Defying Nyquist in Analog to Digital Conversion


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Yonina Eldar, Professor, Technion, Electrical Engineering Department (IEEE SPS DL), Defying Nyquist in Analog to Digital Conversion

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Defying Nyquist in Analog to Digital Conversion

  1. 1. Defying Nyquist in Analog to Digital Conversion Yonina Eldar Department of Electrical Engineering Technion – Israel Institute of Technology Visiting Professor at Stanford, USA In collaboration with my students at the Technion 1/20
  2. 2. Digital Revolution“The change from analog mechanical and electronic technology to digitaltechnology that has taken place since c. 1980 and continues to the present day.” Cell phone subscribers: 4 billion (67% of world population) Digital cameras: 77% of American households now own at least one Internet users: 1.8 billion (26.6% of world population) PC users: 1 billion (16.8% of world population) 2
  3. 3. Sampling: “Analog Girl in a Digital World…” Judy Gorman 99Analog world Digital world Sampling A2D Signal processing Image denoising Reconstruction Analysis… Music Very high sampling rates: Radar D2A hardware excessive solutions Image… High DSP rates (Interpolation) ADCs, the front end of every digital application, remain a major bottleneck 3
  4. 4. Today’s Paradigm Analog designers and circuit experts design samplers at Nyquist rate or higher DSP/machine learning experts process the data Typical first step: Throw away (or combine in a “smart” way) much of the data … Logic: Exploit structure prevalent in most applications to reduce DSP processing rates However, the analog step is one of the costly stepsCan we use the structure to reduce sampling rate + firstDSP rate (data transfer, bus …) as well? 4
  5. 5. Key IdeaExploit structure to improve data processing performance: Reduce storage/reduce sampling rates Reduce processing rates Reduce power consumption Increase resolution Improve denoising/deblurring capabilities Improved classification/source separationGoal: Survey the main principles involved in exploiting “sparse” structure Provide a variety of different applications and benefits 5
  6. 6. Talk OutlineMotivationClasses of structured analog signalsXampling: Compression + samplingSub-Nyquist solutions Multiband communication: Cognitive radio Time delay estimation: Ultrasound, radar, multipath medium identificationApplications to digital processing 6
  7. 7. Shannon-Nyquist Sampling Signal Model Minimal Rate Analog+Digital Implementation ADC DAC Digital Signal Interpolation Processor 7
  8. 8. Structured Analog ModelsMultiband communication: Unknown carriers – non-subspace Can be viewed as bandlimited (subspace) But sampling at rate is a waste of resources For wideband applications Nyquist sampling may be infeasible Question: How do we treat structured (non-subspace) models efficiently? 8
  9. 9. Cognitive Radio Cognitive radio mobiles utilize unused spectrum ``holes’’ Spectral map is unknown a-priori, leading to a multiband modelFederal Communications Commission (FCC)frequency allocation 9
  10. 10. Structured Analog Models Medium identification: ChannelSimilar problem arises in radar, UWBcommunications, timing recovery problems … Unknown delays – non-subspace Digital match filter or super-resolution ideas (MUSIC etc.) (Brukstein, Kailath, Jouradin, Saarnisaari …) But requires sampling at the Nyquist rate of The pulse shape is known – No need to waste sampling resources ! Question (same): How do we treat structured (non-subspace) models efficiently? 10
  11. 11. Ultrasound High digital processing rates Tx pulse Ultrasonic probe Large power consumption(Collaboration with General ElectricIsrael) Rx signal Unknowns Echoes result from scattering in the tissue The image is formed by identifying the scatterers 11
