1. Introduction

Synthetic aperture Ladar (SAL) is a new radar system that combines synthetic aperture technology with Ladar technology. It is an extension of the synthetic aperture technology in the optical frequency band, and its theoretical resolution does not decrease with the increase of detection range [1]. Compared to microwave signals, the Ladar signals have narrow beam width, concentrated energy, short synthetic aperture time, and the frequency characteristics are close to visible light. Therefore, SAL can produce high-resolution, high-data-rate, and refined imaging results, which has potential in the fields of ground observation and space target observation [2–4]. However, the coherent detection technology is very sensitive to vibration. Moreover, because the wavelength of the Ladar signals is on the micron order, the vibrations of airplane and other moving platforms on the micron order can introduce a severe Doppler frequency shift into the echo signals [5,6]. For platform motion sensors, vibration errors on the micron order are difficult to measure. Therefore, vibration errors estimation and compensation are key issues in SAL imaging [7].

The estimation and compensation of vibration errors in SAL imaging mainly focus on two aspects: system design and signal processing. In terms of system design, there are three main types of methods. Some researchers use differential SAL to suppress vibration phase errors [8,9]. The basic idea is to set up two receiving channels along motion direction, apply differential processing to the received signals of two channels, and then integrate the differential signals to compensate for vibration errors. The second type is based on interferometric process along track, which inverts vibration velocity through interferometric phase, and then estimates the vibration errors by integrating the vibration velocity [10]. The third type is the down-looking synthetic aperture imaging, which utilizes autodyne processing to compensate for phase errors caused by vibration [11,12]. In general, the vibration compensation method based on system design requires adding additional receiving channels or changing hardware equipment, which increases system complexity. In terms of signal processing, there are three main types of methods. The first type is the space correlation method which uses the interferometric phase of adjacent pulses to estimate the vibration errors, but it is mainly applicable to the radar transmission signals with high pulse repetition rates [13]. The second type is the phase gradient autofocus algorithm (PGA). Gatt et al. applied PGA to SAL imaging and analyzed the performance boundary [14,15]. Under high resolution conditions, the vibration errors not only defocus azimuth image, but also unavoidably lead to range cell migration, reducing the effectiveness of the method. The third type is the sub aperture method which estimates the vibration errors by dividing sub apertures in the azimuth direction [16]. This method assumes that the vibration velocity within each sub aperture is uniform, so we cannot estimate the vibration velocity of each pulse, reducing the accuracy of vibration error compensation.

This paper focuses on traditional Ladar systems with one receiving channel and compensates for vibration errors based on signal processing methods. Triangular frequency modulated continuous wave (T-FMCW) can effectively reduce range and velocity coupling of targets and produce higher resolution imaging results using lower power. The National Aeronautics and Space Administration has designed a series of T-FMCW Ladar systems which can obtain high-precision speed and range information by setting up multiple receiving channels. The speed and range information can be used to guide autonomous and safe landing of the lander and assist in landing missions on the Moon and Mars [17]. Wang et al. used spectral correlation method to obtain the vibration velocity of T-FMCW signals, which is used to compensate for the vibration errors. This method requires up-sampling of the signal, and it is difficult to accurately estimate the residual phase errors between adjacent pulses, resulting in error transmission and accumulation issues [18,19]. In the field of high-precision ranging, a series of vibration compensation methods were proposed based on the symmetrical characteristics of T-FMCW signals [20–24].

Motivated by the above ideas, we propose an SAL vibration error estimation and compensation method based on triangular interferometric signals. We first establish a de-chirp signal model under vibration environment and analyze the impact of vibration on imaging. Second, according to the symmetrical characteristics of T-FMCW signals and the time-frequency information introduced by the azimuthal vibration phase, we estimate the instantaneous frequency errors that across the range cell by using a two-stage interferometry method. When the system does not suffer from obvious range cell migration, we use a one-stage interferometry method to estimate the instantaneous frequency errors. Then, we use the estimated instantaneous frequency errors to establish a compensation filter, thereby compensating for the vibration errors in the de-chirp signals. Finally, the effectiveness and superiority of the proposed method were verified through experiments. The proposed method designs a signal processing method to compensate for the vibration errors without changing system design, and is suitable for scenarios in which the vibration errors cause obvious range cell migration.

