*2.1. Array Signal Model*

Consider the case of *P* narrow farfield signals *sp*(*t*), *p* = 1, 2, ··· *P* with different DOA θ1, θ2, ... , θ*<sup>P</sup>* arriving at a uniform linear array (ULA) with *M* sensors at discrete time *t*. The DOA of the *p*th source can be written as θ*p*. The received signal of the arrays can be expressed as

$$\mathbf{Z}(t) = \mathbf{A}(\theta)\mathbf{S}(t) + \mathbf{N}(t) \tag{1}$$

where *<sup>N</sup>M*×1(*t*) <sup>=</sup> [**n**1(*t*), **<sup>n</sup>**2(*t*), ... , **<sup>n</sup>***M*(*t*)]*<sup>T</sup>* represents the impulsive noise vector which is not correlated with signals. *<sup>Z</sup>M*×1(*t*) <sup>=</sup> [*z*1(*t*), *<sup>z</sup>*2(*t*), ... , *<sup>z</sup>M*(*t*)]*<sup>T</sup>* is the measurement at time *<sup>t</sup>*, *<sup>A</sup>M*×*P*(*t*) <sup>=</sup> [*a*(θ1), *<sup>a</sup>*(θ2), ... , *<sup>a</sup>*(θ*P*)]*<sup>T</sup>* is array manifold, *<sup>S</sup>P*×1(*t*) <sup>=</sup> [*s*1(*t*),*s*2(*t*), ... ,*sP*(*t*)]*<sup>T</sup>* denotes the acoustic sources matrix, and

$$\mathfrak{a}(\theta\_p) = \left[1, e^{-j\frac{2\pi}{\lambda}d\sin\theta\_p}, \dots, e^{-j\frac{2\pi}{\lambda}(M-1)d\sin\theta\_p}\right] \tag{2}$$

is the steering vector with λ denoting the wavelength of the carrier, and *d* is the array space.
