Unambiguous ISAR Imaging Method for Complex Maneuvering Group Targets

Abstract

In inverse synthetic-aperture radar (ISAR) imaging, it is essential to deal with the Doppler ambiguity of group targets with complex maneuvers in order to avoid the bias of target position towards the actual value. Simultaneously, migration through resolution cell (MTRC) under the Doppler ambiguity is unable to be compensated for as a preprocessing. Traditional ISAR imaging methods for maneuvering targets, however, are undesirable to handle the severe deformation and defocusing in the imaging results induced by the Doppler ambiguity and MTRC. In this paper, we propose a novel and effective ISAR imaging method to improve the imaging quality by removing the Doppler ambiguity and compensating for the MTRC. Specifically, we first model the echo as a multi-component cubic phase signal (m-CPS) and design a high-order instantaneous autocorrelation function–generalized scaled Fourier transform (HIAF–GSCFT) to process the echo. This is to estimate the rotational parameters without MTRC compensation. Then, a maximum weighted contrast algorithm is used to remove the Doppler ambiguity, followed by reconstructing the echo. Compared with the existing method, the proposed method can accurately estimate the rotational parameters under the existing MTRCs and achieves a high-quality ISAR image for group targets, with complex maneuvers without Doppler ambiguity. Experiments of simulated and measured datasets validate its effectiveness and robustness for single target and group targets.

1. Introduction

In traditional ISAR imaging algorithms for maneuvering targets, the difficulty of achieving high-quality imaging results for complex maneuvering group targets is mainly caused by Doppler ambiguity because the Doppler frequency of partial scattering centers exceeds the limit of radar pulse repetition frequency (PRF). For group targets, the Doppler ambiguity generally exists among marginal subtargets, thus yielding the following two problems:

• It results in the difficulty of correcting migration through resolution cell (MTRC) [1] for accurate target rotation parameters estimation. Without the Doppler ambiguity, MTRC can be compensated for by methods such as Keystone [1], and only the Doppler diffusion caused by the higher-order phase is considered. In general, the echo of maneuvering targets can be seen as multi-component linear frequency modulation (LFM) signals in the slow-time domain [2]. ISAR imaging methods for LFM signals can be categorized into two types: time–frequency analysis methods, such as Wigner–Ville distribution [2,3,4], Radon–Wigner transform [5,6], and LV’s distribution (LVD) [7,8], and parameter estimation methods, such as matched Fourier transform (MFT) [9,10] and Chirp–Fourier transform [11,12]. With improvement in the maneuverability of the air targets, the echo can be further modeled as m-CPS [13,14]. Most ISAR imaging methods for m-CPS are based on parameter estimation, including cubic phase function (CPF) [15,16,17,18], high-order ambiguity function (HAF) [19,20,21,22,23], and discrete polynomial phase transformation (DPT) [24]. In recent years, some scholars proposed novel methods based on other well-performing time–frequency distributions [25,26,27]. With the Doppler ambiguity, however, MTRCs can no longer be effectively compensated for by the aforementioned methods. This will further cause difficulty in estimating the echo parameters to compensate rotational motion of targets. Therefore, it is of critical value to design an effective parameter estimation algorithm without corrected MTRC.
• Scattering centers with Doppler ambiguity will appear in an incorrect position [28,29], which highlights the necessity of removing the Doppler ambiguity. To our best knowledge, however, few studies focus on dealing with this issue. Dr. Huang and Dr. Zhang [30] presented a hypothesis that Doppler ambiguity removal can be carried out by sparse reconstruction. However, their proposed method has limited effectiveness for the m-CPS model.

In this study, we propose a novel ISAR imaging method to obtain an unambiguous ISAR imaging result of complex maneuvering group targets. In this method, the echo parameters are estimated to compensate for the rotational motion of targets by high-order instantaneous autocorrelation function–generalized scaled Fourier transform (HIAF–GSCFT), and a maximum weighted contrast (MWC) based algorithm is used to remove the Doppler ambiguity. The echo is then reconstructed to achieve unambiguous ISAR imaging. Experimental results prove that our method has better performance for complex maneuvering targets with MTRCs and Doppler ambiguity than traditional methods and is robust with different SNRs.

2. Materials and Methods

2.1. Signal Model

Generally, the movement of a target consists of translation and rotation. Since in this research, we mainly discuss rotational motion compensation, in this section, we directly analyze the echo after translational motion compensation. In this case, the turntable model of ISAR imaging is shown in Figure 1.