2. De-chirp signal model under vibration environment

Figure 1 is a schematic diagram of the T-FMCW SAL system. The tunable laser source (TLS) generates T-FMCW signals which are divided into two beams through the splitter. One beam separated by the splitter is used as a transmitted signal and transmitted by an optical antenna. Then, the reflected echo is received by the optical antenna when it encounters a target, and the echo signal is coupled with the delayed transmitted signal, and then detected by the photodetector (DM). The heterodyne signal is sampled using a digital acquisition card (DAQ). The above signal acquisition process is a de-chirp reception technology suitable for linear frequency modulation signals, which reduces the signal bandwidth after mixing, thereby reducing the sampling rate [25]. The three small images at the bottom of Fig. 1 show the frequencies associated with the dechirp processing of the LiDAR system. The initial image shows the optical frequency corresponding to the transmitted T-FMCW laser signal. The second image shows the optical frequencies of both the transmitted signal (highlighted by the red line) and the received signal (highlighted by the blue line). The final image shows the dechirp frequency of the dechirp signal. The de-chirp signals in a T-FMCW period obtained from the de-chirp reception are referred to as positive de-chirp signal and negative de-chirp signal, respectively.

 

Fig. 1. Schematic diagram of the T-FMCW SAL system.

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We take the positive frequency modulation signal as an example, and derive the de-chirp reception process. The ideal FMCW transmitted signal can be expressed as:

Then construct a reference signal with reference range ${R_{ref}}$:

When the instantaneous slant range between the Ladar and the target is R, the echo signal can be represented as:

After mixing the echo signal with the reference signal, we can obtain a de-chirp signal:

In the phase of the (4), the first and third terms are constants related to time ${t_r}$, while the second term is a linear phase related to ${t_r}$. Therefore, the above de-chirp signal is a single frequency signal, and we can obtain the range compression result by using the Fourier transform. The range compression result is expressed as:

The first, second, and third terms in the phase of (5) represent azimuth Doppler, the oblique envelope term, and the residual video phase (RVP) term, respectively. The last two terms can be removed by designing a compensation filter:

Because the envelope term has small impact on subsequent analysis, we will not consider this term in the following text. After compensating for the oblique envelope term and RVP term, the de-chirp signal can be expressed as:

The spectrum of the above signal after Fourier transform is a sinc function, and the peak frequency of the target is ${f_0} ={-} 2{K_r}{R_\Delta }\textrm{/}c$. Then, the theoretical estimated peak range of the target can be expressed as:

Assuming that the range and delay with vibration error $\delta R$ are ${R_v} = {R_\Delta } + \delta R$ and $\tau = {\tau _\Delta } + \delta \tau$, respectively, the de-chirp signal can be expressed as:

The phase error is:

Taking the commonly used engineering parameters as an example, ${f_c}$ has a magnitude order of ${10^{14}}$. The first term of the phase error is affected by ${f_c}$ which amplifies the vibration errors many times. Therefore, the first term unavoidably results in severe range cell migration and seriously affect two-dimensional compression in the range and azimuth directions. Because ${f_c}$ is usually much greater than B, the second term of the phase error does not introduce severe range cell migration, but leads to azimuth defocusing. Under the same bandwidth conditions, larger pulse width and smaller frequency modulation rate K lead to greater impact of vibration errors on azimuth focusing.

3. Vibration error estimation using triangular interferometric signals

We categorized the SAL imaging scenarios into two groups based on the impact of vibrations on ranging: one experiencing significant range cell migration due to vibration errors, and the other does not produce obvious range cell migration. Specifically, we consider a SAL image scenario to be affected by range cell migration when the range cell migration introduced by vibration errors is much larger than the ideal range resolution unit. Conversely, we categorize a SAL image scenario as free of range cell migration when the range cell migration introduced by vibration errors is smaller than the ideal resolution unit, and the vibrations only cause a defocusing effect on the azimuth image. To estimate the vibration errors, we propose a two-stage interferometry method for the first scenario and a one-stage interferometry method for the second scenario, respectively.

