Photonic-assisted multi-channel compressive sampling based on effective time delay pattern

Yunhua Liang; Minghua Chen; Hongwei Chen; Cheng Lei; Pengxiao Li; Shizhong Xie

doi:10.1364/OE.21.025700

1. Introduction

Compressive sampling (CS) techniques show great advantages of capturing sparse signals with sub-Nyquist sampling rate [1, 2], and have been applied in many fields such as imaging, spectrum sensing and analog signal acquisition [3–6]. Recently, with the development on broadband optical modulator and stable light sources, photonic techniques have been introduced into CS framework to achieve large instantaneous bandwidth [7–10]. Another photonic compressive sensing system based on spatial light modulation and time-wavelength-space mapping has also been presented to measure sparse radio frequency signal [11]. In these systems, several gigahertz processing bandwidth has been demonstrated by employing single analog-to-digital converter (ADC) with hundreds of MHz bandwidth to digitize the compressed signal. All the above photonic systems are single-channel systems with a single low-speed electronic ADC, while their electronic counterparts are always multi-channel systems employing low-speed ADC array with much lower sampling rate. Hao et al have proposed a photonic CS system with multichannel structure and analyzed its performance by simulation [12]. But in their system, it was required to generate effective high-speed multiple pseudo-random binary sequence (PRBS) signals, which was very complex and no further experimental results have been reported on. In 2012, we have reported a multi-channel photonic CS system with a single PRBS source [13]. However, the uniform time delay employed in the system will degrade the probability of signal recovery.

In this paper, a photonic-assisted multi-channel wideband compressive sampling scheme is proposed and experimentally demonstrated. In this scheme, cost-effective generation of multiple high-speed PRBS signals is achieved by introducing appropriate fiber-induced time delays to the multi-wavelength PRBS-modulated signal, where multi-channel PRBS signals can share one PRBS source and a single modulator. Meanwhile, the restricted isometry property (RIP) of the sensing matrix which guarantees high-probability signal recovery is analyzed. It is found that the proposed approach requires just a bit higher sampling rate to make the sensing matrix satisfy the RIP than that of the conventional method where individual-and-different PRBS is used in each channel, but the complexity of our system is much lower. A four-channel system with 5-GHz bandwidth is set up to testify the effectiveness of this scheme, where four ADCs of 120-MHz bandwidth are used, and a twenty-tone signal is recovered faithfully.

2. Principle

The schematic of proposed photonic-assisted multi-channel compressive sampling system is shown in Fig. 1. Firstly, multiple PRBS signals are generated by modulating PRBS signal on M-wavelength continuous wave (CW) light and then introducing different time delay to each

Fig. 1 The diagram of proposed CS system (CW: continuous wave; MZM: Mach-Zehnder modulator; PD: photodetector; LPF: low-pass filter; DSP: Digital signal processor).

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PRBS signal, and their spectra are shown in Figs. 2(a1) and 2(a2) where f_p is the repetition frequency of PRBS signal. After that, a Mach-Zehnder modulator (MZM) is used to mix RF signal with M-channel PRBS signals, aiming to realize the spectral convolution of RF signal with multi-wavelength PRBS signals. In every wavelength channel, each f_p-width slice of mixed signal’s spectrum is the linear combination of f_p-shifted copies of RF spectrum where the coefficients of combination are determined by the Fourier coefficients of the delayed PRBS signal. When the PRBS signals with different time delays are used in multi-channel signal mixing, each mixed signal’s spectrum is different combination of f_p-shifted copies of RF spectrum X(f), as shown in Figs. 2(b1) and 2(b2) where c_1,_i and c_M_,_i are the ith coefficients of the first and Mth PRBS signals, so each mixed signal contains partial-but-different information of RF signal spectrum to signal recovery. After signal mixing, a photodetector (PD) and a low-pass filter (LPF) with f_LPF bandwidth which depends much on the sparsity of received spectrum and is selected commonly to be times of f_p are employed in each channel to obtain the low-frequency component of electrical mixed signals. An ADC is used to digitize the filtered mixed signal. Finally, the original RF spectrum is reconstructed correctly with signal recovery algorithms [14,15] in a digital signal processor, in condition of that the set of multi-channel mixed signal spectrum supplies enough content of RF spectrum.

Fig. 2 The spectrums of modulated PRBS signals, RF signal and low-pass filtered mixed signal.

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In the following, the signal model and the sensing matrix are derived, and then the RIP of sensing matrix corresponding to optimized time delay patterns is analyzed.