We set the transmitted signal as a linear frequency modulation signal

$$s_T(\widehat{t},t_m)=\mathrm{rect}(\widehat{t}/T_p)e^{j2\pi f_{c}t}{e}^{j\pi \mu {\widehat{t}}^2}$$

(1)

where $T_p$ is the pulse width, $\widehat{t}$ is the fast time, $f_c$ is the carrier frequency, $\mu$ is the chirp rate, $t=\widehat{t}+t_m$ is the total time, and $t_m$ is the slow time, with an integer multiplying the pulse repetition period.

After the stretch processing [31], the echo of the scattering center p is

$$s_p(f_r,t_m)=A_p\mathrm{rect}\left(\frac{f_r}{B}\right)e^{-j\frac{4\pi }{c}(f_c+f_r)\mathsf{\Delta }{R}_p}$$

(2)

where $B$ is the bandwidth, and $f_r$ is the fast time–frequency defined by

$$f_r=\mu \left(\widehat{t}-2R_{ref}/c\right)$$

(3)

where $R_{ref}$ is the reference range, and $c$ is the light speed.

Given that the target performs complex maneuvers, the rotation angle of the scattering center p is

$$\theta \left(t_m\right)=\omega t_m+\frac{1}{2}a{t}_m{}^2+\frac{1}{6}b{t}_m{}^3$$

(4)

where $\omega$ is the angular velocity of rotation, $a$ is the angular acceleration, and $b$ is the second-order angular acceleration.

Based on planar wavefront approximation that is reasonable for ISAR imaging, $\mathsf{\Delta }{R}_p$ in Equation (2) can be expressed as

$$\begin{array}{ll}\mathsf{\Delta }{R}_p(t_m)& =x_p\cos \left(\omega t_m+\frac{1}{2}a{t}_m{}^2+\frac{1}{6}b{t}_m{}^3\right)\\ & +y_p\sin \left(\omega t_m+\frac{1}{2}a{t}_m{}^2+\frac{1}{6}b{t}_m{}^3\right)\end{array}$$

(5)

where $y_p$ and $x_p$ represent the Y–X coordinates of p, respectively, in the coordinate system in Figure 1.

Here, Equation (5) can be approximated by Taylor series expansion as

$$\begin{array}{ll}\mathsf{\Delta }{R}_p(t_m)& =x_p+y_p\omega t_m+\frac{1}{2}\left(y_{p}a-x_p{\omega }^2\right)t_m{}^2\\ & +\frac{1}{6}\left(y_{p}b-3x_{p}a\omega -{\omega }^3\right)t_m{}^3\end{array}$$

(6)

For the simplicity of notation, Equation (6) is written as

$$\mathsf{\Delta }{R}_p(t_m)={\varphi }_{1,p}+{\varphi }_{2,p}{t}_m+\frac{1}{2}{\varphi }_{3,p}{t}_m{}^2+\frac{1}{6}{\varphi }_{4,p}{t}_m{}^3$$

(7)

where ${\varphi }_{1,p}=x_p$, ${\varphi }_{2,p}=y_p\omega$, ${\varphi }_{3,p}=y_{p}a-x_p{\omega }^2$, and ${\varphi }_{4,p}=y_{p}b-3x_{p}a\omega -{\omega }^3$.

Considering the influence of Doppler ambiguity, the echo signal of complex maneuvering group targets containing N scattering centers is

$$\begin{array}{ll}{S}_R(f_r,t_m)& ={\displaystyle \sum _{p=1}^N{s}_p(f_r,t_m)=}{\displaystyle \sum _{p=1}^N{A}_p\mathrm{rect}\left(\frac{f_r}{B}\right)\cdot }\\ & e^{-j\frac{4\pi }{c}(f_c+f_r)\left({\varphi }_{1,p}+\left({\varphi }_{2,p}+da{n}_p\cdot \frac{\lambda }{2}PRF\right)t_m+\frac{1}{2}{\varphi }_{3,p}{t}_m{}^2+\frac{1}{6}{\varphi }_{4,p}{t}_m{}^3\right)}\end{array}$$

(8)

where $da{n}_p\in \left[\dots -2,-1,0,1,2...\right]$ is the number of Doppler ambiguity [30], and $\lambda$ is the wavelength.

It is worth noting that the Doppler ambiguity shifts the unique correspondence between the azimuth information of the scattering centers and its Doppler frequency. In other words, scattering centers with the same Doppler frequency may also have different MTRC slopes, as shown in Figure 2.

