## 4 Numerical Simulations
### Performance of SNR and Sparsity \(K\)  
The results in Figures 4 and 5 indicate the performance of 2 DTV-CES, LASSO, and Improved ESPRIT as a function of sparsity \(K\) and SNR, respectively. Figure 4 shows the detection probability of the 2 DTV-CES, LASSO, and the Improved ESPRIT with the different sparsity \(K\). The performance of three different methods in 2D-DOA's estimation at the range of \([(8+\Delta k)^{\circ},(15+\Delta k)^{\circ}]\), \(\Delta k\in[1:1:9]\), and the sparsity \(K\) at the range of [2:1:10] is evaluated. Each 2D-DOA is obtained by 1000 Monte Carlo experiments. It can be seen that the detection probability of 2 DTV-CES gradually increases up to 98% and tends to stabilize with the increasing of the sparsity; 2 DTV-CES has even better estimation performance in the case of high sparsity than that of lower (\(K<3\)). When \(K\) is lower, the detection probability of LASSO fluctuates slightly between 97 and 100%, and the estimation performance is significantly better than that of the 2 DTV-CES and the Improved ESPRIT. This phenomenon is caused by the weak grouping of an extended source with lower \(K\), and the sparse promotion feature of the LASSO can achieve high resolution. However, with the increasing of \(K\), the extended sources exhibit strongly grouped structural characteristics. Based on the sparsity promotion feature, the 2 DTV-CES utilizes the TV-norm regularization to promote the formation of segmented constant distribution, resulting in a significantly better performance than the other two methods. To summarize, 2 DTV-CES is not sensitive to the change of sparsity when its values are more than 3 and has good robustness.  
Figure 4: The detection probability of different algorithms versus sparsity \(K\).  
Figure 5: The detection probability of different algorithms versus SNR.  
With the same 2D-DOA estimation conditions as shown in Figure 4, the performance of 2 DTV-CES, LASSO, and the Improved ESPRIT in the presence of additive noise at a range of [\(-\)5:5:20] dB SNR is evaluated. The results are shown in Figure 5.  
Figure 5 describes the performance of the three methods under different SNRs. As can be seen, the average detection probability of 2 DTV-CES is up to 75% and is 40% higher than that of LASSO with the increase of SNR. Obviously, the SNR robustness of 2 DTV-CES is higher than that of the other two methods in the whole range of SNR [\(-\)5:1:20]. As described, the performance of SNR robustness of 2 DTV-CES is optimal. In the case of exceeding the limitation of the array element, there is still a high detection probability at a smaller SNR.  
The performance of the proposed method 2 DTV-CES, the LASSO, and the Improved ESPRIT is compared. The spatial average detection probability is taken as the metric, and statistical analysis is carried out from three aspects: sparsity \(K\), signal-to-noise ratio SNR, and computing speed. The results are shown in Table 1.  
As shown in Table 1, the spatial average detection probability of 2 DTV-CES is more than 85% with different sparsity and SNR. The average computing speed is only 29 s. Compared with the other two methods, 2 DTV-CES has the uppermost spatial average detection probability under the same sparsity and SNR conditions and the fastest computing speed with the narrowest range of computing speed variation. Improved ESPRIT has the lowest detection probability and the longest detection time and it is not suitable for sparse DOA estimation of extended sound sources. The results further illustrate the advantages of the proposed method 2 DTV-CES.