Open Access
Add to BibSonomy
Abstract
Existing photonic compressive receivers have the problem of resolution deterioration when applied in wide-range radar detection. In this study, we propose a photonic-assisted space-frequency two-dimensional (2D) compressive radar receiver capable of achieving high-resolution detection in wide-range scenarios. For the space dimension, the compression process is realized by employing a spatially adaptive photonic projection basis, which guarantees complete mapping of arbitrarily delayed echoes—the key to high-resolution wide-range detection. For the frequency dimension, photonic compressive sensing is employed to further compress the bandwidth of the projected sparse signal. Therefore, the proposed system can achieve wide-range radar detection without resolution deterioration with compressed output. Herein, with two channels of 630 MHz outputs, high-resolution distance detection within a range of 21 km with a resolution of up to 2.3 cm is achieved. Moreover, inverse synthetic aperture radar (ISAR) imaging of two sets of four-point turntables distributed within the range of 21 km with a resolution of 2.3 cm × 5.7 cm is realized. The proposed photonic-assisted 2D compressive radar receiver is a viable solution to overcome the tradeoff between detection resolution and range of existing photonic compressive receivers, which indicates a path for the further development of high-resolution wide-range radar detection.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Radar is widely used for long-range detection owing to its all-weather and all-time working capability [1]. It has previously been employed in important applications, including deep-space satellite imaging [2], harbor protection [3] and so on. To improve detection capability, radar is being further developed to achieve higher resolution and wider detection range. In addition, data compression is necessary to alleviate the enormous amount of data required to reduce the requirements of large onboard memory and downlink throughput [4]. However, owing to electronic bottlenecks, the further development of traditional electronic radar faces challenges.
Fortunately, microwave photonic technology provides a viable solution to the aforementioned issues as it exhibits unique features of high frequency, large bandwidth, anti-electromagnetic interference and multi-dimensional multiplexing [5,6]. In contrast to other photonic receivers, photonic fractional Fourier transformers (PFrFter) [7–11] and photonic compressive sensors (PCSor) [12–23] can convert broadband RF signals to narrowband baseband signals. Thus, PFrFter and PCSor can be broadly categorized as “frequency compressive” receivers, which initially have the advantage of data compression.
A PFrFter is a photonic hardware implementation of fractional Fourier transform. The fractional Fourier transform (FrFt) is a powerful mathematical tool for processing time-frequency domain signals [24]; it projects the signal onto a kernel basis, namely, a chirped signal, to realize the rotation of the time-frequency domain [24]. There are three types of photonic FrFt methods: (1) dispersive element (DE) and time lens (TL) [11,25], (2) frequency shift loop (FSL) [10], and (3) LFM modulation [7–9]. For schemes based on DE and TL, the optical signal to be tested should have an extremely broad bandwidth and an extremely narrow pulse width [11]. Thus, these schemes cannot be used on signals with large time-bandwidth products, such as radar signals. However, for the scheme based on FSL, which is limited by the length of the loop, the bandwidth of the processed signal is limited to only a few tens of megahertz (MHz) [10]. Compared with other PFrFter schemes, the LFM modulation-based scheme is more suitable for radar applications. The echo LFM signal is mixed with a duplicated version of the transmitted signal, which acts as the matched order analog projection basis [7–9]. However, certain problems exist in wide-range detection scenarios. Considering that the analog projection basis chirped signal is naturally finite in time and bandwidth, the complete projection of echoes on the analog projection basis cannot be realized unless the relative delay time of the echo is zero. The effective projecting bandwidth decreases as the target is located farther away (the so-called far target condition), resulting in deterioration of the range resolution. This problem can be solved by adjusting the delay of the projection basis signal [26]. However, in the multi-target scenario, simultaneous high-resolution detection of both far and nearby targets cannot be achieved. In addition, the total bandwidth of the electronic analog to digital converter (EADC) is also enlarged, that is, data compression caused by the characteristic of frequency compression vanishes. Therefore, the use of traditional PFrFters in wide-range detection scenario leads to problems, including resolution deterioration and loss of frequency compression.
