Demonstration of speckle-based compressive ... - OSA Publishing

Vol. 26, No. 17 | 20 Aug 2018 | OPTICS EXPRESS 21390

Demonstration of speckle-based compressive sensing system for recovering RF signals GEORGE A. SEFLER, T. JUSTIN SHAW, AND GEORGE C. VALLEY* The Aerospace Corp., P.O. Box 92957, Los Angeles, CA 90009-2957, USA * [email protected]

Abstract: We demonstrate measurement of RF signals in the 2-19 GHz band using a photonic compressive sensing (CS) receiver. The RF is modulated onto chirped optical pulses that then propagate through a multimode fiber that produces the random projections needed for CS via optical speckle. Our system makes 16 independent measurements per optical pulse and we demonstrate several calibration techniques to obtain the CS measurement matrix from these measurements. Then a standard penalized l1 norm method recovers amplitude, phase, and frequency of single-tone and two-tone RF signals with about 100 MHz resolution in a single 4.5 ns pulse. A novel subspace method recovers the frequency to about 20 kHz resolution over 100 pulses in a 2.8 microsecond time window. These experiments use discrete fiber-coupled optical components, but all necessary functions can be realized in photonic and electronic integrated circuits. © 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement OCIS codes: (320.7085) Ultrafast information processing; (070.4560) Data processing by optical means; (250.4745) Optical processing devices.

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

D. L. Donoho, “For most large underdetermined systems of linear equations the minimal l 1-norm solution is also the sparsest solution,” Commun. Pure Appl. Math. 59(6), 797–829 (2006). E. J. Candes, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59(8), 1207–1223 (2006). D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006). Y. C. Eldar and G. Kutyniok, Compressed Sensing: Theory and Applications, (Cambridge University, 2012). S. Foucart and H. Rauhut, A Mathematical Introduction to Compressive Sensing, (Springer, 2013). 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). M. Mishali and Y. Eldar, “From theory to practice: Sub-Nyquist sampling of sparse wideband analog signals,” IEEE J. Sel. Top. Signal Process. 4(2), 375–391 (2010). M. Mishali, Y. C. Eldar, O. Dounaevsky, and E. Shoshan, “Xampling: analog to digital at sub-Nyquist rates,” IET Circuits Dev. Syst. 5(1), 8–20 (2011). M. Mishali and Y. C. Eldar, “Xampling: Compressed sensing for analog signals,” in Compressed Sensing: Theory and Applications, ed. by Y. C. Eldar and G. Kutyniok, (Cambridge University, 2012). G. C. Valley and G. A. Sefler, “Optical time-domain mixer,” Proc. SPIE 7797, 77970F (2010). G. C. Valley and G. A. Sefler, “Systems and methods for converting wideband signals in the optical domain,” U.S. Patent No. 8,026,837, (2011). 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). G. C. Valley, G. A. Sefler, and T. J. Shaw, “Sensing RF signals with the optical wideband converter,” Proc. SPIE 8645, 86450P (2013). G. C. Valley, G. A. Sefler, and T. J. Shaw, “Systems and methods for converting wideband signals into the digital domain using electronics or guided-wave optics,” U.S. Patent No. 8,902,096, (2014). G. C. Valley, G. A. Sefler, and T. J. Shaw, “Optical multi-coset sampling of GHz-band chirped signals,” Proc. SPIE 9362, 93620M (2015). G. A. Sefler, G. C. Valley, and T. J. Shaw, “Photonic compressive sensing of GHz-band RF signals,” Proc. 3rd International Workshop on Compressed Sensing Theory and its Applications to Radar, Sonar and Remote Sensing (CoSeRa) 109–113 (2015). G. C. Valley and G. A. Sefler, “Systems and methods for converting radio frequency signals into the digital domain using multi-mode optics,” U.S. Patent No. 9,413,372, (2016). G. C. Valley, G. A. Sefler, and T. Justin Shaw, “Multimode waveguide speckle patterns for compressive sensing,” Opt. Lett. 41(11), 2529–2532 (2016).

