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Abstract
Photonic compressive sampling (PCS) is an effective method to recover wideband sparse radio frequency (RF) signals. However, the noisy and high-loss photonic link leads to signal-to-noise ratio (SNR) degradation of the RF signal to be tested, which limits the recovery performance of the PCS system. In this paper, a random demodulator-based PCS system with 1-bit quantization is proposed. The system consists of a photonic mixer, a low-pass filter, a 1-bit analog-to-digital converter (ADC), and a digital signal processor (DSP). The 1-bit quantized result is used to recover the spectra of the wideband sparse RF signal with the binary iterative hard thresholding (BIHT) algorithm, which can alleviate the negative impact of the SNR degradation caused by the photonic link. A full theoretical framework of the PCS system with 1-bit quantization is given. Simulation results show that the PCS system with 1-bit quantization can provide better recovery performance than the traditional PCS system under low SNR and stringent bit budget.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Wideband radio frequency (RF) signal acquisition based on analog-to-digital conversion (ADC) is a critical function in ultra-wideband applications such as wideband radar, electronic warfare, and ultra-wideband communications [1–4]. The recently introduced compressive sampling (CS) framework enables signal acquisition systems to take advantage of signal structures beyond bandlimitedness to recover signals at rates well below Nyquist rates [5–8]. Random demodulator and modulated wideband converter are two well-known CS models to realize wideband sparse RF signal acquisition [9,10]. The modulated wideband converter-based CS is preferred for capturing sparse multiband signals with good approximation [9] while the random demodulator-based CS aims to capture sparse multitone signals with simple structure [10]. The measurement process of random demodulator-based CS involves random mixing, integration, and down-sampling. In the random mixing stage, a bipolar pseudo-random binary sequence (PRBS) that alternates between +1 and –1 with a rate at or above the Nyquist rate is always required to mix with the sparse RF signal. Wideband random mixing becomes a challenge in the electrical domain. To overcome this problem, photonic CS (PCS) schemes have been proposed to realize the critical function of random mixing [11–15]. Among them, the most common method is to use two cascaded Mach-Zehnder modulators (MZMs) to modulate the sparse RF signal and PRBS, respectively, and finally realize random mixing by photodetection [11]. However, the noise figure (NF) of microwave photonic links used to achieve optical mixing is very high, which is attributed to the high loss and extra noise in the electrical-optical-electrical conversion process [16]. The SNR of the signal to be tested deteriorates, resulting in a decrease in the PCS system’s compression ratio or recovery accuracy [17]. To address this issue, the most direct method is to reduce the NF of microwave photonic links. Recent studies have shown that the NF of the photonic link can be optimized to below 10 dB [18–21], usually by increasing the link gain with low nonlinearity fibers, high-power lasers, and detectors or decreasing the noise with differential detection or low-biasing. However, it will increase the cost and complexity of the system.
In this letter, 1-bit quantization is introduced into the PCS system to alleviate the influence of SNR degradation caused by photonic mixing. We construct the PCS system with 1-bit quantization (abbreviated as 1-bit PCS) based on the random demodulator model, in which the measurement process consists of photonic random mixing, integration, and a 1-bit ADC, i.e., a comparator. In particular, the sampled signal is quantized to one bit by the 1-bit ADC, and only the symbol is recorded. Compared with the traditional CS, the negative impact of the SNR degradation introduced by the photonic link is reduced in 1-bit PCS since the noise generally will not cause the symbol flip in the 1-bit quantization. In the recovery process, the binary iterative hard threshold (BIHT) algorithm is used to recover the sparse RF signal. The theoretical model of the random demodulator-based 1-bit PCS system is given, especially considering the gain and noise of the photonic link. Simulation results show that the 1-bit PCS system can offer a better recovery performance than the traditional PCS system under low SNR and stringent bit budget. Besides, 1-bit PCS can replace the quantizer with a 1-bit ADC (comparator), reducing the system’s power consumption.