  12. 12. Processing RatesTo increase SNR the reflections are viewed by an antenna arraySNR is improved through beamforming by introducing appropriatetime shifts to the received signals Focusing the received beam by applying delays Xdcr Scan PlaneRequires high sampling rates and large data processing ratesOne image trace requires 128 samplers @ 20M, beamforming to 150points, a total of 6.3x106 sums/frame 12
  13. 13. Resolution (1): RadarPrinciple: A known pulse is transmitted Reflections from targets are received Target’s ranges and velocities are identifiedChallenges: Targets can lie on an arbitrary grid Process of digitizing  loss of resolution in range-velocity domain Wideband radar requires high rate sampling and processing which also results in long processing time 13
  14. 14. Resolution (2): Subwavelength Imaging (Collaboration with the groups of Segev and Cohen) Diffraction limit: Even a perfect optical imaging system has a resolution limit determined by the wavelength λ The smallest observable detail is larger than ~ λ/2 This results in image smearing 462 464 466 468 470 472100 nm 474 Sketch of an optical microscope: 476 the physics of EM waves acts 474 476 478 480 482 484 486 Nano-holes Blurred image as an ideal low-pass filter as seen in seen inelectronic microscope optical microscope 14
  15. 15. Imaging via “Sparse” Modeling Radar: Union method Subwavelength Coherent Diffractive Imaging: Bajwa et al., ‘11462464466 Recovery of468 sub-wavelength images470 from highly truncated472474 Fourier power spectrum476 474 476 478 480 482 484 486 150 nm et al., Nature Photonics, ‘12 Szameit 15
  16. 16. Proposed FrameworkInstead of a single subspace modeling use union of subspacesframeworkAdopt a new design methodology – Xampling Compression+Sampling = Xampling X prefix for compression, e.g. DivXResults in simple hardware and low computational cost onthe DSP Union + Xampling = Practical Low Rate Sampling 16
  17. 17. Talk OutlineMotivationClasses of structured analog signalsXampling: Compression + samplingSub-Nyquist solutions Multiband communication: Cognitive radio Time delay estimation: Ultrasound, radar, multipath medium identificationApplications to digital processing 17
  18. 18. Union of Subspaces (Lu and Do 08, Eldar and Mishali 09)Model:Examples: 18
  19. 19. Union of Subspaces (Lu and Do 08, Eldar and Mishali 09)Model:Standard approach: Look at sum of all subspaces Signal bandlimited to High rate 19
  20. 20. Union of Subspaces (Lu and Do 08, Eldar and Mishali 09)Model:Examples: 20
  21. 21. Union of Subspaces (Lu and Do 08, Eldar and Mishali 09)Model:Allows to keep low dimension in the problem modelLow dimension translates to low sampling rate 21
  22. 22. Talk OutlineMotivationClasses of structured analog signalsXampling: Compression + samplingSub-Nyquist solutions Multiband communication: Cognitive radio Time delay estimation: Ultrasound, radar, multipath medium identificationApplications to digital processing 22
  23. 23. DifficultyNaïve attempt: direct sampling at low rateMost samples do not contain information!!Most bands do not have energy – which band should be sampled? ~ ~ ~ 23
  24. 24. Intuitive Solution: Pre-ProcessingSmear pulse before samplingEach samples contains energyResolve ambiguity in the digital domainAlias all energy to basebandCan sample at low rateResolve ambiguity in the digital domain ~ ~ ~ 24
  25. 25. Xampling: Main IdeaCreate several streams of dataEach stream is sampled at a low rate(overall rate much smaller than the Nyquist rate)Each stream contains a combination from different subspaces Hardware design ideasIdentify subspaces involvedRecover using standard sampling results DSP algorithms 25
  26. 26. Subspace IdentificationFor linear methods: Subspace techniques developed in the context of array processing (such as MUSIC, ESPRIT etc.) Compressed sensing: only for subspace identificationFor nonlinear sampling: Specialized iterative algorithms: quadratic compressed sensing and more generally nonlinear compressed sensing 26
