Photonic-assisted multi-channel compressive sampling based on effective time delay pattern Yunhua Liang, Minghua Chen,* Hongwei Chen, Cheng Lei, Pengxiao Li, and Shizhong Xie Tsinghua National Laboratory for Information Science and Technology (TNList), Department of Electronic Engineering, Tsinghua University, Beijing, 100084, China * [email protected]
Abstract: In this paper, a photonic-assisted multi-channel compressive sampling scheme is proposed with one pseudo-random binary sequence (PRBS) source and Wavelength Division Multiplexing-based time delay. Meanwhile, the restricted isometry property of sensing matrix determined by the optimized time delay pattern is analyzed. In experiment, a fourchannel photonic-assisted system with 5-GHz bandwidth was set up, where four-channel PRBS signals were generated by adding fiber-induced constant time delays to four-wavelength modulated PRBS signal, and a signal composed of twenty tones was recovered faithfully with four analogto-digital converters (ADCs) with only 120-MHz-bandwidth. ©2013 Optical Society of America OCIS codes: (060.2360) Fiber optics links and subsystems; (230.0250) Optoelectronics.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006). R. Baraniuk, “Compressive sensing,” IEEE Signal Process. Mag. 24(4), 118–121 (2007). W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93(12), 121105 (2008). R. Horisaki, X. Xiao, J. Tanida, and B. Javidi, “Feasibility study for compressive multi-dimensional integral imaging,” Opt. Express 21(4), 4263–4279 (2013). J. N. Laska, S. Kirolos, M. F. Duarte, T. Ragheb, R. G. Baraniuk, and Y. Massoud, “Theory and implementation of an analog-to-information conversion using random demodulation,” IEEE Int. Symp. Circuits and Systems (ISCAS), 1959–1962 (2007). M. Mishali and Y. C. Eldar, “From theory to practice: Sub-nyquist sampling of sparse wideband analog signals,” J. Sel. Top. Signal Process. 4(2), 375–391 (2010). J. M. Nichols and F. Bucholtz, “Beating nyquist with light: a compressively sampled photonic link,” Opt. Express 19(8), 7339–7348 (2011). L. Yan, Y. Dai, K. Xu, J. Wu, Y. Li, Y. Ji, and J. Lin, “Integrated Multi-frequency Recognition and Downconversion based on Photonics-assisted Compressive Sampling,” IEEE Photon. J. 4(3), 664–670 (2012). H. Chi, Y. Mei, Y. Chen, D. Wang, S. Zheng, X. Jin, and X. Zhang, “Microwave spectral analysis based on photonic compressive sampling with random demodulation,” Opt. Lett. 37(22), 4636–4638 (2012). P. Timothy, P. McKenna, M. D. Sharp, D. G. Lucarelli, J. A. Nanzer, M. L. Dennis, and T. R. Clark, Jr., “Wideband Photonic Compressive Sampling Analog-to-Digital Converter for RF Spectrum Estimation,” in Proceedings of Optical Fiber Communication Conference, (Anaheim, Calif., 2013), paper OTh3D.1. G. C. Valley, G. A. Sefler, and T. J. Shaw, “Compressive sensing of sparse radio frequency signals using optical mixing,” Opt. Lett. 37(22), 4675–4677 (2012). H. Nan, Y. Gu, and H. Zhang, “Optical analog-to-digital conversion system based on compressive sampling,” IEEE Photon. Technol. Lett. 23(2), 67–69 (2011). Y. Liang, M. Chen, R. Li, H. Chen, and S. Xie, “A photonic-assisted compressive sampling system,” in Proceedings of IEEE Microwave photonics Conference, (Netherlands, 2012), pp. 184–187. E. J. Candès, “The restricted isometry property and its implications for compressed sensing,” C. R. Acad. Sci. Int. 346(9), 717–772 (2008). J. Tropp and A. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53(12), 4655–4666 (2007). M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems,” IEEE J. Sel. Top. Signal Process 1(4), 586– 597 (2007).
#195794 - $15.00 USD Received 15 Aug 2013; revised 4 Oct 2013; accepted 7 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025700 | OPTICS EXPRESS 25700
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-wavelengthspace 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 RF signal PRBS
M-wavelength CW
MZM
Generator of multiple PRBS signals
t
t
t
t Delay
MZM
PD, LPF and Digitizer
Ch1 DSP ChM
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).