#332517 Journal © 2018

https://doi.org/10.1364/OE.26.021401 Received 31 May 2018; revised 17 Jul 2018; accepted 19 Jul 2018; published 6 Aug 2018


19. G. A. Sefler, T. J. Shaw, A. D. Stapleton, and G. C. Valley, “Calibration of a speckle-based compressive sensing receiver,” Proc. SPIE 10103, 101030Z (2017). 20. H. Nan, Y. Gu, and H. Zhang, “Optical analog-to-digital conversion system based on compressive sensing,” IEEE Photonics Technol. Lett. 23(2), 67–69 (2011). 21. J. M. Nichols and F. Bucholtz, “Beating Nyquist with light: a compressively sampled photonic link,” Opt. Express 19(8), 7339–7348 (2011). 22. 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). 23. Y. Liang, M. Chen, H. Chen, C. Lei, P. Li, and S. Xie, “Photonic-assisted multi-channel compressive sampling based on effective time delay pattern,” Opt. Express 21(22), 25700–25707 (2013). 24. H. Chi, Y. Chen, Y. Mei, X. Jin, S. Zheng, and X. Zhang, “Microwave spectrum sensing based on photonic time stretch and compressive sampling,” Opt. Lett. 38(2), 136–138 (2013). 25. Y. Chen, H. Chi, T. Jin, S. Zheng, X. Jin, and X. Zhang, “Sub-Nyquist sampled analog-to-digital conversion based on photonic time stretch and compressive sensing with optical random mixing,” J. Lightwave Technol. 31(21), 3395–3401 (2013). 26. T. 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,” OFC/NFOEC Tech. Dig. OTHD.1 (2013). 27. F. Yin, Y. Gao, Y. Dai, J. Zhang, K. Xu, Z. Zhang, J. Li, and J. Lin, “Multifrequency radio frequency sensing with photonics-assisted spectrum compression,” Opt. Lett. 38(21), 4386–4388 (2013). 28. B. T. Bosworth and M. A. Foster, “High-speed ultrawideband photonically enabled compressed sensing of sparse radio frequency signals,” Opt. Lett. 38(22), 4892–4895 (2013). 29. Y. Chen, X. Yu, H. Chi, X. Jin, X. Zhang, S. Zheng, and M. Galili, “Compressive sensing in a photonic link with optical integration,” Opt. Lett. 39(8), 2222–2224 (2014). 30. B. T. Bosworth, J. R. Stroud, D. N. Tran, T. D. Tran, S. Chin, and M. A. Foster, “Ultrawideband compressed sensing of arbitrary multi-tone sparse radio frequencies using spectrally encoded ultrafast laser pulses,” Opt. Lett. 40(13), 3045–3048 (2015). 31. B. T. Bosworth, J. R. Stroud, D. N. Tran, T. D. Tran, S. Chin, and M. A. Foster, “High-speed flow microscopy using compressed sensing with ultrafast laser pulses,” Opt. Express 23(8), 10521–10532 (2015). 32. J. M. Nichols, F. Bucholtz, C. V. McLaughlin, A. K. Oh, and R. M. Willett, “Fixing basis mismatch in compressively sampled photonic link,” SPIE Sensing Technology + Applications. International Society for Optics and Photonics, (2014). 33. H. Sun, B. T. Bosworth, B. C. Grubel, M. R. Kossey, M. A. Foster, and A. C. Foster, “Compressed sensing of sparse RF signals based on silicon photonic microcavities,” In CLEO: Science and Innovations, pp. SM1O–5. Optical Society of America, (2017). 34. Q. Guo, Y. Liang, M. Chen, H. Chen, and S. Xie, “Compressive spectrum sensing of radar pulses based on photonic techniques,” Opt. Express 23(4), 4517–4522 (2015). 35. J. R. Stroud, B. T. Bosworth, D. N. Tran, T. P. McKenna, T. R. Clark, T. D. Tran, and M. A. Foster, “Continuous 119.2-GSample/s photonic compressed sensing of sparse microwave signals,” In 2015 Conference on Lasers and Electro-Optics (CLEO) IEEE, (2015). 36. T. P. McKenna, J. H. Kalkavage, M. D. Sharp, and T. R. Clark, “Wideband Photonic Compressive Sampling System,” J. Lightwave Technol. 34(11), 2848–2855 (2016). 37. Q. Guo, H. Chen, M. Chen, S. Yang, and S. Xie, “Photonics-assisted compressive sampling systems,” In SPIE/COS Photonics Asia (pp. 100260E–100260E). International Society for Optics and Photonics, (2016). 38. B. Redding and H. Cao, “Using a multimode fiber as a high-resolution, low-loss spectrometer,” Opt. Lett. 37(16), 3384–3386 (2012). 39. B. Redding, S. M. Popoff, and H. Cao, “All-fiber spectrometer based on speckle pattern reconstruction,” Opt. Express 21(5), 6584–6600 (2013). 40. B. Redding, S. F. Liew, R. Sarma, and H. Cao, “Compact spectrometer based on a disordered photonic chip,” Nat. Photonics 7(9), 746–751 (2013). 41. B. Redding, M. Alam, M. Seifert, and H. Cao, “High-resolution and broadband all-fiber spectrometers,” Optica 1(3), 175–180 (2014). 42. B. Redding, S. F. Liew, Y. Bromberg, R. Sarma, and H. Cao, “Evanescently coupled multimode spiral spectrometer,” Optica 3(9), 956–962 (2016). 43. M. Piels and D. Zibar, “Compact silicon multimode waveguide spectrometer with enhanced bandwidth,” Sci. Rep. 7(1), 43454 (2017). 44. M. Piels and D. Zibar, “Compact spectrometer based on a silicon multimode waveguide,” Optical Fiber Communications Conference and Exhibition (OFC), Los Angeles, CA, (2017). 45. A. C. Scofield, G. A. Sefler, T. J. Shaw, A. D. Stapleton, and G. C. Valley, “Demonstration of GHz-band RF receiver and spectrometer using random speckle patterns,” in Conference on Lasers and Electro-optics (CLEO), (2018). 46. A. Saade, F. Caltagirone, I. Carron, L. Daudet, A. Dremeau, S. Gigan, and F. Krzakala, “Random projections through multiple optical scattering: Approximating kernels at the speed of light,” 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), IEEE, 6215–6219 (2016).


