## Abstract

The previously reported photonics-based radar working with a large bandwidth has the advantages of realizing high-resolution imaging of targets with low velocity. However, the high velocity of a target will introduce Doppler dispersion to the echo signals, which severely deteriorates the imaging resolution. This problem becomes more noticeable as the bandwidth increases. In this paper, we propose a radar receiver based on a reconfigurable photonic fractional Fourier transformer (PFrFTer). The order of the PFrFTer can be reconstructed flexibly by changing the optical transform kernel. When the transform order matches the velocity of the target, the chirp echo signals behave as narrow impulses in the fractional Fourier domain, showing the range information with a high resolution. In the experiment, a PFrFTer is established and applied to process the echo signals with a bandwidth of 12 GHz. A lossless range resolution of 1.4 cm is obtained in range profiles and inverse synthetic aperture radar imaging for high-speed targets. This range resolution is much higher than that in the classical optical de-chirping receiver. These results demonstrate the PFrFTer is immune to the Doppler dispersion effect and is excellent for high-resolution imaging of high-speed target. The introduced technique would be of practical interest in the detection and recognition of targets.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Imaging radar plays an essential role in the detection and recognition of targets. It has many applications such as security check [1], foreign-object debris detection in airport runways [2], satellite monitoring [3] and so on [4]. In order to obtain more accurate identification, the imaging radar needs to have higher resolution capability, which demands the radar to work at a larger bandwidth and a higher frequency. However, due to the electronic bottlenecks, it is difficult for the traditional radar to generate and process such signals [5].

Fortunately, microwave photonics technology offers a promising solution to solve this issue for its unique characters of high frequency, wide bandwidth, immunity to electromagnetic interference, etc [6,7]. Due to these merits, many relevant techniques are proposed and demonstrated to generate, distribute and process radio frequency (RF) signals. As for the generation of RF signals, there are several methods including photonic digital-to-analog converter [8], frequency to time mapping [9], and photonic frequency multiplication [10]. As for dissemination of ultra-stable RF signal over a large area, passive compensation scheme and active compensation scheme have been presented, such as photonic microwave phase conjugation, phase-locked loops [11,12]. In terms of processing RF signals, techniques such as photonic filter [13,14], photonic phase shifter [15], photonic analog-to-digital converter (PADC) [16] and photonic de-chirping [17] have been developed extensively. In recent years, the development of these technologies has led to the emergence of high-performance microwave photonic radar systems, including photonic-based inverse synthetic aperture radar (ISAR) [17–21], photonics-based multiple-input and multiple-output radar [22–24], and photonics-based distributed radar [25,26]. Most of these radar systems have high–resolution capability. For example, previously, we proposed a W-band photonics-based radar with a bandwidth of 8 GHz [19]. A toy gun with a width of 18 cm is imaged and recognized by this radar. In [20], a small electric fan is imaged with a frame rate of 100 fps by a K-band photonics-based radar. In [21], a small unmanned aerial vehicle with an arm width of 1.9 cm is imaged and the detail structures of legs and arms are distinguished by a Ka-band photonics-based radar. From these research results, it is obvious that microwave photonics technology has taken radar imaging to the next level.

However, until now, the high-resolution imaging results taken by the photonics-based radar with a large bandwidth are obtained from stationary or low-speed targets. For high-speed targets, Doppler dispersion distorts the echo signals severely and deteriorates the imaging resolution, especially when the Doppler dispersion products $2BTv/c$ is larger than 1, where $v$, $B$, $T$ are the velocity, the bandwidth and the pulse width, respectively [27,28]. For example, in [29], when the velocity is 300 m/s, the spectrum of the de-chirped signal is expanded, resulting in a decrease in resolution. Although some compensation algorithms are proposed to solve this problem, they need to be performed in the digital signal processer with complex computation and take some time, which slow down the imaging rate, especially in the scenario where the imaging accumulation time is long and the amount of data is large [30,31]. Thus, the photonics-based radar mentioned before can work with a large bandwidth, but the imaging resolution would deteriorate if the target moves rapidly.

