# Acoustic Impulsive Noise Based on Non-Gaussian Models: An Experimental Evaluation
## 2 Parameters Estimation
### \(\alpha\)-Stable Parameters Estimation  
Theoretical reasons for statistical modeling using \(\alpha\)-stable distributions are based on the Generalized Central Limit Theorem and the stability property [8]. The Generalized Central Limit Theorem states that if the sum of i.i.d. random variables with or without finite variance converge, the limit distribution must be \(\alpha\)-stable. According to the stability property, \(\alpha\)-stable distributions are closed under convolution, i.e., the sum of two independent random variables with the same characteristic exponent is also \(\alpha\)-stable, keeping the same characteristic exponent [2]. The third reason for using this model is that the measured data exhibits heavy tails and skewness. This behavior may come from a combination of different random variables, which justify the usage of \(\alpha\)-stable model by Generalized Central Limit Theorem.  
There are different parametrizations of \(\alpha\)-stable distribution for different specifications of the characteristic function. We assume the parameters \(\mathbf{\theta}_{\alpha}=(\alpha,\beta,\gamma,\delta)\) and the following characteristic function [2]:  
\[\varphi(\omega;\mathbf{\theta}_{\alpha})=\exp(-\gamma^{\alpha}|\omega|^{\alpha}[1- j\Theta(\omega;\alpha,\beta)]+j\delta\omega), \tag{4}\]  
with  
\[\Theta=\left\{\begin{array}{ll}\beta(\tan\frac{\pi\alpha}{2})(\text{sign }\omega),&\alpha\
eq 1\\ -\beta\frac{2}{\pi}(\ln|\omega|),&\alpha=1,\end{array}\right. \tag{5}\]  
where  
\(\alpha\): is the _characteristic exponent_ such that \(0<\alpha<2\),  
\(\beta\) is the symmetry parameter such that \(-1\leq\beta\leq 1\),\(\gamma\) is the dispersion or scale parameter such that \(\gamma>0\),  
\(\delta\) is the location parameter such that \(-\infty<\delta<\infty\).  
Finally, we also assume a Symmetric \(\alpha\)-Stable (SaS) class, because it has proved to be very useful in modeling impulsive noise [11]. For such distribution class, \(\beta=0\) and \(\delta=0\)[8].  
Considering MLE for \(\alpha\)-stable distribution, the fundamental issue is that there is no known a general closed formula for the probability density. Only when \(\alpha\) has a specific value, there are expressions for densities. This is a problem regarding the calculation of the log-likelihood function in Equation (3).  
However, we apply the direct integration method, detailed in [25], to estimate \(\alpha\) and \(\gamma\) parameters by MLE for the SaS model, with the characteristic function described as  
\[\varphi(\omega;\alpha,\gamma)=\exp(-\gamma^{\alpha}|\omega|^{\alpha}). \tag{6}\]  
The primary parameter, \(\alpha\), describes the heaviness of the distribution tails. The smaller the \(\alpha\), the heavier are the tails; thus, more impulsive is the noise. When the parameter \(\alpha\) is close to 0 or 1, the density function may not be accurate because of numerical issues. On the other hand, the scale parameter \(\gamma\) behaves similarly to the variance of the Gaussian distribution. However, the \(\alpha\)-stable distributions have unbounded variance. The only exception is for \(\alpha=2\) (the Gaussian case) when the \(\alpha\)-stable distribution has a second-order moment.  
The \(\alpha\)-stable distributions have finite moments only for order lower than the parameter \(\alpha\). For instance, assuming a moment of order equal to \(p\), the \(\alpha\)-stable distribution has the following relation with \(\alpha\)  
\[\begin{array}{ll}\alpha<2,&E[X^{p}]\rightarrow\infty\qquad\forall p\geq\alpha \\ \alpha<2,&E[X^{p}]<\infty\qquad\forall 0\leq p<\alpha\\ \alpha=2,&E[X^{p}]<\infty\qquad\forall p\geq 0\end{array} \tag{7}\]