2.1. Signal model and sensing matrix

For MZMs biased at quadrature, the normalized PD output voltage of the ith-channel mixed signal is approximately written as

V_{i} (t) = [1 + π m (t - Δ τ_{i}) / V_{p i 1}] [1 + π x (t) / V_{p i 2}]

where

m (t)

is the waveform of electrical PRBS signal with two levels of + 1,-1,

Δ τ_{i}

is the time delay added to the ith modulated PRBS signal,

x (t)

is the waveform of input RF signal and

V_{p i 1}

and

V_{p i 2}

are the half wave voltage of two MZMs.

Assuming that the length and repetition frequency of PRBS are the odd integer N and f_p respectively, the normalized spectrum of the ith PD’s output could be derived from Eq. (1), which is written as

V_{i} (f) = 1 + π / V_{p i 2} X (f) + M_{i} (f) + π / V_{p i 2} \sum_{k = - N^{'}}^{N^{'}} a_{k} e^{j 2 π k f_{p} Δ τ_{i}} X (f - k f_{P})

where

X (f)

is the RF spectrum,

N^{'} = 0.5 (N - 1)

, and

M_{i} (f)

and

a_{k} e^{j 2 π k f_{p} Δ τ_{i}}

are the spectrum and the kth Fourier coefficient of the ith delayed PRBS signal

π m (t - Δ τ_{i}) / V_{p i 1}

respectively. Because the lowest frequency of RF spectrum is higher than the upper cutoff frequency of the LPF

0.5 L f_{P}

, where L is an odd integer, the output spectrum of the ith LPF can be written as,

V_{i} (f) = 1 + M_{i} (f) + π / V_{p i 2} \sum_{k = - N^{'}}^{N^{'}} a_{k} e^{j 2 π k f_{p} Δ τ_{i}} X (f - k f_{P}), f \in [- 0.5 L f_{P}, 0.5 L f_{P}]

The third term on the right side of Eq. (3) is exactly the component of convolved spectrum used as the measurement for recovery, denoted by

Y_{i} (f)

,

Y_{i} (f) = π / V_{p i 2} \sum_{k = - N^{'}}^{N^{'}} a_{k} e^{j 2 π k f_{p} Δ τ_{i}} X (f - j f_{P})

which is obtained by removing the DC component and the known spectrum

M_{i} (f)

from the LPF output through the digitization and post signal processing in DSP module. The Eq. (4) could be rewritten in matrix form, which is

\underset{Y_{i}}{\underset{︸}{[\begin{matrix} Y_{i} (f - L^{'} f_{p}) \\ ⋮ \\ Y_{i} (f) \\ ⋮ \\ Y_{i} (f + L^{'} f_{p}) \end{matrix}]}} = \frac{π}{V_{p i 2}} \underset{A_{i}}{\underset{︸}{[\begin{matrix} a_{- (N^{'} - L^{'})} e^{- j 2 π (N^{'} - L^{'}) f_{p} Δ τ_{i}} \\ ⋮ \\ a_{- N^{'}} e^{- j 2 π N^{'} f_{p} Δ τ_{i}} \\ ⋮ \\ a_{- (N^{'} + L^{'})} e^{- j 2 π (N^{'} + L^{'}) f_{p} Δ τ_{i}} \end{matrix} \begin{matrix} \dots & a_{N^{'} + L^{'}} e^{j 2 π (N^{'} - L^{'}) f_{p} Δ τ_{i}} \\ ⋱ & ⋰ & ⋮ \\ \dots & a_{- 1} e^{- j 2 π f_{p} Δ τ_{i}} & a_{0} & a_{1} e^{j 2 π f_{p} Δ τ_{i}} & \dots & a_{N^{'}} e^{j 2 π N^{'} f_{p} Δ τ_{i}} \\ ⋰ & ⋱ & ⋮ \\ \dots & a_{(N^{'} - L^{'})} e^{j 2 π (N^{'} + L^{'}) f_{p} Δ τ_{i}} \end{matrix}]}} \underset{X}{\underset{︸}{[\begin{matrix} X (f - N^{'} f_{p}) \\ ⋮ \\ X (f) \\ ⋮ \\ X (f + N^{'} f_{p}) \end{matrix}]}}

where each element of measurement vector Y_i is a f_p-width sub spectrum, the RF spectrum vector X is constructed by N slices of f_p-width spectrum., and A_i is a L × N sensing matrix in the ith channel. For explicit expression, let

L^{'} = 0.5 (L - 1)

in the Eq. (5).

