Algorithm 1 Decoupled joint sparse reconstruction.

Input: Received joint random measurement vector $\mathbf{y}$;

Equivalent random measurement matrix ${\mathsf{\Phi }}_e^h$;

Output: DoA indicating vector $\mathbf{s}$;

1. Estimate the noise level by random sampling in source free scenario, $\gamma =\left|\right|\mathbf{y}|{|}_2^2$;

2. Estimate the common support $\mathbf{T}=\mathbf{supp}\left(\tilde{\mathbf{d}}\right)$ by solving:

$\begin{array}{c}\underset{\tilde{\mathbf{d}}}{min}{\displaystyle \sum _{n=0}^{N-1}}{\displaystyle \sum _{h=1}^H}{d}_h{\left(n\right)}^2\hfill \\ \text{subject to }\left|\right|\mathbf{y}-\mathsf{\Theta }\tilde{\mathbf{d}}{\left|\right|}_2^2\le \gamma ;\hfill \end{array}$

3. Construct the pruned joint reconstruction matrix $\widetilde{\mathsf{{\rm Y}}}$;

$\widetilde{\mathsf{{\rm Y}}}=\left[\mathsf{{\rm Y}}\left[n_1\right],\mathsf{{\rm Y}}\left[n_2\right],\cdots ,\mathsf{{\rm Y}}\left[n_{\left|\mathbf{T}\right|}\right]\right],r\in \mathbf{T}$;

4. Solve the pruned reconstruction problem:

$\begin{array}{c}\underset{{\overline{\mathbf{r}}}_L}{min}{\parallel \mathbf{s}\parallel }_1\text{subject to }\left|\right|\mathbf{y}-\widetilde{\mathsf{{\rm Y}}}\overline{\mathbf{r}}{\left|\right|}_2^2\le \gamma ,\hfill \\ \mathbf{s}\left(\ell \right)={\sum }_{n\in \mathbf{T}}{\mathbf{r}}_{\ell }{\left(n\right)}^2,\overline{\mathbf{r}}={[{\mathbf{r}}^T\left(n_1\right),\cdots ,{\mathbf{r}}^T\left(n_{\left|T\right|}\right)]}^T;\hfill \end{array}$