2. Principle
Figure 1 shows the schematic of the proposed random demodulator-based 1-bit PCS system. The scheme consists of four main parts: a photonic mixer, a low-pass filter (LPF), a 1-bit ADC, and a digital signal processor (DSP). The photonic mixer consists of a laser, a MZM, a dual-output MZM (DO-MZM), and a balanced photodetector (BPD). Firstly, the RF signal to be tested is multiplied by a PRBS through the photonic mixer, where the rate of the PRBS is equal to or higher than the Nyquist rate. The purpose of random mixing with the photonic mixer is to broaden the signal’s spectrum to the entire bandwidth. Then, the LPF filters out the low-frequency part of the mixed signal. Finally, the filtered signal is digitized by the 1-bit ADC with a sub-Nyquist sampling rate. The functions of photonic mixer, LPF, and 1-bit quantization can be understood as the measurement process in 1-bit PCS. Finally, the spectra of the sparse RF signal can be recovered based on the DSP and a CS reconstruction algorithm.
Fig. 1. Schematic of the random demodulator-based 1-bit PCS, MZM: Mach-Zehnder modulator, DO-MZM: dual-output Mach-Zehnder modulators, BPD: balanced photodetector, LPF: low-pass filter, ADC: analog-to-digital converter, DSP: digital signal processor.
In theory, the measurement process of CS can be expressed as
where x is a $N \times \textrm{1}$ vector of Nyquist-rate samples of the wideband sparse RF signal $x(t)$, since x is sparse in the frequency domain, it can be expressed as $x = W\theta$, where W is an $N \times N$ matrix denoting the Fourier orthogonal basis, $\theta $ is an $N \times 1$ vector indicating the sparse spectrum information of x. $\Phi $ is the measurement matrix with dimension $M \times N\,(M < N)$, ${Q_B}$ is a $B$-bit scalar quantization function that maps real-valued CS measurements to the discrete alphabet $\Im$ with $|\Im |= {2^B}$, B is the number of quantization bits, and ${y_B}$ is the measured vector of $M \times \textrm{1}$ with $B$-bit quantization.If the matrix product $\Phi W$ satisfies the restricted isometry property [5], it is possible to recovery $\theta$ from the measured y using a CS algorithm. The challenge is to design hardware that can perform the function of the measurement matrix $\Phi $. In this case, the matrix $\Phi $ is constructed based on the random demodulator model [10,11]. The measurement matrix $\Phi $ in a random demodulator is composed of three parts, which can be expressed as
where R is an $N \times N$ diagonal matrix denoting the random mixing implemented by the photonic mixer with the PRBS $r(t)$, its elements depend on the model of the photonic link and will be described in detail later. H is an $N \times N$ matrix denoting the impulse response of the LPF. The length of the finite impulse response of the LPF is typically chosen to be $N/M$, and the impulse response can be calculated from the measured transfer function (S21) of the LPF. D is an $M \times N$ matrix denoting the down-sampling of the ADC with its elements given by ${D_i}_j = \delta (i - j/{R_{CS}}),\,i = 1\ldots M,\,j = 1\ldots N$, which means the ADC’s sampling rate is reduced to $1/{R_{CS}}$ of the Nyquist rate, ${R_{CS}} = N/M$ is the compression ratio of the sampling rate.To obtain the matrix R, the theoretical model of photonic random mixing is deduced as following. The detected signal after the BPD contains the mixing result of the RF signal and PRBS, which can be expressed as
As a practical matter, the first term $\Omega \rho ({P_{in}}/2){L_T}\sin ({\psi _{PRBS}})$ in (3) can be measured in advance and subtracted from the mixed signal ${V_o}$. In most cases, the RF signal is small enough so that we can make the approximation $\sin ({\psi _{sig}}) \approx {\psi _{sig}}$. Additionally, set the voltage of the PRBS switching between ${V_{\pi 2}}/2$ and $- {V_{\pi 2}}/2$, corresponding to the logic 1 and 0 of the PRBS, respectively. The optical phase shift induced by the PRBS is ${\psi _{PRBS}} ={\pm} \pi /2$. Thus, the random mixed result can be expressed as
By substituting (2) and (4) into (1), the measurement process of random demodulator-based PCS can be expressed as
The random mixing of RF signal and PRBS in the optical domain breaks through the electronic bottleneck and realizes the advantages of large bandwidth, low cost, and small volume. However, we can see from (5) that the signal to be actually tested changes from x to ${x^ \ast } = x\textrm{ + }{n_{ex}}/(g \cdot r)$. Obviously, the SNR of the detected signal ${x^ \ast }$ is smaller than that of input RF signal x due to the additional noise ${n_{ex}}$ and small voltage gain $g$ (usually <1). The SNR degradation of the signal will deteriorate the recovery performance of traditional CS [17].