Another type of frequency compressive receiver, namely PCSor, is mainly implemented based on the principle of random demodulation (RD) [27] or modulated wideband converter (MWC) [27–29]. To realize accurate recovery, the code rate of the random code must be greater than the Nyquist rate of the received signal [27]. Existing photonic compressive sensing methods can be divided into two main categories: frequency-domain control and time-domain control. The frequency domain control method is realized by random coding of the spatial light modulator (SLM) [12,15,19] or multimode waveguides [13,16,18]. However, this type of method has the limitation that it requires the highest code rate [19] to meet the far-field conditions. In addition, the time window of the received signal is limited and cannot be used for reception of a large time-bandwidth product signal. The time-domain control method usually employs a modulator to perform photonic mixing of the received signal and random code [14,20–23]. However, with a further increase in the carrier frequency and bandwidth of the received signal, this type of method is limited by the code rate of the electrically generated random code. In addition, regardless of the frequency or time domain controlling method, existing photonic compressive sensing schemes are mainly used for receiving signals that are naturally sparse in the frequency domain, such as multi-tone signals and extremely narrow bandwidth (10 MHz) LFM [21]. However, when the LFM bandwidth increases to the GHz level, such as K-band LFM signals, which can no longer be considered as sparse in the frequency domain, it is difficult to completely receive the high-frequency wideband waveform. This limitation of the receiving signal bandwidth will limit the detection resolution. This becomes an obstacle for the use of existing PCSors in high-resolution wide-range radar detection.
Thus, the existing frequency compressive receivers, including PFrFter and PCSor, face challenges in the application of high-resolution wide-range radar detection.
Herein, we propose a photonic-assisted space-frequency two-dimensional (2D) compressive radar receiver. For the space dimension, the compression process is realized by employing a spatially adaptive photonic projection basis, which guarantees complete mapping of arbitrarily delayed echoes—the key to high-resolution wide-range detection. Based on this, photonic compressive sensing is employed to realize frequency-dimensional compression. Using the aforementioned 2D compression process, our system can realize high-resolution radar detection for a wide range, without resolution deterioration. The required EADC receiving bandwidth is limited to hundreds of MHz.
In the experiment, with two channels of 630 MHz outputs, our system achieves distance detection within the range of 21 km with a resolution of up to 2.3 cm. In addition, ISAR imaging of two sets of four-point turntables within the range of 21 km is also realized, in which a resolution of 2.3 cm×5.7 cm is achieved. Compared with the existing PFrFter, the proposed system solves the problem of resolution deterioration in wide-range detection while retaining frequency compression. Besides, compared with the existing PCSor, our scheme breaks through the bottleneck of complete nonsparse high-frequency and wideband signal reception, making it applicable to high-resolution wide-range radar detection.
2. Principle
2.1 Principle of photonic-assisted 2D compression
The principle of the proposed photonic-assisted 2D compressive radar receiver is illustrated in Fig. 1. Echo ${x_r}(t )$ is firstly projected onto the effective projection basis $x_b^{effective}(t )$. The effective projection basis $x_b^{effective}(t )$ is the effective part of provided projection basis that works during the projection process. Its time-frequency distribution can adaptively adopt to the delay of the echo, in other words, it has spatial adaptability. Thanks to its spatial adaptability, the echo waveform can be completely projected onto the basis within the entire receiving range window. This is the key to achieving no resolution deterioration in wide-range detection. In this way, spatial domain compression is performed. Sequentially, the projected sparse result is compressed by the modulation of the random code $p(t )$ to realize frequency-domain compression.
After photo-electric conversion, the output photocurrent can be expressed as
We assume that the transmitted signal ${x_t}(t )$ is an LFM signal with a pulse width of ${T_P}$, starting frequency of ${f_0}$, and bandwidth of B. It can be expressed as
According to the time-frequency distribution of an arbitrarily delayed echo signal, to guarantee the spatial adaptiveness of $x_b^{effective}(t )$, the provided projection basis ${x_b}(t )$ is designed as the red solid line $DEFGHI$, as shown in Fig. 2. It can be expressed as
It is composed of three segments: the first segment $DE$ spreads from ${f_0}$ to ${f_0}\textrm{ + }{B / 2}$; the second segment $FG$ spreads from ${f_0}$ to ${f_0} + B$; and the third segment $HI$ spreads from ${f_0}\textrm{ + }{B / 2}$ to ${f_0} + B$.
