Beating Nyquist with light: a compressively sampled photonic link J. M. Nichols* and F. Bucholtz Naval Research Laboratory, Washington, DC 20375, USA *
[email protected]
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 500MS/s, well below the Nyquist rate. © 2011 Optical Society of America OCIS codes: (060.2360) Fiber optics links and subsystems; (000.3870) Mathematics.
References and links 1. J. Campmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007). 2. C. H. Lee, Microwave Photonics (CRC Press, 2007). 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), 520–544 (2010). 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), 83–91 (2008). 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, 121105 (2008). 6. S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17(26), 23920–23946 (2009). 7. O. Katz, Y. Bromberg, and Y. Silberburg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 131110 (2009). 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) (2009). 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), 375–391 (2010). 10. M. Mishali and Y. C. Eldar, “Xampling: Analog Data Compression,” vol. http://doi.ieeecomputersociety.org/10.1109/DCC.2010.39 of Proceedings of the 2010 Data Compression Conference, pp. 366–375 (2010). 11. E. J. Candes and T. Tao, “Decoding by Linear Programming,” IEEE Trans. Inf. Theory 51(12), 4203–4215 (2005). 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), 489–509 (2006). 13. D. L. Donoho, “Compressed Sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006). 14. E. J. Candes and M. B. Wakin, “An Introduction to Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008). 15. J. Romberg, “Imaging Via Compressive Sampling,” IEEE Signal Process. Mag. 25(2), 14–20 (2008). 16. R. G. Baraniuk, “Compressive Sensing,” IEEE Signal Process. Mag. 24, 118–124 (2007). 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), 2076–2089 (2010). 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), 586–597 (2007).
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(C) 2011 OSA
Received 21 Jan 2011; revised 7 Mar 2011; accepted 7 Mar 2011; published 1 Apr 2011
11 April 2011 / Vol. 19, No. 8 / OPTICS EXPRESS 7339
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 50 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 fADC = 1/∆ which, according to the Shannon-Nyquist sampling theorem, guarantees exact recovery of signals possessing frequency content in the interval 0 ≤ f ≤ fNyquist = 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 fADC ∼ O(K log( fNyquist /K)) < fNyquist , 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 240MS/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 500 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 [9,10], 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
