4.1. Simulation Data from an Ideal Point Scatter Target Model
Based on the above analysis, simulation experiments were carried out using the simulation parameters listed in
Table 1. The ideal scatterers, which outline an airplane, are given in
Figure 3. The assumed aspect angles are shown in
Figure 4. The ISAR image (see
Figure 5a) based on the RDA illustrated the defocus in the head and tail of the aircraft and stretch due to non-uniform rotation. Conversely, the scatterers in the central region of cross-range bins remained clear. According to (
4), the central region had small
values and was less affected by non-uniform rotation.
Figure 5b presents the imaging results obtained by using the RID algorithm, which differed from those of the other methods (see
Figure 5c–h) that used time–frequency analysis for imaging. It can be observed in
Figure 5b that the image was manifested as short lines elongated along the azimuthal direction rather than focused points. This was related to the decreased azimuthal resolution, which was caused by the reduced accumulation time with the shorter window length of the STFT.
Figure 5c–f present the imaging results from four different parametric RMC algorithms: the AJTF [
24], the PSO [
25], the LVD [
34], and the ICPF algorithms [
16], respectively. It can be observed that image defocusing still occurred (see the red circles in
Figure 5c) and was caused by ineffective RMC. Moreover, the defocus deteriorated when observed from the central region of the cross-range bins toward the periphery (see the red arrows in
Figure 5f).
Figure 5g shows the imaging results obtained by replacing (
8) with the inverse function from [
27], where it can be seen that the image still exhibited defocusing. In contrast, the proposed method (see
Figure 5h) effectively accomplished RMC for all scatterers to eliminate defocus. Considering that a better-focused image will result in a smaller entropy [
4], we adopted a minimum-entropy criterion to quantitatively assess the imaging quality. The entropy for an image
is defined as
where
. The entropies and computation times for the target when using different algorithms are listed in
Table 2. In addition, considering that a better image will result in a less stretched value, we adopted a novel minimum stretched value criterion to quantitatively assess the imaging quality of 2D ISAR images. The stretched value is defined as
where
represents the image values in the nth range bin after compensation, while
denotes the ideal image values in the nth range bin.
signifies the operation for calculating the norm, and N denotes the total number of range bins. The stretched values from different algorithms are also listed in
Table 2. The proposed algorithm accomplished the smallest values among all, which equaled 6.49 and 11.35. This indicated that the proposed algorithm had the best performance.
ISAR requires a long accumulation time to achieve high resolution in the cross-range dimension, which means that a larger rotation angle results in a better resolution. It was considered that images with different resolutions were affected differently by non-uniform rotation and estimation errors. The entropies versus CPI frames for six typical methods are shown in
Figure 6. Unlike in
Table 1, the angular velocity and angular acceleration were reset to 0.015 rad/s and 0.040 rad/s
2 to prevent spectral aliasing. We first note that as the CPI frames increased, the entropy also increased according to the RD curve, which resulted in more defocused images. Furthermore, the ICPF, LVD, PSO, and AJTF algorithms perform well with small CPI durations. However, as the CPI increased, errors in parameter estimation increasingly impacted the imaging results. In contrast, the algorithm proposed in this study demonstrated strong performance.
Since more accumulated time will result in more estimation errors, the data in the 108th range bin were extracted as an example to demonstrate that the proposed method could effectively deal with non-uniform rotation with large CPI durations and to lay the foundation for the acquisition of an ISAR image of high quality.
Figure 7 shows a comparison of the results when using a CPI duration of 1.4 s for the RDA, the ICPF algorithm, and the proposed method. Specifically, on the one hand, the two scatterers could not be distinguished via the RDA. On the other hand, although the ICPF algorithm could roughly recover the positions of the two scatterers, the spectral lines were broad and did not form distinct peaks. In cases where the scatterers were close to each other, the spectral lines of the ICPF algorithm could exhibit aliasing. Furthermore, the proposed method was able to clearly distinguish the two spectral lines, which is the premise of improving 2D image quality.
