#### 4.1. Algorithm Step Simulations

To prove the validity of the proposed algorithm, we carry out the following simulations.

Figure 1 is the scenario of the simulation. In the simulation, the altitude of the aircraft is

$6000\text{\hspace{0.17em}}\mathrm{m}$, the flight velocity is

$200\text{\hspace{0.17em}}\mathrm{m}/\mathrm{s}$. The carrier frequency, PRF and synthetic aperture time of SAR are

$10\text{\hspace{0.17em}}\mathrm{GHz}$,

$480\text{\hspace{0.17em}}\mathrm{Hz}$, and 1 s, respectively. Set three rotation targets T1, T2 and T3 in a single range cell. Their parameters are in

Table 1. According to the motion state and position of T1, T2 and T3, the echo signal received by SAR is simulated by computer as the raw data. According to

${A}_{\omega}=4\mathsf{\pi}{r}_{0}{f}_{\mathrm{a}}/\lambda $ and

$F=2v{x}_{1}/\lambda {R}_{\mathrm{Jc}}$, the theoretical values of

${A}_{\omega}$ of T1, T2 and T3 come out to be 125.6 Hz, 100.5 Hz and 90.4 Hz, respectively, and their theoretical values of

$F$ are −70.5 Hz, 20 Hz and 40 Hz, respectively. Presume the azimuth echo of the rotating targets are mixed with Gaussian white noise

$n\left({t}_{\mathrm{a}}\right)$, the received azimuth echo can be shown as

Set the signal-to-noise ratio (SNR) as −2 dB.

The autocorrelation result of

$x\left({t}_{\mathrm{a}}\right)$ is shown in

Figure 5a. It is found that periodic component is present. We can get the frequency estimation of periodic component as

${\widehat{f}}_{\mathrm{a}}^{\left(1\right)}=2\text{\hspace{0.17em}}\mathrm{Hz}$. STFT is performed on

$x\left({t}_{\mathrm{a}}\right)$ (the Kaiser window’s width is 45, the same below), and the TF distribution result can be seen in

Figure 5b. The strong reflection component is clearly visible while the weak reflection component is almost impossible to observe. This is the masking phenomenon of weak MM target. Extract the TF curve according to

Section 3.2, and establish the parameter space

$K=\left({A}_{1},{\phi}_{1},{F}_{1}\right)$. Set the maximum Doppler frequency search scope is

$0\sim 240\text{\hspace{0.17em}}\mathrm{Hz}$, the center frequency search scope is

$-100\sim 100\text{\hspace{0.17em}}\mathrm{Hz}$, the two frequency step size in search is 1 Hz, the phase search scope is

$0\sim {360}^{\xb0}$, and the phase step size in search is

${1}^{\xb0}$. The HT is carried out on the extracted TF curve and

$\mathit{Acc}\left({A}_{1},{\phi}_{1},{F}_{1}\right)$ is obtained. The results are illustrated in

Figure 5c,d.

Figure 5c denotes the HT result versus center frequency F within the search scope, thereby obtaining

${\widehat{F}}^{\left(1\right)}=-69\text{\hspace{0.17em}}\mathrm{Hz}$. Hence, the image of

$\mathit{Acc}\left({A}_{1},{\phi}_{1},-69\mathrm{Hz}\right)$ is drawn in

Figure 5d, where the ordinate indicates the maximum m-D amplitude and the abscissa indicates the initial phase value. We can see that the rotating target has formed a peak in

Figure 5d. By seeking the peak in

$\left\{{A}_{1},{\phi}_{1}\right\}$ domain, we get the estimated value of the first target parameter as

$\left({\widehat{A}}_{\omega}^{\left(1\right)},{\widehat{\phi}}_{0}^{\left(1\right)}\right)=\left(125\mathrm{Hz},{120}^{\xb0}\right)$.

According to steps 7–9 in

Section 3.3, the first MM component is removed from

$x\left({t}_{\mathrm{a}}\right)$ and we can get signal

${x}_{1}\left({t}_{\mathrm{a}}\right)$. After performing

${x}_{1}\left({t}_{\mathrm{a}}\right)$ autocorrelation processing, the result is reflected in

Figure 6a. We find the presence of periodic signal component. We can get the frequency estimation of the second periodic component as

${\widehat{f}}_{\mathrm{a}}^{\left(2\right)}=1.5\text{\hspace{0.17em}}\mathrm{Hz}$. STFT result of

${x}_{1}\left({t}_{\mathrm{a}}\right)$ is illustrated in

Figure 6b and the first component has been removed. After performing HT on the extracted TF curve, the results are reflected in

Figure 6c,d. Similarly, we get the estimated values of the second target parameter as

$\left({\widehat{A}}_{\omega}^{\left(2\right)},{\widehat{\phi}}_{0}^{\left(2\right)},{\widehat{F}}^{\left(2\right)}\right)=\left(101\text{\hspace{0.17em}}\mathrm{Hz},{60}^{\xb0},21\text{\hspace{0.17em}}\mathrm{Hz}\right)$.

