## 4 Numerical Simulations and Discussion
### Realization of SAR Imaging
#### 4.1.2 SAR Imaging for Practical Measured Data  
This group of experiments deal with real data, measured by a spotlight SAR. The observed target is a crawler-type vehicle. The radar data are obtained from different azimuth and pitch angles. The angle aperture is \(5^{\circ}\). The center frequency is 9 GHz and the bandwidth is 1 GHz. The number of observation angle samples and frequency samples are both set to 101, thus the number of total pixels of image is \(M=10201\).  
The matrix \(A\) is constructed as the indication of the Equation (3), and \(Q=\lfloor\gamma\cdot M\rfloor\) rows are extracted from it to form a projection matrix. Here, \(\gamma=0.5\) is taken. Note that the number of observations \(Q=O\left(K\log M\right)\), where \(K\) is the sparsity. Therefore, the sparsity can be expressed as \(\lambda Q/\log M\). \(K\) is designed as 200 in this experiment.  
In the following, we investigate the results at different azimuth angles under the condition at a fixed pitch angle firstly, and observe the results at different pitch angles under the condition at a fixed azimuth angle secondly.  
(1) Different azimuth angles  
The pitch angle is fixed at \(50^{\circ}\). The observation data are acquired from three azimuth central angles, \(80^{\circ}\), \(90^{\circ}\) and \(100^{\circ}\), respectively. And the corresponding SAR imaging realizations are shown in Figures 6-8, respectively.  
Figure 6: SAR imaging for practical measured data, \((\phi,\theta)=(80^{\circ},50^{\circ})\).  
The above experiments demonstrate the effectiveness of the proposed method in practical implementation. With the comparison between the selected algorithms, the proposed method obtains the better imaging results. When the azimuth central angle is \(100^{\circ}\), the imaging result of SVA is dim and blurred, but the vehicle contour is clearly visible with the proposed method. For the experiments of different azimuth angles, although the classic matching pursuit approaches, i.e., OMP and CoSaMP, perform much better than SVA, the problem of imaging target ambiguity still exists. For instance, the CoSaMP imaging result has a remarkable bright streak at \(90^{\circ}\) of the azimuth central angle, as shown in Figure 7. Besides, as indicated by the arrows in Figures 6 and 8, CoSaMP shows raster lines over the target image at the central region, which will disturb the recognition of targets significantly. Under the same conditions, the proposed method is able to detect the correct image.  
Figure 8: SAR imaging for practical measured data, \((\phi,\theta)=(100^{\circ},50^{\circ})\).  
Figure 7: SAR imaging for practical measured data, \((\phi,\theta)=(90^{\circ},50^{\circ})\).  
With the comparison between the experiments of different pitch angles, the superiority of the proposed method is demonstrated. As shown in Figure 11, the imaging result of SVA contains many noisy points appearing blurred; OMP outputs several outliers which are marked by white circles; for CoSaMP, a part of the scatter points in the central are lost, which is indicated by arrow C; but the proposed method obtains the clear and accurate imaging result. As shown in Figures 9 and 10, the imaging results of CoSaMP contain raster lines over the targets which are indicated by arrow A and B, respectively; the other three methods achieve the similar performance as demonstrated above. Although the two classic matching pursuit methods can obtain clearer images than SVA, but both of them result in errors of the estimation beyond tolerance. By contrast, the proposed method can achieve the better imaging results. This is true especially as the pitch angle increases.