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Abstract
Photonic compressive sensing (CS) has attracted great research interest for its potentials in the acquisition of wideband sparse signals with relatively low sampling rate. The photonic CS scheme based on optical mixing using a spectral shaper can realize the mixing of a sparse signal with a high-speed pseudo-random bit sequence (PRBS), but avoids the use of high-speed electronics. In this approach, by utilizing the frequency-to-time mapping (FTTM) of chirped pulses, the spectral information on the spatial light modulator (SLM) within a spectral shaper can be projected into the time-domain waveform. However, the generated PRBS in the time domain is a unipolar sequence that alternates between 0 and 1, which leads to a nonzero-mean measurement matrix. This would result in a poorer performance of signal reconstruction compared to that with a zero-mean measurement matrix. Moreover, the length of PRBS that can be recorded in the SLM is also limited by the far-field condition. In this paper, we propose an optical mixer for photonic CS, which utilizes an SLM-based spectral shaper with complementary outputs as well as a balanced photodetector in order to generate bipolar PRBS. The performance of signal reconstruction can be significantly improved owing to the zero-mean measurement matrix induced by bipolar PRBS. In addition, the constraint on the length of PRBS can be greatly alleviated, since the obtained PRBS can still be kept zero-mean even if the PRBS is longer than that the far-field condition demands. Experimental and simulation results are presented to demonstrate the feasibility and advantage of the given approach.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
According to the theory of CS, it is possible to reconstruct a sparse signal with samples far below than that the Nyquist sampling theorem demands [1–3]. The implementation of CS includes a measurement process and a reconstruction process. The measurement process is a linear mapping from the sampling points at or above the Nyquist rate to the measurement results with fewer data points. In the reconstruction process, the original sparse signal can be recovered from the measured results via an optimization method. For the acquisition of time-domain sparse signals, there are two well-established CS frameworks, e.g. the random demodulator (RD) and the modulated wideband converter (MWC) [2–3]. The measurement process in both of the schemes involves the stages of random mixing, low-pass filtering and subsampling despite of their difference in architecture, in which the sparse signal needs to be mixed with a pseudo-random bit sequence (PRBS) firstly, and the mixed product is low-pass filtered and then measured at a sub-Nyquist rate. Photonics-enabled CS shows high potentials in wideband signal acquisition due to the advantage of large bandwidth offered by photonic techniques and devices [4–18].
Photonic CS in a photonic link using a continuous-wave optical carrier has been demonstrated in [4–10], in which electro-optic modulators (EOMs) are employed as the wideband signal modulator as well as the mixer for the signal mixing with a PRBS. There are also many works on photonic CS that use stretched pulses as optical carrier [11–16]. In these approaches, the wideband sparse signal to be measured is modulated on the stretched pulses via an EOM. Optical mixing of a time-domain sparse signal with a PRBS can be implemented by using a conventional EOM, as in [12]. As we know, a short pulse will be stretched in the time domain after propagating through a dispersion element, and the stretched pulse becomes chirped. One of the advantages of using stretched pulses as optical carrier lies in that the technique of pulse stretch/compression in dispersion elements can be effectively applied to speed up or slow down the rate of the signal or the PRBS. In addition, the technique of photonic time stretch can be combined with the photonic CS, which is able to further decrease the sampling rate [13]. The CS-based single pixel imaging can also be realized using the technique of pulse stretch and compression [15–17]. In the approach, a stretched pulse is firstly mixed with a PRBS and then enters into an imaging system, in which the pulse is spatially dispersed to illuminate a target object and the image information is recorded on the spectrum of the reflected pulse. The key function of low-pass filtering in the CS can be realized physically based on the pulse compression in which the chirped pulse is compressed after propagating through a dispersion medium with proper dispersion amount [12].
It is of great interest to use a spatial-light-modulator (SLM)-based spectral shaper as an optical mixer in the photonic CS with stretched pulses, for the acquisition of either time-domain signals or images, which is firstly proposed by Valley et al. in [11]. Note that the time-domain signal acquisition system and the single-pixel imaging system based on spectral shaping and frequency-to-time mapping are similar in the architecture and they share the same theoretical framework [11,18–19]. The optical mixer based on a spectral shaper can realize high-speed mixing but avoids the use of high-speed electronics. The spectral shaper based on liquid crystal SLM allows the setting of arbitrary bit pattern dynamically. In addition, this type of optical mixer also eliminates the complicated synchronization between the bit pattern generator and the receiver, since the arriving time of the generated PRBS is known to the receiver. Despite of the above advantages, the optical mixer based on a spectral shaper has two major disadvantages when it is applied in photonic CS. Firstly, the generated PRBS mixed with the signal is a unipolar sequence, e.g. alternating between 0 and 1, but not +1 and -1, due to the property of intensity modulation and direct detection of optical links. The resulted measurement matrix is not zero-mean, which would lead to a poorer recovery performance than that with a bipolar PRBS [20]. Secondly, the maximum bit rate of the PRBS that can be recorded on an SLM is constrained by the far-field condition of FTTM, while the bit rate of PRBS decides the bandwidth of a CS system [2,3,19,21].
