Beating Nyquist with light: a compressively sampled photonic link

Transcription

1 Beating Nyquist with light: a compressively sampled photonic link J. M. Nichols* and F. Bucholtz Naval Research Laboratory, Washington, DC 2375, USA *jonathan.nichols@nrl.navy.mil Abstract: We report the successful demonstration of a compressively sampled photonic link. The system takes advantage of recent theoretical developments in compressive sampling to enable signal recovery beyond the Nyquist limit of the digitizer. This rather remarkable result requires that (1) the signal being recovered has a sparse (low-dimensional) representation and (2) the digitized samples be incoherent with this representation. We describe an all-photonic system architecture that meets these requirements and then show that 1GHz harmonic signals can be faithfully reconstructed even when digitizing at 5MS/s, well below the Nyquist rate. 211 Optical Society of America OCIS codes: (6.236) Fiber optics links and subsystems; (.387) Mathematics. References and links 1. J. Campmany and D. Novak, Microwave photonics combines two worlds, Nat. Photonics 1(6), (27). 2. C. H. Lee, Microwave Photonics (CRC Press, 27). 3. J. A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, and R. G. Baraniuk, Beyond Nyquist: Efficient Sampling of Sparse Bandlimited Signals, IEEE Trans. Inf. Theory 56(1), (21). 4. M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, Single-pixel Imaging via Compressive Sampling, IEEE Signal Process. Mag. 25(2), (28). 5. 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, (28). 6. S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, Super-resolution and reconstruction of sparse sub-wavelength images, Opt. Express 17(26), (29). 7. O. Katz, Y. Bromberg, and Y. Silberburg, Compressive ghost imaging, Appl. Phys. Lett. 95, (29). 8. D. Yang, H. Li, G. Peterson, and A. Fathy, Compressed Sensing Based UWB Receiver: Hardware Compressing and FPGA Reconstruction, Proceedings of the 43rd Conference on Information Sciences and Systems (CISS) (29). 9. M. Mishali and Y. C. Eldar, From Theory to Practice: Sub-Nyquist Sampling of Sparse Wideband Analog Signals, IEEE J. Sel. Top. Signal Process. 4(2), (21). 1. M. Mishali and Y. C. Eldar, Xampling: Analog Data Compression, vol. of Proceedings of the 21 Data Compression Conference, pp (21). 11. E. J. Candes and T. Tao, Decoding by Linear Programming, IEEE Trans. Inf. Theory 51(12), (25). 12. E. J. Candes, J. Romberg, and T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inf. Theory 52(2), (26). 13. D. L. Donoho, Compressed Sensing, IEEE Trans. Inf. Theory 52(4), (26). 14. E. J. Candes and M. B. Wakin, An Introduction to Compressive Sampling, IEEE Signal Process. Mag. 25(2), 21 3 (28). 15. J. Romberg, Imaging Via Compressive Sampling, IEEE Signal Process. Mag. 25(2), 14 2 (28). 16. R. G. Baraniuk, Compressive Sensing, IEEE Signal Process. Mag. 24, (27). 17. J. M. Nichols, M. Currie, F. Buholtz, and W. A. Link, Bayesian Estimation of Weak Material Dispersion: Theory and Experiment, Opt. Express 18(3), (21). 18. 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), (27). (C) 211 OSA 11 April 211 / Vol. 19, No. 8 / OPTICS EXPRESS 7339