In Figure 2a, the position of the scattering center with MTRC is $x=18 \mathrm{m}$ and $y=-40 \mathrm{m}$. In contrast, the position of the scattering center in Figure 2b is $x=18 \mathrm{m}$ and $y=60 \mathrm{m}$. Although with the same Doppler frequency, the MTRC slopes of these two scattering centers are completely different, making the traditional MTRC correction method fail to achieve an ideal result.

2.2. Parameters Estimation Based on HIAF–GSCFT

Due to Doppler ambiguity, the echo parameters estimation needs to be carried out when the MTRC cannot be compensated for. Thus, HIAF–GSCFT is designed to estimate the echo parameters. Inspired by the autocorrelation function [32], we define HIAF as

$$\begin{array}{ll}R\left(f_r,t_m\right)& =S_R\left(f_r,t_m+{\varsigma }_1+{\varsigma }_2\right)\cdot S_R{}^{*}\left(f_r,t_m-{\varsigma }_1+{\varsigma }_2\right)\cdot \\ & S_R{}^{*}\left(f_r,t_m+{\varsigma }_1-{\varsigma }_2\right)\cdot S_R\left(f_r,t_m-{\varsigma }_1-{\varsigma }_2\right)\end{array}$$

(9)

where ${\varsigma }_1$ and ${\varsigma }_2$ are two delay constants. It is worth noting that $f_r$ and $t_m$ should be interpolated for the sake of the same number of sampling points. Combined with Equation (8), Equation (9) can be written as

$$\begin{array}{ll}R\left(f_r,t_m\right)& ={\displaystyle \sum _{p=1}^N{A}_p^4{e}^{-j\frac{4\pi }{c}\left(f_c+f_r\right)\left(4{\varphi }_{3,p}{\varsigma }_1{\varsigma }_2+4{\varphi }_{4,p}{\varsigma }_1{\varsigma }_2{t}_m\right)}}+\\ & R_{cross}\left(f_r,t_m\right)\end{array}$$

(10)

where $R_{cross}\left(f_r,t_m\right)$ represents the cross-terms created by HIAF. In Equation (10), we ignore $\mathrm{rect}\left(f_r/B\right)$ because in HIAF and phase separation, $\mathrm{rect}\left(f_r/B\right)$ is invariant.

To ensure that GSCFT can effectively estimate the parameters, the phases only related to $t_m$ need to be eliminated. We define this process as phase separation (PS), which can be expressed as

$$\mathrm{PS}\left[R\left(f_r,t_m\right)\right]=\mathrm{F}{\mathrm{T}}_{t_r}\left[\left|{\mathrm{IFT}}_{f_r}\left[R\left(f_r,t_m\right)\right]\right|\right]$$

(11)

where $t_r$ is the corresponding time variable of $f_r$ after inverse Fourier transform, $\mathrm{F}{\mathrm{T}}_{t_r}$ represents the Fourier transform in $t_r$, ${\mathrm{IFT}}_{f_r}$ represents the inverse Fourier transform in $f_r$, and $*$ represents the conjugate operation of the result after IFT.

After PS, the echo phase is

$$\begin{array}{ll}\mathrm{PS}\left[R\left(f_r,t_m\right)\right]& ={\displaystyle \sum _{p=1}^N{A}_p^4{e}^{-j\frac{4\pi f_r}{c}\left(4{\varphi }_{3,p}{\varsigma }_1{\varsigma }_2+4{\varphi }_{4,p}{\varsigma }_1{\varsigma }_2{t}_m\right)}}\\ & +R_{PS-cross}\left(f_r,t_m\right)\end{array}$$

(12)

where $R_{PS-cross}\left(f_r,t_m\right)$ is the cross-terms processed by PS.

According to [15] and [33], GSCFT is defined as

$$\begin{array}{ll}T\left(f_{\left[\xi {\mathsf{\Delta }}_m^e{\mathsf{{\rm Y}}}_m^f\right]},{\mathsf{\Delta }}_m\right)& ={\displaystyle \underset{{\mathsf{{\rm Y}}}_m^f}{\int }\left[g\left({\mathsf{\Delta }}_m\right)e^{j2\pi \psi {\mathsf{\Delta }}_m^e{\mathsf{{\rm Y}}}_m^f}\right]\cdot }\\ & e^{-j2\pi \left(\xi {\mathsf{\Delta }}_m^e{\mathsf{{\rm Y}}}_m^f{f}_{\left[\xi {\mathsf{\Delta }}_m^e{\mathsf{{\rm Y}}}_m^f\right]}\right)}d\left({\mathsf{\Delta }}_m^e{\mathsf{{\rm Y}}}_m^f\right)\\ & =g\left({\mathsf{\Delta }}_m\right)\delta \left(f_{\left[\xi {\mathsf{\Delta }}_m^e{\mathsf{{\rm Y}}}_m^f\right]}-\frac{\psi }{\xi }\right)\end{array}$$