Since Doppler dispersion distorts the echo signals both in the time domain and frequency domain, it is invalid to suppress the Doppler dispersion effect by receiving and processing the echo signals only through methods in one of the domains, such as PADC, photonic de-chirping. Further, using these methods to process the echo signals by waveform design, such as dual-chirp LFM signals, is also invalid because the Doppler dispersion effect will act simultaneously on each chirp LFM signals, instead of cancelling each other out. As a two-dimensional signal processing method in the time-frequency domain, the fractional Fourier transform (FrFT) is an attractive method to suppress the Doppler dispersion effect. FrFT can be interpreted as a rotation in the time-frequency plane. It is a useful tool to analyze signals with the time-dependent spectrum, such as linear frequency modulated wave (LFMW) [32]. For a given LFMW, it behaves as a slant edge in the time-frequency plane. A proper rotation will transform the LFMW to an impulse signal in the fractional Fourier domain (FrFD). The Doppler dispersion effect can only change the chirp rate and center frequency of an LFMW, which still behaves as a slant edge in the time-frequency plane. Therefore, the received LFMW from high-speed targets can still concentrate to narrow pulses in the FrFD when the order of the FrFT matches the velocity, showing the high-resolution range information.

Some methods for realizing photonic fractional Fourier transformer (PFrFTer) have been proposed, and they can be used to process optical signal [33,34] or RF signal [35–38]. The method capable of processing RF signal is mainly summarized into two categories. In the first category, the FrFT of a given RF signal is obtained by coherent addition of temporally delayed replicas of the input signal after modulation by a quadratic phase term [35]. Because of the large time delay, the bandwidth of the signal is limited to tens of MHz in the implementation. Peng et al. proposed an approach to extend the bandwidth and verified the effectiveness by numerical simulation, while the architecture is complex and hard to be implemented [36]. In the second category, the FrFT is mainly realized based on the product of the input RF signal and an LFMW [37,38]. The approach can be used to process signals with large bandwidth and the architecture is simple. However, it lacks tunability.

In this paper, we realize high-resolution imaging of high-speed target based on a reconfigurable PFrFTer. A linear chirp light serves as the transform kernel to rotate the time-frequency plane, resulting in the FrFD. The LFMWs received from high-speed targets are projected onto it and behave as narrow impulses when the order of the PFrFTer matches the velocity of targets. The order can be reconstructed by changing the chirp rate of the light, which can be practically achieved by a dispersion module (DM). Besides, the architecture of the PFrFTer is simple and it is able to process radar signals with broad bandwidth up to tens of GHz. Experiments are performed to verify the effectiveness of the PFrFTer. The transmitted signal is an LFMW with a large bandwidth of 12 GHz. The imaging of moving targets in different scenarios is obtained from the PFrFTer receiver and the classical optical de-chirping receiver. Comparing the two sets of results, it can be seen that the imaging obtained from the PFrFTer receiver is clearer than that from the classical optical de-chirping receiver. The imaging resolution from the PFrFTer receiver keeps lossless as 1.4 cm for high-speed targets, which is almost equal to the theoretical value of 1.3 cm. This technique would be of interest for the detection and recognition of targets.

## 2. Principle

#### 2.1 FrFT of the echo signals from moving targets

The mathematical FrFT expression of a given signal $x(t )$ is written as [32]

*u*is the fractional frequency, $\alpha$ is the transform angle and $p = 2\alpha /\pi$ is the order. The FrFT can be interpreted as a rotation in the time-frequency plane with the angle $\alpha$, which is decided by the kernel. The square modulus of the FrFT is equal to the projection onto the

*u*axis of the time-frequency distribution of the given signal, seen as Fig. 1(a). This characteristic is particularly valuable in processing the LFMW, whose time-frequency distribution is a slant line. At a proper order, the LFMW will concentrate maximally and behave as an impulse in the FrFD.

In a radar system, the transmitted signal is an LFMW, the pulse width, the center frequency and the chirp rate of which are *T*, ${f_1}$ and ${k_1}$, respectively. Assuming the distance and velocity of a target are *R* and *v*, the time delay can be expressed as $\tau (t )= \frac{{2({R\textrm{ - }vt} )}}{{c\textrm{ - }v}} \approx \frac{{2R}}{c} - \frac{{2v}}{c}t$, where *c* is the velocity of the light [39]. Then the received signal of a moving target can be expressed as

When the order does not satisfy the formula $p = 2arc\cot \left( { - {{\left( {1 + \frac{{2v}}{c}} \right)}^2}{k_1}} \right)/\pi$, the echo signals will concentrate and behave as broadened impulses in the FrFD according to Eq. (4). Further, when $|{{k_2} + \cot \alpha } |{T^2} \gg 1$, Eq. (4) can be rewritten as