2 1. Introduction Photonic links have demonstrated great utility for the transport of analog signals [1, 2]. As is well known, photonic links provide extremely large instantaneous signal bandwidth and a mature technology base has been developed to provide stable laser sources, high-speed optical modulators, and large-bandwidth photodetectors. Currently, modulators and photodetectors that operate at RF frequencies in excess of 5 GHz can be obtained commercially. However, after photodetection, it is not uncommon to digitize the signal either directly or after downconversion, a process that often diminishes the large instantaneous bandwidth advantage of the optical system. Conventional digitization occurs at a fixed frequency f ADC = 1/ which, according to the Shannon-Nyquist sampling theorem, guarantees exact recovery of signals possessing frequency content in the interval f f Nyquist = 1/2. However many signals do not possess significant energy outside of K discrete frequencies and are said to be K-sparse in the frequency domain. The emerging field of compressive sampling (CS) has suggested that random projections of such signals, digitized at the much slower frequency f ADC O(K log( f Nyquist /K)) < f Nyquist, can be used to recovered the original signal at the 1/ rate [3]. While the mathematics is clear regarding the conditions under which good recovery is possible, what is not clear is how devices built on CS principles work in practice. To-date, only a few experimental devices have been constructed, most of them in the imaging field. Perhaps the most well-known examples are the single pixel imagers described in [4, 5]. Another recent work by Gazit et al. describes an experimental imager capable of producing sub-wavelength resolution in the reconstructed image [6]. The work of Katz et al. also takes advantage of CS to experimentally demonstrate a significant reduction in the number of measurements required in pseudothermal Ghost Imaging [7]. While CS architectures are used to alleviate spatial sampling requirements in imaging applications, this work is concerned with designing a CS device to overcome temporal sampling restrictions. Along these lines several compressively sampled digitizers have been proposed with all electrical components. Tropp et al. [3] proposed a pseudo-random bit sequence to modulate the signal prior to downsampling while the ultra-wideband radar receiver proposed by Yang et al. [8] suggested using a bank of distributed amplifiers to perform the downsampling. However to-date, the only experimental device we are aware of comes from Mishali and Eldar [9] who report a prototype device that can recover 2GHz signals with only a 24MS/s sampling rate. In this paper we demonstrate an actual sub-nyquist, CS-based digitizer in which intensity modulation of the signal of interest by a pseudo-random bit sequence (PRBS) occurs entirely in the fiber-optical domain. In this initial realization we show successful recovery of a 1 GHz signal after digitization of the randomly-modulated signal at 5 MS/s, that is, at a digitization rate four times below Nyquist. Although compressive digitization with these particular experimental values could perhaps be accomplished entirely with electrical components [9,1], demonstration of the digitizer in the optical domain provides a clear path for digitization of signals well into the high tens of GHz regime - a regime that is practically inaccessible to present-day all-electrical ADCs. 2. Review of compressive sampling Since an understanding of the architecture and operation of the experimental system requires at least a basic understanding of the principles of CS, in this section we provide a brief review of the mathematics. It is now well understood that one can, in principle, recover N pieces of information by collecting M << N linear projections of the signal of interest and applying nonlinear reconstruction algorithms. The fundamentals of this surprising result were developed in the works of Candes & Tao [11], Candes et al. [12], and Donoho [13] (see Candes & Wakin (C) 211 OSA 11 April 211 / Vol. 19, No. 8 / OPTICS EXPRESS 734

