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Abstract
A photonic distributed compressive sampling (PDCS) approach for identifying the spectra of multi-node wideband sparse signals is proposed. The scheme utilizes wavelength division multiplexing (WDM) technology to transmit multi-node signals to a central station, where distributed compressive sampling (DCS) based on the random demodulator (RD) model is employed to simultaneously identify the signal spectrum. By exploiting signal correlations among nodes, DCS achieves a higher compression ratio of the sampling rate than single-node compressive sampling (CS). In a semi-physical simulation experiment, we demonstrate the feasibility of the approach by recovering the spectra of two wideband sparse signals from nodes located 20 km and 10 km away. The spectra of two signals with a mixed support-set sparsity of 2 and 4 are recovered with a compression ratio of 8 and 4, respectively. We further investigate the impact of common parts and the number of nodes on PDCS performance through numerical simulation. The proposed system takes advantage of the ultra-high bandwidth of photonic technology and the low loss of optical fiber transmission, making it suitable for long-distance, multi-node, and large-coverage electromagnetic spectrum identification.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Compressive sampling (CS) is a signal acquisition technique that has gained significant research interest due to its ability to reconstruct sparse signals at sampling rates lower than the Nyquist rate [1–3]. CS typically involves a measurement process and a reconstruction process. In the measurement process, the sparse signal is sampled to obtain a smaller number of samples than traditional sampling techniques, and then the reconstruction algorithm is used to recover the original signal from the measurement results. Two well-known CS models for acquiring wideband sparse signals are the random demodulator (RD) and the modulated wideband converter (MWC) [2,3]. In both models, the measurement process involves random mixing, low-pass filtering (LPF), and down-sampling. In the random mixing stage of CS, a pseudo-random binary sequence (PRBS) with a rate at or above the Nyquist rate is always required to mix with the sparse signal, which becomes a challenge in the electrical domain CS [4]. In recent years, numerous schemes for acquiring ultra-wideband signals using photonic CS with large bandwidth have been proposed [5–19]. Valley et al. initially proposed photonic CS [5,6], which utilized an optical pulse source and an SLM-based pulse shaper to mix RF signals and PRBS in the photonic domain. Subsequently, a photonic CS based on the RD was demonstrated, where the random mixing was implemented with two cascaded Mach–Zehnder modulators (MZMs) driven by the sparse signal and PRBS separately [7,8]. Various optimizations have been made to the random mixing process, such as realizing zero-mean measurement matrices [9,10], photonic down-conversion and recognition [11], utilizing photonics time stretch before random mixing [12,13], and random projection via waveguide speckle [14–16]. In addition, the LPF function of CS can also be implemented in the optical domain, for example, by modulating chirped optical pulses with a sparse signal and PRBS, and then compressing the resulting mixed pulses using a dispersion medium to achieve the LPF operation [17]. Moreover, several photonic CS schemes based on the MWC model have been proposed to recover multiband sparse signals [18,19]. To date, almost all existing photonic CS schemes are designed to recover wideband sparse RF signals at a single node, and have not yet addressed multi-node spectrum recovery scenarios, such as sensor networks for large coverage electromagnetic spectrum identification [20,21]. In a large-scale multi-node spectrum recovery scenario, multiple sensors are usually distributed at different remote locations and receive spectrum sparse signals within their respective coverage areas [21]. In addition to their unique characteristics, these signals often have common frequency components. Radio over fiber technology can take advantage of the low loss of fiber optic transmission to pull the RF signals from remote nodes back to the central station for centralized processing, thereby expanding the coverage range of spectrum sensing [22,23]. On the other hand, distributed compressive sensing (DCS) has been explored to realize distributed compression and joint reconstruction of multiple signals with fewer measurements than separate CS by exploiting both intra- and inter-signal correlation structures [24]. Particularly, Sundman et al. [25] proposed a mixed support-set joint sparse model (JSM) for DCS, in which the common part of the mixed support-set JSM only needs the same frequency location and the amplitude can be different, which is more general and applicable in practical scenarios.
