*2.2.* α *Stable Distribution*

Most of the traditional research methods estimating the DOA are based on Gaussian noise models. In practical situations, such as radar echo and low-frequency atmospheric noise, they consist of impulse noise with a short duration and large amplitude. The performance of the algorithm will drop significantly when the Gaussian noise model is still modeled in an impulse noise environment. The α stable distribution is a good example of such a type with significant spike noise and a Gaussian distribution. The α stable distribution's probability function does not have the closed form, which can be conveniently described by its characteristic function as

$$\phi(t) = e^{\|\text{jat} - \gamma |t|^n \left[1 + j \|\text{s} \text{gm}(t) \omega(t, u)\right] \}\tag{3}$$

where

$$\phi = \begin{cases} \tan\frac{a\pi}{\mathbb{Z}}, \alpha \neq 1\\ \frac{2}{\pi}\log|t|, \alpha = 1 \end{cases} \tag{4}$$

$$\text{sgn}(t) = \begin{cases} -1, t > 0 \\ 0, t = 0 \\ -1, t < 0 \end{cases} \tag{5}$$

α is the characteristic exponent, whose size can affect the degree of impulse and the range is 0 < α ≤ 2. γ is a dispersion parameter whose mean is consistent with the variance of the Gaussian distribution. β is a symmetric parameter, and the distribution at β = 0 is a symmetric α stable (SαS) distribution. *a* is the positional parameter. When α = 2, β = 0, it is a Gaussian distribution model. When α = 1, β = 0, it is the Cauchy distribution model. When α = 1/2, β = −1, it is the Pearson distribution model. A crucial difference between the Gaussian distribution and the α stable distribution is that the latter does not have second-order statistics so that its covariance is inaccurate.