3.1 Two-stage interferometry method

Based on the triangular symmetry of T-FMCW signals, we use triangular interferometry on positive and negative de-chirp signals to estimate vibration errors. The T-FMCW signal is used as the transmitted signal, and the positive and negative de-chirp signals are obtained through de-chirp reception. Within a frequency modulation period, the positive and negative de-chirp signals containing vibration errors are expressed as:

The proposed two-stage interferometry method includes two stages, i.e., the complex multiplication stage (shown in Fig. 2) and the conjugate multiplication stage (shown in Fig. 3). We first introduce the complex multiplication stage. The complex multiplication diagram of positive and negative de-chirp signals is shown in Fig. 2. First, we apply complex multiplication to each pair of positive and negative de-chirp signals of SAL data to obtain the complex interferometric signal ${s_{multi}}$:

 

Fig. 2. Diagram of complex multiplication of positive and negative de-chirp signals.

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Fig. 3. Diagram of conjugate multiplication of positive and negative de-chirp signals.

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The interferometric signal ${s_{multi}}$ after complex multiplication does not contain range information of the target, but only contains phase error introduced by the vibration error $\delta \tau$. The error $\delta \tau$ is multiplied by ${f_c}$, which usually results in severe range cell migration, seriously affecting two-dimensional compression in both range and azimuth directions.

Then, we apply Fourier transform to the interferometric signal ${s_{multi}}$ along range direction to obtain the envelope signal ${S_{multi}}$. The envelope signal does not contain range information of the target, but contains envelope errors introduced by the vibration errors. Using the peak extraction method, we can extract the instantaneous frequency $\Delta {f_{1multi}}$ along azimuth direction which corresponds to the phase error $- 4\pi {f_c}\delta {\tau _{multi}}$.

The second stage is the conjugate multiplication stage, and the conjugate multiplication diagram of positive and negative de-chirp signals is shown in Fig. 3. First, we apply conjugate multiplication to each pair of positive and negative de-chirp signals to obtain the conjugate interferometric signal ${s_{conj}}$:

After conjugate multiplication, the conjugate interferometric signal ${s_{conj}}$ contains not only range information of the target but also phase error introduced by the vibration error $\delta \tau$. In practice, ${f_c}$ is usually much greater than B. Therefore, the residual phase error $- 2\pi K\delta \tau {t_r}$ does not introduce severe range cell migration in the envelope signal, but can cause defocusing of the image in the azimuth direction.

Then, we apply Fourier transform to the interferometric signal ${s_{conj}}$ along range direction to obtain the envelope signal ${S_{conj}}$. The envelope signal does not contain obvious range cell migration, and the envelope curve is located at the true range of the target. Then, we apply short-time Fourier transform (STFT) to the envelope signal of this range cell along azimuth direction and obtain the time-frequency spectra which can accurately depict the frequency errors caused by the residual phase errors. We use the peak extraction method to obtain the instantaneous frequency $\Delta {f_{1conj}}$ and the instantaneous frequency error corresponds to the residual phase error $- 4\pi K{t_r}\delta {\tau _{conj}}$.

3.2 One-stage interferometry method

When the phase error $- 2\pi {f_c}\delta \tau$ is small, the range envelope does not exist severe range cell migration, but it affects the azimuth focusing. In this case, the envelope error estimated in the complex multiplication stage is small, and the frequency errors can be hardly estimated in the conjugate multiplication stage because $|{2\pi {f_c}\delta \tau } |\gg |{2\pi B\delta \tau } |$. Therefore, we use complex multiplication as shown in (14) to amplify the impact of the phase error $- 2\pi {f_c}\delta \tau$, and then use STFT to extract $- 2\pi {f_c}\delta \tau_{multi}$ . The proposed one-stage interferometry method includes one stages, i.e., the complex multiplication stage, and the schematic diagram of conjugate multiplication without obvious range cell migration is shown in Fig. 4.

 

Fig. 4. Schematic diagram of vibration error estimation using one-stage interferometry.