47. J. Dong, S. Gigan, F. Krzakala, and G. Wainrib, “Scaling up Echo-State Networks with multiple light scattering.” arXiv preprint arXiv:1609.05204 (2016). 48. G. C. Valley, T. J. Shaw, A. D. Stapleton, A. C. Scofield, G. A. Sefler, and L. Johannson, “Application of laser speckle to randomized numerical linear algebra,” Proc. SPIE 10551, 105510M (2018). 49. Texas Instruments, http://www.ti.com/product/ADS52J90/description (2018). 50. M. Yang, F. de Hoog, Y. Fan, and W. Hu, “Adaptive sampling by dictionary learning for hyperspectral imaging,” IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens. 9(9), 4501–4509 (2016). 51. R. O. Schmidt, “Multiple Emitter Location and Signal Parameter Estimation,” IEEE Trans. Antenn. Propag. 34(3), 276–280 (1986). 52. I. Loris, “L1Packv2: A Mathematica package for minimizing an l 1-penalized functional,” Comput. Phys. Commun. 179(12), 895–902 (2008).

1. Introduction The field of compressive sensing (CS) is now more than a decade old [1–3] and a wide range of exciting results have been reported [4,5]. A particularly attractive application for CS is sensing RF signals in the GHz band because Nyquist rate sampling with electronic ADCs generates a huge amount of data and requires more size, weight and power (SWAP) than are practical for many applications; furthermore, for many applications the GHz band is sparsely occupied. These considerations motivated a lot of research into reduced sampling rate, electronic systems for detecting GHz-band signals [6-9 and references therein]. In these electronic CS systems the GHz signal of interest must be multiplied by analog waveforms that constitute the rows of the CS measurement matrix. These analog waveforms must contain frequency content at more than twice the highest frequency of the input signal, and they must be highly stable, carefully calibrated, and capable of being generated with electronic devices of small SWAP. This issue motivated us [10–19] and other researchers [20–37] to consider photonic devices for generating and applying the CS measurement matrix. Most optical CS systems to date have been single-channel systems that apply the rows of the measurement matrix serially in time and have used free-space or fiber-coupled optical components. Neither property is desirable for general applications. Multiple channels, such as discussed for Xampling and the modulated wideband converter [7–9], are needed to achieve small time windows, and integrated optical components are needed to achieve small SWAP. These two objectives have led us to consider using optical speckle in a planar waveguide to perform the random projections needed for compressive sensing [17–20]. Optical spectrometers using speckle in multimode fibers and waveguides have recently been demonstrated by several groups [38–44], and we have shown that if one modulates an RF signal directly onto a single-frequency laser, the speckle spectrometer technique can resolve radio frequencies to an accuracy of 100 MHz [45]. This latter work used a silicon-oninsulator (SOI) planar waveguide whose output was imaged onto an InGaAs CCD camera having a frame rate of 60-100 Hz and as such was not capable of real-time measurement of RF signals. Other related work shows that optical speckle may be useful for performing the random projections needed for randomized numerical linear algebra calculations and machine learning [46–48]. For the canonical CS problem, a sparse input signal x (dimension N) is recovered from a measurement vector y (dimension M) with M