(13)

where $\xi$ represents the zoom factor to avoid spectrum aliasing, $f_{\left[\xi {\mathsf{\Delta }}_m^e{\mathsf{{\rm Y}}}_m^f\right]}$ is the scaled frequency domain with respect to ${\mathsf{{\rm Y}}}_m^f$, $e$ and $f$ are constant power, $g\left({\mathsf{\Delta }}_m\right)$ is a function of ${\mathsf{\Delta }}_m$, and $\psi$ is an empirical parameter.

According to Equation (12), the symbol in Equation (13) can be replaced by

$${\mathsf{\Delta }}_m=f_r,{\mathsf{{\rm Y}}}_m=t_m,e=f=1$$

(14)

$$g\left({\mathsf{\Delta }}_m\right)e^{j2\pi \psi {\mathsf{\Delta }}_m^e{\mathsf{{\rm Y}}}_m^f}=\mathrm{PS}\left[R\left(f_r,t_m\right)\right]$$

(15)

To this end, the result of GSCFT is

$$\begin{array}{ll}T\left(f_r,f_{\left[\xi f_r{t}_m\right]}\right)& ={\displaystyle \sum _{p=1}^N{A}_p^4}\delta \left(f_{\left[\xi f_r{t}_m\right]}+\frac{8{\varphi }_{4,p}{\varsigma }_1{\varsigma }_2}{c\xi }\right)e^{-j\frac{16\pi }{c}{\varphi }_{3,p}{\varsigma }_1{\varsigma }_2{f}_r}+\\ & T_{cross}\left(f_r,f_{\left[\xi f_r{t}_m\right]}\right)\end{array}$$

(16)

where $T_{cross}\left(f_r,f_{\left[\xi f_r{t}_m\right]}\right)$ is the cross-terms processed by GSCFT.

Clearly, the first term of Equation (16) on the right side achieves energy focusing in $f_{\left[\xi f_r{t}_m\right]}$ domain, and the positions of the peaks are only related to ${\varphi }_{4,p}$. After inverse Fourier transform in $f_r$ domain, $T\left(f_r,f_{\left[\xi f_r{t}_m\right]}\right)$ can be expressed as

$$\begin{array}{ll}{\mathrm{IFT}}_{f_r}\left[T\left(f_r,f_{\left[\xi t_r{t}_m\right]}\right)\right]& ={\displaystyle \sum _{p=1}^N{A}_p^4}\delta \left(f_{\left[\xi f_r{t}_m\right]}+\frac{8{\varphi }_{4,p}{\varsigma }_1{\varsigma }_2}{c\xi }\right)\cdot \\ & \delta \left(t_r-\frac{8{\varphi }_{3,p}{\varsigma }_1{\varsigma }_2}{c}\right)+\\ & T_{FT-cross}\left(t_r,f_{\left[\xi f_r{t}_m\right]}\right)\end{array}$$

(17)

In Equation (17), the echo of each individual scattering center has formed a peak on the $t_r-f_{\left[\xi f_r{t}_m\right]}$ domain. Simultaneously, the cross-terms cannot achieve energy accumulation through HIAF–GSCFT (see a detailed analysis of the cross-term in Appendix A). Therefore, the peak detection to estimate ${\varphi }_{3,p}$ and ${\varphi }_{4,p}$ is defined by

$$\left\{t_{rm},f_{{\left[\xi f_r{t}_m\right]}_m}\right\}=\arg \max \left|{\mathrm{IFT}}_{f_r}\left[T\left(t_r,f_{\left[\xi f_r{t}_m\right]}\right)\right]\right|$$

(18)

$${\widehat{\varphi }}_{3,pm}=\frac{c{t}_{rm}}{8{\varsigma }_1{\varsigma }_2},{\widehat{\varphi }}_{4,pm}=-\frac{c\xi f_{{\left[\xi f_r{t}_m\right]}_m}}{8{\varsigma }_1{\varsigma }_2}$$

(19)

where $t_{rm}$ and $f_{{\left[\xi f_r{t}_m\right]}_m}$ represent the coordinates of the maximum value, respectively.

