## 3 The Proposed Algorithm  
### Mutual Coupling Elimination  
To successfully implement the noncircular source-based sparse representation method, we firstly eliminate the effect of mutual coupling. Exploiting the structure of mutual coupling matrices \(\mathbf{C}_{t}\) and \(\mathbf{C}_{r}\), we define two selection matrices as follows:  
\[ \begin{split}\mathbf{\Gamma}_{1}&=[\mathbf{0}_{(M-2k) \times k}\quad\mathbf{I}_{(M-2k)\times(M-2k)}\quad\mathbf{0}_{(M-2k)\times k}] \\ \mathbf{\Gamma}_{2}&=[\mathbf{0}_{(N-2k)\times k} \quad\mathbf{I}_{(N-2k)\times(N-2k)}\quad\mathbf{0}_{(N-2k)\times k}]\end{split}  \tag{8}\]  
where \(\mathbf{\Gamma}_{1}\in\mathbb{C}^{(M-2k)\times M}\), and \(\mathbf{\Gamma}_{2}\in\mathbb{C}^{(N-2k)\times N}\). \(\mathbf{\Gamma}_{1}\) and \(\mathbf{\Gamma}_{2}\) can be used to choose the central rows of the mutual coupling matrices and then transform the transmit-receive mutual coupling matrix \(\mathbf{C}\) into a diagonal one. Thus, let \(z_{p}=e^{j\pi\sin(\theta_{p})}\), by multiplying the selection matrix \(\mathbf{\Gamma}_{1}\) on the left side of the transmit steering vector with mutual coupling; the transformation of the \(i\)-th element in \(\mathbf{\Gamma}_{1}\mathbf{C}_{i}\mathbf{a}_{t}(\theta_{p})\in\mathbb{C}^{(M -2k)\times 1}\) can be derived as:  
\[[\mathbf{\Gamma}_{1}\mathbf{C}_{t}\mathbf{a}_{t}(\theta_{p})](i) =c_{tk}z_{p}^{i-1}+c_{t(k-1)}z_{p}^{i}+\ldots+z_{p}^{i+k-1}+\ldots +c_{tk}z_{p}^{i+k-1+k}\] \[=(\sum_{j=-k}^{k}c_{t|j|}z_{p}^{j+k})z_{p}^{i-1} \tag{9}\] \[=\omega_{tp}z_{p}^{i-1}\]  
where \(\omega_{tp}\) is a scalar that just contains the information of \(\theta_{p}\) and mutual coupling in the transmit array. Let \(\bar{M}=M-2k\); thus, for \(i=1,2,\ldots,\bar{M}\), we have \(\mathbf{\Gamma}_{1}\mathbf{C}_{t}\mathbf{a}_{t}(\theta_{p})=\omega_{tp} \bar{\mathbf{a}}_{t}\), in which \(\bar{\mathbf{a}}_{t}\) is the new transmit steering vector and can be expressed as \(\bar{\mathbf{a}}_{t}=[1,z_{p},z_{p}^{2}\ldots,z_{p}^{\bar{M}-1}]\), \(p=1,2,\ldots,P\). Similar to Equation (9), we use the other selection matrix \(\mathbf{\Gamma}_{2}\) to extract the mutual coupling coefficients in \(\mathbf{C}_{r}\mathbf{a}_{r}(\theta_{p})\) into a scalar, that is:  
\[ \begin{split}[\mathbf{\Gamma}_{1}\mathbf{C}_{r}\mathbf{a}_{r}( \theta_{p})](i)&=c_{rk}z_{p}^{i-1}+c_{r(k-1)}z_{p}^{i}+\ldots+z_{p }^{i+k-1}+\ldots+c_{rk}z_{p}^{i+k-1+k}\\ &=(\sum_{j=-k}^{k}c_{r|j|}z_{p}^{j+k})z_{p}^{j-1}\\ &=\omega_{rp}z_{p}^{i-1}\end{split}  \tag{10}\]  
where \(p=1,2,\ldots,P\). The value of \(\omega_{rp}\) depends on the mutual coupling coefficients and \(\theta_{p}\). Then, for \(i=1,2,\ldots,\bar{N}\), we further have \(\mathbf{\Gamma}_{2}\mathbf{C}_{r}\mathbf{a}_{r}(\theta_{p})=\omega_{rp}\bar{ \mathbf{a}}_{r}\) with \(\bar{\mathbf{a}}_{r}=[1,z_{p},z_{p}^{2}\ldots,z_{p}^{\bar{N}-1}]\) being the new receive steering vector. As a result, exploiting the property of the Kronecker product operator, let \(\mathbf{\Gamma}=\mathbf{\Gamma}_{1}\otimes\mathbf{\Gamma}_{2}\in\mathbb{C}^ {\bar{M}\bar{N}\times M\bar{N}}\) be the selection matrix of transmit-receive array; we deduce that:  
\[ \begin{split}\mathbf{\Gamma}\mathbf{C}\mathbf{a}(\theta_{p})& =[\mathbf{\Gamma}_{1}\mathbf{C}_{t}\mathbf{a}_{t}(\theta_{p})] \otimes[\mathbf{\Gamma}_{2}\mathbf{C}_{r}\mathbf{a}_{r}(\theta_{p})]\\ &=\omega_{tp}\omega_{rp}\bar{\mathbf{a}}(\theta_{p})\end{split}  \tag{11}\]  
where \(\bar{\mathbf{a}}(\theta_{p})=\bar{\mathbf{a}}_{t}(\theta_{p})\otimes\bar{ \mathbf{a}}_{r}(\theta_{p})\in\mathbb{C}^{\bar{M}\bar{N}\times 1}\) is the new transmit-receive steering vector.