**4. Improved Algorithm for Likelihood Function**

In the practical engineering application, to guarantee the real-time and effectiveness of the estimation, the observation matrix of the array is obtained with a limited number of snapshots. Assuming L observations at time *k*, the array covariance matrix is calculated as *R*ˆ *<sup>k</sup>* = *X*(*tk*)*X*(*tk*) *<sup>H</sup>*/*L*. We assume that the noise vector *N*(*t*) is independent to the target signal and has a SαS distribution with a characteristic exponent of α. From [25], if the array observation matrix *Z<sup>k</sup>* at time *<sup>k</sup>* is obtained, the FLOM matrix is defined as

$$\psi\_{i,j} = \mathbb{E}\left\{ \mathbf{Z}\_{i,j}(k) \Big| \mathbf{Z}\_{j,i}(k) \Big|^{p-2} \mathbf{Z}\_{j,i}^\*(k) \right\} \\ 1 < p < a \le 2 \tag{19}$$

where ψ*i*,*<sup>j</sup>* represents the (*i*, *j*)th element of **Ψ***k*, and (·) <sup>∗</sup> represents conjugate operation. The dimension of matrix **Ψ***<sup>k</sup>* is *M* × *M*. In [25], the authors derived the form of the FLOM matrix as

$$\mathbf{\varUpsilon}\_{k} = \mathfrak{a}(\theta\_{k}) \mathbf{R}\_{\mathfrak{s}} \mathbf{a}^{H}(\theta\_{k}) + r \mathbf{I}\_{M} \tag{20}$$

where *R<sup>s</sup>* and *r* represent the source and additive noise of the FLOM matrix, respectively. As can be seen from Equation (20), the (*i*, *j*)th FLOM matrix element is defined as

$$\psi\_{i,j} = \frac{\sum\_{l=1}^{L} \mathbf{Z}\_i(k) \left| \mathbf{Z}\_j(k) \right|^{p-2} \mathbf{Z}\_j^\*(k)}{L} \tag{21}$$

Fractional moment *p* must satisfy 1 < *p* < α ≤ 2. The FLOM is used to replace the covariance matrix of the signal in impulse noise, and then the eigendecomposition is performed on **Ψ***<sup>k</sup>* in the MUSIC algorithm to obtain the noise subspace *Un*. The form of the FLOM-MUSIC spatial spectrum estimation function is

$$\log(\mathbf{Z}\_k | \mathbf{X}\_k) = P\_{\text{FLO-MULIC}}(\mathbf{X}\_k) = \left| \frac{1}{a^{\text{fl}}(\mathbf{C} \mathbf{X}\_k) \mathbf{U}\_{\text{ll}} \mathbf{U}\_{\text{ll}}^{\text{H}} a(\mathbf{C} \mathbf{X}\_k)} \right|^{\zeta} \tag{22}$$

where **<sup>C</sup>** <sup>=</sup> [**1**, **<sup>0</sup>**], and the **<sup>C</sup>***X<sup>k</sup>* represents source azimuth information. *<sup>a</sup>*(·) is a space vector, and <sup>ζ</sup> <sup>∈</sup> **<sup>R</sup>**<sup>+</sup> represents an exponential weighting of the spatial spectrum. The response of the traditional MUSIC spatial spectral beamformer in an impulse noise environment is distorted, which can result in a significant degradation in the performance of the resampling step. After being weighted, the particles can be moved to the high likelihood region to the resampling performance.