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First, we apply Fourier transform to the complex interferometric signal ${s_{multi}}$ along range direction to obtain the envelope signal ${S_{multi}}$. The envelope signal does not contain obvious range cell migration. Then, we apply STFT to the envelope signal of this range cell along azimuth direction to obtain ${S_{multi}}({{t_a},{f_a}} )$. Finally, we extract the instantaneous frequency which corresponds to the phase error $- 4\pi {f_c}\delta {\tau _{multi}}$ by using the peak extraction method.

3.3 Vibration error compensation for SAL signals

After estimating the instantaneous frequency errors, we design the vibration compensation filters based on the estimated instantaneous frequency to compensate for the vibration errors. Based on the proposed triangular wave interferometry method, we estimate the instantaneous frequency:

Then, a vibration compensation filter ${H_v}$ can be designed based on the estimated phase error:

We eliminate the vibration errors from the SAL signals, and obtain the clean de-chirp signal ${s_{com}}({{t_r},{t_a}} )$ after compensating for the vibration errors:

The workflow of the proposed vibration error estimation and compensation method is shown in Fig. 5, and the main steps are summarized as follows:

 

Fig. 5. Workflow of the vibration error estimation and compensation method based on triangular interferometric signals.

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4. Numerical experiments

This section reports two numerical experiments in the scenarios with and without obvious range cell migration to verify the effectiveness and superiority of the proposed method. Both experiments include line target test and complex target test. The main system parameters are shown in Table 1.

Table 1. Main System Parameters

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4.1 Experiments with range cell migration

We first used a line target test to verify the effectiveness of the proposed method in which the vibration errors caused range cell migration. The system parameters are shown in Table 1, with a sweep period of 200 µs. The ideal SAL imaging result is shown in Fig. 6. We added sinusoidal vibration errors with a frequency of 60 Hz and an amplitude of 15 µm to the ideal echo signals. We first applied Fourier transform to the de-chirp signals with vibration errors along range direction. Figures 7 (a) and 7 (b) show the range compression images of positive and negative de-chirp signals, respectively. Owing to the vibration errors, there is a severe range migration in the range compression result. Then, we applied SAL imaging to the de-chirp signals with vibration errors, and Figs. 7 (c) and 7 (d) show the SAL imaging results of positive and negative de-chirp signals, respectively. As shown in Figs. 7 (c) and 7 (d), the vibration errors lead to severe defocusing in both range and azimuth directions of the SAL images, making it difficult to identify the line targets.

 

Fig. 6. Ideal SAL imaging results.

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Fig. 7. Imaging results with vibration errors. Rang compression results of positive de-chirp signal (a) and negative de-chirp signal (b), SAL imaging results of positive de-chirp signal (c) and negative de-chirp signal (d) with vibration errors.

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We used the proposed method to compensate for the vibration errors of the line target. Since the vibration errors lead to an obvious range cell migration in the range compression results, we used the two-stage interferometry method to extract the instantaneous frequency. First, we applied complex multiplication to the positive and negative de-chirp signals to obtain the interferometric signal according to the workflow as shown in Fig. 2, and applied Fourier transform to the interferometric signal along range direction to obtain the envelope signal, as shown in Fig. 8 (a). The envelope does not contain range information of the target, but contains envelope errors introduced by the vibration. The instantaneous frequency was extracted along azimuth direction using the peak extraction method. Then, we applied conjugate multiplication to each positive and negative de-chirp signals according to the workflow as shown in Fig. 3, and the range compression result is shown in Fig. 8 (b). The interferometric signal contains residual phase errors introduced by vibration which do not cause severe range cell migration but lead to azimuth defocusing. We used STFT to obtain the time-frequency spectra of the conjugate signal along azimuth direction, and the results are shown in Fig. 8 (c). The time-frequency spectra can depict the instantaneous frequency containing residual phase errors, and the instantaneous frequency can be extracted by using the peak extraction method. Then, the instantaneous frequency introduced by the vibration errors was calculated by the filtering method, and the vibration compensation filter was designed to compensate for the vibration errors. Finally, we applied SAL imaging to the clean positive de-chirp signal and obtained a focused SAL image, as shown in Fig. 8 (d). Compared with Fig. 7 (c), Fig. 8 (d) shows that the SAL image after compensating for the vibration errors does not have the defocusing phenomenon along azimuth and range directions, which is similar to the ideal SAL imaging results as shown in Fig. 6. The results indicate that the proposed method effectively compensate for the range cell migration and the residual phase errors caused by the vibration errors.

