## 4 Numerical Simulations
### Point-like Target Simulation  
As shown in Figure 12, 11 points with equal energy are set with equal spacing of 5 m in the range direction. These points cover a width of 50 m in total. In addition, these points are surrounded by Gaussian noise to simulate the pure-water environment. The signal-to-noise ratio (SNR) is set as 0 dB for the original echoes. In all, the whole scene covers 10 km width in the range direction, and the reference receiving window covers the same width of 3 km as the transmitting signal. The results are shown in Figure 13. The first row of the figure presents the results corresponding to the current framework; in the second row, our results are presented. This simulation is to verify the feasibility of the framework in the range direction. Thus, only the imaging results in the range direction are presented. The situation in the azimuth direction is given in the next simulation. The light green rectangles in the left two columns of the figure are to indicate the area of targets. The first column includes the imaging results of the whole scene, the second column consists of the enlarged images of the target, and the last column contains the interpolated images of the point located at the center to measure the resolution metric. For visual comparison, the results of the two frameworks are interpolated to the same length. Figure 11: Change of image entropy with the azimuth resolution. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Platform velocity & 200 m/s \\ Channel configuration & 1 transmitting channels \\  & 4 receiving channels \\ Channel spacing & 0.16 m \\ Carrier frequency & 9.6 GHz \\ Signal bandwidth & 150 MHz \\ Pulse width & 3 km \\ PRF & 625 Hz \\ Antenna aperture length & 2 m \\ Center slant range & 50 km \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Simulation parameters. Figure 12: Targets of 11 points with 5 m equal spacing in the range direction. **(a)** The whole scene; **(b)** enlarged target’s area. Figure 13: Imaging results in the range direction. **(a)** Result of the current framework; **(b)** enlarged target’s area in **(a)**; **(c)** interpolated peak of the center point in **(b)**; **(d)** result of the proposed framework; **(e)** enlarged target’s area in **(d)**; **(f)** interpolated peak of the center point in **(e)**. although the whole scene covers 10 km width, the area of targets is only 50 m long, indicating the strong sparsity of the scene, which is around only 0.5%. However, following the current framework, the whole scene's echoes are received. The oversampling ratio is 1.2, and the resulting sampling frequency is 180 MHz. Thus, in total, 12,000 sampling points are received and stored. Meanwhile, for the proposed framework, as seen in Figure 13d, only echoes of 3 km width are received and processed, and the adequate bandwidth corresponding to the area of targets is 2.5 MHz. Using the same oversampling ratio, the resulting sampling frequency is 3 MHz, only around 1.7% of the current one; and the number of sampling points is 60, which is only 0.5% of the current one. To validate the resolution performance, as shown in Figure 13c,f, we choose the center point to calculate its resolution (3 dB mainlobe width), and the results are 0.961 m and 0.928 m, respectively. This indicates that the proposed framework can still obtain the same resolution result, even with much less resource consumption. Moreover, the SNR is increased by around 30 dB, the as same as the current framework. This indicates that the proposed framework does not lose the processing gain. Next, we rearrange these 11 points along the azimuth direction to verify the proposed framework's performance in the azimuth direction. The simulated situation is shown in Figure 14. In the figure, Figure 14a is the whole scene's image of 10 km in width, and the targets are centered at 5 km with 5 m equal spacing. In Figure 14b, the area of targets is enlarged.