#195794 - $15.00 USD Received 15 Aug 2013; revised 4 Oct 2013; accepted 7 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025700 | OPTICS EXPRESS 25701
Fig. 2. The spectrums of modulated PRBS signals, RF signal and low-pass filtered mixed signal.
PRBS signal, and their spectra are shown in Figs. 2(a1) and 2(a2) where fp 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 fp-width slice of mixed signal’s spectrum is the linear combination of fp-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 fp-shifted copies of RF spectrum X(f), as shown in Figs. 2(b1) and 2(b2) where c1,i and cM,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 fLPF bandwidth which depends much on the sparsity of received spectrum and is selected commonly to be times of fp 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. 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
Vi (t ) = [1+π m(t − Δτ i ) / V pi1 ][1+π x (t ) / V pi 2 ]
(1)
#195794 - $15.00 USD Received 15 Aug 2013; revised 4 Oct 2013; accepted 7 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025700 | OPTICS EXPRESS 25702
where m(t ) is the waveform of electrical PRBS signal with two levels of + 1,-1, Δτ i is the time delay added to the ith modulated PRBS signal, x (t ) is the waveform of input RF signal and V pi1 and V pi 2 are the half wave voltage of two MZMs. Assuming that the length and repetition frequency of PRBS are the odd integer N and fp respectively, the normalized spectrum of the ith PD’s output could be derived from Eq. (1), which is written as Vi ( f ) = 1 + π / V pi 2 X ( f ) + M i ( f ) + π / V pi 2
where X ( f ) is the spectrum and the kth respectively. Because frequency of the LPF can be written as,
N′
ae
k =− N ′
j 2π kf p Δτ i
k
X ( f − kf P )
(2)
j 2π kf Δτ
p i RF spectrum, N ′ = 0.5( N − 1) , and M i ( f ) and ak e are the Fourier coefficient of the ith delayed PRBS signal π m(t − Δτ i ) / V pi1 the lowest frequency of RF spectrum is higher than the upper cutoff 0.5Lf P , where L is an odd integer, the output spectrum of the ith LPF
Vi ( f ) = 1 + M i ( f ) + π / V pi 2
N′
ae
j 2π kf p Δτ i
k
k =− N ′
X ( f − kf P ), f ∈ [ −0.5Lf P ,0.5Lf P ] (3)
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 Yi ( f ) , Yi ( f ) = π / V pi 2
N′
ae
k =− N ′
j 2π kf p Δτ i
k
X ( f − jf P )
(4)
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 ′
Y ( f − L f ) π Y(f) = V Y ( f + L ′f ) i
p
i
pi 2
i
p
Yi
′ ′ − j 2 π ( N − L ) f Δτ
′ ′ j 2 π ( N − L ) f Δτ
a e a e X ( f − N ′f ) a e X ( f ) (5) a e a ae a e a e X ( f + N ′f ) a e p
i
p
′ ′ −( N −L )
′ − j 2 π N f Δτ p
−N
i
′ ′ N +L
- j 2 π f Δτ
i
p
′
′ ′ − j 2 π ( N + L ) f Δτ p
j 2 π f Δτ
i
p
0
−1
p
′ j 2 π N f Δτ
i
1
p
N
i
′
′ ′ j 2 π ( N + L ) f Δτ
i
p
′ ′ −(N +L )
i
p
′ ′ ( N −L )
X
Ai
where each element of measurement vector Yi is a fp-width sub spectrum, the RF spectrum vector X is constructed by N slices of fp-width spectrum., and Ai 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 Yi (i = 1...M) and X meets the equations Y = AX unfolded as Y A X ( f − N ′f ) Y = A X( f ) YM AM X ( f + N ′f ) 1
1
i
i
Y
A
p
(6)
p
X
#195794 - $15.00 USD Received 15 Aug 2013; revised 4 Oct 2013; accepted 7 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025700 | OPTICS EXPRESS 25703
where A is L*M × N sensing matrix constructed by the sub sensing matrices Ai 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 fp-width Δf resolution discrete spectrum of Yi ( f ) for i = 1…M obtained by fast Fourier transform (FFT), each element of Y is actually a 1 × N ′ vector, N ′ = f p / Δf , 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 fp-width Δf -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 fp-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 ) ≤
2
AX X
2 2
≤ (1 + δ K )
(7)
2
and δ K ∈ (0, 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 ak and Δτ 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-anddifferent PRBS for each PRBS signal and the proposed method using the optimized nonuniform 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/fp and c2(i-1)/fp 2 AX 2 for different values of where c1 and c2 are the constant 0.031 and 0.25. The values of 2 X 2 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 ∈ (0, 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.
