#### 4.1. Simulation

Table 1 lists the simulated parameters.

Figure 4 shows the distribution of the 24 point targets for simulation, where the near range

${R}_{min}$, center range

${R}_{c}$, and far range

${R}_{\mathrm{max}}$ are 10 m, 500 m, and 1000 m, respectively.

In this simulation, we select the center range ${R}_{c}$ as the reference range.

First, we analyze the errors of our algorithm with the given parameters.

Figure 5 shows the curve of differential RCM

${R}_{\mathrm{dif}}$ with respect to range, according to (

29). We can see that the differential RCM reaches its maximum at the near range, but it still does not exceeds half a range cell, so, in this simulation, the range re-sampling step in the range and angular frequency domain can be skipped for efficiency.

The phase error of target at center range is zero, because the matched filter is accurately formulated for the reference range. Here, the phase errors that are calculated with (

33) of targets at near range and far range are shown in

Figure 6. We can see that, even for the near target at 10 m, the phase error is still far smaller than

$\pi /4$ and, for far range target, the phase error is even smaller. Therefore, our algorithm is very accurate for both near and far range imaging, and no segmenting strategy is needed.

To analyze the phase error that are caused by reference plane imaging, we substitute the simulation parameters into (

41). For an optimal focus, the threshold that is calculated of the elevation angle is approximately 7.25°. That is to say, if the depression angle

${\alpha}_{\mathrm{p}}$ exceeds the value, then a new imaging plane should be selected.

Figure 7 shows the imaging results of the simulation with our algorithm. The image is in polar coordinates, and the whole 360° scanning data are focused just for one imaging process, and no segmenting strategy is used. To clearly show the imaging results of near, center, and far range, targets are shown separately.

Figure 8 shows the image details, as well as the range and aspect profiles. In polar coordinates, the point spread function is approximately a 2-D sinc function. The point spread function in polar coordinates for the whole swath is the same, because the angular resolution and the range resolution are constant with range.

For comparison,

Figure 9 also shows the detailed target images and profiles of Lee’s algorithm. In Lee’s algorithm, the 2-D matched filter is in time domain, and it is formulated for a selected reference range. However, in the formulation, Taylor expansion with respect to the aspect angle is used, and the terms that are higher than quadratic are ignored. This approximation may cause defocusing, even for targets at the reference range under wide beam or near range condition. Defocusing is even more severe for targets not at the reference range.

We choose the BP algorithm as a benchmark. BP is based on pixel-by-pixel processing, and it is very accurate, although time consuming.

Figure 10 compares the angular profiles of these three different algorithms. It shows that the image quality of our method is close to that of BP.

When comparing our results with that of Lee’s, we can see that, with our method, the whole swath is well focused. While, with Lee’s method, the near range targets are severely defocused, and the main lobe is split.

The point spread function of ArcSAR in polar coordinates is approximately a 2-D sinc function. We measure the 3 dB impulse response width (IRW), peak sidelobe ratio (PSLR), and integrated sidelobe ratio (ISLR) of the three targets marked with red rectangles in

Figure 7 in order to test the imaging quality quantitatively. Because the algorithm induced defocusing occurs in angular direction, the quality test is only done in an angular direction for a concise purpose.

Table 2 shows the results of imaging quality test. With (

8), the angular resolution is calculated as 0.5056°, and the 3 dB pulse width should be that value multiplied by 0.886, and that is 0.4479°. The tested angular resolution is about 0.4656° with our algorithm, and it is very close to the theoretical value. From (

8), we also know that the azimuth resolution is the angular resolution multiplied by the target range. Thus, in our simulation example, the azimuth resolution of the near range, center range, and far range is 0.08 m, 4.06 m, and 8.12 m, respectively. The image quality is tested for BP result as a benchmark. The quality test results are also compared with that of Lee’s method. Need to mention that, for the near range target, the main lobe is split with Lee’s method, so the image quality is not measured. From the table, we can see that our method is very accurate for the whole swath.

The computing efficiency is also tested under the same environment. The codes of the three algorithms (in MATLAB language) have been implemented on the computer with 1.8-GHz i7-8550U CPU. The running times of our method, Lee’s method, and BP are 1.2414 s, 1.0108 s, and 693.5731 s, respectively.

#### 4.2. Real Data

The ArcSAR system that is used in this field test was developed by our lab. It is still an experimental system, and the rotating angle is not 360, but about ±80°. This system uses frequency-modulated continuous wave (FMCW) and intermediate frequency (IF) receiver.

Table 3 lists the main parameters of the system, where

${f}_{s}$ denotes the analogue to digital (AD) sampling rate and

${T}_{p}$ denotes the pulse duration. The unambiguous range is equal to

${f}_{s}{T}_{p}C/\left(4{B}_{r}\right)$, and that is about 900 m.

In this experiment, the observed scene is a bridge under construction.

Figure 11 shows the photo of the experimental ArcSAR system and the observed scene. During data acquisition, only the arm rotates in order to scan the scene.

With our algorithm, the echo data can be focused accurately and fast. The left column of

Figure 12 shows the output images of three algorithms in polar coordinates. We can see the 2-D sinc spread function of strong point-like targets in the image with our algorithm, indicating an optimal focus. For a better display, the polar coordinates image is transformed into the Cartesian coordinates, as shown in the right column. The deterioration of the image in Cartesian coordinates is not due to the focusing method, but rather to the limitation of

${\theta}_{syn}$ introduced earlier in

Figure 1b. The correspondence between the ArcSAR image and the photo are labelled with the same number. Label 1 and label 3 represent the two towers of the bridge, respectively, and label 2 represents the slogan made of iron. The region between the two yellow lines in the figure indicates ±30° field of view limit that is covered by linear-scan GBSAR with 60° beamwidth antenna.

We can see that the results of our method are close to that of BP. Both near-range target (in the blue rectangle) and far-range target (in the green rectangle) are well focused. The image is defocused with Lee’s method, especially for the near-range.

In the same environment mentioned in the simulation section, the running times of our method, Lee’s method, and BP are 0.2448 s, 0.1833 s, and 46.1571 s, respectively.