In addition, as the data is usually processed by a field programmable gate array (FPGA) or digital signal processing (DSP) chip in practice, we used an eight-core TMS320c6678 chip produced by Texas Instruments (TI) as an example to build a hardware platform, and tested the computing time of different algorithms.
4.1. Simulation
For the simulation, the original scene consisted of three groups of adjacent targets at distances of 5600, 5360, and 5160 m, as well as one island target at a distance of 4930 m, as shown in
Figure 2. The antenna pattern was a
function, defined as
, with beam width of 3.5°. Therefore, according to the Rayleigh criterion, the minimum distinguishable distances at the range bin of adjacent targets were
,
, and
m, respectively. The radar worked in X-band, the scanning region was
, and the scanning speed was
.
The simulation results are shown in
Figure 3. From the prior information, we know that the intervals of adjacent targets were
,
, and
m, respectively. They are undistinguishable in the real-beam echo.
Figure 3a is the real beam echo polluted by white Gaussian white noise (GWN), with signal-to-noise ratio (SNR) of 20 dB, where the SNR is defined as
It can be seen that all the adjacent targets could not be distinguished.
Figure 3b–i are the results processed by different methods. From the results, WF only could distinguish adjacent targets at 5600 m, with limited improvement in resolution. TSVD and REGU achieved limited resolution improvement. Although adjacent targets at 5600 and 5360 m could be distinguished, the targets at 5160 m could not be distinguished, as shown by
Figure 3c,d. From
Figure 3e,f, we can see that ML and IAA achieved better resolution improvement than WF, TSVD, and REGU; however, their performance was not as good as SFMM, SBA, and the proposed method. It is obvious that SFMM, SBA, and the proposed method can evidently improve the resolution, as shown by
Figure 3g–i. All adjacent targets are distinguished, and the noise is suppressed. Comparing
Figure 3i with
Figure 3h, it can be seen that the proposed method had almost no performance loss compared to traditional SBA, as it is difficult to discern the differences between them.
The profiles of the adjacent targets at 5160 m are shown in
Figure 4. From the profiles, it is obvious that SFMM, SBA, and FSBA can improve the azimuth resolution better than the other methods. Adjacent targets are completely distinguished and the noise is also suppressed. As for SBA and FSBA, it can be seen that their profiles basically coincide.
In addition, the mean square error (MSE), beam sharpening ratio (BSR), and image entropy were utilized to quantitatively analyze the performances of the different algorithms. The MSE is used to measure the similarity between the processing result and the original scene, which is defined as
The BSR is defined as the ratio of beam width before and after processing of a single target, which is used to measure beam sharpening performance. According to the principle of minimum entropy, the image entropy can be used to measure the clarity of an image, as the entropy increases with the increase of image blurring level [
44]. The results are shown in
Table 1.
As can be seen from the table, the MSEs of the processing results of SFMM, SBA, and FSBA were very similar, and much smaller than those of the other methods, which shows that the processing results of these three methods were closer to the original scene. SFMM, SBA, and FSBA achieved the same beam sharpening ability, also far better than the other methods, which is conducive to distinguishing adjacent targets. In addition, the image entropies of the processing results of SFMM, SBA, and FSBA were far less than those of the other methods, which indicates that the quality of their processing results was higher than that of other methods. In conclusion, SFMM, SBA, and FSBA had similar performance, being better than WF, REGU, TSVD, RL, and IAA. As for SFMM, SBA, and FSBA, their performance difference is mainly reflected in computing time. The computing time was tested on the hardware platform; the results are given in the following subsection.
Then, in order to verify the performance of resolution improvement at low SNR, we repeated above simulation with SNR = 10 dB. The simulation results are shown in
Figure 5. It can be seen that, under the conditions of low SNR, WF, REGU, TSVD, RL, and IAA could not distinguish adjacent targets at 5160 m. SFMM, SBA and FSBA still had good resolution ability, and all adjacent targets are clearly distinguishable. Similarly, the profiles of the adjacent targets at 5160 m are shown in
Figure 6. It can be clearly seen that SFMM, SBA, and FSBA could distinguish adjacent targets, while the other methods could not. The relevant performance indices are shown in
Table 2. From the view of MSE, SBA, and image entropy, we obtained almost the same conclusions as those under the condition of high SNR.
The simulations have demonstrated the superior performance of the proposed FSBA. Having said that, some small approximations have been made in the acceleration process. In the condition of high SNR, these approximations hardly affect the performance of the algorithm. As shown in
Figure 3 and
Table 1, the results of FSBA and SBA are almost the same. In the condition of low SNR, strong noise will have a slight impact on the performance. From
Figure 5 and
Table 2, the results of FSBA and SBA are different in MSE and entropy, but these differences are very small and acceptable.