In this paper, we present an optical mixer based on an SLM-based spectral shaper with dual outputs for realizing photonic CS, which is based on the theoretical framework of RD in [2]. A spectrally sparse RF signal to be analyzed is modulated on stretched and chirped pulses via an EOM. An SLM-based spectral shaper with dual outputs is employed as the optical mixer, in which two complementary PRBSs are recorded in the SLM corresponding to two output ports. By using a balanced-photodetector (BPD), a desired mixing term between the sparse signal and a bipolar PRBS can be obtained, which is necessary in constructing a zero-mean measurement matrix and helps to improve the recovery performance. More importantly, the restriction imposed by the far-field condition on the length of PRBS can be largely alleviated in our scheme. Using the given configuration for optical mixing, the length of PRBS can be much longer than previous approaches, which means a higher system bandwidth. Experimental demonstration is implemented, in which the measurement and recovery of single-tone and two-tone signals are presented. Further simulation results are presented to show the performance of the proposed approach.
2. Principle
The schematic illustration of the proposed photonic CS scheme is shown in Fig. 1. The pulses emitted from a mode-locked laser (MLL) is stretched after propagating through a dispersion element. A spectrally sparse RF signal to be analyzed is modulated on the broadened pulses via an EOM. Then the modulated pulses are incident into a spectral shaper with dual outputs, in which the SLM within the spectral shaper is carved by two predefined complementary PRBSs, each for one output. Due to the chirping property of the stretched pulses, the frequency-domain information recorded in the SLM is converted into a time-domain waveform. A BPD is employed at the outputs of the spectral shaper to achieve the mixing of the sparse signal with a bipolar PRBS. Then a low-pass filter (LPF) and a digitizer are applied to realize the functions of integration and down-sampling required in CS. By means of a sparse recovery algorithm, the input sparse signal can be recovered by using the obtained samples. Note that the setup here is similar to the works in [11,21], but in this work we employ a novel configuration with a dual-output spectral shaper and a BPD to achieve an equivalent bipolar PRBS in order to improve the CS performance.
Fig. 1. Schematic illustration of photonic CS based on optical mixing using a spectral shaper with complementary outputs. MLL: mode-locked laser; EOM: electro-optic modulator; LPF: low-pass filter; ADC: analog-to-digital converter; DSP: digital signal processing.
The CS measurement process under the framework of RD in [2] can be modeled as $\textbf{y} = \boldsymbol{\mathrm{\Psi}} \textbf{x} = {\textbf{DHRx}}$, where $\textbf{x}$ is a $N \times 1$ vector which represents the sparse signal sampled at or above Nyquist rate and $\boldsymbol{\mathrm{\Psi}}$ represents the measurement matrix with dimension $M \times N$, $\textrm{y}$ is a $M \times 1$ ($M \ll N$) vector representing the measurement results. The measurement matrix $\boldsymbol{\mathrm{\Psi}} = {\textbf{DHR}}$ in RD includes a $N \times N$ diagonal matrix $\textbf{R}$ representing the PRBS, a $N \times N$ matrix $\textbf{H}$ representing the impulse response of the LPF, and a $M \times N$ matrix $\textbf{D}$ representing the subsampling process. The performance of signal reconstruction from $\textrm{y}$ to $\textrm{x}$ is affected by several factors, like the length of signal N, the compressive ratio $N/M$, the signal-to-noise ratio (SNR) of the system. And it has been proved that a zero-mean measurement matrix is conducive to the signal reconstruction, which demands a zero-mean PRBS in the RD framework [20,22].