Moreover, GSCFT, ${\widehat{\varphi }}_{3,pm}$, and ${\widehat{\varphi }}_{4,pm}$ can also be used to estimate ${\varphi }_{1,pm}$ and ${\varphi }_{2,pm}$. We first compensate for the higher-order phase of the dominant scattering center and perform PS by

$$\mathsf{\Phi }\left(f_r,t_m\right)=e^{-j\frac{4\pi }{c}\left(f_r+f_c\right)\left(\frac{1}{2}{\widehat{\varphi }}_{3,pm}{t}_m{}^2+\frac{1}{6}{\widehat{\varphi }}_{4,pm}{t}_m{}^3\right)}$$

(20)

$$\begin{array}{ll}{S^{\prime }}_{R-PS}\left(f_r,t_m\right)& =\mathrm{PS}\left[S_R\left(f_r,t_m\right)\cdot \mathsf{\Phi }\left(f_r,t_m\right)\right]\\ & =A_{pm}^{}{e}^{-\frac{4\pi }{c}\left({\varphi }_{1,pm}{f}_r+{\varphi }_{2,pm}{f}_r{t}_m\right)}+{S^{\prime }}_{others}\left(f_r,t_m\right)\end{array}$$

(21)

where ${S^{\prime }}_{others}\left(f_r,t_m\right)$ represents the echo components of nondominant scattering centers.

After GSCFT and Fourier transform, ${S^{\prime }}_{R-PS}\left(f_r,t_m\right)$ becomes

$$\begin{array}{ll}{T^{\prime }}_R\left(t_r,f_{\left[\xi f_r{t}_m\right]}\right)& ={\mathrm{IFT}}_{f_r}\left[\mathrm{GSCFT}\left[{S^{\prime }}_R\left(f_r,t_m\right)\right]\right]\\ & =A_{pm}^{}\delta \left(t_r-\frac{2{\varphi }_{1,pm}}{c}\right)\delta \left(f_{\left[\xi f_r{t}_m\right]}+\frac{2{\varphi }_{2,pm}}{c\xi }\right)+\\ & {T^{\prime }}_{others}\left(t_r,f_{\left[\xi f_r{t}_m\right]}\right)\end{array}$$

(22)

where ${T^{\prime }}_{others}\left(t_r,f_{\left[\xi f_r{t}_m\right]}\right)$ is ${S^{\prime }}_{others}\left(f_r,t_m\right)$ processed by GSCFT and Fourier transform (detailed analysis is provided in Appendix A).

Similarly, peak detection is used to obtain ${\widehat{\varphi }}_{1,pm}$ and ${\widehat{\varphi }}_{2,pm}$ by

$$\left\{t_{r{m}^{\prime }},f_{{\left[\xi f_r{t}_m\right]}_{m^{\prime }}}\right\}=\arg \max \left|{T^{\prime }}_R\left(t_r,f_{\left[\xi f_r{t}_m\right]}\right)\right|$$

(23)

$${\widehat{\varphi }}_{1,pm}=\frac{c{t}_{r{m}^{\prime }}}{2},{\widehat{\varphi }}_{2,pm}=-\frac{c\xi f_{{\left[\xi f_r{t}_m\right]}_{m^{\prime }}}}{2}$$

(24)

In order to ensure the accuracy of parameter estimation, we use a grid-searching manner for $f_{\left[\xi f_r{t}_m\right]}$ in $\left[-PRF/2,PRF/2\right]$.

2.3. Doppler Ambiguity Removal Based on MWC

Given the Doppler ambiguity, the relationship between ${\widehat{\varphi }}_{2,pm}$ and the actual value is

$${{\varphi }^{\prime }}_{2,pm}={\widehat{\varphi }}_{2,pm}+da{n}_{pm}\cdot \frac{\lambda }{2}PRF$$

(25)

where $da{n}_{pm}$ is the Doppler ambiguity number of the dominant scattering center.

MWC is thus proposed to determine $da{n}_{pm}$. We first construct Equation (26) to multiply with the echo and transform the result into the $t_r-f_d$ domain

$$f_4\left(f_r,t_m,dan\right)=e^{j\frac{4\pi }{c}{f}_r\left(\left({\widehat{\varphi }}_{2,pm}+dan\cdot \frac{\lambda }{2}PRF\right)t_m+\frac{1}{2}{\widehat{\varphi }}_{3,pm}{t}_m{}^2+\frac{1}{6}{\widehat{\varphi }}_{4,pm}{t}_m{}^3\right)}$$

(26)

$${\mathit{I}}_{da}\left(t_r,f_d,dan\right)=\left|{\mathrm{FFT}}_{2D}\left[S_R(f_r,t_m)\cdot f_4\left(f_r,t_m,dan\right)\right]\right|$$

(27)

where $dan\in \left[\dots -2,-1,0,1,2...\right]$ is used to determine $da{n}_{pm}$.