The distance is derived as

According to Eq. (5) and Eq. (6), the resolution of $u\csc \alpha$ is $|{{k_2} + \cot \alpha } |T$, so the range resolution is approximately calculated as $|{{k_2} + \cot \alpha } |{T^2}c/({2B} )$. The resolution becomes worse as the absolute value of the sum of ${k_2}$ and $\cot \alpha$ is greater. Therefore, in order to get the highest resolution of the imaging, the angle should satisfy $\cot \alpha ={-} {k_2}$, meaning the order should be $p = 2arc\cot \left( { - {{\left( {1 + \frac{{2v}}{c}} \right)}^2}{k_1}} \right)/\pi$. Then the FrFT of the echo signal is calculated as

It can be seen that the echo signals from stationary or moving targets are transformed into sinc-type pulses in the FrFD, as shown in Fig. 1(b). The distance is also derived as Eq. (6). According to Eq. (6) and Eq. (7), the resolution of $u\csc \alpha$ becomes $1/T$, so the range resolution is approximately calculated as , which is determined by the bandwidth of the transmitted signal. It is obvious that the resolution is lossless for targets with any velocity and is equal to the ideal resolution for an LFMW with a bandwidth of $B$[40].

#### 2.2 Model of the proposed radar receiver based on the PFrFTer

Since the target's velocity is varying, the order of the FrFT must be reconstructed to match the velocity for high-resolution imaging. Here, a reconfigurable PFrFTer is proposed to realize it. The schematic diagram of the radar receiver based on the PFrFTer is shown in Fig. 2. The PFrFTer mainly consists of four components: light source, electro-optical conversion, opto-electronic conversion and Fourier transform. The light source includes an optical carrier and a coherent linear chirp light. Its optical field can be expressed as

Making a variable substitution: ${k_s}\textrm{ = } - \cot \alpha ,\;\;f = u\csc \alpha - {f_s}$, Eq. (10) can be written as

The order of the PFrFT is derived as

It can be seen that the order can be reconstructed by changing the chirp rate of the linear chirp light. Specifically, it can be reconstructed by setting the dispersion. According to the discussion in section 2.1, a high-resolution imaging would be obtained when we adjust the order and make it equal to $2arc\cot ({ - {k_2}} )/\pi$. Besides, according to Eq. (6), the distance of the target can be rewritten as

In order to make the order equal to $2arc\cot ({ - {k_2}} )/\pi$, we only need to set the dispersion to

It is worth noting that the previous analysis is also right for the single upper sideband modulation. In this case, the dispersion is needed to be set as

Here, we can see that the dispersion with the same value can be used to process the targets with the same absolute velocity in opposite direction by single lower sideband modulation or single upper sideband modulation.

## 3. Experiment and discussion

To investigate the performance of the PFrFTer in the radar system, an experiment is carried out based on the schematic diagram in Fig. 2. An arbitrary waveform generator (AWG) is performed as the transmitter. Besides, it is also used to generate the simulated echo signals according to Eq. (3). Most studies about the detection of moving targets are all based on such experimental approach [29]. Although distortion will occur in the echo signals due to the electromagnetic scattering characteristic of the targets, transport characteristics of the environment in actual applications, it can be eliminated to a certain extent by some compensation approaches [42]. Besides, it does not affect the effectiveness of the proposed method to suppress the Doppler dispersion effect and realize high-resolution imaging. Therefore, these influences are not considered in the echo signals generated from the AWG. A light centered at 1545 nm with a power of 16 dBm is generated from an LD (keysight 81606A). SSB modulation is implemented by a Mach-Zehnder modulator (MZM) (Fujitsu FTM7938E) and an optical band-pass filter (OBPF) (WaveShaper 1000s). The MZM is biased at the quadrature point, resulting in dual-sideband modulation, and the OBPF is used to filter out one of the sidebands. The DM is realized by the dispersion compensation fiber (DCF). The electro-optical conversion is completed by an MZM (Photline MXAN-LN-40), which is also biased at the quadrature point. The opto-electronic conversion is achieved by a photodiode with a bandwidth of 3 GHz. A digital storage oscilloscope (Keysight UXR0134A) is used to sample the signal generated from the electro-optical conversion and the FT is performed by a digital signal processor.