Assume the dispersion amount is ${\Phi _2}$ (in the unit of $\textrm{p}{\textrm{s}^\textrm{2}}$) and the two frequency-domain complementary PRBSs recorded in the SLM of the spectral shaper are ${R_1}(f)$ and ${R_2}(f)$ (alternating between 0 and 1), the corresponding time-domain sequences are ${r_1}(t) = {R_1}(t/{\Phi _2})$ and ${r_2}(t) = {R_2}(t/{\Phi _2})$, respectively, according to the principle of FTTM, if the far-field condition is satisfied [19]. The photocurrents of the two photodetectors (PDs) in the BPD are ${i_1}(t) = P \cdot \mathrm{\Re } \cdot {r_1}(t) \cdot [{1 + \alpha x(t)} ]$ and ${i_2}(t) = P \cdot \mathrm{\Re } \cdot {r_2}(t) \cdot [{1 + \alpha x(t)} ]$, where $x(t)$ denotes the sparse signal to be measured, $\alpha$ is the modulation coefficient, $P$ is the optical power and $\mathrm{\Re }$ is the responsivity of the PDs. Due to the complementarity of the two sequences, we have ${r_1}(t) = 1 - {r_2}(t)$. The output of the BPD is as,
Since ${r_1}(t)$ and ${r_2}(t)$ are a pair of complementary unipolar sequences, alternating between 0 and 1, their difference ${r_1}(t) - {r_2}(t)$ is a bipolar sequence alternating between ±1, which is denoted by ${r_b}(t)$ hereafter. In order to remove the dc component from the signal, we can record the bipolar sequence $P \cdot \mathrm{\Re } \cdot {r_b}(t)$ in a calibration process prior to the measurement, which will be subtracted from the obtained differential current in the following measurement. Then we can obtain the mixing product:
In the above discussion, we assume a perfect mapping from the frequency-domain sequences ${R_1}(f)$ and ${R_2}(f)$ to the time-domain sequences ${r_1}(t) = {R_1}(t/{\Phi _2})$ and ${r_2}(t) = {R_2}(t/{\Phi _2})$, respectively, when the time-domain far-field condition is satisfied. The time-domain far-filed condition corresponds to the Fraunhofer condition in the spatial optics, under which the real-time Fourier transform or FTTM occurs. The time-domain far-field condition requires [19]:
where $\Delta {t_0}$ is the temporal width of the optical pulse after spectral shaping and $\ddot{\Phi }$ is the dispersion value. Based on the FTTM, the PRBS set in the spectral shaper can be converted into a temporal waveform. It has been demonstrated in [21] that this far-field condition is equivalent to the condition that the length N of PRBS that can be recorded in the SLM is constrained as $N \ll {N_{\max }} = B{(\pi |\ddot{\Phi }|/2)^{1/2}}$, where B is the optical bandwidth, ${N_{\max }}$ is the upper-limit of the PRBS length. It has been verified that this condition can be changed to be $N < {N_{\max }} = B{(\pi |\ddot{\Phi }|/2)^{1/2}}$, where the sign much greater is relaxed to the sign greater, under which an acceptable mapping from the frequency domain to the time domain can be obtained [19,23–24]. Nevertheless, the length of PRBS still should be less than the upper-limit ${N_{\max }} = B{(\pi |\ddot{\Phi }|/2)^{1/2}}$. Equivalently, the bit rate of the generated PRBS is also limited, which can be expressed as ${f_{br}} < {N_{\max }}/(\textrm{2}\pi B|\ddot{\Phi }|)$. In a typical system with parameters of $\ddot{\Phi } = \textrm{450 p}{\textrm{s}^2}$ and $B = \textrm{3520 GHz}$, the upper-limits of the length and bit rate of the generated PRBS are ${N_{\max }} = \textrm{93}$ and ${f_{br}} = \textrm{9}\textrm{.3 GHz}$, respectively. According to the theory of RD, the maximum frequency component of the input sparse signal should be less than ${f_{br}}/2$, which means the electrical bandwidth of the photonic CS system is limited by the maximum PRBS length fundamentally. It is highly desired to break or alleviate this constraint in order to improve the bandwidth of the SLM-based CS system.Let’s consider the performance of the generated PRBS in a qualitative way. With the above parameters, two examples of the generated PRBS are shown in Fig. 2, where the bandwidth of the PDs is set as 25 GHz. The length of the PRBS in the first example is 50, which is less than the upper-limit. The generated waveforms using a single-ended PD and a BPD are shown in Figs. 2(a) and 2(b) respectively. Since the far-field condition is satisfied, the performance of the PRBS projection from the frequency domain to the time domain is pretty good without much distortion. When the length of the recorded PRBS is increased to be 180, which is larger than the upper-limit, the generated unipolar time-domain waveforms is shown in Fig. 2(c). It is seen the generated sequence has poorer extinguish ratio compared to the former case, owing to the imperfect FTTM when the far-field condition is not satisfied. The deviation from the distribution between 0 and 1 of the PRBS means a poorer distribution of the entries in the measurement matrix and therefore a poorer recovery performance [20,22]. The generated bipolar PRBS is shown in Fig. 2(d). In this case, in spite of the uneven peak values, the generated PRBS has a zero mean, which implies that we still can achieve a measurement matrix with a zero mean and normally distributed entries [20,22]. Therefore, we not only can obtain a bipolar PRBS using the proposed configuration, but also may alleviate the constraint on the PRBS length imposed by the far-field condition, which will be demonstrated in the next experimental part.