Based on the parameters in
Table 1, we reset the angular velocity to 0.015 rad/s, and the angular acceleration was set from 0.035 to 0.015 rad/s
2.
Figure 8 shows the entropies versus angular acceleration for different methods. The RD curve showed that the entropy gradually decreased with the angular acceleration due to the decrease in the azimuthal resolution. In contrast, it can be seen that the entropies resulting from the ICPF, LVD, PSO, and AJTF algorithms slightly increased with the angular acceleration due to their similar principles for estimating parameters. They estimated the rotation parameters in descending order from complex to simple according to the complexity of the target’s non-uniform rotation. For example, if the target presented a form of motion with high angular acceleration and low angular velocity, the estimation error of the higher-order parameters affected the estimation results of the lower-order parameters to a lesser extent. However, if the target presented a motion form with comparable angular acceleration and angular velocity, the estimation error of the higher-order parameters seriously affected the estimation results of the lower-order parameters, resulting in a defocused image. However, the proposed algorithm adopted the method of an independent estimation procedure for rotational parameters, allowing the influence of estimation errors to be avoided.
According to (
4), a larger target represents a larger
, which results in more estimation errors. To verify the algorithm’s performance in this situation, we applied scaling factors to the target (see
Figure 3).
Figure 9a shows the geometric model of the smallest target size after applying the scaling factor. In contrast,
Figure 9b shows the geometric model of the largest target size. We uniformly set 20 target sizes ranging from the smallest size to the largest one.
Figure 10 shows the entropy of the images when using six algorithms with different target sizes. The RD curve showed that the entropy gradually increased with the target size. This is because a larger target occupies a larger proportion of images at a constant azimuth resolution. With the increase in target size, the ICPF, LVD, PSO, and AJTF algorithms performed worse and worse since larger targets resulted in more estimation errors. In contrast, the proposed algorithm performed better when using the independent estimation procedure for rotational parameters.
Furthermore, we expanded (
2) up to the third order to validate the performance of the proposed algorithm in the presence of motion with third-order rotation. The target rotation consisted of three parts, rotation angular velocity, angular acceleration, and angular acceleration rate, and they were set to 0.015 rad/s, 0.017 rad/s
2, and 0.022 rad/s
3, respectively.
Figure 11a shows the image obtained by using the RD algorithm. Comparing it with
Figure 5a, it was found that due to the more complex motion, the defocusing (stretching) of the image was more obvious. Since the highest order of target motion equaled 3, the number of parameters for the search was reset to three, and the inverse function was accordingly refined. Then, the proposed algorithm was used to generate an ISAR image, as shown in
Figure 11b. Furthermore, in order to demonstrate the performance of the proposed algorithm in this case, the entropy, time, and stretched value of ISAR images were calculated before and after compensating using (
21) and (
22). The results were as follows: The entropy of the image before compensating was 8.3247, and after, it was 6.5958. The stretched value of the image before compensating was 158.7457, and after, it was 12.1112. The consumed time in this case was 0.8322s. Obviously, it can be seen that the proposed algorithm still worked well in this case. Therefore, the proposed algorithm can be characterized as a more attractive candidate for obtaining high-quality ISAR images.
4.2. Full-Wave Electromagnetic Simulation Data of an Airplane 3D Conducting Body Model
In this subsection, for a more realistic evaluation of the proposed RMC method, an electromagnetic (EM) scattering prediction technique is used to calculate the full-wave electromagnetic simulation data of an airplane 3D conducting body model.
Figure 12 shows the target model, a Boeing 737 aircraft. The simulation parameters are listed in
Table 3. The polarization mode used is linear polarization. The center frequency of the transmitted signal is 10 GHz, corresponding to a wavelength of 0.03 m. Given its size, the Boeing 737 is classified as an electrically large target, and the large-element PO technique is adopted to calculate complex RCS.