3 [14], Romberg [15], or Baraniuk [16] for a more general overview of the subject). Accurate recovery depends heavily on the signal model and the nature of the projections. Assume we are interested in recovering a real-valued signal x(t) at a discrete set of temporally sampled points x x i = x((i 1) ) i = 1 N where is the sampling interval at which we hope to eventually reconstruct the signal. In compressive sampling, is potentially much smaller than the time interval δ at which the modulated signal is actually digitized. It is common to model the signal as a linear combination of a set of N basis vectors W i j i, j = 1 N i.e. x i = N j=1 W i j θ j, (1) or in matrix form, x = Wθ. Using this representation the signal is completely specified by the coefficients θ j. Compressive sampling takes advantage of situations where the signal model W is chosen such that K << N basis vectors provide an accurate representation of the signal. Such a signal is said to be K sparse in W and the recovery problem becomes one of measuring or estimating the θ j. That a signal can often be described with only a few coefficients is not surprising and underlies data compression methods. However in data compression one records all N observations x i and then solves for θ j, storing only the K coefficients. Compressive sampling circumvents the compression step by directly recording only M linear projections of the signal where K < M << N. That is we record y = Φx = ΦWθ, (2) and then recover θ j. The matrix Φ is an M N matrix that takes the unknown N vector Wθ and projects it onto the M samples y. If W is a standard ortho-basis (as will be the case in this work), and Φ consists of random entries drawn from a suitable probability distribution, then the rows of Φ and columns of W are incoherent and the matrix product ΦW satisfies the restricted isometry property (RIP) [14]. In this case, CS theory holds that full recovery of θ is possible from the M vector y by solving the minimization problem ˆθ : min θ { y ΦWθ τ θ 1 } giving the signal estimate ˆx = W ˆθ. That is, we find the coefficients such that the reconstructed data matches our observations in a least-squares sense, and also possesses the smallest possible L 1 norm, θ 1 N j=1 θ j. The parameter τ is a sparsity-promoting term that penalizes complex signal models. The mathematics imply that we can digitize y at a coarse sampling interval δ = N M and still recover x. The challenge is to design hardware that can perform the function of the projection matrix Φ. Our system realizes this projection through a combination of random optical modulation, low-pass filtering, and electrical digitization at frequency 1/δ. 3. System architecture Tropp et al. [3] suggested a projection matrix Φ with random ±1 entries (i.e. a PRBS). This was accomplished in their work by simulating a circuit that multiplied the incoming data by a stream of random ±1 digits, spaced apart in time and then sampling the result at the slow time-scale δ. Our system also randomly modulates the signal at the fast time-scale, but does so optically. Additionally, the system proposed in [3] was assumed to employ a sample and hold circuit. Each digitized sample, spaced δ apart, carries information about the signal at the desired fast time-scale. This behavior is necessary to satisfy the RIP property. (3) (C) 211 OSA 11 April 211 / Vol. 19, No. 8 / OPTICS EXPRESS 7341

4 (a) (b) Fig. 1. (a) Overview of the approach. Both the signal voltage and the pseudo-random bit sequence (PRBS) voltage applied to the photonic compressive-sampling system modulate optical power and yield a mixing term in the output voltage. (b) Detailed system layout of the compressively sampled, photonic link. Electro-optical components are shown as shaded boxes. PRBS = pseudo-random bit sequence, DFB = Distributed-feedback laser, RFA = RF Amplifier, MZM = Mach-Zehnder modulator, 3-dB = RF power divider, PD = photodetector, LPF = low-pass filter, and 1-dB=1 db attenuator. The compressed samples y(t) are recorded on Channel 4. The other channels were recorded for calibrating the system and testing the fidelity of our signal model. A problem with implementing this architecture directly is the unavailability of sample-andhold devices operating at repetition frequencies of 5 MHz, or above. As was suggested in [3], we substituted a low-pass filter (LPF) for the integrating sample-and-hold. The general layout of the compressively-sampled, photonic ADC is shown schematically in Fig. 1. The PRBS was generated by a Pulse Pattern Generator (Anritsu MP1763B). For the data presented here we used a sequence length. An Analog Signal Generator (Agilent E8257B) provided the clock for the Pattern Generator. The signal itself was produced by a Vector Signal Generator (Agilent E8267C). A distributed-feedback (DFB) laser (EM4 EM ) provided light to single-mode optical fiber and two Mach-Zehnder modulators (MZM). The first MZM (JDS Uniphase ) used the signal voltage to modulate the light intensity while the second MZM (JDS Uniphase OC ) modulated the light a second time, this time with the PRBS voltage. In both cases, the MZMs were held at optical quadrature manually. Modulated light was then detected using a photodetector (Discovery Semiconductor DSC (C) 211 OSA 11 April 211 / Vol. 19, No. 8 / OPTICS EXPRESS 7342

