## 3 The Proposed Algorithm
### Joint Reweighted Sparse Representation-Based DOA Estimation Scheme  
In order to successfully apply the sparse representation theory to estimate the DOAs, we bring in the SVD technique of **Z**, i.e., \(\textbf{Z}=\textbf{U}\textbf{\Lambda}\textbf{V}^{\mathbb{H}}\), where \(\textbf{\Lambda}=\text{diag}(\omega_{1},\omega_{2},\dots,\omega_{\tilde{M}+ \tilde{N}-1})\), and \(\omega_{1}\geq\omega_{2}\geq\cdots\geq\omega_{\tilde{M}+\tilde{N}-1}\) are singular values. Let \(\textbf{V}_{s}\) be composed of the \(P\) vectors in **V** that correspond to \((\omega_{1},\omega_{2},\dots,\omega_{P})\). With \(\textbf{V}_{s}\), the dimension of the observation matrix can be further reduced [9], which is conducive to the signal reconstruction, that is:  
\[\textbf{Z}_{v}=\left[ \begin{array}{c}\textbf{F}\textbf{B}\textbf{D}\textbf{ \Lambda}_{p}\textbf{S}_{r}\textbf{V}_{s}\\ \textbf{F}\textbf{B}^{*}\textbf{D}\textbf{\Lambda}_{p}^{*}\textbf{S}_{r}\textbf {V}_{s}\end{array} \right]+\left[ \begin{array}{c}\textbf{N}_{d}\textbf{V}_ {s}\\ \textbf{N}_{d}^{*}\textbf{V}_{s}\end{array} \right]=\left[ \begin{array}{c} \textbf{B}\textbf{T}\\ \textbf{B}\textbf{T}\end{array} \right]+\left[ \begin{array}{c}\textbf{N}_{ ds}\\ \textbf{N}_{ds}\end{array} \right] \tag{18}\]  
where \(\textbf{Z}_{v}=\textbf{Z}\textbf{V}_{s}\in\mathbb{C}^{2(\tilde{M}+\tilde{N}-1) \times P}\), \(\tilde{\textbf{B}}=\textbf{F}\textbf{B}\), \(\tilde{\textbf{T}}=\textbf{D}\textbf{\Lambda}_{p}\textbf{S}_{r}\textbf{V}_{s}\), \(\tilde{\textbf{B}}=\textbf{F}\textbf{B}^{*}\), \(\tilde{\textbf{T}}=\textbf{D}\textbf{\Lambda}_{p}^{*}\textbf{S}_{r}\textbf{V}_ {s}\), \(\tilde{\textbf{N}}_{ds}\) and \(\tilde{\textbf{N}}_{ds}\) are the new noise matrices. \(\textbf{Z}_{v}\) can be divided into the following two parts; the first \((\tilde{M}+\tilde{N}-1)\) rows in \(\textbf{Z}_{v}\) make up \(\textbf{Z}_{v1}\in\mathbb{C}^{(\tilde{M}+\tilde{N}-1)\times P}\), and the remaining rows compose \(\textbf{Z}_{v2}\in\mathbb{C}^{(\tilde{M}+\tilde{N}-1)\times P}\), namely \(\textbf{Z}_{v}=[\textbf{Z}_{v1}^{\mathbb{T}},\textbf{Z}_{v2}^{\mathbb{T}}]^{ \mathbb{T}}\). By constructing two complete dictionaries \(\tilde{\textbf{B}}_{\tilde{\theta}}=\textbf{F}[\textbf{b}(\tilde{\theta}_{1}), \textbf{b}(\tilde{\theta}_{2}),\dots,\textbf{b}(\tilde{\theta}_{L})]\)and \(\bar{\mathbf{B}}_{\hat{\theta}}=\mathbf{F}[\mathbf{b}^{*}(\hat{\theta}_{1}),\mathbf{b} ^{*}(\hat{\theta}_{2}),\ldots,\mathbf{b}^{*}(\hat{\theta}_{L})]\) with \(\{\hat{\theta}_{l}\}_{l=1}^{L}(L\gg P)\) being the discretized sampling grids of all potential DOAs, the corresponding sparse representation models can be separately formulated as:  
\[\mathbf{Z}_{v1} =\mathbf{B}_{\hat{\theta}}\mathbf{T}_{\hat{\theta}}+\mathbf{N}_{ ds} \tag{19}\] \[\mathbf{Z}_{v2} =\mathbf{B}_{\hat{\theta}}\mathbf{T}_{\hat{\theta}}+\mathbf{N}_{ ds}\]  
where \(\bar{\mathbf{T}}_{\hat{\theta}}\)\(\in\mathbb{C}^{L\times P}\) and \(\bar{\mathbf{T}}_{\hat{\theta}}\)\(\in\mathbb{C}^{L\times P}\) hold the same row support with \(\bar{\mathbf{T}}\) and \(\bar{\mathbf{T}}\), respectively. For the two models in Equation (19), if they are solved independently by conventional sparse representation methods that are based on the \(l_{1}\)-norm penalty [9], the signal noncircularity cannot be utilized to extend the array aperture. In order to make the best of signal noncircularity, we design a joint sparse vector \(\mathbf{r}\)\(\in\)\(\mathbb{C}^{L\times 1}\) whose \(k\)-th element is \(\mathbf{r}(k)=\sqrt{(\bar{\mathbf{T}}_{\hat{\theta}}^{(l_{2})}(k))^{2}+(\bar {\mathbf{T}}_{\hat{\theta}}^{(l_{2})}(k))^{2}}\), then the sparsities of \(\bar{\mathbf{T}}_{\hat{\theta}}\) and \(\bar{\mathbf{T}}_{\hat{\theta}}\) are combined. In \(\mathbf{r}(k)\), \(\bar{\mathbf{T}}_{\hat{\theta}}^{(l_{2})}(k)\) and \(\bar{\mathbf{T}}_{\hat{\theta}}^{(l_{2})}(k)=\parallel\bar{\mathbf{T}}_{\hat {\theta}}(k,:)\parallel_{2}\) and \(\bar{\mathbf{T}}_{\hat{\theta}}^{(l_{2})}(k)=\parallel\bar{\mathbf{T}}_{\hat {\theta}}(k,:)\parallel_{2}\), respectively.