2}r_{1}^{0.5\alpha_{1}(\mu+1)-1}r_{2}^{0.5\alpha_{2}(\mu+1 )-1}r_{2}^{-\alpha_{1}(\mu-1)/2}}{\Gamma(\mu)(\sqrt{(\beta_{1}/w_{1})\Gamma_{ 1})^{0.5\alpha_{1}}((\beta_{2}/w_{2})\Gamma_{2})^{0.5\alpha_{2}}(\mu)}(1-\rho_ {N})}\\ &\times\exp\left[-\frac{1}{1-\rho_{N}}\left(\frac{r_{1}^{\alpha_{1}}\rho_{1}^{ 0.5\alpha_{1}}}{(w_{1})\Gamma_{0.5\alpha_{1}}}+\frac{r_{2}^{\alpha_{2}}\rho_{ 0.5\alpha_{2}}^{0.5\alpha_{2}}}{(w_{2}\epsilon_{2})^{0.5\alpha_{2}}}\right) \right]I_{H}-1\left[\frac{2}{(1-\rho_{N})}\sqrt{\frac{\rho_{N}r_{1}^{\alpha_{1} }r_{2}^{2}\rho_{1}^{0.5\alpha_{1}}\rho_{2}^{0.5\alpha_{2}}}{(w_{1}\Gamma_{0.5 \alpha_{1}})^{0.5\alpha_{1}}(w_{2}\Omega_{2})^{0.5\alpha_{2}}}}\right],\end{array}  \tag{14}\] \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{2}{c}{\multirow{2}{*}{
\begin{tabular}{c} **Channel** \\ **Parameters** \\ \end{tabular} }} & \multicolumn{4}{c}{\(N=8\)} & \multicolumn{4}{c}{\(N=20\)} \\ \cline{3-10}  & & \multicolumn{2}{c}{**Mixed IGA**} & \multicolumn{2}{c}{**Mixed \(\mathbf{a}\)-\(\mu\)**} & \multicolumn{2}{c}{**Mixed IGA**} & \multicolumn{2}{c}{**Mixed \(\mathbf{a}\)-\(\mu\)**} \\ \cline{3-10}  & & **MSE** & **KLD** & **MSE** & **KLD** & **MSE** & **KLD** & **MSE** & **KLD** \\ \hline \(\alpha=2\), & \(n=2\) & 0.001158 & 0.007285 & 1.148 \(\times\)\(10^{-5}\) & 0.009336 & 9.947 \(\times\)\(10^{-5}\) & 0.001056 & 3.008 \(\times\)\(10^{-8}\) & 6.149 \(\times\)\(10^{-4}\) \\ \(\mu=2\) & \(n=20\) & 0.03936 & 0.4076 & 2.766 \(\times\)\(10^{-5}\) & 0.009583 & 0.008485 & 0.09145 & 2.933 \(\times\)\(10^{-19}\) & 1.816 \(\times\)\(10^{-9}\) \\ \hline \(\alpha=2\), & \(n=1\) & 0.045102 & 1.5389 & 4.759 \(\times\)\(10^{-11}\) & 2.399 \(\times\)\(10^{-5}\) & 0.02487 & 0.5276 & 6.964 \(\times\)\(10^{-20}\) & 8.243 \(\times\)\(10^{-10}\) \\ \(n=5\) & \(n=5\) & 1.491 \(\times\)\(10^{-5}\) & 2.3875 \(\times\)\(10^{-4}\) & 1.258 \(\times\)\(10^{-6}\) & 4.8473 \(\times\)\(10^{-4}\) & 1.886 \(\times\)\(10^{-8}\) & 3.608 \(\times\)\(10^{-6}\) & 1.047 \(\times\)\(10^{-10}\) & 1.048 \(\times\)\(10^{-5}\) \\ \hline \(\alpha=2\), & \(n=1\) & 0.1449 & 0.3154 & 2.345 \(\times\)\(10^{-12}\) & 5.906 \(\times\)\(10^{-6}\) & 0.03569 & 0.08006 & 2.536 \(\times\)\(10^{-7}\) & 2.433 \(\times\)\(10^{-8}\) \\ \(n=5\) & \(n=5\) & \(2.526\times\)\(10^{-5}\) & 8.651 \(\times\)\(10^{-4}\) & 2.366 \(\times\)\(10^{-4}\) & 0.007668 & 1.012 \(\times\)\(10^{-4}\) & 1.378 \(\times\)\(10^{-4}\) & 8.094 \(\times\)\(10^{-7}\) & 4.4059 \(\times\)\(10^{-4}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Comparisons of the MSE and the KLD for the exact PDF and the mixed PDFs with different channel parameters (\(\alpha\), \(\mu\), and \(n\)) and the truncated term (\(N\)). Figure 2: Impacts of different system and channel parameters on the exact PDF in (7), the mixed \(\alpha\)-\(\mu\) PDF in (10), and the mixed IGA PDF in (13). where \(\alpha_{i}\) is the nonlinear fading parameter, \(\rho_{N}\) is the power correlation coefficient between \(R_{1}^{2}\) and \(R_{2}^{2}\), and \(I_{\mu-1}(x)\) represents the modified Bessel function of the first kind and order (\(\mu-1\)) defined in [36] (Equation (9.210/1)), \(\beta_{i}=~{}\Gamma(\mu+2/\alpha_{i})/\Gamma(\mu)\).