5 3S). Below we calculate the transfer function of this analog link and show that the output voltage contains a term proportional to the product of the signal voltage and the PRBS voltage. It is this mixing term that is necessary for CS digitization. With both modulators operating at quadrature, the PRBS modulator acted as an on-off modulator. As discussed earlier a lowpass filter (Mini-Circuits SBLP-933) was used to perform the integration function. Prior to taking CS measurements, the RF transfer function (S21) of the filter was measured using a network analyzer and, via Fourier transform, the time-domain response of the filter was calculated. Two RF amplifiers were also employed. The first device (Picosecond Pulse Lab 5865 Driver) amplified the PRBS voltage to levels sufficient to drive the second modulator to on and off states. The second amplifier (Picosecond Pulse Lab 584 Amplifier) raised the output voltage levels sufficiently so that noise internal to the 4-Channel Digitizer (LeCroy WaveMaster 85A) was negligible compared to signal voltages. Note also that the LeCroy digitizer recorded replicas of both the PRBS voltage and the signal voltage (Channels 2 & 3 in Fig. 1(b)). Although not required for signal reconstruction, having access to these voltages (a) was necessary for determining certain coefficients in the system model, (b) greatly aided troubleshooting when the system was first brought online and (c) was necessary for testing the fidelity of the signal model. 4. Signal model Signal recovery is predicated on the ability to discretely model (1) the random modulation step and (2) the filtering step. The relationship between applied voltages (signal & PRBS) and the output voltage of the photodetector is obtained as follows. For an ideal MZM at quadrature, the relationship between input voltage V i, input optical power P i and output optical power P o is well-known [2]: ( ) P o = (P i /2) 1+sin(ψ) (4) where ψ = πv i /V π is the optical phase shift in the MZM due to the applied voltage V i and V π is the half-wave voltage. At the photodetector, the output voltage is, to a very good approximation, linearly proportional to the optical power. An expression for output voltage more appropriate for an actual (nonideal) device is ( ) v out = a+bsin(ψ) (5) where the coefficients a and b are determined through experimental calibration. In the case of two modulators in series, the overall output is just the product of two expressions of the type shown in Eq. (5). Let φ denote the phase shift in the first MZM due to the signal voltage and let γ correspond to the phase shift in the second MZM due to the PRBS voltage. Then the output voltage of the photodetector, including PRBS and signal voltages is ( )( ) v out (t) = a+bsin(φ(t)) c+d sin(γ(t)) (6) where φ(t) = π v sig (t)/v π sig, γ(t) = π v PRBS (t)/v π PRBS, and v sig (t) and v PRBS (t) are, respectively, the signal and PRBS voltages, and V π sig and V π PRBS are the half-wave voltages for the signal and PRBS modulators, respectively. Note that v sig (t) is the signal we wish to recover. As a practical matter, the photodetector output is AC-coupled and, hence the term in Eq. (6) proportional to ac is not present at the output. Then the output voltage can be written v out (t) = ad sin(γ(t))+[bc+bd sin(γ(t))]sin(φ(t)) (7) (C) 211 OSA 11 April 211 / Vol. 19, No. 8 / OPTICS EXPRESS 7343

6 or, denoting the input and output voltages x(t) and y(t) respectively, y(t) = Asin(π v PRBS (t)/v π PRBS )+ [ B+C sin(π v PRBS (t)/v π PRBS ) ] sin(πx(t)/v π s ) (8) where A = ad,b = bc,c = bd are constants to be estimated. The two half-wave voltages were measured and found to be V π PRBS = 7.97 V and V π sig = 4.2 V. The other three parameters were estimated from a single set of N = 12 observations y using a Bayesian approach. Starting with vague priors on each of the variables, we used a simple Markov Chain Monte Carlo algorithm with Gibbs sampling to generate values from each parameter s posterior distribution [17]. We then used the maximum a posterior values as the final estimate. These estimated values were A =.26 V, B =.27 V, and C =.49 V. Equation (8) is clearly nonlinear in x, however for this type of optical link, the signals are small enough that we can make the approximation sin(πx/v π sig ) πx/v π sig. Additionally, the first term on the right hand side of Eq. (8) is known and can be subtracted from the observations y. Thus, our signal model becomes ỹ = y Asin(π v PRBS (t)/v π PRBS ) = π V π sig [ B+C sin(π vprbs (t)/v π PRBS ) ] x. (9) The problem can now be put in the form of Eq. (2) by describing both the random modulation and filtering processes as linear matrix operations. The random modulation can be modeled as a multiplication of the input x with the N N matrix ( π [ R = diag B+C sin(π vprbs (t)/v π PRBS ) ] ). (1) V π sig The filtering operation is modeled as a convolution of y(t) with the filter s impulse response function h(n) n = 1 N. In order to mimic the behavior of the sample-and-hold device, the filter can be selected such that the impulse response decays quickly relative to the coarse sampling interval δ. In discrete terms this means we only need to retain L << N coefficients to accurately predict the action of the filter on the signal. The randomly modulated, filtered signal is therefore given by the product HRx where H = h(1) h(2) h(1). h(l) h(l 1) h(1) h(l) h(l 1) h(1). h(l) h(l 1) h(1) The final step is to model the downsampling effect of the digitizer which simply takes every N/M samples. This matrix is given explicitly by (11) D i j = δ(i j/m) i = 1 M, j = 1 N (12) The projection matrix in Eq. (2) is therefore given by Φ = DHR and serves as a model for the modulation, filtering, and sampling steps. Upon substituting the sparse signal representation, x = Wθ, our overall compressed signal model becomes ỹ = DHRWθ. (13) (C) 211 OSA 11 April 211 / Vol. 19, No. 8 / OPTICS EXPRESS 7344