After the second MM component is removed from

${x}_{1}\left({t}_{\mathrm{a}}\right)$, we can get signal

${x}_{2}\left({t}_{\mathrm{a}}\right)$. The autocorrelation result of

${x}_{2}\left({t}_{\mathrm{a}}\right)$ is shown in

Figure 7a. We find that periodic signal component is still present. The frequency estimation of the third periodic component is

${\widehat{f}}_{\mathrm{a}}^{\left(3\right)}=1.2\text{\hspace{0.17em}}\mathrm{Hz}$.

Figure 7b demonstrates the STFT result of

${x}_{2}\left({t}_{\mathrm{a}}\right)$. We can see that the second component is successfully removed and the weak reflection component is clearly displayed in the TF distribution. Elimination of strong targets prove to be a useful method in weak target enhancement. The results of HT are illustrated in

Figure 7c,d. Hence, we can get the estimated values of the third target parameter as

$\left({\widehat{A}}_{\omega}^{\left(3\right)},{\widehat{\phi}}_{0}^{\left(3\right)},{\widehat{F}}^{\left(3\right)}\right)=$$\left(89\text{\hspace{0.17em}}\mathrm{Hz},{29}^{\xb0},43\text{\hspace{0.17em}}\mathrm{Hz}\right)$.

After the third MM component is removed from

${x}_{2}\left({t}_{\mathrm{a}}\right)$, we can get signal

${x}_{3}\left({t}_{\mathrm{a}}\right)$. The autocorrelation result and STFT result of

${x}_{3}\left({t}_{\mathrm{a}}\right)$ are shown in

Figure 8. We cannot find any periodic signal component from the results, so we stop detecting. So far, all the three rotating targets set in the simulation have been detected. We make a comparison between the estimated values of the parameters and their theoretical values. The results are shown in

Table 2. We find that the estimated values of parameters are in accordance with the theoretical values. The final processed results show that we successfully achieve the detection of large dynamic reflection coefficient MM targets. Therefore, it proves the correctness of our algorithm.

To compare our algorithm with the one proposed in [

16], we use the IRT algorithm to detect the MM target T1.

Figure 9a demonstrates the IRT result. The IRT result is defocused, so we fail to obtain any information about T1. This is because the azimuth echo generated by T1 is a non-centered SFM signal (

${F}^{\left(1\right)}=-70.5\text{}\mathrm{Hz}\ne 0$). If the rotational center coordinates of T1 is changed to

$\left(0,8000\text{\hspace{0.17em}}\mathrm{m},0\right)$, the detection result of T1 is shown in

Figure 9b. It can be seen that the IRT algorithm can accurately detect T1, since the azimuth echo generated by T1 is a centered SFM signal (

${F}^{\left(1\right)}=0$) this time. Similarly, the center frequencies of T2 and T3 are not zero, so the IRT algorithm cannot detect T2, T3. Therefore, our algorithm has a broader scope of applications for MM target detection compared with the IRT algorithm.

The algorithm in [

15] is used to detect T1, T2 and T3, and the detection results are shown in

Figure 10. It can be seen that the algorithm in [

15] can accurately detect T1 and T2, but it is not possible to obtain any information about T3. This is because T3 is too weak, and the TF curve of T3 cannot be extracted by the algorithm in [

15]. Therefore, compared with the algorithm in [

15], the algorithm in this paper is able to detect the MM targets with larger reflection coefficient difference.

