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Abstract
An approach to expanding the instantaneous bandwidth of a photonic sampling analog-to-digital converter (ADC) for receiving linear frequency modulation waveforms (LFMWs) is proposed and experimentally demonstrated based on up-sampling and filtering in the fractional Fourier domain. Through twice zero interpolation, the equivalent sampling rate is quadrupled, which also quadruples the nominal instantaneous bandwidth of the photonic sampling ADC. In addition, with the assistance of bandpass filtering in the fraction Fourier domain, the image signals and the harmonic distortions generated in the interpolation process are filtered out. As a result, the effective instantaneous bandwidth of the photonic sampling ADC is doubled. In the experiment, the instantaneous bandwidth of a photonic sampling ADC with a sampling rate of 5 GSa/s for receiving LFMWs is increased from 2.5 GHz to 5 GHz by using the proposed method. Input LFMWs within the frequency range of 24–27 GHz and 30–33 GHz, i.e., with an instantaneous bandwidth of 3 GHz, are digitized without frequency-domain aliasing. Besides, the ability of the proposed method to enhance the ranging accuracy in a broadband radar system is demonstrated. This method reduces the hardware complexity of the photonic sampling ADC for receiving broadband LFMWs in radar systems.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
A linear frequency modulation waveform (LFMW) is the most commonly used wideband radar signal since long-range and high-resolution detection can be achieved due to its large pulse width to maintain the energy of the transmitted signal and its large instantaneous bandwidth to achieve pulse compression [1–3]. The instantaneous bandwidth of a LFMW is the key factor to determine the radar ranging accuracy. In modern high-resolution radar systems, a LFMW with a high center frequency and a large instantaneous bandwidth is essential. For example, the Haystack Ultrawideband Satellite Imaging Radar (HUSIR) operates in the frequency range of 92 GHz to 100 GHz, where the largest instantaneous bandwidth of the transmitted LFMW reaches 8 GHz [4,5]. This kind of high-frequency and broadband signal can hardly be digitized with a high accuracy directly through using electronic analog-to-digital converters (ADCs) whose analog input bandwidth and sampling rate suffer from limited carrier mobility and large time jitter [6]. In general, dechirping processing, which uses a reference signal to transform the echo signal into an intermediate-frequency (IF) signal, is a compromised solution to avoid the use of a high-speed ADC in receiving a broadband LFMW. Nevertheless, the detection range is segmented in dechirping receiving, where the range segmentation is determined by the relative delay between the reference signal and the transmitted signal. Hence, this kind of processing is not beneficial for real-time full-range detection.
Photonic ADCs are promising candidates to achieve high-speed digitization of broadband microwave signals [7–9]. Thereinto, photonic sampling ADC is regarded as the most practical one to realize high-precision digitization, and has already been employed to achieve broadband LFMW reception in real-time full-range radar detection [10–12]. The photonic sampling ADC utilizes an ultra-short optical pulse source with a high repetition frequency and a low time jitter, together with a broadband electro-optic Mach-Zehnder modulator (MZM), to achieve high-speed and high-precision sampling of broadband microwave signals [13–16]. Then, the high-speed sampled optical pulse train is demultiplexed into low-speed sampled optical pulse trains which are subsequently digitized by using a parallel low-speed electronic ADC array with a high quantization accuracy after photoelectric conversion [17–19]. The most prominent advantage of the photonic sampling ADC lies in its large analog input bandwidth, where microwave signals in the frequency bands up to tens of gigahertz or even beyond 100 GHz can be directly digitized [20–22]. It is precisely because the photonic sampling ADC has a large analog input bandwidth that, when using it to digitize broadband LFMWs, it is important to pay attention to the frequency-domain aliasing issue. Frequency-domain aliasing occurs in the following two situations. The first one is that the instantaneous bandwidth of the photonic sampling ADC, i.e., half of the sampling rate, is smaller than that of the LFMW. The second situation is that the frequency range of the LFMW spans any two neighboring Nyquist bands regardless of its instantaneous bandwidth. The general solution to circumvent the frequency-domain aliasing issue is increasing the sampling rate, which, however, inevitably increases the hardware complexity of the photonic sampling ADC.