In this paper, we propose a novel approach for identifying the spectra of multi-node wideband sparse signals using photonic distributed compressive sampling (PDCS). The proposed method employs MZMs to modulate spectrum sparse RF signals at multiple remote nodes. These signals are carried on different optical carriers through the wavelength division multiplexing (WDM) technique. The modulated signals are then transmitted over optical fibers of varying lengths and mixed with the PRBSs at the central station. The mixed terms are obtained through AC-coupled photodetectors (PDs) and subsequently filtered and down-sampled to produce multiple sets of measurements. To recover the multiple signals simultaneously, a joint reconstruction CS algorithm is applied. In a semi-physical simulation experiment, the proposed scheme successfully recognized two sets of signals satisfying the mixed support-set JSM after being transmitted over 10 km and 20 km fiber, respectively. The reconstruction performance of the proposed PDCS method is found to be better than that of the single-node photonic CS system. The proposed scheme not only benefits from the large bandwidth of photonic CS, but also greatly expands the number and coverage of monitoring nodes by leveraging radio over fiber and WDM technology.
2. Principle
The schematic of the proposed PDCS system based on the RD model is shown in Fig. 1. The system includes a central station and multiple remote nodes, which are composed of one fiber and multiple wavelengths to form a unidirectional transmission WDM ring network. At the central station, a multi-wavelength light source provides continuous light at wavelengths of ${\lambda _1}$, ${\lambda _2}$, …, up to ${\lambda _J}$. The multi-wavelength optical carrier is transmitted clockwise through a single-mode fiber to the first remote node, where the wavelength of ${\lambda _1}$ is extracted through a demultiplexer, and the other optical carriers are passed directly. The signal ${x_\textrm{1}}(t)$ received by the antenna is modulated to the optical carrier ${\lambda _1}$ through the MZM. Then, the modulated optical signal is combined with other optical wavelengths by a wavelength division multiplexer for further transmission on the fiber. Similarly, at the $j\textrm{th}$ remote node, the signal ${x_j}(t)$ to be tested is modulated on the optical carrier ${\lambda _j}$. The multi-wavelength optical carrier carrying the RF signals of J nodes is transmitted back to the central station and then decomposed into J channels by the demultiplexer. Each modulated optical carrier is mixed with a PRBS ${r_j}(t)$ by a MZM. It should be noted that the bit rate of the PRBS ${r_j}(t)$ should be at or above the Nyquist rate ${f_s}$ of the applied sparse RF signal. The mixed signal of each channel is converted into electrical signals by the PD, and then pass through an LPF and is finally digitized at a sampling rate of ${f_s}/{R_{DCS}}$, where ${R_{DCS}}$ is the compression ratio of the sampling rate. Finally, the sparse signal of each node can be recovered jointly by using the samples obtained from J channels and a DCS joint reconstruction algorithm. The spectral variation of the signals in the PDCS system is also shown in Fig. 1.
Fig. 1. Schematic of the proposed PDCS system based on the RD model. WDM: wavelength division multiplexer, MZM: Mach–Zehnder modulator, PRBS: pseudo-random binary sequence, PD: photodetector, LPF: low pass filter, DSP: digital signal processing. A1, D1∼F1: the spectral variations of the electrical signal in first channel, B1, C1: the spectral variations of the optical signal in first channel, AJ, DJ∼FJ: the spectral variations of the electrical signal in Jth channel, BJ, CJ: the spectral variations of the optical signal in Jth channel.
