# Online Sparse DOA Estimation Based on Sub-Aperture Recursive LASSO for TDM-MIMO Radar
## 3 Results
### Simulation Results  
The azimuth of two unrelated sources with equal power is assumed to be \([-5^{\circ},0^{\circ}]\). The scan range is between \(-90^{\circ}\) and \(90^{\circ}\), and the number of points is K=512. The SNR of the signal is \(\text{SNR}=10\) dB. A MIMO radar system with two transmitters and four receivers (carrier frequency \(f_{c}=77\) GHz; receiving array element spacing: \(d_{r}=\lambda/2=1.9\) mm; transmitting array element spacing: \(d_{t}=M_{t}\cdot d_{r}=7.6\) mm) is investigated.  
Figure 4a illustrates the results of 10 Monte Carlo experiments with the traditional DAS method. Although the average processing time is only 0.0002 s, effectively distinguishing between the two adjacent targets is impossible. Figure 4b shows the results of the 10 Monte Carlo experiments with the IAA method. The average processing time is 0.0879 s. The results of the 10 IAA experiments show lower side lobes, and the adjacent targets are basically distinguished and located, which is better than that shown in Figure 4a.  
However, the IAA method also has drawbacks, such as an unsatisfactory resolution, a high side lobe, and high computational complexity. Figure 4c shows the results of the 10 LASSO experiments. It is seen that they can well distinguish the adjacent target directions, and the angular estimation error is highly non-significant. Nonetheless, the LASSO method has a high algorithmic complexity and requires the support of complete aperture data, thus  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(M_{r}\times M_{t}\)Method & \begin{tabular}{c} **DAS** \\ **(second)** \\ \end{tabular} & \begin{tabular}{c} **IAA** \\ **(second)** \\ \end{tabular} & \begin{tabular}{c} **LASSO** \\ **(second)** \\ \end{tabular} & \
\begin{tabular}{c} **Proposed Method** \\ **(second)** \\ \end{tabular} &
\begin{tabular}{c} **Speedup Ratio** \\ **(vs. LASSO)** \\ \end{tabular} \\ \hline
10 & \(2.0760\times 10^{-4}\) & 0.0200 & 0.0292 & 0.0147 & 1.9863 \\ \hline
100 & \(6.0470\times 10^{-4}\) & 0.3034 & 2.4910 & 0.1501 & 16.5956 \\ \hline
200 & \(9.8360\times 10^{-4}\) & 1.0196  
exhibiting poor real-time performance. Furthermore, its average processing time is 0.2403 s. As shown in Figure 4d, the 10 results of the proposed method are nearly consistent with the LASSO results. Compared with the results of the traditional DAS and IAA methods, the results of the proposed method provide better resolving of adjacent targets and side lobe suppression.  
The average processing time for each sub-aperture data update takes only 0.0215 s. Furthermore, Figure 5 illustrates the DOA result at the arrival of each sub-aperture data block under simulation conditions by the proposed method. The real-time DOA results are progressively improved as the MIMO sub-aperture data are received, and the best performance is achieved at the receipt of the last data block (the DOA performance of the LASSO method). Figure 6 shows the RMSE and CRB of angle estimation by the four methods for SNRs ranging from \(-5\) to 30 dB [44].  
Moreover, we can reduce the estimation error of DOA by compensation in two ways. One is the compensation for mutual coupling of array elements [45; 46]. The other is the motion compensation: the Doppler phase needs to be considered for the moving platform [47; 48]. Good research has been carried out on the above two compensation ways. Because this article mainly focuses on real-time signal processing algorithms, it will not be further discussed here.  
Figure 4: Simulated DOA results and analysis. (**a**) The DAS method, (**b**) the IAA method, (**c**) the LASSO method, (**d**) the proposed method.