In this paper, an approach to expanding the instantaneous bandwidth of a photonic sampling ADC for receiving LFMWs is proposed and demonstrated based on up-sampling and filtering in the fractional Fourier domain. This method doubles the instantaneous bandwidth through twice zero interpolation and fractional Fourier domain signal processing without requiring additional hardware complexity. In the experiment, the instantaneous bandwidth of a photonic sampling ADC with a sampling rate of 5 GSa/s for receiving LFMWs is increased from 2.5 GHz to 5 GHz. LFMWs within the frequency range of 24-27 GHz and 30-33 GHz, i.e., with an instantaneous bandwidth of 3 GHz, are digitized without frequency-domain aliasing. In addition, the feasibility of the proposed method to enhance the ranging accuracy in a broadband radar system is verified.
2. Operation principle
Figure 1 shows the schematic diagram of a typical single-channel photonic sampling ADC and the proposed instantaneous bandwidth expansion method. The operation principle of a single-channel photonic sampling ADC can be described as follows. Firstly, an ultra-short optical pulse train with a repetition frequency of fs from an optical pulse source is modulated by the input LFMW with an instantaneous bandwidth of fb via an electro-optic MZM biased at its quadrature point to achieve photonic sampling. Then, a photodetector with an operation bandwidth larger than fs/2 is used to convert the sampled optical pulse train into a sampled electrical pulse train. Subsequently, a low-pass filter with a bandwidth of fs/2 is employed to broaden the sampled electrical pulse train and eliminate the out-of-band noise beyond the first Nyquist band. Hence, the input LFMW is down-converted into the first Nyquist band, i.e., 0- fs/2. Finally, an electronic ADC with a sampling rate of fs is utilized to achieve digitization of the down-converted signal. If the instantaneous bandwidth of the input LFMW, i.e., fb, is larger than that of the photonic sampling ADC, i.e., fs/2, or the frequency range of the input LFMW spans any two neighboring Nyquist bands, inevitable frequency-domain aliasing occurs as shown in Fig. 2(a). Therefore, the data from the electronic ADC must be sent to a digital signal process (DSP) module to retrieve the LFMW. As shown in Fig. 1, the DSP module is composed of two parts. The first one is twice zero interpolation, which is used to quadruple the nominal instantaneous bandwidth of the photonic sampling ADC. The second part is filtering in the fractional Fourier domain, which is used to filter out the image signals and the harmonic distortions generated by the twice zero interpolation.
Fig. 1. Schematic diagram of a typical single-channel photonic sampling ADC and the proposed instantaneous bandwidth expansion method for receiving LFMWs. LFMW: linear frequency modulation waveform; MZM: Mach-Zehnder modulator; PD: photodetector; LPF: low-pass filter; EADC: electronic analog-to-digital converter.
Fig. 2. Operation principle of the proposed instantaneous bandwidth expansion method and simulation examples for receiving broadband LFMWs. (a) Principle and simulation examples of photonic sampling and down-conversion. (b) Principle and simulation examples of twice zero interpolation. (c) Principle and simulation examples of filtering in the fractional Fourier domain. LFMW: linear frequency modulation waveform; PADC: photonic analog-to-digital converter; FrFT: fractional Fourier transform; iFrFT: inverse fractional Fourier transform.
The operation principle of the instantaneous bandwidth expansion method is detailed in Fig. 2 through taking two input LFWMs as examples, where the instantaneous bandwidths of the two LFWMs are not larger than the sampling rate, i.e., fb ≤ fs. The simulation examples at the right column of Fig. 2 are carried out by using a photonic sampling ADC with a sampling rate of 10 GSa/s, i.e., with an instantaneous bandwidth of 5 GHz. As shown in Fig. 2(a), the input LFWM with its instantaneous frequency sweeping from 22 GHz to 27 GHz represents the case that the frequency range of the input LFMW spans two neighboring Nyquist bands. On this condition, the first and the second parts of the LFWM in the frequency range of 22 GHz to 25 GHz and 25 GHz to 27 GHz are down-converted to the frequency range of 2 GHz to 5 GHz and 5 GHz to 3 GHz, respectively. The other input LFWM with its instantaneous frequency sweeping from 36 GHz to 44 GHz represents the case that the instantaneous bandwidth of the input LFMW is larger than that of the photonic sampling ADC. In such a case, the first and the second parts of the LFWM in the frequency range of 36 GHz to 40 GHz and 40 GHz to 44 GHz are down-converted to the frequency range of 4 GHz to 0 GHz and 0 GHz to 4 GHz, respectively.
