## 2 Statistical Characteristics of \(\alpha\)-\(\mu\)/IGA Distribution
### Univariate Case
Thus, by using (6) and applying the change of variables, the corresponding PDF of \(\gamma\) can be obtained as  
\[f_{\gamma}(\gamma)=\frac{\alpha\gamma^{-1}}{2\Gamma(\mu)\Gamma(n)}H_{1,1}^{1,1 }[\lambda\gamma^{0.5\alpha}]^{(1-n,0.5\alpha)}_{(\mu,1)}, \tag{7}\]  
where \(\lambda=(\beta/(n-1)\overline{\gamma})^{0.5\alpha}\). Obviously, (7) can reduce to [39] (Equation (2)) as \(\alpha=2\) by using the identity [38] (Equation (1.60)). Invoking the definition of the univariate Fox's H-function and using the similar derived process of (6), the corresponding CDF and MGF of \(\gamma\) can be obtained, respectively, as  
\[F_{\gamma}(\gamma)=\int_{0}^{\gamma}f_{\gamma}(\gamma)d\gamma=\frac{1}{\Gamma( \mu)\Gamma(n)}H_{2,2}^{1,2}[\lambda\gamma^{0.5\alpha}]^{(1-n,0.5\alpha),(1,1) }_{(\mu,1),(0,1)}], \tag{8}\]  
\[MGF_{\gamma}(s)=\int_{0}^{\infty}\exp(-s\gamma)f_{\gamma}(\gamma)d\gamma= \frac{\alpha}{2\Gamma(\mu)\Gamma(n)}H_{2,1}^{1,2}[\lambda s^{-0.5\alpha}]^{(1 -n,0.5\alpha),(1,0.5\alpha)}_{(\mu,1)}]. \tag{9}\]  
It is noted here that (8) and (9) can also reduce to [40] (Equation (2)), [41] (Equation (2)) as \(\alpha=2\), respectively. Interestingly, the Fisher-Snedecor \(\mathcal{F}\) model is also a special case of the \(\alpha\)-\(\mathcal{F}\) distribution. In [32], the \(\alpha\)-\(\mathcal{F}\) model was deduced from the \(\mathcal{F}\) model by using a power transformation, and its signal envelope \(R\) is expressed as \(R^{\alpha}=\sum\limits_{i=1}^{\mu}Z\big{(}X_{i}^{2}+Y_{i}^{2}\big{)}\). From the right-hand side of (1), the \(\alpha\)-\(\mu\)/IGA and \(\alpha\)-\(\mathcal{F}\) models have the identical mathematical form and physical parameters including \(\alpha\), \(\mu\), and \(n\). However, they have different physical interpretations from the view of the physical processes of the signal envelopes. The former represents the mean of the non-linear multipath envelopes is subject to the shadowing effects following the IGA distribution, whereas the latter can be considered as the composite envelopes following the \(\mathcal{F}\) distribution encounter the non-linear propagation in a nonhomogeneous environment. For the sake of comparison, we evaluated the impacts of the physical parameters on the PDFs of both models in Figure 1. It can be observed from Figure 1 that as \(\alpha=2\), both models reduce to the \(\mathcal{F}\) one, while as \(\alpha\
eq 2\) and \(n=5\) (moderate shadowing), they diverge greatly from each other. With the increase in the value of shadowing parameters (\(n=5{\rightarrow}50\)), they tend to have similar results, which indicates the shadowing effects decrease, and they reduce to the \(\alpha\)-\(\mu\) model. Moreover, the parameter \(\mu\) shows the same effect on the PDFs of both models when \(\mu\) grows. This is because the nonlinear feature of the \(\alpha\)-\(\mathcal{F}\) model affects both \(\mu\) and \(n\), while only \(\mu\) is affected by its nonlinear parameter in the \(\alpha\)-\(\mu\)/IGA model. As expected, the PDF curves of both models converge to around the average SNR (\(\overline{\gamma}=1\)) when the values of \(\alpha\) and \(\mu\) get larger, which means good channel conditions. Recently, several mixed PDF models were presented to provide some accurate approximated expressions via the Gaussian-Laguerre quadrature approximation and overcome certain mathematical difficulties, such as the mixture Gamma (MG) distribution [41] and the mixture inverse Gaussian distribution [7]. By applying similar approaches, the exact closed-form expression of (6) can also be approximated as a mixed form of the IGA or the \(\alpha\)-\(\mu\) distribution. In [26], the mixture IGA model has been considered to approximate the \(\alpha\)-\(\mu\)/IGA model just as the MG distribution was used to approximate the \(\alpha\)-\(\mu\)/GA model in [6]. However, it was reported in [26] that the number of the truncation term is large and goes up to 45 in order to achieve the error \(<\)10\({}^{-5}\). To this end, we consider an alternative new form to approximate the \(\alpha\)-\(\mu\)/IGA model.