For the provided projection basis ${x_b}(t )$, the effective part that works in the projection process is the part that overlaps with the echo in time, as indicated by the red solid line $PEFQ$ in Fig. 3(a) and $PGHQ$ in Fig. 3(b). For an arbitrarily delayed echo in the delay window of $[{\textrm{0, }{T_p}} ]$, without changing the provided projection basis, the corresponding effective part of the projection basis $x_b^{effective}(t )$ automatically adapts to the time-frequency distribution of the echo. This guarantees the echo’s complete projection within the range window of ${R_w} = {{c{T_p}} / 2}$ . Hence, the detection resolution ${R_{reso}}$ remains ${c / {2B}}$ within the entire receiving range window, as shown in Fig. 4(b). However, for traditional PFrFters, the range resolution deteriorates as the target is located farther away, as shown in Fig. 4(a). Therefore, within the viewing angle of our system, the target at any position within the receiving range window has the characteristics of “near” targets in the viewing angle of the traditional PFrFters. This is equivalent to space-domain compression.
Fig. 3. Basis projection process and projected result. The delay $\tau$ of the echo features (a) $0 \le \tau < {{{T_p}} / 2}$ and (b) ${{{T_p}} / \textrm{2}} \le \tau < {T_p}$.
Fig. 4. Variation in range resolution when detected by (a) traditional PFrFters; (b) our proposed photonic-assisted 2D compressive radar receiver.
Next, the characteristics of the projected results are discussed. We assume that the projection result is $\alpha (t )= {x_r}(t )x_b^\ast (t )$. When $0 \le \tau < {{{T_p}} / 2}$, the condition shown in Fig. 3(a), the projected result $\alpha (t )$ can be expressed as
When ${{{T_p}} / \textrm{2}} \le \tau < {T_p}$, the condition shown in Fig. 3(b), the projected result can be expressed as
For the multitarget condition, we assume that the target number is n. The echo signal has the form
This means that the projected result $\alpha (t )$ in the multitarget condition is the summation of each target’s projected result ${\alpha _i}(t )$.
Through the aforementioned projection process, the high-frequency wideband echo is converted to sparse signals within a bandwidth of merely $[{\textrm{ - }{B / 2}, {B / 2}} ]$.This makes the following compressive sensing possible and reduces the demand for the code rate of the following random code to $B$ for successive sparse recovery. Based on this, photonic compressive sensing is employed to further compress the bandwidth of the projected result to hundreds of MHz.
2.2 Principle of high-resolution detection
To realize wide-range radar detection without resolution deterioration, waveform recovery is digitally performed further.
First, according to the output of the photonic-assisted 2D compressive radar receiver, using the compressive sensing recovery algorithm from [27], the recovered sparse projected result $\alpha ^{\prime}(t )\textrm{ = }{x_r}(t )x_b^\ast (t )$ is obtained. Then, the recovered echo waveform ${x_r}^{\prime}(t )$ can be obtained by multiplying the recovered projected result $\alpha ^{\prime}(t )$ and the known provided projection basis ${x_b}(t )$, which is expressed as
Eventually, owing to the characteristic of space-frequency 2D compression, with bandwidth of several hundred megahertz of EADC, any arbitrarily delayed echoes within the distance window of ${R_w}$ can be completely acquired. Thus, high-resolution wide-range radar detection without resolution deterioration can be achieved.
3. Scheme
3.1 Experimental scheme
A schematic of the experiment is shown in Fig. 5. The K-band (18-26 GHz) radar echo ${x_r}(t )$ is sent to the MZM (Fujitsu, FTM7938) for sampling. Two K-band analog projection basis signals with a ${90^ \circ }$ phase difference, namely $x_b^I(t )$ and $x_b^Q(t )$, are respectively sent to one sub-MZM of DPMZM in the I/Q channel (Fujitsu, FTM7961EX). Two random codes, namely ${p_I}(t )$ and ${p_Q}(t )$, have a code rate of 10 GHz and are filtered by 5 GHz low pass filters (LPF). They are sent to another sub-MZM of the DPMZM in the I/Q channel. An erbium-doped fiber amplifier (EDFA) is used to compensate for the insertion loss, and an optical bandpass filter (OBPF) is used to eliminate the amplified spontaneous emission (ASE) noise. The filtered optical signal is then evenly divided into I and Q channels using an optical coupler (OC). Before being injected into the DPMZMs, the optical time-delay line (OTDL) is used to ensure the frequency match of the projected results of the I/Q channel, and a variable optical attenuator (VOA) is used to guarantee the amplitude match. The DPMZM outputs undergo photoelectric conversion using photodetectors (PD). Both pass through post electronic processing modules, including microwave amplifiers (AMP), phase shifters (PS), and 630 MHz lowpass filters (LPF). Ultimately, the two channels of the narrowband 2D compression outputs are sampled by the EADC.