The first step of the instantaneous bandwidth expansion method is twice zero interpolation. Compared with other kinds of interpolation, the most prominent advantage of zero interpolation lies in that it can be used to retrieve the desired signals in the enlarged nominal instantaneous bandwidth without extra filtering effect. After interpolation, the equivalent sampling rate is increased from 10 GSa/s to 40 GSa/s. Hence, the nominal instantaneous bandwidth of the photonic sampling ADC is quadrupled as shown in Fig. 2(b), which effectively solves the frequency-domain aliasing issue for the two input LFWMs. Nevertheless, image signals with an opposite chirp and harmonic distortions with an identical chirp are also generated in the interpolation process, which are overlapped with the desired signals in either the time or the frequency domains. Specifically, it should be pointed out that, although the sampling rate is quadrupled, the effective instantaneous bandwidth of the photonic sampling ADC is only doubled due to the fact that the frequency-domain aliasing can be completely eliminated when the instantaneous bandwidths of the input LFMWs are smaller than twice that of the photonic sampling ADC, while it may not be eliminated for an input LFMW with its instantaneous bandwidth larger than twice that of the photonic sampling ADC.
In order to eliminate these image signals and harmonic distortions, fractional Fourier transform (FrFT) is performed on the data after interpolation. In essential, the FrFT can be considered as a rotation of the time-frequency plane. For a LFWM, the rotation angle is α=β+π/2, where β is the angle between frequency-time distribution of the LFWM and the time axis of the time-frequency plane. After FrFT, the desired LFMWs are transformed to spikes in the rotated frequency domain, i.e., u-axis in Fig. 2(c), which can be represented as [23]
It can be seen from the simulation examples in Fig. 2(b) and (c) that the effective instantaneous bandwidth of the photonic sampling ADC for receiving LFMWs is doubled. This effective instantaneous bandwidth can be further expanded through interpolating more times, which is favorable for reducing the hardware complexity of the wideband radar receiver.
3. Experiment and discussion
An experiment is carried out to verify the feasibility of the proposed instantaneous bandwidth expansion method. The structure of the single-channel photonic sampling ADC used in the experiment is presented in Fig. 3, in which the ultra-short optical pulse train with a repetition frequency of 5 GHz is generated by using a time-lens-based optical pulse source [24]. Hence, the sampling rate of the photonic ADC is 5 GSa/s, which corresponds to an instantaneous bandwidth of 2.5 GHz. In the optical pulse source, continuous-wave (CW) light at 1555.6 nm from a distributed feedback laser diode (DFB-LD) is carved into an optical pulse train with a repetition frequency of 5 GHz via an electro-optic intensity modulator (AX-0MSS-20-PFA-PFA-LV) biased at its low point (i.e., below the quadrature point) and driven by a single-tone microwave signal with a frequency of 5 GHz [20]. Then, the optical pulse enters two electro-optic phase modulators (PM-5S5-20-PFAPFA-UV) driven by single-tone microwave signals at 5 GHz and with an identical power of 33 dBm. The phase of the two microwave signals applied to the phase modulators are finely tuned to guarantee that the peak or the valley of the single-tone microwave signals aligns with each optical pulse in the phase modulators. Hence, an approximately linear chirp is introduced into each optical pulse. Finally, a dispersion compensation module (AD-SM-C-2.19-FC/APC-98/98A-10) with a group velocity dispersion value of -33.95 ps/nm@1550 nm is used to compensate for the linear chirp and compress the pulse width to picosecond scale. In the photonic sampling, another electro-optic intensity modulator (AX-0MVS-40-PFA-PFA) with a 3-dB bandwidth of 36 GHz and biased at its quadrature point is used to sample the input LFMWs. The sampled optical pulse train is then detected by using a photodetector with a bandwidth of 16 GHz. A low-pass filter with a cut-off frequency of 2.5 GHz is cascaded after the photodetector to select the down-converted signals. After that, the down-converted signals are digitized by using an electronic ADC with a sampling rate of 5 GSa/s. Finally, the proposed method is implemented in the DSP module.
Fig. 3. Structure of the single-channel photonic sampling ADC used in the experiment. LD: laser diode; MZM: Mach-Zehnder modulator; PM: phase modulator; PA: power amplifier; PS: phase shifter; DCM: dispersion compensation module; LFMW: linear frequency modulation waveform; PD: photodetector; LPF: low-pass filter; ADC: analog-to-digital converter; DSP: digital signal processing.
Figure 4(a) and (b) show the temporal waveform and the optical spectrum of the ultra-short optical pulse train from the time-lens-based optical pulse source. The temporal waveform and the optical spectrum are measured by using an optical sampling oscilloscope (EXFO PSO-102) with an analog bandwidth of 500 GHz and an optical spectrum analyzer (YOKOGAWA AQ6370D), respectively. It can be seen from Fig. 4 that the pulse width and the spectrum width are 3.3 ps and 1.9 nm, respectively, indicating that the generated ultra-short optical pulse train, together with the broadband electro-optic intensity modulator, can be used to sample LFWMs in the frequency bands up to 40 GHz.