The signals received by remote nodes satisfying the mixed support-set model can be expressed as
The measurement process of the signal received by the $j\textrm{th}$ remote node in the PDCS system can be modeled as
After combining the measurement results of all nodes, the complete measurement matrix can be expressed as
where $X = [{{x_1},{x_2},\ldots ,{x_J}} ]\in {{\mathbb R}^{JN}}$ is the set of signals that satisfy the mixed support-set, $Y = [{{y_1},{y_2},\ldots ,{y_J}} ]\in {{\mathbb R}^{JM}}$ is the measurement results of J nodes, and $\Phi \in {{\mathbb R}^{JM \times JN}}$ is a diagonal matrix consisting of the measurement matrix of each signal ${x_j}$. ${W^ \ast } \in {{\mathbb R}^{JN \times JN}}$ is a diagonal matrix consisting of J Fourier orthogonal basis matrices W, and ${\theta ^ \ast } = [{{\theta_1},{\theta_2},\ldots ,{\theta_J}} ]\in {{\mathbb R}^{JN}}$ represents the sparse spectral information of the J signals. To simultaneously reconstruct the signal set X and its spectrum ${\theta ^ \ast }$ from Y, the sparsity adaptive matching pursuit for DCS (DCS-SAMP) algorithm is adopted [26], which does not require the knowledge of sparsity in advance. The stopping criterion for the DCS-SAMP algorithm is defined as the point at which the relative improvement in the energy of the recovered signal between two consecutive stages becomes smaller than a specific threshold, which is associated with the noise level present in the signals.The signal at each node is sparse in the frequency domain as in Eq. (1), we can measure and reconstruct each signal using the photonic CS system based on the RD model. In fact, for an $N$-sample signal that $K$-sparse, roughly $M = C \cdot K$ measurements are required to guarantee reconstruction with high probability ($C$ is a constant). Therefore, If J signals are acquired individually, and the sparsity of each signal is K, it requires $J \cdot C \cdot K$ measurements and J independent reconstruction processes. However, due to the existence of the common part, the number of measurements required for the PDCS system is significantly reduced, especially when the common part dominates. In the same setting, the DCS method just requires $C \cdot {K_{total}}$ measurements and only one reconstruction procedure to recover J signals simultaneously.
3. Results and discussion
A semi-physical simulation experiment is implemented to demonstrate the proposed PDCS approach. The PDCS system consists of one central station and two remote nodes. The signals of the two nodes are measured sequentially, allowing for sharing a set of laser diode (LD), signal generator, MZM, PRBS generator, and PD. The RF signal under test is generated by an arbitrary waveform generator (M8190A, 12 GS/s), while the PRBS is generated by a programmable pulse generator (Anritsu MP2100). An LD with a 100-kHz linewidth emits continuous-wave light with an output power of 16 dBm. The optical carrier is modulated by the RF signal at the remote node using a MZM (MXLAN-LN-40) biased at the orthogonal point, and then transmitted through an optical fiber with a length of 20 km to the central station. Another MZM (Fujitsu FTM7928FB) biased at the orthogonal bias point is used to achieve random mixing with the PRBS and the RF signal. The electrical mixed result is realized by an AC-coupled PD with a 3 dB bandwidth of 10 GHz, and captured by a real-time oscilloscope (R&S RTP084, 8 GHz). The function of the low-pass filtering (LPF) and down-sampling is implemented by post-processing the captured data. The LPF is realized by accumulating the mixed signal over a time period of $N/M$ samples, and the down-sampling is accomplished by sampling the signal after LPF every predetermined period of $N/M$ samples. Once the data acquisition of the first remote node is completed, we change the fiber length between the two MZMs to 10 km for simulating the second remote node. After changing the RF signal and PRBS, the same experimental steps are carried out to complete the data acquisition of the second remote node.