#195794 - $15.00 USD Received 15 Aug 2013; revised 4 Oct 2013; accepted 7 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025700 | OPTICS EXPRESS 25704
2
AX X
2 2 2
Fig. 3. For random vectors X, the values of
2
AX X
2 2
achieved with: (a) Multiple individual-and-
2
different PRBSs; (b) PRBS signals of nonuniform time delays; (c) PRBS signals of uniform time delays.
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, which PRBS
λ1 : m(t − Δτ1 )
t
λ4 : m(t − Δτ 4 )
t
MZM
4-wavelength DFB laser array Ch1 ADC
MZM
PD
Amplifier
Ch1
X=A-1Y DSP
RF signal
Fig. 4. Experimental setup (DFB: Distributed-feedback laser; MUX: Multiplexer; DeMUX: Demultiplexer; PD: Photodetector;; DSP: Digital signal processor)
#195794 - $15.00 USD Received 15 Aug 2013; revised 4 Oct 2013; accepted 7 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025700 | OPTICS EXPRESS 25705
Fig. 5. The waveforms of four-channel PRBS signals.
was proposed in [16] and is one of the most popular algorithms for the recovery of a highdimensional sparse signal based on a small number of noisy linear measurements. 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. 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 the
#195794 - $15.00 USD Received 15 Aug 2013; revised 4 Oct 2013; accepted 7 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025700 | OPTICS EXPRESS 25706
Fig. 6. (a) Original RF spectrum; (b) four-channel compressed spectrum; (c) and (d) recovered spectrums for L = 3 and L = 5.
results 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.
#195794 - $15.00 USD Received 15 Aug 2013; revised 4 Oct 2013; accepted 7 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025700 | OPTICS EXPRESS 25707
Abstract: In this paper, a photonic-assisted multi-channel compressive sampling scheme is proposed with one pseudo-random binary sequence (PRBS) source and Wavelength Division Multiplexing-based time delay. Meanwhile, the restricted isometry property of sensing matrix determined by the optimized time delay pattern is analyzed. In experiment, a fourchannel photonic-assisted system with 5-GHz bandwidth was set up, where four-channel PRBS signals were generated by adding fiber-induced constant time delays to four-wavelength modulated PRBS signal, and a signal composed of twenty tones was recovered faithfully with four analogto-digital converters (ADCs) with only 120-MHz-bandwidth. ©2013 Optical Society of America OCIS codes: (060.2360) Fiber optics links and subsystems; (230.0250) Optoelectronics.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006). R. Baraniuk, “Compressive sensing,” IEEE Signal Process. Mag. 24(4), 118–121 (2007). W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93(12), 121105 (2008). R. Horisaki, X. Xiao, J. Tanida, and B. Javidi, “Feasibility study for compressive multi-dimensional integral imaging,” Opt. Express 21(4), 4263–4279 (2013). J. N. Laska, S. Kirolos, M. F. Duarte, T. Ragheb, R. G. Baraniuk, and Y. Massoud, “Theory and implementation of an analog-to-information conversion using random demodulation,” IEEE Int. Symp. Circuits and Systems (ISCAS), 1959–1962 (2007). M. Mishali and Y. C. Eldar, “From theory to practice: Sub-nyquist sampling of sparse wideband analog signals,” J. Sel. Top. Signal Process. 