Fig. 4. Measured (a) temporal waveform and (b) spectrum of the ultra-short optical pulse train from the time-lens-based optical pulse source.
Figure 5 shows the experimental results of digitizing an LFMW with its instantaneous frequency sweeping from 24 GHz to 27 GHz. The LFWM with a period of 4 µs and a time width of 3 µs is generated by using an arbitrary waveform generator (AWG, Keysight M8194A). As shown in Fig. 5(a), after photonic sampling ADC, the first and the second parts of the LFWM in the frequency range of 24 GHz to 25 GHz and 25 GHz to 27 GHz are down-converted to the frequency range of 1 GHz to 0 GHz and 0 GHz to 2 GHz, respectively, representing the case that frequency aliasing occurs at the low frequency edge of the first Nyquist band. After once zero interpolation, the nominal instantaneous bandwidth is doubled to 5 GHz as shown in Fig. 5(b). After twice zero interpolation, the nominal instantaneous bandwidth is quadrupled to 10 GHz as shown in Fig. 5(c). After FrFT, the desired broadband LFMW is transformed to a spike along the u-axis in Fig. 5(d), which is marked as ①. In addition, the image signals and the harmonic distortions generated during the interpolation are converted to wide-spread noises (③, ④, ⑤ in Fig. 5(d)) and distinguishable spurs (② in Fig. 5(d)) along the u-axis, respectively. Then, band-pass filtering with a bandwidth value of 80 points in the rotated frequency domain is applied to select the desired broadband LFMW. Figure 5(e) shows the result after filtering in the rotated frequency domain. Finally, the retrieved LFMW without frequency-domain aliasing is obtained by performing iFrFT on the data after filtering. Figure 5(f) exhibits the spectrogram of the retrieved LFWM, indicating that the input LFMW with its instantaneous frequency sweeping from 24 GHz to 27 GHz is down-converted to the frequency range of 4 GHz to 7 GHz without frequency aliasing. Figure 6 shows the experimental results of digitizing an LFMW with its instantaneous frequency sweeping from 30 GHz to 33 GHz. As shown in Fig. 6(a), after photonic sampling ADC, the first and the second parts of the LFWM in the frequency range of 30 GHz to 32.5 GHz and 32.5 GHz to 33 GHz are down-converted to the frequency range of 0 GHz to 2.5 GHz and 2.5 GHz to 2 GHz, respectively, representing the case that frequency aliasing occurs at the high frequency edge of the first Nyquist band. After processing by using the proposed method, it can be seen from Fig. 6(f) that this broadband LFMW is successfully down-converted to the frequency range of 5 GHz to 8 GHz without frequency aliasing.
Fig. 5. Experimental results of digitizing an LFMW with its instantaneous frequency sweeping from 24 GHz to 27 GHz. (a) Spectrogram after photonic sampling ADC. (b) Spectrogram after once interpolation. (c) Spectrogram after twice interpolation. (d) Spectrum after FrFT. (e) Spectrum after band-pass filtering in the fraction Fourier domain. (f) Spectrogram after iFrFT.
Fig. 6. Experimental results of digitizing an LFMW with its instantaneous frequency sweeping from 30 GHz to 33 GHz. (a) Spectrogram after photonic sampling ADC. (b) Spectrogram after once interpolation. (c) Spectrogram after twice interpolation. (d) Spectrum after FrFT. (e) Spectrum after band-pass filtering in the fraction Fourier domain. (f) Spectrogram after iFrFT.
Figure 7 shows the normalized instantaneous phase of the retrieved LFMW in Fig. 6(f), which is obtained by carrying out Hilbert transform to the data output from the DSP module. The quadratic fit curve (red dashed line) corresponding to the calculated phase curve (blue solid line) is also presented in Fig. 7. The two curves fit in with each other, where the correlation coefficient R-square is 0.9999. This result indicates that the phase of the LFMW after being processed by using the proposed method is maintained.
Fig. 7. Calculated phase of the retrieved LFMW (blue solid line) and its corresponding quadratic fit curve (red dashed line).