In Equation (27), when $dan=da{n}_{pm}$, the MTRC will be completely compensated for. In this case, the image quality of ${\mathit{I}}_{da}\left(t_r,f_d,dan\right)$ is better than $dan\ne da{n}_{pm}$.

To prevent the interference caused by the nondominant scattering centers, we define the weighted contrast as

$$\mathrm{WC}\left(dan\right)=\sqrt{\frac{{\displaystyle \mathsf{\iint }\left(k\left(t_r,f_d\right){{\mathit{{\rm I}}}^{\prime }}_{da}\left(t_r,f_d,dan\right)-\mu \right)d{t}_{r}d{f}_d}}{N_r{N}_d{\mu }^2}}$$

(28)

$${{\mathit{{\rm I}}}^{\prime }}_{da}\left(t_r,f_d,dan\right)=I_{da}\left(t_r+\frac{2{\widehat{\varphi }}_{1,pm}}{c},f_d+\frac{2{\widehat{\varphi }}_{2,pm}}{\lambda },dan\right)$$

(29)

$$\mu =\frac{1}{N_r{N}_d}{\displaystyle \mathsf{\iint }k\left(t_r,f_d\right){{\mathit{I}}^{\prime }}_{da}\left(t_r,f_d,dan\right)d}{t}_{r}d{f}_d$$

(30)

where $k\left(t_r,f_d\right)$ is the weighting coefficient defined by

$$k\left(t_r,f_d\right)=\frac{1}{\sqrt{2\pi }\sigma \left(t_r\right)}{e}^{\frac{f_d{}^2}{2\sigma {\left(t_r\right)}^2}}$$

(31)

where $\sigma \left(t_r\right)$ is proportional to the standard deviation of the data in each range cell.

Now, the dominant scattering center is moved to the middle of ${\mathit{I}}_{da}\left(t_r,f_d,dan\right)$ and has a larger weight than the other scattering centers. Thus, $da{n}_{pm}$ can be determined by the maximum of $\mathrm{WC}\left(dan\right)$ by

$$\left\{d\widehat{a}{n}_{pm}\right\}=\max \left\{\mathrm{WC}\left(dan\right)\right\}$$

(32)

and the estimated Doppler frequency is

$${{\widehat{\varphi }}^{\prime }}_{2,pm}={\widehat{\varphi }}_{2,pm}+d\widehat{a}{n}_{pm}\cdot \frac{\lambda }{2}PRF$$

(33)

2.4. ISAR Imaging Method Based on HIAF–GSCFT and Doppler Ambiguity Removal

In this section, a HIAF–GSCFT- and MWC-based ISAR imaging method is introduced for complex maneuvering group targets with the following procedure:

Step 1

Obtain $S_R(f_r,t_m)$ by processing the original echo signal using stretch processing;

Step 2

Estimate ${\varphi }_{3,pm}$ and ${\varphi }_{4,pm}$ using HIAF–GSCFT;

Step 3

Estimate ${\varphi }_{1,pm}$ and ${\varphi }_{2,pm}$ with PS, GSCFT, inverse Fourier transform, and the compensation function constructed by the estimated echo parameters;

Step 4

Remove the Doppler ambiguity by using the MWC and determine the actual Doppler frequency of the dominant scattering center;

Step 5

Estimate the amplitude of the dominant scattering center by the least-squares method as

$${\widehat{A}}_{pm}=\frac{S_R(f_r,t_m)\cdot {\mathsf{\Theta }}^{*}(f_r,t_m)}{{\left|\mathsf{\Theta }(f_r,t_m)\right|}^2}$$

(34)

$$\mathsf{\Theta }(f_r,t_m)=e^{-j\frac{4\pi }{c}\left(f_c+f_r\right)\left({\widehat{\varphi }}_{1,pm}+{{\widehat{\varphi }}^{\prime }}_{2,pm}{t}_m+\frac{1}{2}{\widehat{\varphi }}_{3,pm}{t}_m{}^2+\frac{1}{6}{\widehat{\varphi }}_{4,pm}{t}_m{}^3\right)}$$

(35)