7 1 (a).15 (b) H(f) Frequency (Hz) x 1 9 Impulse response h(τ) Time (s) x 1 8 Fig. 2. (a) Frequency domain and (b) Time domain representation of the filter, sampled at the 1GHz rate. The random construction of Φ ensures that the RIP property is satisfied [3] and, provided that θ is indeed sparse in W, Eq. (3) can now be solved for θ even though M < N. A number of algorithms have been proposed for solving the minimization problem [Eq. (3)] given the signal model [Eq. (13)], as specified by Φ,W, and the observed data y(t). In each of the algorithms the key parameter is the sparsity promoting constant τ. We have found that proper selection of this parameter is essential for good results. However, once the value has been selected it can be fixed for subsequent experiments. In this work we use the Gradient Projection for Sparse Reconstruction algorithm of Figueiredo et al. [18] for solving Eq. (3). For the signal lengths considered in this work (N O(ksamples)) the algorithm is extremely fast, taking << 1 s to recover θ on a single 3GHz processor. For our experiments the value τ =.75 was chosen as the value that minimized the reconstruction error. 5. Experimental results In this section we present results that demonstrate operation of the system. First, we recover a 1GHz sinusoid at a 1/1GHz sampling interval using compressed measurements acquired at only 833MHz. This represents a compression factor of 1e9/.833e9 = 12. Ideally one would recover the signal at the PRBS clock rate of 2.5MS/s (the highest frequency at which signal information is present). However, in order to accurately model the low-pass filter s impulse response, the 1GS/s recovery time-scale was needed. A better choice of low-pass filter in future designs will perhaps eliminate this wasted bandwidth. Filter characterization was accomplished by measuring the magnitude and phase of the S21 transfer function using a network analyzer (Agilent 872ES S-Parameter Network Analyzer). The complex transfer function was then inverse Fourier transformed to obtain the real impulse response needed in constructing H. Both the frequency and time-domain descriptions of the filter are shown in Fig. (2). In order to test the accuracy of our signal model Eq. (13) we digitized the 1GHz tone at 1GHz ( = 1/1e9) giving N = 12 observations x((i 1) ) i = 1 N. We also acquired M = 83 compressed samples ỹ(( j 1)δ) j = 1 M with sampling interval δ = 1/.833e9 s. Figure 3 compares the model to the acquired data. Data and model show fairly good agreement, however there are clearly inaccuracies stemming directly from our inability to perfectly model the PRBS sequence and digitally mimic the operation of the filter. (C) 211 OSA 11 April 211 / Vol. 19, No. 8 / OPTICS EXPRESS 7345

