# Machine Learning-Assisted Secure Random Communication System
## 3 Allied Concepts
### Alpha Stable (\(\alpha\)-Stable) Distribution  
\(\alpha\)-stable noise, a well-known random noise model, has been generally identified as a severe channel impairment in communication systems because of its unique properties. In contrast to AWGN, the heavy-tailed distribution of \(\alpha\)-stable noise can cause significant disruption to communication channels. \(\alpha\)-stable noise simulates impulsive channel conditions. It is highly relevant in contexts such as vehicular networks, space communications, wireless systems, and IoT scenarios, which are susceptible to excessive interference [53; 54; 55].  
1. The \(\alpha\)-stable distribution includes several subcategories of \(\alpha\)-stable noise: symmetric \(\alpha\)-stable (S\(\alpha\)S) and skewed \(\alpha\)-stable (Sk\(\alpha\)S). The overall statistical parameters of the distribution are denoted by \(S_{\alpha}\) (\(\beta\), \(\gamma\), \(\mu\)), where a random variable R follows R-\(S_{\alpha}\) (\(\beta\), \(\gamma\), \(\mu\)). The parameters responsible for defining the distribution are as follows.
2. The characteristic exponent (\(\alpha\)) controls the impulsiveness of the distribution and varies within \(\alpha\in(0,\,2]\).
3. The skewness (\(\beta\)) governs the asymmetry of the distribution, with \(\beta=-1\) creating a leftward skew and \(\beta=1\) creating a rightward skew, falling within \(\beta\in[-1,\,1]\).  
The scale parameter (\(\gamma\)) is the dispersion or scaling of the distribution within \(\gamma\in(0,\,\infty)\).
5. The location parameter (\(\mu\)) shifts the distribution along the horizontal axis with \(\mu\in(-\infty,\infty)\).  
The basic structure of the \(\alpha\)-stable noise R\(-S_{\alpha}\) (\(\beta,\,\gamma,\,\mu\)) is dependent on its characteristic function derived in [56]:  
\[\phi(\theta)=\left\{\begin{array}{l}exp\{j\mu\theta-\gamma^{\alpha}\;|\theta |^{\mu}(1-j\beta sign(\theta)tan\left(\frac{\alpha\pi}{2}\right))\};\,if\, \alpha\
eq 1\\ exp\{j\mu\theta-\gamma\;|\theta|\big{(}1+j\beta\frac{2}{\pi}sign(\theta)ln \left(\frac{\alpha\pi}{2}\right)\big{)}\};\,if\,\alpha=1\end{array}\right\} \tag{1}\]  
In this study, \(\alpha\)-stable noise was produced via the approach described in [57]. A range of probability density functions (PDFs) for R\(-S_{\alpha}\) (\(\beta,\,\gamma,\,\mu\)) were produced, as illustrated in Figure 1, by systematically varying the parameters \(\alpha\) and \(\beta\) within their respective ranges while keeping \(\mu=0\) and \(\gamma=1\) constant. This was performed to understand the basic roles of \(\alpha\) and \(\beta\) in dealing with \(\alpha\)-stable noise.  
Note: Specific cases within the \(\alpha\)-stable family, such as R\(-S_{\alpha=2}\) (\(\beta=0,\,\gamma,\,\mu\)), R\(-S_{\alpha=1}\) (\(\beta=0,\,\gamma,\,\mu\)), and R\(-S_{\alpha=0.5}\) (\(\beta=1,\,\gamma,\,\mu\)), represent Gaussian, Cauchy, and Levy noises, respectively, as shown in Figure 2. In contrast, Sk\(\alpha\)5 noise is represented by R\(-S_{\alpha}\) (\(+\beta/-\beta\), \(\gamma,\,\mu\)), and its characteristics are illustrated in Figure 1.  
When \(\alpha<2\), the corresponding distribution has a second-order moment-generating function. However, a first-order moment-generating function does not exist under such conditions. As shown in [56; 57], \(\alpha\)-stable noise accurately models heavy-tailed impulsive events that are orders of magnitude larger than the channel impairments induced by AWGN.  
Figure 1: PDF of \(\alpha\)-stable noise for parameters \(\alpha\) and \(\beta\), where \(\gamma=1.0\) and \(\mu=0\).  
Figure 2: PDF of \(\alpha\)-stable noise for various special noises.  
This characteristic makes it virtually undetectable by illegitimate receivers or eavesdroppers, while also conferring natural immunity to channel distortions. Consequently, \(\alpha\)-stable noise is an appealing candidate for physical layer covert communications.