We first consider the I-channel optical path as an example. Suppose that ${m_r}$ is the modulation index of the MZM and ${\varphi _r}$ is the phase bias of the MZM. $x_b^I(t )$ is the provided projection basis and ${p_I}(t )$ is the provided random code.$m_b^I$ and $m_p^I$ are the modulation indexes of the two sub-MZMs of the DPMZM.$\varphi _b^I$ and $\varphi _p^I$ are the phase biases of the two sub-MZMs. $\varphi _{}^I$ is the phase bias between two sub-MZMs. When the bias points of the MZM and DPMZM are in the state of ${\varphi _r}\textrm{ = }{{ - \pi } / 4}, \varphi _b^I\textrm{ = }\varphi _p^I = {\pi / 2},\varphi _{}^I = 0$, the PD output $I_I^0(t )$ can be expressed as:
It is assumed that the impulse response of the following electric devices, including the amplifier, PS, and LPF, is ${h_e}$. Thus, the ultimate output of the I channel is
However, in Eq. (10), the DC component in the first bracket $\{{1\textrm{ + }\sin [{2{m_r}{x_r}(t )} ]} \}$ can bring $\cos [{2m_p^I{p_I}(t )} ]$ and other interference components into the low-frequency band of $I_I^0(t )$. These interference components enter the passband of the LPF and interfere with the effective component for compressive sensing, namely, $\sin [{2{m_r}{x_r}(t )} ]\sin [{m_b^Ix_b^I(t )} ]\sin [{m_p^I{p_I}(t )} ]$.This type of interference can be eliminated by digitally calibrating $s_I^0(t )$ using the output signal of the system itself in the state of no echo, namely $s_I^{comp}(t )$.
When there is no echo sent to the system and the bias points are maintained at the same aforementioned state, the PD output of the system is
Assuming that the projected result $\alpha _I^{}(t )$ in channel I is expressed as $\alpha _{\; I}^{}(t )\textrm{ = }{x_r}(t )x_b^I(t )$, the calibrated output is
Similarly, assuming that the projected result in the Q channel is $\alpha _Q^{}(t )\textrm{ = }{x_r}(t )x_b^Q(t )$, the narrowband calibrated output of channel Q is
where $x_b^Q(t )$ is shifted ${90^ \circ }$ relative to $x_b^I(t )$.Based on the calibrated compressed results, namely ${s_I}(t )$ and ${s_Q}(t )$, through the sparse recovery algorithm, the projected results $\alpha _I^{}(t )$ and $\alpha _Q^{}(t )$ can be recovered respectively. Subsequently, the complex projected result $\alpha (t )\textrm{ = }\alpha _I^{}(t )+ j\alpha _Q^{}(t )$ is acquired. Then, the complete echo waveform is further recovered according to section 2.2. Therefore, combined with the distance detecting and ISAR imaging algorithm, high-resolution radar detection without resolution deterioration within a certain distance window can be achieved.
3.2 Experimental result
A prototype to demonstrate the principle of the proposed photonic-assisted 2D compressive receiver is shown in Fig. 6. We assume that the transmitted signal is an 18-26 GHz LFM with a pulse width of 140 µs and a duty cycle of 50%. Echo signal is the superposition of arbitrarily delayed replicas of the transmitted signal. It is emulated through upconverting the baseband signal provided by an arbitrary waveform generator (AWG). The provided projection basis signals have the same time-frequency distribution as mentioned in Section 2.1. They first spread from 18 to 22 GHz, then from 18 to 26 GHz, and finally from 22 to 26 GHz. They are also obtained through upconverting the baseband version provided by the other two channels of the AWG. By regulating the phase difference of the oscillator, the phase difference of ${90^ \circ }$ is guaranteed to be between each other.