  27. 27. Compressed Sensing (Candès, Romberg, Tao 2006) (Donoho 2006) 27
  28. 28. Compressed Sensing 28
  29. 29. Compressed Sensing and Hardware Explosion of work on compressed sensing in many digital applications Many papers describing models for CS of analog signals Have these systems made it into hardware? CS is a digital theory – treats vectors not analog inputs Standard CS Analog CSInput vector x analog signal x(t)Sparsity few nonzero values ?Measurement random matrix real hardwareRecovery convex optimization need to recover analog input greedy methods We use CS only after sampling and only to detect the subspace Enables efficient hardware and low processing rates 29
  30. 30. Optimal Xampling Hardware Sampling Reconstruction(det. by ) White Gaussian noise We derive two lower bounds on the performance of UoS estimation: Fundamental limit – regardless of sampling technique or rate Lower bound for a given sampling rate Allows to determine optimal sampling method Can compare practical algorithms to bound Ben-Haim, Michaeli, and Eldar (2010), “Performance Bounds and Design Criteria for Estimating Finite Rate of Innovation Signals,” to appear in IEEE Trans. Inf. Theory. Sampling with Sinusoids is Optimal 30
  31. 31. Xampling Hardware - periodic functions sums of exponentialsThe filter H(f) allows for additional freedom in shaping the tonesThe channels can be collapsed to a single channel 31
  32. 32. Talk OutlineMotivationClasses of structured analog signalsXampling: Compression + samplingSub-Nyquist solutions Multiband communication Time delay estimation: Ultrasound, radar, multipath medium identificationApplications to digital processing 32
  33. 33. Signal Model (Mishali and Eldar, 2007-2009) ~ ~ ~1. Each band has an uncountable 2. Band locations lie on the continuum number of non-zero elements 3. Band locations are unknown in advance no more than N bands, max width B, bandlimited to 33
  34. 34. Rate Requirement Theorem (Single multiband subspace) (Landau 1967) Average sampling rate Theorem (Union of multiband subspaces) (Mishali and Eldar 2007)1. The minimal rate is doubled.2. For , the rate requirement is samples/sec (on average). 34
  35. 35. The Modulated Wideband Converter ~ ~ ~ 35
  36. 36. Recovery From Xamples ~ ~ ~Spectrum sparsity: Most of the are identically zeroFor each n we have a small size CS problemProblem: CS algorithms for each n  many computationsSolution: Use the ``CTF’’ block which exploits the joint sparsity andreduces the problem to a single finite CS problem 36
  37. 37. A 2.4 GHz Prototype (Mishali, Eldar, Dounaevsky, and Shoshan, 2010)Rate proportional to the actual band occupancyAll DSP done at low rate as well2.3 GHz Nyquist-rate, 120 MHz occupancy280 MHz sampling rateWideband receiver mode: 49 dB dynamic range, SNDR > 30 dB over all input rangeADC mode: 1.2 volt peak-to-peak full-scale, 42 dB SNDR = 6.7 ENOB 37
  38. 38. Sub-Nyquist Demonstration Carrier frequencies are chosen to create overlayed aliasing at baeband + + Overlayed sub-NyquistFM @ 631.2 MHz AM @ 807.8 MHz Sine @ 981.9 MHz MWC prototype aliasing around 6.171 MHz 10 kHz 100 kHz 1.5 MHzReconstruction (CTF) FM @ 631.2 MHz AM @ 807.8 MHz Mishali et al., 10 38
  39. 39. Online Demonstrations GUI package of the MWCVideo recording of sub-Nyquist sampling + carrier recovery in lab SiPS 2010 2010 IEEE Workshop on Signal Processing Systems 39
  40. 40. Talk OutlineBrief overview of standard samplingClasses of structured analog signalsXampling: Compression + samplingSub-Nyquist solutions Multiband communication Time delay estimation: Ultrasound, radar, multipath medium identificationApplications to digital processing 40
  41. 41. Streams of Pulses (Gedalyahu, Tur, Eldar 10, Tur, Freidman, Eldar 10) is replaced by an integratorCan equivalently be implemented as a single channel withApplication to radar, ultrasound and general localization problems such as GPS 41
