Table 1.

A novelty direction of arrival (DOA) estimation approach. MUSIC: multiple signal classification.

Input	The received signal vector ${\tilde{\mathbf{x}}}_{\mathbb{S}}\left[k\right]={\mathbf{A}}_{\mathbb{S}}\mathbf{s}\left[k\right]+\mathbf{n}\left[k\right],k=1,2,\cdots ,K$.
Output	DOA Estimation.
Step 1	Compute the covariance matrix ${\tilde{\mathbf{R}}}_{\mathbb{S}}=\sum _{k=1}^K{\tilde{\mathbf{x}}}_{\mathbb{S}}\left[k\right]{\tilde{\mathbf{x}}}_{\mathbb{S}}^H\left[k\right]$.
Step 2	Reshape ${\tilde{\mathbf{R}}}_{\mathbb{S}}$ to get the signal vector of coarray ${\mathbf{x}}_{\mathbb{D}}=\mathbf{F}\mathrm{vec}\left({\mathbf{R}}_{\mathbb{S}}\right)$.
Step 3	Optimize (14) to get the covariance matrix ${\tilde{\mathbf{R}}}_{\mathbb{V}}$.
Step 4	Optimize (18) or (19) to get the denoised covariance matrix ${\mathbf{R}}_{\mathbb{V}}$.
Step 5	Perform eigenvalue decomposition of ${\mathbf{R}}_{\mathbb{V}}$ and construct ${\mathbf{U}}_{N_{\mathbb{V}}}=[{\mathbf{u}}_1,\cdots ,{\mathbf{u}}_{|{\mathbb{V}}^{+}|-Q}]$ where $\{{\mathbf{u}}_i,i=1,\cdots ,|{\mathbb{V}}^{+}|-Q\}$ is the eigenvector corresponding to the $|{\mathbb{V}}^{+}|-Q$ smallest eigenvalues.
Step 6	Compute $P_{MUSIC}=\frac{1}{{\mathbf{a}}_{\mathbb{V}}^H\left(\theta \right){\mathbf{U}}_{N_{\mathbb{V}}}{\mathbf{U}}_{N_{\mathbb{V}}}^H{\mathbf{a}}_{\mathbb{V}}\left(\theta \right)}$ and find the Q largest peaks which correspond to the estimation of DOAs.