8 (a) Compressed samples Modeled ỹ [V ] Time x 1 7 Fig. 3. Acquired vs. predicted compressed measurements ỹ. 4 (a).4 (b) Original Recovered 3.2 θ 2 x(t) Frequency (GHz) Time (s) x 1 8 Fig. 4. (a) The reconstructed cosine basis coefficients θ and (b) the associated signal reconstruction ˆx. Nonetheless, Fig. 4 shows good signal recovery in both the frequency and time domains. The reconstructed signal consists of N = 12 points, despite only using the M = 83 points shown in Figure 3. While there are clearly non-zero coefficients other than the one associated with the 1GHz signal, these are small in magnitude. As a second test we changed the frequency of the signal to 1.1GHz to ensure it was a noninteger multiple of the 1GHz reconstruction frequency. We also explored the limits of the approach in terms of the degree of compression that could be obtained. Given appropriate settings on the sparsity-promoting term, we were able to recover the 1.1GHz tone while sampling at only 5MS/s. That is to say, using only 51 compressed samples spaced δ = 1/5e6 seconds apart, we were able to recover 12 samples of the 1.1GHz tone at sampling interval = 1/1e9 seconds. The results of this recovery are shown in the frequency domain in Fig. 5. There are certainly low-frequency components that are present in the response, however the reconstruction clearly emphasizes the 1.1GHz tone. While this work has not addressed questions of optimality (e.g., optimal low-pass filter design), it clearly represents a step forward in the practical implementation of a compressively sampled system. (C) 211 OSA 11 April 211 / Vol. 19, No. 8 / OPTICS EXPRESS 7346

9 4 (a) 3 θ Frequency (GHz) Fig. 5. (a) The N = 12 cosine basis coefficients θ spanning 5GHz, recovered from only M = 51 data points collected at 5MS/s. 6. Discussion In this work we have designed, built, and tested a compressively sampled photonic link. The system randomly modulates, then filters the signal prior to digitization. We have shown that recovering a single tone is possible when digitizing at a frequency well below the Nyquist rate. Our goal in this work was not necessarily to sample the received signal in any particular photonics link but rather to show that compressive sampling could be performed successfully in the photonic domain using readily-available technology developed for such links. The photonic link thus served as a well-established tool for demonstration. That is, if a system transmits and receives signals that are known a priori to be suitable for compressive sampling (i.e. sparse), then a photonic-link architecture offers a viable approach. Importantly, this work established a clear technological path toward a realistic system capable of high-bit rate digitization in frequency regimes currently inaccessible to electrical digitizers. For example, with current, offthe-shelf components it should be possible to sample 2 GHz sparse signals at a rate of 4 GS/s or greater and at 1-bits resolution using a photonic link employing a 4 GHz MZM for random PRBS modulation and with a 4 GS/s, 8-bit electrical digitizer on the output. Both the 4 GHz modulator and the 8-bit, 4 GS/s digitizer are available today. However, obeying the Nyquist criteria in the electrical domain, such sampling would require a 8-bit, 4 GS/s electrical digitizer - a device not readily available at present. Significant work remains in order to understand and optimize the practical implementation of CS both in general and for photoniclink CS. One significant area involves the modulation sequence and the low-pass filter that work in tandem as the measurement matrix - a matrix that must exhibit the RIP property with respect to the reconstruction basis. At present, very little work has been done to find optimum modulation/filter pairs for given recovery bases. Specifically for photonic links, numerous alternate architectures to the one presented in this paper are possible and a comprehensive understanding of the trade-offs found in each architecture is needed. Acknowledgments The authors would like to acknowledge Chris McDermitt and Colin Mclaughlin for help in setting up the experimental system. We would also like to acknowledge helpful technical dis- (C) 211 OSA 11 April 211 / Vol. 19, No. 8 / OPTICS EXPRESS 7347

10 cussions with Vince Urick and Marc Currie. Subsequent to the initial submission of this paper, the authors have had in-depth technical discussion with Rebecca Willett and Waheed Bajwa of Duke University and some of the insights gained from these discussions appear in the final version. (C) 211 OSA 11 April 211 / Vol. 19, No. 8 / OPTICS EXPRESS 7348