3.2.1 Waveform recovery
First, we demonstrate the complete waveform recovery ability of our photonic-assisted 2D compressive radar receiver. Here, we assume that this is a single target scenario. Based on the 630 MHz narrowband outputs of the I/Q channels, the sparse recovered projected results are obtained, as shown in Fig. 7(a). The recovered projected results are comprised of two frequencies, 1.665 GHz and 2.335 GHz. The zoomed in versions of spectrum in Fig. 7(a) are shown in Figs. 7(b),(c). The complex version of the recovered projected results, comprised of −2.335 GHz and 1.665 GHz, is obtained, as shown in Fig. 7(d). Each tone has a single-sided spectrum, and the spectral slices in the vicinity of the mirror frequency, 2.335 GHz and -1.665 GHz, are converted to the noise floor. The zoomed in versions of the spectral slices in the vicinity of -1.665 GHz and 2.335 GHz in Fig. 7(d) are shown in Figs. 7(e),(f). Combined with the provided projection basis, the complete K-band waveform of the echo is further digitally recovered, as shown in Fig. 7(g). The pulse compression result between the recovered waveform and ideal transmitted signal is shown in Fig. 7(h). The 3-dB main lobe width of the pulse compression result is approximately 1.64 cm. This is consistent with the autocorrelation of the ideal signal, equal to 1.64 cm. This demonstrates the high performance of the complete waveform recovery of the proposed receiver.
Fig. 7. Experimental complete waveform recovery. (a) Recovered sparse projected results of I and Q channel. (b),(c) Zoomed in versions of (a). (d) Complex version of sparse projected result. (e),(f) Zoomed in versions of the spectral slices in the vicinity of -1.665 GHz and 2.335 GHz in (d). (g) Time-frequency distribution of the recovered waveform. (h) Pulse compressing result.
To further demonstrate the ability of complete waveform recovery among the entire receiving range window ${R_\textrm{w}} = 21km$, we adjust the delay of the AWG baseband signal to emulate the target’s location increasing every 2 km from 100 m. The pulse compression process is carried out between the recovered waveform and ideal LFM signal. The main lobe of the pulse-compression results corresponding to each target’s location is shown in Fig. 8.
Fig. 8. Pulse compression result of recovered echoes (the target’s location increases every 2 km from 100 m).
The main lobe widths are further plotted as red asterisks in Fig. 9, with the blue dashed line being the main lobe width of the autocorrelation of the ideal LFM, which is equal to 1.64 cm. Within the receiving range window of 21 km, the main lobe widths are close to that of the ideal LFM. This demonstrates the complete waveform recovery ability of our system, which is the foundation for high-resolution and wide-range radar detection.
Fig. 9. Red asterisks: the main lobe widths of pulse compression experimental results; Blue dashed line: the main lobe width of ideal LFM autocorrelation.
3.2.2 High-resolution distance detection in wide range
Next, we demonstrate our system on high-resolution distance detection in a wide-range and multi-target scenario. We consider two pairs of targets, with one pair located near 129 m and the other pair located near 20.9 km. After sparse recovery, sparse projected results are obtained, as shown in Fig. 10(a). The zoomed in versions of Fig. 10(a) are shown in Figs. 10(b),(c). The recovered results are comprised of two pairs of signals—one pair appearing at 0.027 GHz and 3.973 GHz, and the other pair appearing at 0.049 GHz and 3.951 GHz. The frequencies of each pair are complementary to those in ${B / 2}$. The obtained complex version is shown in Fig. 10(d). As can be seen, the complex version of the first pair appears at 0.027 GHz and -3.973 GHz, and the other pair appears at -0.049 GHz and 3.951 GHz, as shown in Figs. 10(e)-(h). Based on this, the K-band echo waveform is recovered; the pulse compression results are shown in Figs. 10(i),(j). Both pairs of double targets can be distinguished, demonstrating a resolution of 2.3 cm in the range of ∼21 km with only two channels of 630 MHz outputs.
Fig. 10. Experimental results of distance detection. (a): Recovered sparse projected result of I and Q channel. (b),(c): The zoomed in versions of (a). (d): Complex version of sparse projected result. (e)-(h): The zoomed in versions of (d). (i),(j): Pulse compressing result.