Step 6

Delete the echo of the dominant scattering center from $S_R(f_r,t_m)$ as

$$S_{rm}\left(f_r,t_m\right)={\widehat{A}}_p{e}^{-j\frac{4\pi }{c}\left(f_c+f_r\right)\left({\widehat{\varphi }}_{1,pm}+{{\widehat{\varphi }}^{\prime }}_{2,pm}{t}_m+\frac{1}{2}{\widehat{\varphi }}_{3,pm}{t}_m{}^2+\frac{1}{6}{\widehat{\varphi }}_{4,pm}{t}_m{}^3\right)}$$

(36)

$$S_R(f_r,t_m)=S_R(f_r,t_m)-S_{rm}\left(f_r,t_m\right)$$

(37)

Step 7

Repeat steps (2)–(6) till the residual energy of the echo reach the energy threshold;

Step 8

Reconstruct the echo by

$$S_{rec}(f_r,t_m)={\displaystyle \sum _{p=1}^N{\widehat{A}}_{pm}{e}^{-j\frac{4\pi }{c}{\widehat{\varphi }}_{1,pm}{f}_r}{e}^{-j\frac{4\pi }{\lambda }{{\widehat{\varphi }}^{\prime }}_{2,pm}{t}_m}}$$

(38)

Eventually, we interpolate the slow time grid and perform a 2D Fourier transform on $S_{rec}(f_r,t_m)$ to obtain an ISAR imaging result as

$$S_{rec}(t_r,f_d)={\displaystyle \sum _{p=1}^N{\widehat{A}}_{pm}\delta \left(t_r+\frac{2{\widehat{\varphi }}_{1,pm}}{c}\right)\delta \left(f_d+\frac{2{{\widehat{\varphi }}^{\prime }}_{2,pm}}{\lambda }\right)}$$

(39)

The flowchart of the proposed method is shown in Figure 3.

3. Results

In this section, we verify the proposed method from three aspects: the performance of parameter estimation, the capability of single target imaging, and the effectiveness of imaging for complex maneuvering group targets.

3.1. Validation of the Parameters Estimation Method Based on HIAF–GSCFT

As an m-CPS has two components, its signal model is similar to (8), and the rotational parameters are set as

Component 1: ${\varphi }_{1,1}=20$, ${\varphi }_{2,1}=30$, ${\varphi }_{3,1}=15$, ${\varphi }_{4,1}=25$;

Component 2: ${\varphi }_{1,2}=-25$, ${\varphi }_{2,2}=-15$, ${\varphi }_{3,2}=-20$, ${\varphi }_{4,2}=-25$.

After HIAF–GSCFT for the m-CPS, the result is shown in Figure 4.

In Figure 4, CR and QCR represent the chirp rate and the quadratic chirp rate, respectively.

The two components achieve two individual energy accumulation. According to Figure 4, we estimate the parameters as ${\widehat{\varphi }}_{3,1}=14.91$, ${\widehat{\varphi }}_{4,1}=24.94$, ${\widehat{\varphi }}_{3,2}=-19.88$, and ${\widehat{\varphi }}_{4,2}=-24.94$, corresponding to relative estimation errors as 0.6%, 0.24%, 0.6%, and 0.24%, respectively. This is caused by discrete sampling intervals. Considering the small scale, however, these errors can be ignored.

Using the estimated parameters to compensate for the high-order phases of component 1 and component 2, respectively, and perform GSCFT, the results under two compensation cases are shown in Figure 5.

In Figure 5a, only component 1 achieves effective energy focusing because its high-order phases are compensated for correctly. Thus, we have ${\widehat{\varphi }}_{1,1}=19.88$ and ${\widehat{\varphi }}_{2,1}=29.96$. Similarly, the frequency and the initial phase of component 2 can be estimated from Figure 5b as ${\widehat{\varphi }}_{1,2}=-14.91$ and ${\widehat{\varphi }}_{2,2}=-24.94$ respectively. The relative errors of these estimated parameters are 0.6%, 0.24%, 0.6%, and 0.24%, respectively.

From these results, it is fundamental that our proposed HIAF–GSCFT-based parameter estimation method has a strong capability of achieving accurate parameter estimation of m-CPS, with high-order coupling between $f_r$ and $t_m$.

3.2. Validation of the Capability of Single Target Imaging

We adopted Yak-42 measured data to validate the imaging ability for a single target of the proposed method. The outline of Yak-42 is shown in Figure 6.

Parameters of the radar are 5.52 GHz carrier frequency, 100 Hz pulse repetition frequency, and 400 MHz bandwidth. We added white Gaussian noise (SNR = 5 dB and 0 dB) and MTRC to the data and compared our proposed method with range-Doppler (RD) [1], HAF–ICPF [21], and integrated parametric cubic phase function-reversing Wigner–Ville distribution (IPCPF–RWVD) [15], where RD is a classical ISAR imaging algorithm, HAF–ICPF is an effective imaging method for complex maneuvering target, and IPCPF–RWVD is a robust imaging method for a complex maneuvering target. The results of the above four methods are in Figure 7 and Figure 8.

