Algorithm 1   The proposed sparse method using denoised FOCs vector
1)	Input: $\mathbf{x}$, $\mathbb{S}$, and d.
2)	Initialization: r, $\mathbf{\alpha }$, $\rho$, $ϵ$, and $\tau$.
3)	According to the associated scenarios, calculate ${\widehat{\mathbf{c}}}_x$ by (19) or (29).
4)	According to the associated scenarios, calculate $\widehat{\mathbf{Q}}$ by (27), (28), (30), or (31).
5)	Calculate $\mathbf{J}$, ${\tilde{\mathbf{A}}}_{\mathbb{D}}$, ${\tilde{\mathbf{A}}}_{\mathbb{D}}^{\prime }$, and ${\tilde{\mathbf{c}}}_x$ by Definition 2, (10), (11), and (14), respectively.
6)	Repeat the following:
	a)  Calculate $\mathbf{\mu }$ and $\Sigma$ by (38) and (39), respectively.
	b)  Update $\mathbf{\alpha }$ by (40).
	c)  Update $\mathbf{\beta }$ by (42).
	d)  If $\parallel {\mathbf{\alpha }}^{\kappa +1}-{\mathbf{\alpha }}^{\kappa }{\parallel }_2/{\parallel {\mathbf{\alpha }}^{\kappa }\parallel }_2<\tau$ or the maximum number of iterations is reached, iteration terminates.
7)	Find the grid indices of the highest peaks of (43), and output the DOAs by (44).