## 4 Related Remarks  
**Remark 1**. : _In the problem of the sparse signal reconstruction that is based on the \(l_{1}\)-norm minimization, the chosen values of the regularization parameters \(\eta_{1}\) and \(\eta_{2}\) are important for the estimation accuracy. In the proposed algorithm, noise matrix \(\mathbf{N}\) in Equation (4) is complex white Gaussian with zero mean. By mutual coupling elimination and reduced dimensional transformation, the corresponding selection matrix \(\mathbf{\Gamma}\) and transformation matrix \(\mathbf{R}\) transform \(\mathbf{N}\) into \(\bar{\mathbf{N}}_{d}\). According to the designed matrix structure of \(\mathbf{\Gamma}\) and \(\mathbf{R}\), \(\mathbf{\Gamma}\mathbf{\Gamma}^{B}=\mathbf{I}_{\bar{M}\bar{N}}\) and \(\mathbf{R}\mathbf{R}^{B}=\mathbf{I}_{M+\bar{N}-1}\) can be obtained._  
_Based on the invariance speciality of linear transformation in an asymptotic normal distribution [29], if \(\mathbf{n}\) is asymptotic normal with zero-mean and covariance matrix \(\sigma^{2}\mathbf{I}_{MN}\), \(\mathbf{R}\mathbf{\Gamma}\mathbf{n}\) complies with a zero-mean asymptotic normal distribution with covariance matrix \(\mathbf{R}\mathbf{\Gamma}(\sigma^{2}\mathbf{I}_{MN})(\mathbf{R}\mathbf{\Gamma })^{B}=\sigma^{2}\mathbf{I}_{\bar{M}+\bar{N}-1}\). Therefore, \(\bar{\mathbf{N}}_{d}=[\bar{\mathbf{n}}_{d1},\bar{\mathbf{n}}_{d2},\ldots, \bar{\mathbf{n}}_{dI}]\) in Equation (15) is a zero-mean complex Gaussian white noise matrix._  
_In [9], it has been verified that if the noise \(\bar{\mathbf{n}}_{d}\) is independent and identically distributed (i.i.d.) Gaussian, then \(\parallel\bar{\mathbf{N}}_{ds}\parallel_{F}^{2}\) has approximately a chi-square distribution with \((\bar{M}+\bar{N}-1)P\) degrees of freedom upon normalization by the variance of \(\bar{\mathbf{n}}_{d}\), where \(\bar{\mathbf{N}}_{ds}=\bar{\mathbf{N}}_{d}\mathbf{V}_{s}\) with \(\mathbf{V}_{s}\) being the signal subspace of \(\mathbf{Z}\)[9]. As a result, \(\eta_{1}\) and \(\eta_{2}\) can be chosen as the upper limit value of \(\parallel\bar{\mathbf{N}}_{ds}\parallel_{F}^{2}\) and \(\parallel\bar{\mathbf{N}}_{ds}\parallel_{F}^{2}\) upon \(\sigma^{2}\), with a high probability \(1-\varepsilon\) confidence interval, and \(\varepsilon=0.001\) is enough. In addition, \(\sigma^{2}\) can be estimated by the average of the squares of \(2(\bar{M}+\bar{N}-1)P\) smallest singular values of \(\mathbf{Z}\). By using MATLAB software, the function chi\(2 inv(1-\varepsilon,(\bar{M}+\bar{N}-1)P)\sigma^{2}\) can be used to compute the regularization parameters._  
**Remark 2**. : _The computational complexity of the proposed algorithm mainly focuses on eliminating the effect of mutual coupling, constructing the reweighted matrix and obtaining the sparse solution from Equation (25), which require \(O\{\bar{M}\bar{N}MN\}\), \(O\{[4(\bar{M}+\bar{N}-1)(2(\bar{M}+\bar{N}-1)-P)+4(\bar{M}+\bar{N}-1)(2(\bar{ M}+\bar{N}-1)-P)+8(\bar{M}+\bar{N}-1)|\bar{L}\}\) and \(O\{\bar{L}^{3}P\}\) calculation burden, respectively, where \(\bar{L}\) is the number of the discretized sampling grids. Therefore, the total computational complexity of the proposed algorithm is \(O\{\bar{M}\bar{N}MN\}+[4(\bar{M}+\bar{N}-1)(2(\bar{M}+\bar{N}-1)-P+1)]\bar{L} +\bar{L}^{3}P\}\). Although the constructions of the extended data matrix and the reweighted matrix increase the computational complexity, they enlarge the virtual array aperture and enhance the solution of the sparse reconstruction. As a result, the angle estimation performance of the proposed method is improved._  
**Remark 3**. : _The prior knowledge of the number of emitting sources is important. This is because in the absence of a perfect knowledge of the target number \(P\), either adding spurious sources or missing actual signals may lead to the deviations of the signal and the noise subspaces, which further cause the deviations of extended observation matrix \(\mathbf{Z}_{v}\) in Equation (18) and reweighted matrix \(\mathbf{W}_{r}\) in Equation (24).