#### 4.3. Field Experiment

In this part, we conduct experiments to check into our algorithm using X-band SAR. The SAR is equipped on Yun-8 aircraft (

Figure 12a) with a flying altitude of 6000 m and a flying velocity of 168 m/s. The carrier frequency of SAR is 9.8 GHz, the PRF is 400 Hz, and the synthetic aperture time is 1.5 s. In the experiment, SAR operates in side-view mode. A symmetrical rotating angle reflector P1, P2 and a fixed angle reflector P3 are placed in the scene, wherein all reflectors are of size

$0.25\text{\hspace{0.17em}}\mathrm{m}\times 0.25\text{\hspace{0.17em}}\mathrm{m}\times 0.18\text{\hspace{0.17em}}\mathrm{m}$, the angle reflectors P1, P3 are made of aluminum, and the angle reflector P2 is made of reinforced plastics. The effective radii of the two rotating reflectors are both 0.3 m and their rotational frequencies are 1.5 Hz. The angle reflector P3 serves as a positioning angle reflector, and the angle reflectors P1 and P2 are the MM targets to be detected. Although P1 and P2 are of the same size, the reflection coefficient of P1 is stronger than that of P2. After SAR gets the echo, we use R-D algorithm for imaging. The on-site shooting of the scene is shown in

Figure 12b and the imaging result of SAR is illustrated in

Figure 13. We can see that the positioning angle reflector focuses well, while the rotating targets produce azimuth defocusing, which is caused by the micro-Doppler effect.

Before processing the real echo data, we use the computer to obtain the simulation echo data of P1 and P2 in the scene. Then the azimuth echo of the rotation targets is extracted, and the rotation frequency can be estimated to be 1.5 Hz by autocorrelation method. Next, we use the algorithm to process the simulation echo signal and the results are shown in

Figure 14.

Figure 14a is the STFT result of the azimuth echo (the Kaiser window’s width is 37),

Figure 14b,c is the HT result of the strongest component TF curve,

Figure 14d is the STFT result of the residual azimuth echo, and

Figure 14e,f is the HT result of the residual signal. Therefore, the estimated values of MM parameters of P1 and P2 can be obtained as

$\left({\widehat{A}}_{\omega}^{\left(1\right)},{\widehat{\phi}}_{0}^{\left(1\right)},{\widehat{F}}^{\left(1\right)}\right)=\left(188\text{\hspace{0.17em}}\mathrm{Hz},{52}^{\xb0},1\text{\hspace{0.17em}}\mathrm{Hz}\right)$ and

$\left({\widehat{A}}_{\omega}^{\left(2\right)},{\widehat{\phi}}_{0}^{\left(2\right)},{\widehat{F}}^{\left(2\right)}\right)=\left(188\text{\hspace{0.17em}}\mathrm{Hz},{232}^{\xb0},2\text{\hspace{0.17em}}\mathrm{Hz}\right)$ respectively.

The real echo data processing is carried out below. Initially, we extract the azimuth echo where rotating targets exist and obtain the rotational frequency as 1.5 Hz. The TF distribution of the azimuth echo acquired by STFT (the Kaiser window’s width is 37) is shown in

Figure 15a. According to the result, the component of P1 is apparent while the component of P2 is hardly visible. Then, the strongest TF curve extracted in

Figure 14a is processed by HT, and the results can be seen in

Figure 15b,c. The existence of a peak can be clearly seen in HT results. Hence, the parameter estimation of P1 can be obtained as

$\left({\widehat{A}}_{\omega}^{\left(1\right)},{\widehat{\phi}}_{0}^{\left(1\right)},{\widehat{F}}^{\left(1\right)}\right)=\left(189\text{\hspace{0.17em}}\mathrm{Hz},{53}^{\xb0},1\text{\hspace{0.17em}}\mathrm{Hz}\right)$. Next, the component of P1 is removed from the original azimuth echo. After processing the residual signal by autocorrelation method, we find there is also a periodic signal with frequency of 1.5Hz in the residual signal. The residual signal is processed by the STFT (the Kaiser window’s width is 37) and the TF distribution can be seen in

Figure 15d. The component of P1 has been removed, and the component of P2 is highlighted. Then, we perform HT on the TF curve extracted in

Figure 14d, and the results can be seen in

Figure 15e,f. Thus, the parameter estimation of P2 is

$\left({\widehat{A}}_{\omega}^{\left(2\right)},{\widehat{\phi}}_{0}^{\left(2\right)},{\widehat{F}}^{\left(2\right)}\right)=\left(188\text{\hspace{0.17em}}\mathrm{Hz},{232}^{\xb0},0\right)$. According to

${A}_{\omega}=4\mathsf{\pi}{r}_{0}{f}_{\mathrm{a}}/\lambda $, we can get the rotational radii of P1 and P2 as 0.307 m and 0.305 m respectively with the estimated values of the rotational frequency and the maximal m-D frequency. Moreover, the initial phase difference between P1 and P2 is

${179}^{\xb0}$, which generally accords with the size of the rotating angle reflectors. It is shown that both rotating targets with large reflection coefficient differences are detected successfully and the parameters are estimated accurately. Again, we verify the correctness of the proposed algorithm.