To ease this problem, 1-bit quantization is employed, where B is 1 and only the sign symbol of measured results is recorded as
We can use the BIHT algorithm proposed in [22] to impose an energy constraint ${||{{\theta^ \ast }} ||_2} = 1$ on ${\theta ^ \ast}$ that the amplitude information is lost, search for and reconstruct the sparse spectra $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \theta} $ that is consistent with the measured value ${y_1}$. It is demonstrated that the BIHT performs better than the previous algorithms and is robust to sign flips. The BIHT algorithm can be thought of trying to solve the problemCompared to traditional PCS, 1-bit PCS only takes the symbol of the measured value. The measured value that changes within a specific range will not affect the quantization results so that the impact of SNR degradation can be alleviated.
3. Simulation and discussion
Numerical simulation is implemented to verify the effectiveness of the proposed 1-bit PCS. In the simulation, RF signals with a spectral sparsity of 5, a power of 0.5 mW, and a bandwidth of 10 GHz are tested. The bit rate of the PRBS is set as 20 Gb/s. The diagonal matrix R denoting the random mixing is constructed according to the parameters of the microwave photonic link listed in Table 1. The matrix H is constructed based on an ideal integrator that sums ${R_{CS}}$ discrete values of the mixed signal. The down-sampling matrix D is an $M \times N$ matrix with its elements given by ${D_i}_j = \delta (i - j/{R_{CS}}),\,i = 1\ldots M,\,j = 1\ldots N$. In the recovery process, 1-bit PCS uses BIHT algorithm [22], and the traditional PCS with other quantization bits uses the basis pursuit denoising (BPDN) algorithm which achieves high performance under low SNR [23].
Table 1. Parameters of the microwave photonic link for numerical simulation
Firstly, we measured the SNR of the detected signal ${x^ \ast}$. The power of the extra noise introduced by the microwave photonic link is calculated to be –63.4 dBm. When the SNR of the input RF signals x is set to 5, 10, 15, and 20 dB, respectively, the SNR of the detected signals ${x^ \ast}$ is calculated to be 1.9, 5.1, 7.5, 9.3 dB, respectively. Obviously, the SNR of the signal to be tested is deteriorated by the photonic link. Figure 2 shows the recovered spectra with the proposed 1-bit PCS. In this simulation, the length of signal is $N = 1000$ and the compression ratio is ${R_{CS}} = 2$. It can be seen that the position and the number of tones are accurately recovered, except for amplitude errors. The reconstruction results of the four groups of input RF signals have little difference, mainly attributed to the strong anti-noise performance of 1-bit quantization. Furthermore, the amplitude errors of the lower frequency components are smaller than that of the higher frequency components. The reason may be that the lower frequency components obtain more sampling points in one period under the same compression ratio.
Fig. 2. The original and recovered spectra of 1-bit PCS for four groups of input RF signals. The SNR of the detected signal is (a) 1.9 dB, (b) 5.1 dB, (c) 7.5 dB, and (d) 9.3 dB.
