## 2 Statistical Characteristics of \(\alpha\)-\(\mu\)/IGA Distribution  
### Univariate Case  
In the \(\alpha\)-\(\mu\)/IGA composite fading environments, the amplitude of the multipath envelopes of the \(\alpha\)-\(\mu\) distribution suffers from random fluctuations induced by an IGA shadowing RV. Thus, the received signal envelope \(R\) can be expressed as  
\[R^{2}=ZR_{\alpha-u}^{\alpha}=\sum_{i=1}^{\mu}Z(X_{i}^{2}+Y_{i}^{2}), \tag{1}\]  
where \(R_{\alpha-\mu}\) denotes the envelope of \(\alpha\)-\(\mu\) distribution, \(Z\) follows the IGA distribution, \(\alpha\) is the non-linear parameter and \(\alpha>0\), \(\mu\) represents the number of multipath clusters, and \(\mu\geq 1/2\), \(X_{i}\), and \(Y_{i}\) are mutually independent Gaussian RVs with zero-mean and average power \(\sigma^{2}\), which represent the in-phase and quadrature components of cluster \(i\). Hence, the PDF of \(R\) can be obtained by averaging the conditional PDF of the \(\alpha\)-\(\mu\) fading process over the statistics of the IGA shadowing RV. Mathematically,  
\[f_{R}(r)=\int_{0}^{\infty}f_{R|Z}(r|z)f_{Z}(z)dz, \tag{2}\]where \(f_{R|Z}(r|z)\) is the conditional PDF of the \(\alpha\)-\(\mu\) fading distribution in [36] with respect to RV \(z\), as follows  
\[f_{R|Z}(r|z)=\frac{\alpha\beta^{0.5\alpha\mu}r^{\alpha\mu-1}}{\Gamma(\mu)(z \Omega)^{0.5\alpha\mu}}\exp[-\frac{\beta^{0.5\alpha}r^{\alpha}}{(z\Omega)^{0.5 \alpha}}], \tag{3}\]  
where \(\mathbb{E}\big{[}r^{2}\big{]}=\Omega\), \(\beta=\Gamma\big{(}\mu+\frac{2}{\alpha}\big{)}/\Gamma(\mu)\) represents the power scale factor, \(\mathbb{E}[\cdot]\) denoting the statistical mean, and \(\Gamma(\cdot)\) is the gamma function defined in ([37], Equation (8.310.1)). In (2), \(f_{Z}(z)\) is the PDF of the IGA distribution to model shadowing fading in [15], and can be given by  
\[f_{Z}(z)=\frac{(n-1)^{n}}{\Gamma(n)}z^{-(n+1)}\exp(-\frac{n-1}{z}), \tag{4}\]  
where \(n\) denotes the shape parameter and \(n>1\), \(\mathbb{E}[z]=1\). Note that \(n\to 1\) denotes the received signals are subject to the heavy shadowing; on the contrary, \(n\to\infty\) represents the absence of shadowing. Substituting (3) and (4) in (2), the PDF of the received signal envelope \(R\) can be expressed as  
\[f_{R}(r)=\frac{\alpha\beta^{0.5\alpha\mu}(n-1)^{n}r^{\alpha\mu-1}}{\Gamma(\mu )\Gamma(n)\Omega^{0.5\alpha\mu}}\int_{0}^{\infty}z^{-(n+0.5\alpha\mu+1)}\exp( -\frac{n-1}{z})\exp[-\frac{\beta^{0.5\alpha}r^{\alpha}}{(z\Omega)^{0.5\alpha }}]dy. \tag{5}\]  
To obtain the closed-form expression of the PDF of the composite envelope \(R\), we represent the exponential form \(\exp(-x)\) in terms of [PERSON]'s H function \(\mathrm{H}_{0,1}^{1,0}\big{[}x|_{(0,1)}^{-}\big{]}\) by using the identity ([38], Equation (1.125)). After a simple variable transformation and based on the definition of [PERSON]'s H function in [38] (Equation (1.2)), (5) can be rewritten as  
\[f_{R}(r)=\frac{\alpha r^{-1}}{\Gamma(\mu)\Gamma(n)}H_{1,1}^{1,1}\big{[}\frac{ \beta^{0.5\alpha}r^{\alpha}}{\big{[}(n-1)\Omega\big{]}^{0.5\alpha}}\big{]}^{(1 -n,0.5\alpha)}\big{]}, \tag{6}\]  
where \(H_{p,q}^{m,n}[\cdot|\cdot]\) denotes a univariate [PERSON]'s H-function. Similar to the \(\eta\)-\(\mu\)/IGA and \(\kappa\)-\(\mu\)/IGA models, the \(\alpha\)-\(\mu\)/IGA composite distribution is also a generalized composite distribution. As \(\alpha=2\), this model is equivalent to the Fisher-Snedecor \(\mathcal{F}\) composite fading model proposed in [28]. As \(\mu=1\), the \(\alpha\)-\(\mu\)/IGA distribution reduces to Weibull/IGA. By setting \(\alpha=2\) and \(\mu=1\), this model results in Rayleigh/IGA, whilst \(n\to\infty\), the \(\alpha\)-\(\mu\) model becomes its special case. Here, we let the instantaneous SNR \(\gamma=r^{2}E_{s}/N_{0}\), where \(E_{s}\) is the average energy per symbol and \(N_{0}\) denotes the single-sided power spectral density, then the average SNR can be given as \(\overline{\gamma}=\ \Omega E_{s}/N_{0}\).