Specifically, the RD method was severely defocused, especially on the body and wings of the aircraft. The HAF–ICPF method slightly improved the imaging quality, but the scattering centers of the body were still defocused. The IPCPF–RWVD method had better performance than HAF–ICPF because the IPCPF–RWVD method had a lower degree of nonlinearity. However, the result of the IPCPF–RWVD method was still defocused because MTRC affects the accuracy of parameter estimation. In contrast, our method effectively overcame the negative influence of MTRC in parameter estimation and obtained clear imaging results, as shown in Figure 7d and Figure 8d. In summary, this experiment proved the strong ability for high-quality ISAR imaging for a single target when MTRC is uncalibrated.

3.3. Validation of Imaging for Complex Maneuvering Group Targets

Next, we used synthetic data to validate the effectiveness of the proposed method for complex maneuvering group targets. As shown in Figure 9a, the simulated aircraft consists of 330 scattering centers, with a length of 18.67 m and a wingspan of 24 m. The amplitude of the scattering centers obeys the Rayleigh distribution, with $\sigma =0.5$. The group targets were composed of five simulated aircraft, and the spatial distribution is given in Figure 9b.

Parameters of radar and group targets are shown in

Table 1. Parameters of radar.

Frequency	9.6 GHz
Bandwidth	2 GHz
PRF	200 Hz
Pulse width	60 μs
Sampling rate	13 MHz
Sampling number	512

and

Table 2. Parameters of group targets.

Target Code	Target 1, 5	Target 2, 3, 4
Velocity/m/s	453.78	471.24
A = Acceleration/m/s2	69.81	74.17
Jerk/ m/s3	43.63	47.99
Rotation angle/◦	5.77	6.04

, respectively.

For SNR = 5 dB, the ISAR imaging result obtained with the proposed method is shown in Figure 10f. In comparison, the results by RD, RD–Keystone, SPWVD, and HAF–ICPF, as well as the results reported in [30], are provided in Figure 10a–e, respectively. SPWVD is an effective imaging method for the LFM signal model, and the method in [30] is an imaging method for group targets.

When using the RD method, the Doppler ambiguity resulted in targets 1 and 5 appearing in the wrong positions, and the results were completely defocused. The result of RD–Keystone was even worse than RD because Keystone is invalid with MTRC under the Doppler ambiguity. SPWVD and HAF–ICPF methods improved the imaging results at different but limited levels of target defocusing; however, they did not have any effects on removing the Doppler ambiguity. The method in [30] was unable to remove the Doppler ambiguity of every scattering center because [30] is not ideal when applied to the m-CPS model. In contrast, our proposed method completely eliminated the Doppler ambiguity and defocus, as shown in Figure 10f. In summary, the proposed method effectively achieved the best quality imaging result for complex maneuvering group targets among all of the compared methods.

Finally, we validated the imaging quality of the proposed method under different SNRs. We adopted entropy and contrast to measure the quality of imaging results. After 50 Monte Carlo experiments, the SNR versus entropy curves and SNR versus contrast curves of the above six methods were drawn, which are shown in Figure 11.

We found that the imaging quality of RD and RD–Keystone were invalid. This is because the two methods were unable to compensate for the influence caused by the nonuniform rotation. SPWVD and HAF had better performance than the above two methods but were still not ideal because of the existence of Doppler ambiguity and MTRC. The performance of the method in [30] was the closest to that of the proposed method. However, the method was unable to correct the Doppler ambiguity of complex maneuvering group targets. In comparison, the imaging quality of our proposed method is significantly superior to all the other methods with different SNRs. This experiment effectively revealed the robustness of our method against different noise levels.

4. Conclusions

In this research, we presented a novel and effective ISAR imaging method for complex maneuvering group targets. In this method, the echo parameters were estimated via HIAF–GSCFT, and the MWC algorithm was used to remove the Doppler ambiguity. The proposed method effectively estimated rotational parameters in the presence of MTRC and improved the accuracy of Doppler ambiguity removal. Moreover, our method could be applied for complex maneuvering group targets. Experimental results demonstrated the effectiveness and robustness of this method with different SNRs. In future research, we plan to further improve the proposed method, by reducing the nonlinearity of the algorithm to improve the algorithm’s noise robustness and reduce the computational complexity.