4.2. Real Data Processing
Next, airborne data was processed to verify the performance of the proposed FSBA in practice. In this experiment, a X-band radar was mounted on a transport plane. The beam width was 5.1° and the scanning range was –.
The optical scene, which is shown in
Figure 7, was captured from Google earth. It can be seen that the original scene is an airport runway with five co-operative airplanes (marked by the red rectangle). Furthermore, the red circle covers two adjacent airplanes. After antenna scanning, pulse compression, and range walk correction, the real beam echo with low resolution is shown in
Figure 8a. In
Figure 8a, we cannot distinguish the exact number of planes on the runway. For the real data, the noise is also assumed as GWN and the SNR is defined in Equation (30).
Figure 8b–d show that the WF, REGU, and TSVD methods achieved poor resolution improvement and noise suppression. RL and IAA can further improve the resolution, but have limited performance in noise suppression, as shown by
Figure 8e,f; in particular, false targets appear in the IAA result, which makes it impossible to determine the number and location of real targets.
Figure 8g–i are the processing results of SFMM, SBA, and FSBA, respectively. It can be seen that they have higher resolution and that not only adjacent planes are distinguished, but also the noise is greatly suppressed. Therefore, we can clearly see that the target area has five strong points; that is, the locations of the real targets. Comparing
Figure 8h,i, it can be seen that FSBA and SBA had almost the same performance; that is to say, after acceleration, the performance of the SBA had almost no degradation.
The profiles of adjacent airplanes are plotted in
Figure 9. It is obvious that SFMM, SBA, and FSBA had a better processing effect than the other methods. Not only are the two targets clearly distinguished, but the noise is also greatly suppressed. For SBA and FSBA, the profiles of their processing results are almost overlapped, which also proves that, after acceleration, the performance of the algorithm has almost no degradation.
In order to quantitatively evaluate the performance of different algorithms, the BSR and entropy are shown in
Table 3. Furthermore, the peak side lobe ratio (PSLR) was used to measure the side lobe suppression ability, which is defined as
where
denotes the signal amplitude and
denotes the side-lobe amplitude. From the table, we can see that SFMM, SBA, and the proposed FSBA had better performance in beam sharpening and side lobe suppression than the other methods. Additionally, the entropy of their processing results was smaller than that of the other methods.
It should be noted that, for the processing of real data, it can be seen that although SBA and FSBA can effectively improve the resolution and suppress the noise, they cannot completely separate the target from the noise, and the processing performance is not as good as that with the simulated data. This is due to the assumption of ideal GWN in the simulation; the real world experiment highlights the pitfalls of assuming idealistic noise and clutter sources.
4.3. Assessment of Computing Time
We used the TMS320c6678 chip to test the real-time ability of the proposed FSBA. The main frequency was 1 GHz and the memory was 2 GB. Under this hardware condition, we first give the approximate maximum echo dimension allowed for each algorithm, as shown in
Table 4. It can be seen that the maximum echo dimension allowed for FSBA is much larger than other methods except WF and RL. Although the maximum echo dimension allowed for RL and FSBA seems similar, but for airborne radar data, usually we can process each range bin data in turn, and the difficulty of processing depends on the number of azimuth points. That is, the more azimuth data points, the more difficult it is to process. Therefore, the performance of the proposed FSBA to process large-scale data is higher than RL. The maximum data dimension allowed for WF is larger than FSBA, but its resolution improvement is extremely limited.
For above experiments, the dimension of data is
. Based on the hardware platform, the computing times of the different algorithms are listed in
Table 5. The results show that the computing time of the proposed FSBA is much less than that of other methods, except for WF and RL. Although the computing time of WF and RL was less than that of FSBA, it can be seen, from the above experiments, that the performance of WF and RL for resolution improvement was not as good as that of FSBA. Furthermore, the computing time of FSBA can meet real-time requirements. Although SFMM, SBA and FSBA can provide similar results, it is obvious that the computing time of FSBA is much less than SBA and SFMM. In particular, the proposed FSBA only needs about
to complete processing, while the traditional SBA needs about
and SFMM needs
. It can be found that the computational efficiency of the proposed FSBA is about 51 times that of SBA and 10 times that of SFMM. As for REGU, TSVD and IAA, their performance of resolution improvement is not as good as that of FSBA, and their computational efficiency is also lower than FSBA. In general, FBSA can perfectly meet the needs of airborne radar real-time resolution improvement in practical applications.
Because the proposed FSBA is accelerated by avoiding matrix inversion, the larger the dimension of echo, the greater the advantage in computational efficiency. To verify this conclusion, the computing time of different algorithms for processing larger dimension echo (
) is shown in
Table 6. The computing time of the proposed FSBA is also much less than that of other methods, except for WF and RL. In particular, we can obtain that the computational efficiency of the proposed FSBA is about 77 times that of SBA and 14 times that of SFMM.