For better analyzing the effectiveness of the PFrFTer in the radar system, a controlled experiment of classical optical de-chirping processing is also carried out. Most of the high-resolution photonics imaging radars presented before are based on classical optical de-chirping processing technology [19–21]. The detail introduction about the optical de-chirping receiver can be found in our previous work [25]. The transmitted signal is an LFMW centered at 14 GHz with 12 GHz bandwidth and 250 µs pulse width, and its spectrum is shown in Fig. 3(a). The OBPF is set to filter out the upper sideband, resulting in single lower sideband modulation. The spectrum of the light is shown in Fig. 3(b).

Four targets with different velocities are assumed and the echo signals from them are generated by the AWG. The four targets are all single scattering point targets. In other words, there is only one strong scattering point in each target. Their velocities are 0 m/s, 56.91m/s, 113.45m/s and 151.40 m/s, as shown in Table 1. The optical de-chirping process is performed to the echo signals, and the range profile results of the target are obtained. Figures 4(a)-(d) are the range profiles of the targets 1-4 respectively. As can be seen from the graphs, the −3 dB main-lobe width increases and the peak side-lobe ratio (PSLR) decreases with larger velocity, which means the range resolution deteriorates when the velocity gets larger. These results are caused by the Doppler dispersion effect. For a stationary target, the de-chirped signal is a single frequency signal with a narrow spectrum. However, for a moving target, the de-chirped signal is an LFMW thanks to the Doppler dispersion effect and its spectrum expands with the increase of the velocity, making the deterioration of the range resolution. The PFrFTer receiver is also used to detect targets 1-4. According to Eq. (12), the order of the PFrFTer is reconstructed by setting the dispersion of the DCF to 0 ps/nm, −1986.9 ps/nm, −3960.7 ps/nm and −5285.4 ps/nm, respectively, which is shown in Table 1. The echo signals from the targets are all converted into narrow pulses in the FrFD at the proper order, thus the range profiles are obtained and shown in Fig. 4(e)-(h). From these graphs, it can be seen that the −3 dB main-lobe width keeps 1.1 cm and the PSLR keeps 13.2 dB when the velocity varies. This −3 dB main-lobe width is basically the same as the theoretical value of $B$ is the bandwidth of the transmitted signal). Obviously, the PFrFTer is immune to the Doppler dispersion effect and achieves lossless-resolution range profiles of moving targets.

To further validate the range resolution, it is assumed that there are two targets in space with a velocity of 113.45m /s. The range profiles are obtained from the optical de-chirping receiver and the PFrFTer receiver when the targets are separated by different distances. Figures 5(a) and 5(b) show the range profiles from the optical de-chirping receiver when the targets are separated by 5 cm and 1.4 cm, respectively. It is hard to distinguish the two targets for the optical de-chirping receiver when the distance between the targets is less than 5 cm. Figures 5(c) and 5(d) show the corresponding range profiles from the PFrFTer receiver, in which the dispersion is set to −3960.7 ps/nm. From these graphs, we can see the two targets can be distinguished clearly by the PFrFTer even if they are separated by 1.4 cm. The distances between them are measured to be 4.9 cm and 1.6 cm when they are separated by 5 cm and 1.4 cm, respectively. The measurement error of the distances is mainly caused by the noise of the receiver and the interference between the frequency spectrums of the echo signals from different targets. This experiment shows the resolution of the PFrFTer can be as high as 1.4 cm for high-speed targets, which is basically consistent with the theoretical value of 1.3 cm.

The two-dimensional imaging capability of PFrFTer is verified by ISAR imaging experiments. The target is a small quadcopter with four scattering centers. These scattering centers are evenly distributed on a circle with a diameter of 12 cm, as shown in Fig. 6(a). First, the distance ${R_0}$ between the target and the radar is 100 m. The target moves at a 45-degree angle to the initial line of the radar's sight with a speed of ${v_1} = 160.44$m/s. The radial velocity relative to the radar is calculated as 113.45 m/s. The accumulation time of ISAR imaging is 0.3 s. According to it, the rotation angle $\delta \theta$ is calculated as 14.25°.