Fig. 2. The generated unipolar (a) and bipolar (b) PRBS when N=50 (the far-field condition is satisfied). The generated unipolar (c) and bipolar (d) PRBS when N=180 (the far-field condition is not satisfied). In both the examples, ${N_{\max }} = \textrm{93}$.
3. Results and discussions
To verify the feasibility and advantage of the proposed scheme, experimental demonstrations of the proposed approach are implemented. In the experiment, a passive mode-locked laser (Menlo ELMA100) with a repetition rate of 100 MHz is employed to generate short pulses. A four-port spectral shaper (Finisar 4000A) with a bandwidth of 5 THz, a resolution of 1 GHz is applied to perform spectral modulation, in which complementary PRBS are recorded for two outputs. The optical bandwidth of the system is around 3519 GHz. The optical spectra are recorded by an optical spectrum analyzer (Yokogawa AQ6370D). A coil of single-mode fiber with a length of 20 km is employed as the dispersion medium, whose dispersion amount is around $\textrm{450 p}{\textrm{s}^2}$. The time-domain waveforms are captured and recorded by an oscilloscope (Anritsu MP2100B). The upper-limit ${N_{\max }}$ of the length of PRBS, given by the far-field condition, is calculated to be 93 [21]. Typical two complementary PRBSs with a length of 48, recorded in the spectral shaper, are shown in Fig. 3.
Fig. 3. Typical two complementary PRBSs recorded in the spectral shaper (solid: output 1; dotted: output 2).
In the first experiment, the length of PRBS set in the spectral shaper is N=48, which meets the requirement of the far-field condition. Meanwhile, a single-tone sinusoidal signal with frequency of 900 MHz is applied as an input sparse signal to test the performance of the system. The mixed products between the signal and the complementary PRBSs from the two output ports of the spectral shaper are detected by a BPD with a bandwidth of 40 GHz and captured by the sampling oscilloscope. For comparison, the mixed product from a single output port of the spectral shaper is also detected by using a single-ended PD and recorded by the oscilloscope. The low-pass filtering is realized by the accumulation of the mixed signal in a time period (N/M bits) and the down-sampling is accomplished by sampling the signal after LPF every predetermined period (N/M bits), both in an off-line computer program. In the signal reconstruction stage, the l1-ls algorithm with for solving l1-regularized least squares problem, developed by Koh and Kim [25], is used to recover the input sparse signal with a regularization parameter of $\textrm{1}{\textrm{0}^{\textrm{ - 5}}}$. We test the recovery performance with the measurements of M=12 and M=8, which means the compressive ratios are N/M=4 and N/M=6, respectively. The reconstructed spectra, as well as the original spectra are shown in Fig. 4, where the results based on the unipolar PRBS are given in Figs. 4(a) and 4(b), and those based on the bipolar PRBS are given in Figs. 4(c) and 4(d). It is seen that when using unipolar PRBS, the recovery performance under a compression ratio of 4 is acceptable, but that under a compression ratio of 6 is much poorer and not acceptable. And the reconstruction performance degrades when the value of N/M reaches 8, which is not shown. In Fig. 4, the performance of the recovered spectra under bipolar PRBS is much better than that under unipolar PRBS, if the compression ratio is the same. Especially, the performance under a compression ratio of 6 when using bipolar PRBS is pretty good, as compared with the case using unipolar one. When the compressive ratio is 6, the root mean square error of the reconstructed spectrum under bipolar PRBS is 0.05, while that under unipolar PRBS is 0.12. Note that the peak location of the reconstructed spectra is slightly different from the original one, which is induced by the basis mismatch [26]. Therefore, it is evident that our approach can improve the recovery performance, which owes to the zero-mean measurement matrix induced by bipolar PRBS.
Fig. 4. The original (solid) and reconstructed (dotted) spectra when N=48 (${N_{\max }} = 93$). (a) unipolar PRBS, N/M=4; (b) unipolar PRBS, N/M=6; (c) bipolar PRBS, N/M=4; (d) bipolar PRBS, N/M=6.