 

Fig. 8. Vibration compensation results of two-stage interferometry. Rang compression results of complex interferometric signal (a) and conjugate interferometric signal (b), time-frequency spectra of conjugate interferometric signal (c) and the SAL imaging result (d).

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In the second test, we verified the effectiveness of the proposed method for complex targets. The ideal image of the target is “XDU”, as shown in Fig. 10 (a). We added vibration errors with a frequency of 60 Hz and an amplitude of 15 µm to the ideal echo signal, and applied Fourier transform along range direction to compress the positive de-chirp signal with vibration errors. Figure 9 (a) shows the range compression result of the positive de-chirp signal. Owing to the vibration errors, there is a severe range migration in the range compression result. Figure 9 (b) shows the SAL imaging result of the positive de-chirp signal with vibration errors. As shown in Fig. 9 (b), the vibration errors lead to a severe range cell migration in the SAL image as well as severe defocusing in azimuth direction, making it difficult to identify the “XDU” target. Then, we used the proposed method to compensate for the vibration errors. Since the vibration errors result in obvious range cell migration in the range compression result, we used the two-stage interferometry method to estimate the instantaneous frequency. According to the workflow as shown in Fig. 5, we designed a compensation filter to compensate for the vibration errors. Finally, we applied imaging processing to the clean positive de-chirp signal to obtain the SAL image, as shown in Fig. 10 (b). For comparison, we also used the spectral correlation method to compensate for the vibration errors, and the corresponding SAL imaging result is shown in Fig. 10 (c).

 

Fig. 9. Range compression result (a) and SAL imaging result (b) of positive de-chirp signals.

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Fig. 10. SAL imaging results with range cell migration. The ideal SAL image (a); the proposed method (b) and spectral correlation method (c).

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In order to quantitatively analyze the effectiveness and superiority of the proposed method, we used the image entropy and image contrast to evaluate the quality of SAL images [27,28]. For a SAL image $s({m,n} )$, the image entropy can be defined as:

The image contrast is defined as:

According to (18)-(20), better SAL imaging results have smaller image entropy and greater image contrast. We calculated the entropy and contrast of the imaging results as shown in Figs. 9(b) and 10, and the results are shown in Table 2. Compared with Fig. 9 (b), the proposed method and the spectral correlation method effectively compensated for the vibration errors, and obtained two-dimensional focused SAL images. Compared with the spectral correlation method, the proposed method compensates for not only the range migration error but also the small residual azimuth phase error between pulses. Therefore, the azimuthal focusing result of the proposed method is better than that of the spectral correlation method.

Table 2. Evaluation of SAL Imaging Results

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4.2 Experiments without obvious range cell migration

We used two tests to verify the effectiveness of the proposed method when the vibration errors do not introduce severe range cell migration. The first one is a line target test, and the ideal SAL imaging result is shown in Fig. 11 (a). The system parameters are shown in Table 1, with a sweep period of 20 µs. We added sinusoidal vibration errors with a frequency of 60 Hz and an amplitude of 15 µm as well as sinusoidal vibration errors with a frequency of 200 Hz and an amplitude of 5 µm, to the ideal echo signal.

 

Fig. 11. (a) Ideal SAL imaging result; (b) Range compression result of positive de-chirp signal; (c) SAL imaging result without vibration compensation.

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The range compression image was obtained by applying Fourier transform to the positive de-chirp signals along range direction, as shown in Fig. 11 (b). Then, we applied SAL imaging to the positive de-chirp signals containing vibration errors, and the result is shown in Fig. 11 (c), in which the vibration errors lead to severe defocusing in the azimuth direction of the SAL image.

