## 3 The Proposed Algorithm
### Joint Reweighted Sparse Representation-Based DOA Estimation Scheme
More specifically, \(\mathbf{T}_{\hat{\theta}}^{(l_{2})}(k)\) is equal to the \(l_{2}\)-norm of the \(k\)-th row in \(\bar{\mathbf{T}}_{\hat{\theta}}\); similarly, \(\bar{\mathbf{T}}_{\hat{\theta}}^{(l_{2})}(k)\) is equal to the \(l_{2}\)-norm of the \(k\)-th row in \(\bar{\mathbf{T}}_{\hat{\theta}}\). Let \(\hat{\mathbf{r}}\) represent the solution vector of the sparse signal reconstruction, \(\hat{\mathbf{r}}(k)\geq\mathbf{r}(k)\), \(k=1,2,\ldots,L\). As a result, based on the \(l_{1}\)-norm constrained minimization, a joint sparse representation scheme is designed as follows:  
\[\text{min}\parallel\hat{\mathbf{r}}\parallel_{1},\quad\text{s.t.}\parallel \mathbf{Z}_{v1}-\bar{\mathbf{B}}_{\hat{\theta}}\bar{\mathbf{T}}_{\hat{\theta} }\parallel_{F}^{2}\leq\eta_{1}\quad\parallel\mathbf{Z}_{v2}-\bar{\mathbf{B}}_{ \hat{\theta}}\bar{\mathbf{T}}_{\hat{\theta}}\parallel_{F}^{2}\leq\eta_{2} \tag{20}\]  
where \(\eta_{1}\) and \(\eta_{2}\) are regularization parameters that set the error amount. To make the \(l_{1}\)-norm penalty better approximate the \(l_{0}\)-norm penalty, we introduce the reweighted matrix in the following. Let \(\mathbf{U}_{n}\) be the noise subspace and composed of the \((\bar{M}+\bar{N}-1-P)\) vectors in \(\mathbf{U}_{n}\) that correspond to \((\omega_{p+1},\omega_{p+2},\ldots,\omega_{\bar{M}+\bar{N}-1})\). According to the subspace principle [13], for the true target \(\theta_{p}\), \([(\omega_{tp}\omega_{rp}e^{j\varphi_{p}}\mathbf{Fb}(\theta_{p}))^{\mathsf{T}},(\omega_{tp}\omega_{rp}e^{-j\varphi_{p}}\mathbf{Fb}(\theta_{p})^{*})^{ \mathsf{T}}]^{\mathsf{T}}\) in \(\mathbf{Z}\) is orthogonal to the noise subspace \(\mathbf{U}_{n}\), that is:  
\[Q(\theta_{p},\varphi_{p},\omega_{tp},\omega_{rp})=[\omega_{tp}\omega_{rp}e^{j \varphi_{p}}\quad\omega_{tp}\omega_{rp}e^{-j\varphi_{p}}]\bar{\mathbf{Q}}( \theta_{p})\left[ \begin{array}{c}\omega_{tp}\omega_{rp}e^{j\varphi_{p}}\\ \omega_{tp}\omega_{rp}e^{-j\varphi_{p}}\end{array} \right]\to 0 \tag{21}\]  
where:  
\[\bar{\mathbf{Q}}(\theta_{p})=\left[ \begin{array}{cc}\mathbf{Fb}(\theta_{p}) &\mathbf{0}_{(\bar{M}+\bar{N}-1)\times 1}\\ \mathbf{0}_{(\bar{M}+\bar{N}-1)\times 1}&\mathbf{Fb}^{*}(\theta_{p})\end{array}  \right]^{\mathsf{H}}\mathbf{U}_{n}\mathbf{U}_{n}^{\mathsf{H}}\left[ \begin{array} []{cc}\mathbf{Fb}(\theta_{p})&\mathbf{0}_{(\bar{M}+\bar{N}-1)\times 1}\\ \mathbf{0}_{(\bar{M}+\bar{N}-1)\times 1}&\mathbf{Fb}^{*}(\theta_{p})\end{array} \right] \tag{22}\]  
Since \(Q(\theta_{p},\varphi_{p},\omega_{tp},\omega_{rp})\to 0\), \(\omega_{tp}\), \(\omega_{rp}\) and \(e^{j\varphi_{p}}\) are all non-zero scalars and the rank of \(\mathbf{U}_{n}\mathbf{U}_{n}^{\mathsf{H}}\) satisfies \(\text{rank}(\mathbf{U}_{n}\mathbf{U}_{n}^{\mathsf{H}})>2\), for the true DOA \(\theta_{p}\), the rank reduction of \(\bar{\mathbf{Q}}(\theta_{p})\) will take place, that is \(\text{det}[\bar{\mathbf{Q}}(\theta_{p})]\to 0\). Consequently, a reweighted vector is designed as:  
\[\mathbf{w}(i)=\text{det}[\bar{\mathbf{Q}}(\hat{\theta}_{i})] \tag{23}\]  
for \(i=1,2,\ldots,L\). Thus, the reweighted matrix is:  
\[\mathbf{W}_{r}=\text{diag}(\mathbf{w})/\text{max}(\mathbf{w}) \tag{24}\]  
where \(\mathbf{W}_{r}\)\(\in\)\(\mathbb{C}^{L\times L}\) is a diagonal matrix. Large weights in \(\mathbf{W}_{r}\) can punish the entries that are more likely to be zeros in \(\mathbf{w}\), while small weights reserve the larger entries. Hence, \(\mathbf{W}_{r}\) can enhance the solution of the sparse recovery problem in Equation (20).