  42. 42. Unknown Pulses (Matusiak and Eldar, 11)Output corresponds to aliased version of Gabor coefficientsRecovery by solving 2-step CS problem Row-sparse Gabor Coeff. 42
  43. 43. Noise Robustness MSE of the delays estimation, versus integrators approach (Kusuma & Goyal ) L=2 pulses, 5 samples L=10 pulses, 21 samples 40 40 20 20 0 0 -20 -20MSE [dB] MSE [dB] -40 -40 -60 -60 -80 -80 proposed method proposed method -100 integrators -100 integrators 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 SNR [dB] SNR [dB] The proposed scheme is stable even for high rates of innovation! 43
  44. 44. Application:Multipath Medium Identification (Gedalyahu and Eldar 09-10) LTV channel propagation paths pulses per periodMedium identification (collaboration with National Instruments): Recovery of the time delays Recovery of time-variant gain coefficientsThe proposed method can recover the channel parameters fromsub-Nyquist samples 44
  45. 45. Application: RadarEach target is defined by: (Bajwa, Gedalyahu and Eldar, 10) Range – delay Velocity – dopplerTargets can be identified with infiniteresolution as long as the time-bandwidthproduct satisfiesPrevious results required infinite time-bandwidth product 45
  46. 46. Xampling in Ultrasound Imaging Wagner, Eldar, and Friedman, ’11A scheme which enables reconstruction of a two dimensionalimage, from samples obtained at a rate 10-15 times below NyquistThe resulting image depicts strong perturbations in the tissueObtained by beamforming in the compressed domainStandard Imaging Xampled beamforming1662 real-valued samples, per sensor 200 real-valued samples, per sensor per per image line image line (assume L=25 reflectors per line) 46
  47. 47. Nonlinear Sampling Michaeli & Eldar, ’12Results can be extended to include many classes of nonlinearsampling Example: Nonlinear SamplingIn particular we have extended these ideas to phase retrievalproblems where we recover signals from samples of the Fouriertransform magnitude (Candes et. al., Szameit et. al., Shechtman et. al.)Many applications in optics: recovery from partially coherent light,crystallography, subwavelength imaging and more 47
  48. 48. Structure in Digital Problems Union of subspaces structure can be exploited in many digital models Subspace models lead to block sparse recovery Block sparsity: algorithms and recovery guarantees in noisy environments (Eldar and Mishali 09, Eldar et. al. 10, Ben-Haim and Eldar 10) Hierarchical models with structure on the subspace level and within the subspaces (Sprechmann, Ramirez, Sapiro and Eldar, 10)Noisy merged Missing pixels Separated 48
  49. 49. Source Separation Cont.Texture Separation: 49
  50. 50. Subspace LearningPrior work has focused primarily on learning a single subspace(Vidal et. al., Ma et. al., Elhamifar …)We developed methods for multiple subspace learning from trainingdata (Rosenblum, Zelnik-Manor and Eldar, 10)Subspace learning from reduced-dimensional measured data: Blindcompressed sensing (Gleichman and Eldar 10)Current work: Extending these ideas to more practical scenarios (Carin, Silva,Chen, Sapiro) 50
  51. 51. 50% Missing PixelsInterpolation by learning the basisfrom the corrupted image 51/20
  52. 52. Conclusions Compressed sampling and processing of many signals Wideband sub-Nyquist samplers in hardware Union of subspaces: broad and flexible model Practical and efficient hardware Many applications and many research opportunities: extensions to other analog and digital problems, robustness, hardware … Exploiting structure can lead to a new sampling paradigm which combines analog + digitalMore details in:M. Duarte and Y. C. Eldar, “Structured Compressed Sensing: From Theory to Applications,” Review for TSP.M. Mishali and Y. C. Eldar, "Xampling: Compressed Sensing for Analog Signals", book chapter available at 52
  53. 53. Xampling Website C. Eldar and G. Kutyniok, "Compressed Sensing: Theory and Applications",Cambridge University Press, to appear in 2012 53
  54. 54. Thank you 54