In the first experiment, the signals of two nodes are set with a common sparsity ${K_c} = 1$ and innovation sparsity ${K_1} = {K_2} = 1$. The signal frequencies of the first node are set to 1.1 GHz and 1.8 GHz, with input powers of 1.1 dBm and 4.9 dBm, respectively. After passing through a 20 km fiber optical link and a PD, the received RF powers are measured as –35.8 dBm and –31.9 dBm, respectively. The signal frequencies of the second node are set to 1 GHz and 1.8 GHz, with input powers of 3.7 dBm and 2.3 dBm, respectively. After passing through a 10 km fiber optical link and a PD, the received RF powers are measured as –31 dBm and –32.6 dBm, respectively. Each PRBS has a bit rate of 4 Gb/s and a timing jitter of 1 ps. The pattern length of the PRBS is set to be 1000, which is equal to the data length N. The signals after the PD are sequentially low-pass filtered and down-sampled with a compression ratio of 8. Finally, the two sparse measurements are separately reconstructed (separate reconstruction does not take advantage of signal correlation) and jointly reconstructed using the DCS-SAMP algorithm. The algorithm parameters are kept consistent for each run. Figure 2 displays the spectra of both the input signals and the recovered signals. To facilitate comparison, the spectra of the input signal are measured at the output of the PD without modulating the PRBS signal. The outcomes of the separate reconstructions for the two signals are presented in Fig. 2(a) and Fig. 2(c), whereas the results of the joint reconstruction are illustrated in Fig. 2(b) and Fig. 2(d). It can be observed that, when the compression ratios are the same, the separate reconstruction exhibits a higher noise component in the recovered signal. The presence of noise in the low-frequency band is evident, and this phenomenon may be attributed to the noise-folding effect in CS [27]. Note that there are still amplitude errors between the input signal and the recovery signal. These errors can be mitigated by employing a smaller compression ratio.
Fig. 2. Experimental results, $N\textrm{ = 1000}$, ${K_c} = 1$, ${K_1} = {K_2} = 1$. (a), (c) separate recovery $({R_{DCS}} = 8)$. (b), (d) joint recovery $({R_{DCS}} = 8)$.
In the second experiment, the common sparsity of the two signals is increased to 3, while the innovation sparsity remains 1. The signal frequencies of the first node are set to 0.4 GHz, 1.1 GHz, 1.5 GHz, and 1.8 GHz, with input powers of –3.9 dBm, –1.8 dBm, –2.5 dBm, and –1 dBm, respectively. After passing through a 20 km fiber optical link and a PD, the received RF powers are measured as –39.7 dBm, –37.3 dBm, –37.9 dBm, and –36.7 dBm, respectively. The signal frequencies of the second node are set to 0.4 GHz, 1 GHz, 1.5 GHz, and 1.8 GHz, with input powers of –3.9 dBm, –0.7 dBm, –2.5 dBm, and –2 dBm, respectively. After passing through a 10 km fiber optical link and a PD, the received RF powers are measured as –38.2 dBm, –35.8 dBm, –37.6 dBm, and –37.3 dBm, respectively. The bit rate of PRBS, data length N, and recovery algorithm remain consistent with the first experiment. We investigate the recovery performance of the two signals separately or jointly with a compression ratio of 4. Figure 3 shows the reconstructed results. It can be observed that it fails to recover the frequency points with separate reconstruction, whereas all the frequency points are recovered by the joint reconstruction. The advantage of joint reconstruction is more pronounced, which can be attributed to the increased common sparsity. The recovery performance of the joint reconstruction is significantly superior to that of separate reconstruction, thereby validating the advantages of the proposed scheme.
Fig. 3. Experimental results, $N\textrm{ = 1000}$, ${K_c} = 3$, ${K_1} = {K_2} = 1$. (a), (c) separate recovery $({R_{DCS}} = 4)$. (b), (d) joint recovery $({R_{DCS}} = 4)$.
