## 4 Numerical Simulations
### Performance of TV-Norm Regularization  
The results in Figures 2 and 3 indicate qualitatively the performance of total variation regularization (TV-norm) of 2D-DOA estimation in the presence of two kinds of spatially extended sources in Case 1 and Case 2 under high-SNR (SNR = 20) and sparsity (\(K\) = 10).  
Figure 2 shows the 2D-DOA's estimations of spatial spectrum distribution and DOA's map of 2 DTV-CES and LASSO in Case 1. The results demonstrate the performance of total variation regularization (TV-norm) to promote the block sparsity of solution.  
Figure 2a shows the spatial spectrum of the spatially extended sources with the equal amplitudes in Case 1. The amplitude of the solution coefficients is strict with the constant value 1 dB, and each focusing direction can be clearly distinguished. The solution coefficient is divided into two groups distinctly. It is clear that the TV-norm can promote the piecewise constant solution so that the solution coefficients are closely related to its neighbors, which reflects the block sparse characteristics of the spatially extended sources.  
Figure 2b,c shows the 2D-DOA's estimation maps of 2 DTV-CES and LASSO respectively. It can be seen that the 2 DTV-CES can accurately estimate all focusing directions with TV-norm regularization. There are 19 focusing directions of 21 directions that can be accurately distinguished, and only 2 focusing directions with \(1^{\circ}\) deviation. They are also within the allowable error range \(\pm 2^{\circ}\), and can be regarded as effective estimation. Moreover, the two groups of focusing direction can be effectively separated. Relatively  
Figure 2: In Case 1, the 2D-DOA’s estimation with and without TV-norm. (**a**) Spatial Spectrum of 2 DTV-CES, (**b**) DOA’s map of 2 DTV-CES, (**c**) DOA’s map of LASSO.  
speaking, the estimation accuracy of LASSO is poor. There are only 11 focusing directions out of 21 directions that can be identified and are unable to distinguish between different groups. This is because LASSO can only promote solution sparsity in the data domain but not in the gradient domain.  
In contrast, 2 DTV-CES not only promotes the sparsity in the data domain but in the gradient domain with TV-norm. Therefore, 2 DTV-CES has a better capacity for the promotion of block sparsity.  
Figure 3 depicts the 2D-DOA's estimations of spatial spectrum distribution and DOA's map of 2 DTV-CES and LASSO in Case 2. The results show 2D-DOA's estimates of complex extended sources mixed with point sources, to demonstrate the promotion of sparsity and clustering by the 2 DTV-CES method.  
Figure 3a shows the spatial spectrum of 2D-DOA estimation. It can be seen that when there are point sources and extended sources at the same time, the constant solution can be obviously separated into three groups by \(\ell_{1}\)-norm and the TV-norm regularization of 2 DTV-CES. The amplitude of the solution coefficients is strict with the constant value 1 dB, and each focusing direction can be clearly distinguished. The method can effectively identify the focusing directions of extended sources and two single-point sources.  
Figure 3b,c demonstrates the 2D-DOA maps of the method 2 DTV-CES and LASSO, respectively. The results show that 2 DTV-CES can estimate all focusing directions. Especially, there are 19 directions with zero deviation, and 2 directions have 1\({}^{\circ}\) deviation. Only  
Figure 3: In Case 2, the 2D-DOA’s estimation with and without TV-norm. (**a**) Spatial Spectrum of 2 DTV-CES, (**b**) DOA’s map of 2 DTV-CES, (**c**) DOA’s map of LASSO.  
11 of 21 directions can be estimated by LASSO with more than 1\({}^{\circ}\) deviation. Both methods can estimate two peaks within the allowable \(\pm 2^{\circ}\) deviation. Compared with the DOA's estimation, the 2 DTV-CES is better than LASSO, since 2 DTV-CES introduces and utilizes TV-norm regularization on the basis of LASSO, which promotes piecewise constant and peak solutions.