Figure 13 shows the ISAR imaging results from the Boeing 737 aircraft target, and the corresponding entropies and computation time are listed in
Table 4. Because the stretched values in (
22) cannot be calculated in this subsection, they are not listed in
Table 4. Consistently with
Figure 5a,
Figure 13a shows the RDA imaging result, from which it can be seen that due to the non-uniform rotation, the plane was stretched along the azimuth direction. For instance, the tip of the left wing appeared as a line, and the tail was indistinguishable. Although the image quality of other typical methods (see
Figure 13b–e) improved compared with that in
Figure 13a, detailed parts of the images were still not adequately compensated, as the left wing and tail of the target remained somewhat stretched. However, the airplane image was good when using the proposed algorithm, as illustrated in
Figure 13f.
Complex zero-mean white Gaussian noise with a variance of
was added to demonstrate the algorithm’s robustness. The tested SNRs varied from 0 to 30 dB with a step size of 1 dB. For each given SNR, a total of 100 Monte Carlo experiments were processed. In
Figure 14, the entropy values of the ISAR images increased for all methods as the SNR decreased. However, the entropy of the ISAR images using the proposed method was lower than that of the others across all SNRs. Consequently, the proposed RMC method was superior to the other existing methods for the removal of the non-uniform rotational motion in a noisy environment.
4.3. Actual Radar Measurement Data from an Aircraft Target
In this subsection, a set of measured data from the Yak-42 aircraft were utilized to further validate the effectiveness of the proposed algorithm. The Yak-42 is a medium-sized jet aircraft with a length of 36.68 m, a wingspan of 34.88 m, and a height of 9.83 m. The ground-based ISAR operated at a center frequency of 5.5 GHz with a bandwidth of 400 MHz. The dataset comprised approximately 100,000 PRIs, from which we extracted 4000 PRIs, each containing 256 sampling points [
35].
Figure 15a first presents the pulse compression results along the range bins, from which it can be seen that severe range migration occurred. After performing the TMC procedure, the motion trajectories of all scatterers were precisely concentrated in their own range bins, as depicted in
Figure 15b. After the effective motion compensation, the basic aircraft was outlined using the RD algorithm, as shown in
Figure 15c. Furthermore,
Figure 15d depicts the phase in the 199th range bin of the echo data. It should be noted that the phase of the signal varied linearly, which meant that the FT could accomplish the imaging procedure in the azimuthal bin.
To evaluate the effectiveness of the proposed method in real situations, we extracted 1024 PRIs from 4000 PRIs using a non-uniform sampling procedure, where the sampling ratio between the angular velocity and angular acceleration was 3:4, and we used the RD algorithm to generate an ISAR image, as shown in
Figure 16a. The results revealed a stretching phenomenon in the cross-range dimension due to non-uniform rotation (non-uniform sampling). Similarly, employing different algorithms for RMC, we obtained different results, as shown in
Figure 16b–e. Although there was a noticeable improvement compared with
Figure 16a, enabling a rough distinction of the airplane’s contour, the stretching phenomenon in the azimuth direction still existed. Furthermore,
Figure 16f shows the ISAR image obtained via the proposed method, which outlined the airplane vividly, especially the crucial parts, such as the wings and tail. To quantify the ISAR image quality,
Table 5 also lists the entropies and computation time of the images in
Figure 16, showing that the proposed method was able to generate clear and focused images with the lowest entropy, which was 8.0015. To sum up, by combining both the ISAR images and quantitative indices, the proposed method outperformed the other existing methods in terms of computation time and entropy. The latter index was more worthy of attention because, for parameter estimation, the iteration method was based on simple numerical calculations, while the existing methods relied on solving complex equations and lacked feasibility in terms of their implementation in engineering.
Additionally, we use the same method, i.e., non-uniform sampling, to extract echo data of two other motion trajectories from the 100,000 PRIs [
35]. Subsequently, we applied the proposed algorithm, resulting in the outcomes depicted in
Figure 17 and
Figure 18.
Figure 17a and
Figure 18a represent the results of the RD algorithm, while
Figure 17b and
Figure 18b illustrate the imaging results using the proposed algorithm. These results show that the proposed algorithm effectively compensated for the impact from non-uniform rotation and generated clear images.