Computer simulations are conducted to comprehensively compare the performance of PDCS under separate and joint reconstruction. Successful recovery is defined as all frequency components of the sparse signal being successfully recovered and the normalized amplitude of the maximum noise frequency component is less than 0.2. The recovery success rate is calculated as the number of successful recoveries divided by 100 total simulation times. In the simulations, the frequency components of the input signals are randomly distributed within a 10 GHz bandwidth. The jointly sparse signals adhere to the mixed support-set model. The data length is set to N = 1000, the bit rate of the PRBS is configured at 20 Gb/s, and the recovery algorithm used is consistent with that employed in the experiments. Firstly, we investigate the recovery success rate under different compression ratios. The common sparsity, innovative sparsity, number of nodes, and signal-to-noise ratio (SNR) are 1, 1, 2, and 20 dB, respectively. As depicted in Fig. 4(a), it can be observed that under the same compression ratio, joint reconstruction exhibits a higher recovery success rate compared to separate reconstruction, particularly when the compression ratio exceeds 10, highlighting the pronounced advantage of joint reconstruction. Subsequently, the common sparsity is set to 3 while keeping other parameters constant. Figure 4(b) also shows that, at the same compression ratio, joint reconstruction exhibits a higher recovery success rate than separate reconstruction, with the advantage of joint reconstruction becoming more pronounced for compression ratios exceeding 8. A comparison between Fig. 4(a) and Fig. 4(b) reveals that the advantage of joint reconstruction becomes increasingly apparent as the common sparsity of the signals increases. Next, we evaluate the recovery success rate at different SNR levels, with a compression ratio of 10. Figure 4(c) shows that separate reconstruction results in a lower success rate, indicating that the original signals cannot be reliably identified in such scenarios. In contrast, joint reconstruction exhibits a nearly 100% success rate when the SNR is greater than 15 dB. Finally, we investigate recovery performance for different numbers of nodes while keeping the common sparsity, innovative sparsity, and SNR unchanged. In Fig. 4(d), the recovery success rate of signal 1 is depicted as the number of node changes. It is evident that from 1 to 7 nodes, the joint reconstruction consistently demonstrates a higher recovery success rate in comparison to separate reconstruction. Note that joint reconstruction achieves the best results when the number of nodes is equal to 4. It indicates that the number of nodes for joint recovery needs to be optimized, since too many nodes will cause the total sparsity of the signal set to be too high and place higher demands on the joint recovery algorithm. In conclusion, joint reconstruction significantly outperforms separate reconstruction in terms of recovery success rate, confirming the advantages of the proposed approach.
Fig. 4. Simulation results. Recovery success rate with different compression ratios for $J = 2$, ${K_j} = 1$, SNR = 20 dB, (a) ${K_c} = 1$, (b) ${K_c} = 3$. (c) Recovery success rate with different SNRs for $J = 2$, ${K_c} = 3$, ${K_j} = 1$, ${R_{DCS}} = 10$. (d) Recovery success rate of signal 1 with different numbers of nodes for ${K_c} = 3$, ${K_j} = 1$, SNR = 20 dB.
The PDCS scheme utilizes long-distance optical fibers for remote signal transmission. As the transmission distance and the frequency of the signal to be tested increase, it is necessary to further consider the impact of power fading caused by dispersion. Furthermore, optical link parameters for different nodes should be measured in advance to avoid distortion of the joint measurement matrix in practical applications. It is worth noting that the measurement matrix in this scheme is non-zero mean, and the performance of the PDCS system can be further improved by using a zero-mean measurement matrix based on a dual-output modulator and a balanced detection structure [9].
4. Conclusion
In conclusion, we have proposed a PDCS system that enables spectrum identification of long-distance multi-node wideband sparse signals. In the semi-physical simulation experiment, joint recovery of wideband sparse signals from two remote nodes located at 10 km and 20 km are realized, using a compression ratio of 8 and 4 for a mixed support-set sparsity of 2 and 4, respectively. Experimental and simulation results demonstrate that PDCS offers significant advantages over single-node photonic CS, particularly when the inter-node signals satisfy the mixed support-set model and the common part is large. This scheme holds great potential for applications in long-distance, multi-node, and large-coverage space electromagnetic spectrum identification.
Funding
National Natural Science Foundation of China (62101168, 62001148, 62375071); Project of Key Laboratory of Radar Imaging and Microwave Photonics (Nanjing University of Aeronautics and Astronautics), Ministry of Education (NJ20220006).
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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