In the second experiment, the length of PRBS is increased to be N=96, which is longer than the upper-limit given by the far-field condition. In this case, a two-tone RF signal with frequencies of 2 GHz and 2.7 GHz is under test. We investigate the recovery performance with the measurements of M=24 and M=16, which corresponds to the compressive ratios of N/M=4 and N/M=6, respectively. The results of reconstruction are shown in Fig. 5. Again, the results based on bipolar PRBS is much better than those based on unipolar one under the same compressive ratio. When the compressive ratio is 6, the root mean square errors of the reconstructed spectra in the cases of bipolar and unipolar PRBS are 0.06 and 0.11, respectively. Note that the length of PRBS recorded in the SLM within the spectral shaper in this case is longer than the upper-limit given by the far-field condition. Therefore, we can say that the given approach alleviates the constraint of the far-field condition on the length of PRBS. In the above example, the bit rate of the generated PRBS is around 4.8 Gb/s. In this example, the bit rate is increased to be 9.6 Gb/s as the length of PRBS is increased while other parameters keep unchanged. Therefore, the given approach can increase the equivalent bit rate of the generated PRBS, and the electrical bandwidth of the CS system.
Fig. 5. The original (solid) and reconstructed (dotted) spectra when N=96 (${N_{\max }} = 93$). (a) unipolar PRBS, N/M=4; (b) unipolar PRBS, N/M=6; (c) bipolar PRBS, N/M=4; (d) bipolar PRBS, N/M=6.
In order to further investigate the potential of longer PRBS, we use computer simulations to compare the recovery performance between the unipolar and bipolar PRBS. In this case, the length of PRBS is increased to be N=192, while other parameters keep the same as the experiment. The equivalent bit rate of the PRBS is estimated to be 19.1 Gb/s. A two-tone signal with frequencies of 3.4 GHz and 3.6 GHz is applied as the sparse signal. The signal-to-noise ratio (SNR), which is defined as the ratio of the signal power to the noise power, at the output of the BPD, is set to be 20 dB. The compression ratio N/M is set as 6. The reconstructed time-domain and frequency-domain signals when using unipolar and bipolar PRBS are shown in Fig. 6. It is seen, as expected, the performance of the reconstructed signal in the case of bipolar PRBS is much better than that with unipolar one. Next, we implement more simulations to test the proposed approach with longer PRBS. It is found that under the same system parameters (optical bandwidth, dispersion amounts and SNR) as the above example, the maximum PRBS length is around 204, in which a two-tone signal can be recovered with an acceptable performance (the spectrum of the signal is at least 5 dB higher than the level of the noise). Note that the upper-limit ${N_{\max }}$ given by the far-field condition is still 93. This means the proposed approach can mitigate the constraint of the far-field condition on the length of PRBS that can be recorded in the SLM to some extent, which not only greatly improves the performance of the signal reconstruction, but also significantly increases the bandwidth of the CS system with SLM-based optical mixing.
Fig. 6. The original (solid) and reconstructed (dotted) signals in the time (a) and frequency (b) domains with unipolar PRBS when N/M=6. The original (solid) and reconstructed (dotted) signals in the time (c) and frequency (d) domains using bipolar PRBS when N/M = 6.
It should be noted that the work in [2] on RD and the work in [3] on MWC are two CS frameworks for the time-domain sparse signal acquisition. In both the work in [11] and our scheme, the RD framework is employed and a SLM-based spectral shaper is utilized for random mixing where the PRBS is recorded in the SLM within the spectral shaper. This work and the work in [21] are on the same CS architecture. The work in [21] presented theoretical and simulation results on the maximum length of PRBS that can be recorded in the SLM constrained by the far-field condition, as well as its influence on the CS performance. However, this work proposes a novel method to realize bipolar PRBS and therefore a zero-mean measurement matrix in order to improve the CS performance.
4. Conclusion
In summary, we have proposed a photonic CS scheme based on optical mixing using a spectral shaper with dual outputs to achieve bipolar coding. An SLM-based spectral shaper with complementary outputs, as well as a BPD, is employed to generate bipolar PRBS. It is shown that the recovery performance is greatly improved compared with that uses the conventional spectral-shaper-based optical mixing owing to the zero-mean measurement matrix induced by the bipolar PRBS that alternates between ±1. More importantly, the PRBS length that can be recorded in the SLM can break the upper-limit imposed by the time-domain far-field condition to some extent in the given approach, which means a higher bandwidth of the CS system. The presented experimental and simulation results verify the feasibility and advantage of the proposed scheme.
Funding
National Key Research and Development Program of China (2019YFB2203200); National Natural Science Foundation of China (61975048,61901148,41905024); Natural Science Foundation of Zhejiang Province (LZ20F010003, LQ20F010008).
Disclosures
The authors declare no conflicts of interest.
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