Considering a M-channel CS system, the relationship between the measurement vector Y composed by the vectors Y_i (i = 1...M) and X meets the equations $Y = A X$ unfolded as

\underset{Y}{\underset{︸}{[\begin{matrix} Y_{1} \\ ⋮ \\ Y_{i} \\ ⋮ \\ Y_{M} \end{matrix}]}} = \underset{A}{\underset{︸}{[\begin{matrix} A_{1} \\ ⋮ \\ A_{i} \\ ⋮ \\ A_{M} \end{matrix}]}} \underset{X}{\underset{︸}{[\begin{matrix} X (f - N^{'} f_{p}) \\ ⋮ \\ X (f) \\ ⋮ \\ X (f + N^{'} f_{p}) \end{matrix}]}}

where A is L*M × N sensing matrix constructed by the sub sensing matrices A_i for i = 1…M. By inverting the Eq. (6), the N-length unknown spectrum vector X is finally recovered from the measurement vector Y. Since each element of the vector Y is a slice of f_p-width

Δ f

-resolution discrete spectrum of

Y_{i} (f)

for i = 1…M obtained by fast Fourier transform (FFT), each element of Y is actually a 1 ×

N^{'}

vector,

N^{'} = f_{p} / Δ f

, constructed by the spectrums at different-frequency positions. Consequently, it is known from Eq. (6) that each element of the recovered vector X is also a 1 ×

N^{'}

vector representing a f_p-width

Δ f

-resolution discrete spectrum of

x (t)

. Furthermore, the resolution of the recovered spectrum X could be improved by increasing the sampling time.

2.2. Condition of signal recovery

According to the CS theory [13], to achieve high recovery probability of unknown K-nonzero vector X where K refers to the number of f_p-width RF spectrum slices that overlap after signal mixing, the sensing matrix A should own the RIP of order K with the constant $δ_{K}$ satisfying

(1 - δ_{K}) \leq \frac{{‖ A X ‖}_{2}^{2}}{{‖ X ‖}_{2}^{2}} \leq (1 + δ_{K})

and

δ_{K} \in (0, \sqrt{2} - 1)

for all random K-sparse vectors X.

As shown in Eq. (5), the RIP of the matrix of our proposed photonic compressive sampling system is determined by a_k and $Δ τ_{i}$ , which are corresponding to the PRBSs and their relative time delays. In theory, choosing different PRBS signal for each channel is optimal to satisfy RIP, but this will make the photonic CS system complex. Therefore, in our proposed system, the optimized time delay is used to guarantee the RIP. Here $K = 6$ is taken as an example to illustrate that the sensing matrix used by the experimental four-channel CS system using a 127-length PRBS, which is a 4L × 127 matrix, satisfies the RIP. Meanwhile, the RIP constant $δ_{K}$ achieved by the conventional approach employing an individual-and-different PRBS for each PRBS signal and the proposed method using the optimized non-uniform and uniform time delay patterns are compared. In the latter two cases, the relative time delay between the ith and the first channel is proposed to be c₁(i-1)³/f_p and c₂(i-1)/f_p where c₁ and c₂ are the constant 0.031 and 0.25. The values of $\frac{{‖ A X ‖}_{2}^{2}}{{‖ X ‖}_{2}^{2}}$ for different values of 4L are measured with 1,000,000 random sparse vectors X input, and the results are shown in Fig. 3. It is observed in Fig. 3 that to guarantee the RIP constant $δ_{6} \in (0, \sqrt{2} - 1)$ , L required in three cases are 5, 6 and 6 respectively, which means that our proposed method requires additional 20% sampling rate to satisfy the RIP for K = 6. In addition, it is found that for other larger values of K, the RIP could also be satisfied by increasing L, but it needs to optimize time delay pattern further so that the value of L required can be reduced to certain extent and then the ADC’s sampling rate will drop down.

Fig. 3 For random vectors X, the values of $\frac{{‖ A X ‖}_{2}^{2}}{{‖ X ‖}_{2}^{2}}$ achieved with: (a) Multiple individual-and-different PRBSs; (b) PRBS signals of nonuniform time delays; (c) PRBS signals of uniform time delays.