We used the proposed method to compensate for the vibration errors of the line target. Since the vibration errors do not cause obvious range cell migration, and there is obvious defocus in the azimuth direction, we use the one-stage interferometry method to extract the instantaneous frequency. First, we applied complex multiplication to the positive and negative de-chirp signals according to the workflow as shown in Fig. 4. We used STFT to obtain the time-frequency spectra of the interference signals along azimuth direction as shown in Fig. 12 (a). Because STFT can characterize the instantaneous frequency containing vibration errors, we extracted the instantaneous frequency using peak extraction method. According to the filtering method, we calculated the instantaneous frequency error introduced by the vibration errors, and then designed a vibration compensation filter to compensate for the vibration errors. Finally, we applied SAL imaging to the clean positive de-chirp signals, and the result is shown in Fig. 12 (b). Compared with Fig. 11 (c), the SAL image after applying the proposed method is well focused in azimuth direction, which is close to the ideal SAL imaging result as shown in Fig. 11 (a). The results indicate that the proposed method can effectively compensate for the azimuthal phase errors caused by the vibration errors of line targets.

Table 3. Evaluation of SAL Imaging Results

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Fig. 12. (a) Time-frequency spectra of the complex interferometric signal; (b) SAL imaging result after one-stage interferometry.

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Next, we verified the effectiveness of the method for complex targets. The ideal image of the target is “XDU”, and the SAL imaging result is shown in Fig. 14 (a). The Fourier transform was applied to the positive de-chirp signal along range direction to produce range compression results. The result is shown in Fig. 13 (a), and the vibration errors do not cause severe range cell migration. We directly applied SAL imaging to the positive de-chirp signal containing vibration errors, and the imaging result is shown in Fig. 13 (b). As shown in Fig. 13 (b), the vibration errors lead to severe defocusing in the azimuth direction of the SAL image. Then, we used the proposed method to compensate for the vibration errors. Because the vibration errors do not cause obvious range cell migration and the azimuthal defocus is obvious, we used the one-stage interferometry method to estimate the instantaneous frequency. According to the workflow as shown in Fig. 5, we designed a vibration compensation filter based on the estimated instantaneous frequency and compensated for the vibration errors. Finally, we applied SAL imaging to the clean positive de-chirp signal, and the result is shown in Fig. 14 (b). For comparison, we also used the spectral correlation method to compensate for the vibration errors, and the SAL imaging result is shown in Fig. 14 (c). We calculated the image entropy and image contrast of the SAL images as shown in Figs. 13(b) and 14, and the results are shown in Table 3. Compared with Fig. 14 (c), the SAL images are well focused in the azimuth direction using both the proposed method and the spectral correlation method. From the imaging results and quantitative analysis, we can see that the proposed method estimates the residual azimuth phase errors between pulses more accurately compared to the spectral correlation method. Therefore, the SAL imaging result corresponding to the proposed method is better focused in the azimuth direction than that of the spectral correlation method.

 

Fig. 13. (a) Range compression result of positive de-chirp signal; (b) SAL imaging result without vibration error compensation.

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Fig. 14. SAL imaging results without range cell migration. The ideal SAL image (a); the proposed method (b) and spectral correlation method(c).

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5. Conclusion

Because the laser wavelength is very short, the vibration errors on the micron order inevitably cause azimuthal defocusing and range cell migration, reducing SAL imaging resolution. We first establish a de-chirp signal model under vibration environment and analyze the impact of vibration on SAL imaging. In order to reduce the impact of vibration on SAL imaging, we propose a SAL vibration error estimation and compensation method based on triangular interferometric signals that does not require changing the system design. Theoretical analysis and numerical experiments indicate that the proposed method can effectively eliminate the range cell migration and azimuthal phase errors introduced by vibration errors. Compared with the conventional spectral correlation method, the proposed method compensates for not only the range migration error but also the small residual azimuth phase error between pulses, thereby produces SAL images with higher resolution. The proposed method is also robust to noise. In the complex multiplication stage, we obtain the peak value of the range compression result after matched filtering process which produces results with a high signal-to-noise ratio (SNR). In conjugate multiplication stage, the Doppler frequency is extracted through time-frequency analysis of the range compression results, contributing to its noise immunity to a certain extent.