The photonic sampling ADC is used as the receiver in radar detection. Figure 8 shows the experimental setup for detecting two cuboid targets. The two targets are with an identical size of 10 cm ×13 cm ×13 cm, and are placed at distances of 100 cm and 150 cm from the antennas, respectively. In the experiment, an LFMW with its instantaneous frequency sweeping from 14 GHz to 18 GHz is generated by using the AWG, and is then amplified by using an electrical amplifier with an operation frequency range from 2 GHz to 20 GHz and a saturation output power of 22 dBm before sending to the transmitting antenna (HD-140SGAH20S, 12-18 GHz). The echo signal received by the receiving antenna (HD-140SGAH20S, 12-18 GHz) is firstly amplified by using another electrical amplifier with an operation frequency range from 2 GHz to 20 GHz and a saturation output power of 22 dBm, and is then directly digitized by using the photonic sampling ADC. After computing the autocorrelation of the echo signals, the distance between the two targets can be obtained. Since the sampling rate of the photonic ADC is 5 GSa/s, the echo signals in the frequency range of 14 GHz to 15 GHz, 15 GHz to 17.5 GHz and 17.5 GHz to 18 GHz are down-converted to the frequency range of 1 GHz to 0 GHz, 0 GHz to 2.5 GHz and 2.5 GHz to 2 GHz, respectively. After processing by using the proposed method, the echo signal is successfully down-converted to the frequency range of 4 GHz to 8 GHz without frequency aliasing. The bandwidth value of the FrFT filter here is set to be 3000 points. Figure 9(a) and (b) present the ranging results without and with using the proposed method, respectively. It can be seen that the ranging accuracy is seriously deteriorated as a result of severe frequency-domain aliasing in Fig. 9(a). In contrast, it is largely improved owing to the expanded effective bandwidth after eliminating frequency-domain aliasing, as shown in Fig. 9(b). The upper bound of the ranging accuracy, which is determined by the temporal resolution, is calculated to be 3 cm for a sampling rate of 5 GSa/s. Although the temporal resolution is further enhanced after interpolation in Fig. 9(b), the ranging accuracy is not further enhanced, indicating that it is ultimately limited by the bandwidth of the transmitting LFMW.
Fig. 8. Experimental setup for detecting two cuboid targets. AWG: arbitrary waveform generator; ADC: analog-to-digital converter.
Fig. 9. Experimental results for measuring the distance between the two cuboid targets (a) without using the proposed method, and (b) through using the proposed method.
Finally, it should be pointed out that the proposed method can be used for other radar signals through replacing FrFT and iFrFT with proper transform pairs. The essence of FrFT is fast Fourier transformation (FFT) with an additional transformation factor of quadratic phase, i.e., linear chirp. Hence, the input LFMW can be transformed to a single-tone signal in the rotated spectrum when the rotation angle α is properly set to compensate for the linear chirp of the input LFMW. In radar systems, the format and the parameters of the transmitting waveforms are known in advance. Therefore, the additional transformation factor can be precisely set to guarantee that the input signals can be transformed to single-tone signals in the rotated spectrum. For example, nonlinear frequency modulation waveforms (NLFMWs) can be transformed to a single-tone signal in the rotated spectrum by adding an additional transformation factor of opposite nonlinear phase, i.e., opposite nonlinear chirp, in the FFT process. In this circumstance, the proposed method based on interpolation and filtering in the rotated spectrum can also be used to increase the instantaneous bandwidth of the photonic sampling ADC for receiving such radar signals.
4. Conclusion
In summary, we have proposed and experimentally demonstrated an approach to expanding the instantaneous bandwidth of a photonic sampling ADC for receiving LFMWs based on up-sampling and filtering in the fractional Fourier domain. This method doubles the instantaneous bandwidth through twice zero interpolation and fractional Fourier domain signal processing. In the experiment, input LFMWs with instantaneous frequency sweeping from 24 GHz to 27 GHz and from 30 GHz to 33 GHz, i.e., with an instantaneous bandwidth of 3 GHz, were digitized without frequency-domain aliasing by using a photonic sampling ADC with a sampling rate of 5 GSa/s, i.e., with an instantaneous bandwidth of 2.5 GHz, where the phase of the signal after being processed was maintained. In addition, the distance between two targets was measured by using the photonic sampling ADC to directly digitize the LFMW with its instantaneous frequency sweeping from 14 GHz to 18 GHz. The frequency aliasing was eliminated by using the proposed method, which largely enhances the ranging accuracy. Hence, the proposed method can be used to reduce the hardware complexity of the photonic sampling ADC for receiving broadband radar signals.
Funding
National Key Research and Development Program of China (2019YFB2203800); National Natural Science Foundation of China (61927821); Fundamental Research Funds for the Central Universities (ZYGX2020ZB012).
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 may be obtained from the authors upon reasonable request.
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