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3. Experimental setup and results

An experimental setup of four-channel system with 5-GHz bandwidth is shown in Fig. 4. Four-wavelength Distributed-feedback (DFB) laser array is used as light sources with the wavelengths of 1546.12, 1547.72, 1550.12, and 1551.72 nm respectively. The 10.16-Gbps 127-length PRBS signal, generated by signal quality analyzer (Anritsu MP1800A), is modulated on the four optical waves simultaneously by a Mach-Zehnder modulator (MZM). Then, a wavelength demultiplexer (DeMUX) is used to separate four-wavelength PRBS signal, and different time delay is individually introduced to each wavelength by different length of fibers. After that, four delayed signals are combined together by the wavelength multiplexer (MUX). The relative time delay between the PRBS signal in 1547.12 nm channel and other wavelength channels are approximately 0.34, 2.5 and 8.41 ns, which is shown in Fig. 5. Another MZM is used to mix input RF signal with four-channel PRBS signals. The mixed signal is demultiplexed and then detected by a narrow-band photodetector (PD) in each channel. After passing through a low-noise amplifier and a low-pass filter, the low frequency components of the mixed signal are quantized by a four-channel Digital Phosphor Oscillator (Tektronix 72004B) which is used as an ADC array with 8 bits of quantization. Finally, original spectrum is estimated by the orthogonal matching pursuit (OMP) algorithm, whichwas proposed in [16] and is one of the most popular algorithms for the recovery of a high-dimensional sparse signal based on a small number of noisy linear measurements.

Fig. 4 Experimental setup (DFB: Distributed-feedback laser; MUX: Multiplexer; DeMUX: Demultiplexer; PD: Photodetector;; DSP: Digital signal processor)

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Fig. 5 The waveforms of four-channel PRBS signals.

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To test whether unknown sparse signal could be correctly recovered and observe the noise characteristic of recovered signal, a twenty-tone signal, generated by arbitrary waveform generator (Tektronix 7122B) and amplified by a wideband amplifier, is used as a test sample. The frequencies of tones are 251.02, 492.08, 733.18, 974.32, 1215.50, 1456.72, 1697.98, 1939.28, 2180.62, 2422.00, 2663.42, 2904.88, 3146.38, 3387.92, 3629.50, 3871.12, 4112.78, 4354.48, 4596.22, and 4838.00 MHz respectively. The original spectrum is shown in Fig. 6(a), measured by the spectrum analyzer (Anritsu MS2668C) with 10-kHz resolution.

Fig. 6 (a) Original RF spectrum; (b) four-channel compressed spectrum; (c) and (d) recovered spectrums for L = 3 and L = 5.

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The sampling time for acquiring 4-channel mixed signals is 100 us, and their spectrums with 10-kHz resolution achieved by FFT are shown in Fig. 6(b). It is observed Fig. 6(b) that the in-band signal-to-noise ratio (SNR) of mixed signal degrades to about 26 dB, which primarily results from the noise folding in the process of signal mixing. To compare the recovery performance achieved with different ADC bandwidth, two cases for 120-MHz and 200-MHz bandwidth are observed, which means that L is selected to be 3 and 5 respectively.By inverting the Eq. (6) with M = 4 and N = 127, which are corresponding to number of channels and the length of PRBS, the original spectrum with 10-kHz resolution is reconstructed and theresults are shown in Figs. 6(c) and 6(d). The 20-tone signal is recovered successfully in both cases, and their SNR will increase after signal recovery process by about 10.0 and 11.5 dB respectively in two cases, which are close to the theoretical limit of 10log(4L), i.e. 10.8 and 13.0dB. It is because that the noises which are folded in spectrum convolution in each channel are correlative strong and are mostly suppressed in the signal recover process.

4. Conclusion

A photonic-assisted multi-channel CS scheme is proposed, where multi-channel PRBS signals are generated by introducing time delay to multi-wavelength modulated PRBS signal. By using optimized time delay pattern, this system could achieve stable signal recovery. Twenty tones covering 5-GHz bandwidth are recovered in experiment by a four-channel CS system utilizing 120-MHz-bandwidth spectrum of mixed signals. Moreover, the SNR improvement from compressed signal to recovered signal is quite close to the theoretical limit. Benefiting from the structure, this scheme could be reconfigured flexibly to acquire different-sparsity signals with low-bandwidth ADCs.

Acknowledgments

This work is supported by National Program on Key Basic Research Project (973) under Contract 2012CB315703, NSFC under Contract 61271134 and 61120106001.

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Photonic-assisted multi-channel compressive sampling based on effective time delay pattern

Abstract

1. Introduction

2. Principle

2.1. Signal model and sensing matrix

2.2. Condition of signal recovery

3. Experimental setup and results

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Equations (7)

Optics Express