4(2), 375–391 (2010). J. M. Nichols and F. Bucholtz, “Beating nyquist with light: a compressively sampled photonic link,” Opt. Express 19(8), 7339–7348 (2011). L. Yan, Y. Dai, K. Xu, J. Wu, Y. Li, Y. Ji, and J. Lin, “Integrated Multi-frequency Recognition and Downconversion based on Photonics-assisted Compressive Sampling,” IEEE Photon. J. 4(3), 664–670 (2012). H. Chi, Y. Mei, Y. Chen, D. Wang, S. Zheng, X. Jin, and X. Zhang, “Microwave spectral analysis based on photonic compressive sampling with random demodulation,” Opt. Lett. 37(22), 4636–4638 (2012). P. Timothy, P. McKenna, M. D. Sharp, D. G. Lucarelli, J. A. Nanzer, M. L. Dennis, and T. R. Clark, Jr., “Wideband Photonic Compressive Sampling Analog-to-Digital Converter for RF Spectrum Estimation,” in Proceedings of Optical Fiber Communication Conference, (Anaheim, Calif., 2013), paper OTh3D.1. G. C. Valley, G. A. Sefler, and T. J. Shaw, “Compressive sensing of sparse radio frequency signals using optical mixing,” Opt. Lett. 37(22), 4675–4677 (2012). H. Nan, Y. Gu, and H. Zhang, “Optical analog-to-digital conversion system based on compressive sampling,” IEEE Photon. Technol. Lett. 23(2), 67–69 (2011). Y. Liang, M. Chen, R. Li, H. Chen, and S. Xie, “A photonic-assisted compressive sampling system,” in Proceedings of IEEE Microwave photonics Conference, (Netherlands, 2012), pp. 184–187. E. J. Candès, “The restricted isometry property and its implications for compressed sensing,” C. R. Acad. Sci. Int. 346(9), 717–772 (2008). J. Tropp and A. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53(12), 4655–4666 (2007). M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems,” IEEE J. Sel. Top. Signal Process 1(4), 586– 597 (2007).
#195794 - $15.00 USD Received 15 Aug 2013; revised 4 Oct 2013; accepted 7 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025700 | OPTICS EXPRESS 25700
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-wavelengthspace 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 RF signal PRBS
M-wavelength CW
MZM
Generator of multiple PRBS signals
t
t
t
t Delay
MZM
PD, LPF and Digitizer
Ch1 DSP ChM
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).
#195794 - $15.00 USD Received 15 Aug 2013; revised 4 Oct 2013; accepted 7 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025700 | OPTICS EXPRESS 25701
Fig. 2. The spectrums of modulated PRBS signals, RF signal and low-pass filtered mixed signal.
PRBS signal, and their spectra are shown in Figs. 2(a1) and 2(a2) where fp 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 fp-width slice of mixed signal’s spectrum is the linear combination of fp-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 fp-shifted copies of RF spectrum X(f), as shown in Figs. 2(b1) and 2(b2) where c1,i and cM,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 fLPF bandwidth which depends much on the sparsity of received spectrum and is selected commonly to be times of fp 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. 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
Vi (t ) = [1+π m(t − Δτ i ) / V pi1 ][1+π x (t ) / V pi 2 ]
(1)
#195794 - $15.00 USD Received 15 Aug 2013; revised 4 Oct 2013; accepted 7 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025700 | OPTICS EXPRESS 25702
where m(t ) is the waveform of electrical PRBS signal with two levels of + 1,-1, Δτ i is the time delay added to the ith modulated PRBS signal, x (t ) is the waveform of input RF signal and V pi1 and V pi 2 are the half wave voltage of two MZMs. Assuming that the length and repetition frequency of PRBS are the odd integer N and fp respectively, the normalized spectrum of the ith PD’s output could be derived from Eq. (1), which is written as Vi ( f ) = 1 + π / V pi 2 X ( f ) + M i ( f ) + π / V pi 2
where X ( f ) is the spectrum and the kth respectively. Because frequency of the LPF can be written as,
N′
ae
k =− N ′
j 2π kf p Δτ i
k
X ( f − kf P )
(2)
j 2π kf Δτ
p i RF spectrum, N ′ = 0.5( N − 1) , and M i ( f ) and ak e are the Fourier coefficient of the ith delayed PRBS signal π m(t − Δτ i ) / V pi1 the lowest frequency of RF spectrum is higher than the upper cutoff 0.5Lf P , where L is an odd integer, the output spectrum of the ith LPF
Vi ( f ) = 1 + M i ( f ) + π / V pi 2
N′
ae
j 2π kf p Δτ i
k
k =− N ′
X ( f − kf P ), f ∈ [ −0.5Lf P ,0.5Lf P ] (3)