Two groups of experiments are implemented based on the optical de-chirping and the PFrFTer. The range-Doppler algorithm is applied to process the data. The range profiles are obtained at each pulse firstly. Figures 7(a) and 7(d) show the range profiles of the first pulse from the optical de-chirping receiver and the PFrFTer receiver, respectively. The latter can clearly distinguish four scattering centers, while the former cannot, which again proves the resolution deterioration of the optical de-chirping. Then range alignment is conducted on the range profiles and the results are shown in Fig. 7(b) and 7(e). Obviously, the range alignment result based on the optical de-chirping is worse than that based on the PFrFTer because of the deteriorated range resolution. Further, the error of the range alignment will affect the azimuth compression. The ISAR imaging based on the optical de-chirping is shown in Fig. 7(c). This image is blurred, and it is difficult to recognize the shape of the target. However, due to the high-resolution range profile based on the PFrFTer, high precision range alignment can be achieved and it further guarantees effective azimuth compression. As a result, clear imaging with high resolution in range and cross-range direction is obtained, as shown in Fig. 7(f). Generally, the image entropy is applied to measure the focus quality of the ISAR image [43]. The smaller the value of the image entropy, the better the focus quality of the ISAR image. Here, the image entropies obtained from the ISAR images showed in Fig. 7(c) and 7(f) are 7.1 and 6.2, respectively. These results show the PFrFTer is a promising technology in the recognition of moving targets.

In order to get the imaging of high-speed targets with a lossless and high resolution based on the PFrFTer, the order of the PFrFTer must satisfy the relationship:$p = 2arc\cot \left( { - {{\left( {1 + \frac{{2v}}{c}} \right)}^2}{k_1}} \right)/\pi$. From this property, we can see that the proposed system is more suitable to be applied to the detection and recognition of a large single target or multiple targets with the relatively same speed, such as missiles [3,44]. Besides, we should obtain the velocity of the target firstly before getting the imaging of the target. As for cooperative targets, it is easy to obtain the velocity. As for non-cooperative targets, the velocity measurement can be realized by many simple systems such as lidar [45]. With these systems, the radar system based on the PFrFTer can realize high-resolution imaging of a moving target.

The order of the PFrFTer is reconstructed by changing the chirp rate of the light, which can be practically realized by a DM. From Eq. (14) and Eq. (15), it is known that the required dispersion value is determined by the target velocity, the chirp rate of the transmitted signal and the wavelength of the light. A simulation is conducted to show this relationship. In the simulation, the wavelength of the light is set to a normal value, 1545 nm. Figure 8 shows the required dispersion as a function of the chirp rate of the transmitted signal and the target velocity under single lower sideband modulation. From the graph, we can see that the absolute dispersion value is less than 6×10^{4} ps/nm when the velocity is smaller than Mach 10, and the chirp rate is larger than 0.1 GHz/µs, which are feasible for most scenarios and imaging radar systems. The dispersion can be provided by the tunable dispersion compensation module, DCF or other dispersive devices, the dispersion of which can be as large as −70790 ps/(nm·km) [46].

In the actual application of the PFrFTer in radar imaging, there are errors between the obtained velocity and the real velocity of the target, as well as between the set dispersion and the calculated ideal dispersion. These errors will deteriorate the resolution. From [27], it is known that the −3 dB main-lobe width of the range profile will hardly expand when the velocity is less than $c/2BT$. It means the Doppler dispersion effect can be ignored when the deviation between the set dispersion and the optimal dispersion is less than $({4/{k_1}{\lambda^2}} )\cdot ({c/2BT} )$ according to Eq. (14) and (15), or in other words, when the set dispersion *D* and the real velocity *v* satisfy following equation

In the ISAR experiments, the radial velocity of the target is continuously increasing in the detection processing. During the accumulation time, the velocity and the dispersion satisfy Eq. (17), so the imaging still keeps a high resolution, which can be seen in Fig. 7(d) and 7(f).

## 4. Conclusions

In conclusion, we have proposed and demonstrated a radar receiver based on a reconfigurable PFrFTer, aiming to process the wideband echo signals, eliminate the Doppler dispersion effect and obtain high-resolution imaging of targets with high speed. A receiver based on the PFrFTer is established and investigated in the experiments. Advantages of the proposed PFrFTer compared with the traditional de-chirping receiver are demonstrated through detection and imaging of different targets with high speed. The proposed technique would be of interest in the detection and recognition of targets.

## Funding

National Key Research and Development Program of China (2019YFB2203301); National Natural Science Foundation of China (61690191, 61690192, 61621064).

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but maybe obtained from the authors upon reasonable request.

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