Secondly, we compared the performance of 1-bit PCS and traditional PCS with 8-bit quantization under the same bit budget. The bit budget is defined as $\Re = N/{R_{CS}} \times B$, that is, under the same bit budget, the compression ratio of 8-bit quantization is 8 times that of 1-bit quantization. The two detected RF signals with SNRs of 9.3 dB and 1.9 dB are used for recovery, and the bit budget is set $\Re = N/2$. Here, we use the reconstructed signal-to-noise ratio (RSNR) to characterize the recovery performance of the system [22]. The RSNR is expressed as $RSNR = 10\lg ({||x ||_2^2/||{x - \hat{x}} ||_2^2} )$. Each group of input RF signals with different bit budgets is recovered 100 times, and the RSNR is calculated and averaged. Figure 3 shows the RSNR for 1-bit PCS and traditional PCS under different bit budgets. When the bit budget is low, the performance of 1-bit PCS is significantly better than that of traditional PCS with 8-bit quantization. The 1-bit PCS has good performance even when the bit budget is small. The influence of SNR on 1-bit PCS is significantly weaker than that of traditional PCS. The results show that 1-bit PCS is suitable for the case of small SNR and stringent bit budget.
Fig. 3. The recovery performance of 1-bit PCS and traditional PCS with 8-bit quantization under different bit budgets
Furthermore, we analyze the recovery performance of PCS under different numbers of quantization bits. Set the parameters $N = 1000$, and $\Re = 500$. The number of quantization bits B is changed from 1 to 8. It can be seen from Fig. 4 that when SNR ≤ 7.5, the recovery performance of 1-bit quantization is significantly better than other quantization bits. In the case of 1-bit quantization, the RSNR does not increase with the increase of the input SNR, which is different from other quantization bits. This is because 1-bit PCS is insensitive to the input SNR changing within a certain range as long as 1-bit quantization preserves the sign of the measured value. As the input SNR increases, the recovery performance of 1-bit quantization may be surpassed by that of other quantization bits.
Finally, we evaluated the success rate of reconstruction at different numbers of quantization bits. It is considered that the signal is recovered successfully if RSNR > 0, and the success rate is calculated by counting the 100 times simulation results. Figure 5 shows the results. When the number of quantization bits is larger than 3, the success rate decreases with the SNR. When the number of quantization bits is smaller than 3, the success rate almost keeps at 100%. The results show that the PCS system with smaller quantization bits has a higher success rate and stability. As an extreme case, 1-bit quantization can use comparators instead of quantizers to improve quantization speed and reduce the system’s power consumption.
For a signal with a length of 1000 and a bit budget of 500, the computational time of the 1-bit PCS system and the traditional PCS system on a computer with a 2.3 GHz processor is about 0.5 s and 0.2 s, respectively. To meet the requirement of real-time applications, highly efficient and robust algorithms for 1-bit PCS systems are still desired. Note that a small-signal model in line with most practical situations is used to evaluate the performance of 1-bit PCS. If the power of the signal to be tested is relatively large, it will cause nonlinearity in the modulation process. Moreover, it is impossible to realize a PRBS signal with an ideal rectangle waveform as well as eliminate completely the relative intensity noise in a real system, which will also lead to the deterioration of the performance of PCS. However, compared with traditional PCS, the nonlinearity and non-ideal factors will have a less negative impact on 1-bit PCS, since 1-bit quantization is insensitive to small amplitude fluctuations.
4. Conclusion
We propose a random demodulator-based 1-bit PCS system. While maintaining the advantages of large bandwidth and anti-electromagnetic interference of the photonic link, the scheme uses 1-bit quantization to alleviate the negative impact of the SNR degradation caused by the noisy and high-loss photonic link. The 1-bit PCS features large bandwidth, anti-noise, and low power consumption, which is suitable for wideband sparse RF signal acquisition under low SNR and stringent bit budget.
Funding
National Natural Science Foundation of China (62101168, 61975048, 61901148); 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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