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 Yi ( f ) , Yi ( f ) = π / V pi 2
N′
ae
k =− N ′
j 2π kf p Δτ i
k
X ( f − jf P )
(4)
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 ′
Y ( f − L f ) π Y(f) = V Y ( f + L ′f ) i
p
i
pi 2
i
p
Yi
′ ′ − j 2 π ( N − L ) f Δτ
′ ′ j 2 π ( N − L ) f Δτ
a e a e X ( f − N ′f ) a e X ( f ) (5) a e a ae a e a e X ( f + N ′f ) a e p
i
p
′ ′ −( N −L )
′ − j 2 π N f Δτ p
−N
i
′ ′ N +L
- j 2 π f Δτ
i
p
′
′ ′ − j 2 π ( N + L ) f Δτ p
j 2 π f Δτ
i
p
0
−1
p
′ j 2 π N f Δτ
i
1
p
N
i
′
′ ′ j 2 π ( N + L ) f Δτ
i
p
′ ′ −(N +L )
i
p
′ ′ ( N −L )
X
Ai
where each element of measurement vector Yi is a fp-width sub spectrum, the RF spectrum vector X is constructed by N slices of fp-width spectrum., and Ai 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 Yi (i = 1...M) and X meets the equations Y = AX unfolded as Y A X ( f − N ′f ) Y = A X( f ) YM AM X ( f + N ′f ) 1
1
i
i
Y
A
p
(6)
p
X
#195794 - $15.00 USD Received 15 Aug 2013; revised 4 Oct 2013; accepted 7 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025700 | OPTICS EXPRESS 25703
where A is L*M × N sensing matrix constructed by the sub sensing matrices Ai 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 fp-width Δf resolution discrete spectrum of Yi ( f ) for i = 1…M obtained by fast Fourier transform (FFT), each element of Y is actually a 1 × N ′ vector, N ′ = f p / Δf , 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 fp-width Δf -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 fp-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 ) ≤
2
AX X
2 2
≤ (1 + δ K )
(7)
2
and δ K ∈ (0, 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 ak and Δτ 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-anddifferent PRBS for each PRBS signal and the proposed method using the optimized nonuniform 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/fp and c2(i-1)/fp 2 AX 2 for different values of where c1 and c2 are the constant 0.031 and 0.25. The values of 2 X 2 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 ∈ (0, 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.
#195794 - $15.00 USD Received 15 Aug 2013; revised 4 Oct 2013; accepted 7 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025700 | OPTICS EXPRESS 25704
2
AX X
2 2 2
Fig. 3. For random vectors X, the values of
2
AX X
2 2
achieved with: (a) Multiple individual-and-
2
different PRBSs; (b) PRBS signals of nonuniform time delays; (c) PRBS signals of uniform time delays.
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, which PRBS
λ1 : m(t − Δτ1 )
t
λ4 : m(t − Δτ 4 )
t
MZM
4-wavelength DFB laser array Ch1 ADC
MZM
PD
Amplifier
Ch1
X=A-1Y DSP
RF signal
Fig. 4. Experimental setup (DFB: Distributed-feedback laser; MUX: Multiplexer; DeMUX: Demultiplexer; PD: Photodetector;; DSP: Digital signal processor)
#195794 - $15.00 USD Received 15 Aug 2013; revised 4 Oct 2013; accepted 7 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025700 | OPTICS EXPRESS 25705
Fig. 5. The waveforms of four-channel PRBS signals.
was proposed in [16] and is one of the most popular algorithms for the recovery of a highdimensional sparse signal based on a small number of noisy linear measurements. 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. 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 the
#195794 - $15.00 USD Received 15 Aug 2013; revised 4 Oct 2013; accepted 7 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025700 | OPTICS EXPRESS 25706
Fig. 6. (a) Original RF spectrum; (b) four-channel compressed spectrum; (c) and (d) recovered spectrums for L = 3 and L = 5.
results 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.
#195794 - $15.00 USD Received 15 Aug 2013; revised 4 Oct 2013; accepted 7 Oct 2013; published 21 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.025700 | OPTICS EXPRESS 25707