3.2.3 High-resolution ISAR imaging in wide range
Next, an ISAR imaging experiment is conducted to further highlight the superiority of the proposed photonic-assisted 2D compressive radar receiver. The scenario is assumed to have two sets of 4-point turntables, with one set located near 129 m and the other located near 20.93 km. The rotating speeds of both the rotators are 5π rad/s, and both the rotation radii are 30 cm. The K band echo signal is also emulated through upconverting the baseband signal provided by AWG. The integration time for ISAR imaging is set to 7.6 ms, corresponding to a cross-range resolution of ∼5.7 cm.
The resulting ISAR images of the two sets of 4-point turntables in the range of 21 km are shown in Figs. 11(a) and (b). The resolution is 2.3 cm×5.7 cm. This demonstrates the ability of our system to perform high-resolution ISAR imaging in a wide-range scenario.
Fig. 11. ISAR imaging results of two sets of 4-point turntables, with (a) one set located near 129 m and (b) other set located near 20.9 km.
4. Conclusion
In summary, we have presented a photonic-assisted space-frequency 2D compressive radar receiver to overcome the bottleneck of high-resolution detection over a wide range with compressive output. A spatially adaptive photonic projection basis is designed to guarantee complete mapping of arbitrarily delayed echoes, which is the key to high-resolution detection in wide-range scenarios. Simultaneously, photonic compressive sensing is utilized to further compress the bandwidth of the projected sparse signal, which ensures low amount of data. In the experiment, with only two channels of 630 MHz outputs, distance detection with a resolution of 2.3 cm and ISAR imaging with a resolution of 2.3 cm × 5.7 cm within the range of 21 km are realized. The experimental results validate the ability of high-resolution detection in wide-range scenario with low amount of data. The proposed system will provide a new avenue for future high-resolution and wide-range radar detection.
Funding
National Key Research and Development Program of China (2019YFB2203301); National Natural Science Foundation of China (61690191).
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
References
1. V. C. Chen and M. Martorella, Inverse Synthetic Aperture Radar Imaging (SciTech Publishing, 2014), 55–56.
2. M. G. Czerwinski and J. M. Usoff, “Development of the haystack ultrawideband satellite imaging radar,” Lincoln Laboratory Journal 21, 28–44 (2014).
3. A. Capria, E. Giusti, C. Moscardini, M. Conti, D. Petri, M. Martorella, and F. Berizzi, “Multifunction imaging passive radar for harbour protection and navigation safety,” IEEE Aerosp. Electron. Syst. Mag. 32(2), 30–38 (2017). [CrossRef]
4. M. T. Alonso, P. López-Dekker, and J. J. Mallorquí, “A novel strategy for radar imaging based on compressive sensing,” IEEE Trans. Geosci. Remote Sensing 48(12), 4285–4295 (2010). [CrossRef]
5. Y. Tong, “Advanced Photonics-Based Radar Signal Generation Technology for Practical Radar Application,” J. Lightwave Technol. 39(11), 3371–3382 (2021). [CrossRef]
6. S. Pan and Y. Zhang, “Microwave Photonic Radars,” J. Lightwave Technol. 38(19), 5450–5484 (2020). [CrossRef]
7. S. Peng, S. Li, G. Han, X. Xue, and X. Zheng, “High-resolution imaging of a high-speed target based on a reconfigurable photonic fractional Fourier transformer,” Opt. Express 29(13), 19985–19997 (2021). [CrossRef]
8. G. Han, S. Li, X. Xue, and X. Zheng, “Photonic chirp rates estimator for piecewise linear frequency modulated waveforms based on photonic self-fractional Fourier transform,” Opt. Express 28(15), 21783–21791 (2020). [CrossRef]
9. G. Han, S. Li, X. Xue, and X. Zheng, “Photonic fractional Fourier transformer for chirp radar with ghost target elimination,” Opt. Lett. 45(15), 4228–4231 (2020). [CrossRef]
10. C. Schnébelin and H. G. de Chatellus, “Agile photonic fractional Fourier transformation of optical and RF signals,” Optica 4(8), 907–910 (2017). [CrossRef]
11. C. Cuadrado-Laborde, A. Carrascosa, A. Díez, J. L. Cruz, and M. V. Andres, “Photonic fractional Fourier transformer with a single dispersive device,” Opt. Express 21(7), 8558–8563 (2013). [CrossRef]
12. H. Chi, H. Zhou, S. Yang, J. Ou, Y. Zhai, and B. Yang, “Compressive sensing based on optical mixing using a spectral shaper with bipolar coding,” Opt. Express 29(11), 16422–16431 (2021). [CrossRef]
13. D. B. Borlaug, S. Estrella, C. T. Boone, G. A. Sefler, T. J. Shaw, A. Roy, L. Johansson, and G. C. Valley, “Photonic integrated circuit based compressive sensing radio frequency receiver using waveguide speckle,” Opt. Express 29(13), 19222–19239 (2021). [CrossRef]
14. B. Yang, S. Yang, Z. Cao, J. Ou, Y. Zhai, and H. Chi, “Photonic compressive sensing of sparse radio frequency signals with a single dual-electrode Mach–Zehnder modulator,” Opt. Lett. 45(20), 5708–5711 (2020). [CrossRef]
15. Z. Zhu, H. Chi, T. Jin, S. Zheng, X. Yu, X. Jin, and X. Zhang, “Photonics-enabled compressive sensing with spectral encoding using an incoherent broadband source,” Opt. Lett. 43(2), 330–333 (2018). [CrossRef]
16. G. A. Sefler, T. J. Shaw, and G. C. Valley, “Demonstration of speckle-based compressive sensing system for recovering RF signals,” Opt. Express 26(17), 21390–21402 (2018). [CrossRef]
17. H. Chi and Z. Zhu, “Analytical Model for Photonic Compressive Sensing With Pulse Stretch and Compression,” IEEE Photonics J. 11, 5500410 (2019). [CrossRef]
18. G. C. Valley, G. A. Sefler, and T. J. Shaw, “Multimode waveguide speckle patterns for compressive sensing,” Opt. Lett. 41(11), 2529–2532 (2016). [CrossRef]
19. Z. Zhu, H. Chi, S. Zheng, T. Jin, X. Jin, and X. Zhang, “Analysis of compressive sensing with optical mixing using a spatial light modulator,” Appl. Opt. 54(8), 1894–1899 (2015). [CrossRef]
20. Q. Guo, Y. Liang, M. Chen, H. Chen, S. Yang, and S. Xie, “Time-interleaved 20-ghz modulated wideband converter based on random optical sampling,” IEEE Photonics Technol. Lett. 27(9), 1022–1025 (2015). [CrossRef]
21. Q. Guo, Y. Liang, M. Chen, H. Chen, and S. Xie, “Compressive spectrum sensing of radar pulses based on photonic techniques,” Opt. Express 23(4), 4517–4522 (2015). [CrossRef]
22. Q. Guo, H. Chen, Z. Weng, M. Chen, S. Yang, and S. Xie, “Compressive sensing based high-speed time-stretch optical microscopy for two-dimensional image acquisition,” Opt. Express 23(23), 29639–29646 (2015). [CrossRef]
23. B. T. Bosworth, J. R. Stroud, D. N. Tran, T. D. Tran, S. Chin, and M. A. Foster, “Ultrawideband compressed sensing of arbitrary multi-tone sparse radio frequencies using spectrally encoded ultrafast laser pulses,” Opt. Lett. 40(13), 3045–3048 (2015). [CrossRef]
24. L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42(11), 3084–3091 (1994). [CrossRef]
25. A. W. Lohmann and D. Mendlovic, “Fractional Fourier transform: photonic implementation,” Appl. Opt. 33(32), 7661–7664 (1994). [CrossRef]
26. X. Xiao, S. Li, S. Peng, L. Xing, X. Xue, X. Zheng, and B. Zhou, “A large-range autofocus microwave photonic radar based on adaptive spatial filtering along the range direction,” Opt. Commun. 477, 126354 (2020). [CrossRef]
27. M. Mishali, Y. C. Eldar, and A. J. Elron, “Xampling: Signal acquisition and processing in union of subspaces,” IEEE Transactions on Signal Processing 59(10), 4719–4734 (2011). [CrossRef]
28. D. Cohen, K. V. Mishra, and Y. C. Eldar, “Spectrum Sharing Radar: Coexistence via Xampling,” IEEE Trans. Aerosp. Electron. Syst. 54(3), 1279–1296 (2018). [CrossRef]
29. M. Mishali and Y. C. Eldar, “From Theory to Practice: Sub-Nyquist Sampling of Sparse Wideband Analog Signals,” IEEE J. Sel. Top. Signal Process. 4(2), 375–391 (2010). [CrossRef]
