Defying Nyquist in Analog to Digital Conversion
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The document discusses the advancements in analog to digital conversion, particularly focusing on the concept of 'xampling', which combines compression and sampling to efficiently process data at reduced rates. It highlights the need for innovative approaches in handling structured analog signals and explores various applications, including cognitive radio, ultrasound imaging, and radar. The overarching goal is to leverage sparse structures to enhance data processing performance while minimizing power consumption and improving resolution.
![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 -20
MSE [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!
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Defying Nyquist in Analog to Digital Conversion
- 1. Defying Nyquist in Analog to Digital Conversion Yonina Eldar Department of Electrical Engineering Technion – Israel Institute of Technology Visiting Professor at Stanford, USA http://www.ee.technion.ac.il/people/YoninaEldar yonina@ee.technion.ac.il In collaboration with my students at the Technion 1/20
- 2. Digital Revolution “The change from analog mechanical and electronic technology to digital technology 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. Sampling: “Analog Girl in a Digital World…” Judy Gorman 99 Analog 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. 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 steps Can we use the structure to reduce sampling rate + first DSP rate (data transfer, bus …) as well? 4
- 5. Key Idea Exploit 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 separation Goal: Survey the main principles involved in exploiting “sparse” structure Provide a variety of different applications and benefits 5
- 6. Talk Outline Motivation Classes of structured analog signals Xampling: Compression + sampling Sub-Nyquist solutions Multiband communication: Cognitive radio Time delay estimation: Ultrasound, radar, multipath medium identification Applications to digital processing 6
- 7. Shannon-Nyquist Sampling Signal Model Minimal Rate Analog+Digital Implementation ADC DAC Digital Signal Interpolation Processor 7
- 8. Structured Analog Models Multiband 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. Cognitive Radio Cognitive radio mobiles utilize unused spectrum ``holes’’ Spectral map is unknown a-priori, leading to a multiband model Federal Communications Commission (FCC) frequency allocation 9
- 10. Structured Analog Models Medium identification: Channel Similar problem arises in radar, UWB communications, 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. Ultrasound High digital processing rates Tx pulse Ultrasonic probe Large power consumption (Collaboration with General Electric Israel) Rx signal Unknowns Echoes result from scattering in the tissue The image is formed by identifying the scatterers 11
- 12. Processing Rates To increase SNR the reflections are viewed by an antenna array SNR is improved through beamforming by introducing appropriate time shifts to the received signals Focusing the received beam by applying delays Xdcr Scan Plane Requires high sampling rates and large data processing rates One image trace requires 128 samplers @ 20M, beamforming to 150 points, a total of 6.3x106 sums/frame 12
- 13. Resolution (1): Radar Principle: A known pulse is transmitted Reflections from targets are received Target’s ranges and velocities are identified Challenges: 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. 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 462 464 466 468 470 472 100 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 in electronic microscope optical microscope 14
- 15. Imaging via “Sparse” Modeling Radar: Union method Subwavelength Coherent Diffractive Imaging: Bajwa et al., ‘11 462 464 466 Recovery of 468 sub-wavelength images 470 from highly truncated 472 474 Fourier power spectrum 476 474 476 478 480 482 484 486 150 nm et al., Nature Photonics, ‘12 Szameit 15
- 16. Proposed Framework Instead of a single subspace modeling use union of subspaces framework Adopt a new design methodology – Xampling Compression+Sampling = Xampling X prefix for compression, e.g. DivX Results in simple hardware and low computational cost on the DSP Union + Xampling = Practical Low Rate Sampling 16
- 17. Talk Outline Motivation Classes of structured analog signals Xampling: Compression + sampling Sub-Nyquist solutions Multiband communication: Cognitive radio Time delay estimation: Ultrasound, radar, multipath medium identification Applications to digital processing 17
- 18. Union of Subspaces (Lu and Do 08, Eldar and Mishali 09) Model: Examples: 18
- 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. Union of Subspaces (Lu and Do 08, Eldar and Mishali 09) Model: Examples: 20
- 21. Union of Subspaces (Lu and Do 08, Eldar and Mishali 09) Model: Allows to keep low dimension in the problem model Low dimension translates to low sampling rate 21
- 22. Talk Outline Motivation Classes of structured analog signals Xampling: Compression + sampling Sub-Nyquist solutions Multiband communication: Cognitive radio Time delay estimation: Ultrasound, radar, multipath medium identification Applications to digital processing 22
- 23. Difficulty Naïve attempt: direct sampling at low rate Most samples do not contain information!! Most bands do not have energy – which band should be sampled? ~ ~ ~ 23
- 24. Intuitive Solution: Pre-Processing Smear pulse before sampling Each samples contains energy Resolve ambiguity in the digital domain Alias all energy to baseband Can sample at low rate Resolve ambiguity in the digital domain ~ ~ ~ 24
- 25. Xampling: Main Idea Create several streams of data Each 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 ideas Identify subspaces involved Recover using standard sampling results DSP algorithms 25
- 26. Subspace Identification For linear methods: Subspace techniques developed in the context of array processing (such as MUSIC, ESPRIT etc.) Compressed sensing: only for subspace identification For nonlinear sampling: Specialized iterative algorithms: quadratic compressed sensing and more generally nonlinear compressed sensing 26
- 27. Compressed Sensing (Candès, Romberg, Tao 2006) (Donoho 2006) 27
- 28. Compressed Sensing 28
- 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 CS Input vector x analog signal x(t) Sparsity few nonzero values ? Measurement random matrix real hardware Recovery 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. 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. Xampling Hardware - periodic functions sums of exponentials The filter H(f) allows for additional freedom in shaping the tones The channels can be collapsed to a single channel 31
- 32. Talk Outline Motivation Classes of structured analog signals Xampling: Compression + sampling Sub-Nyquist solutions Multiband communication Time delay estimation: Ultrasound, radar, multipath medium identification Applications to digital processing 32
- 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. 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. The Modulated Wideband Converter ~ ~ ~ 35
- 36. Recovery From Xamples ~ ~ ~ Spectrum sparsity: Most of the are identically zero For each n we have a small size CS problem Problem: CS algorithms for each n many computations Solution: Use the ``CTF’’ block which exploits the joint sparsity and reduces the problem to a single finite CS problem 36
- 37. A 2.4 GHz Prototype (Mishali, Eldar, Dounaevsky, and Shoshan, 2010) Rate proportional to the actual band occupancy All DSP done at low rate as well 2.3 GHz Nyquist-rate, 120 MHz occupancy 280 MHz sampling rate Wideband receiver mode: 49 dB dynamic range, SNDR > 30 dB over all input range ADC mode: 1.2 volt peak-to-peak full-scale, 42 dB SNDR = 6.7 ENOB 37
- 38. Sub-Nyquist Demonstration Carrier frequencies are chosen to create overlayed aliasing at baeband + + Overlayed sub-Nyquist FM @ 631.2 MHz AM @ 807.8 MHz Sine @ 981.9 MHz MWC prototype aliasing around 6.171 MHz 10 kHz 100 kHz 1.5 MHz Reconstruction (CTF) FM @ 631.2 MHz AM @ 807.8 MHz Mishali et al., 10 38
- 39. Online Demonstrations GUI package of the MWC Video recording of sub-Nyquist sampling + carrier recovery in lab SiPS 2010 2010 IEEE Workshop on Signal Processing Systems 39
- 40. Talk Outline Brief overview of standard sampling Classes of structured analog signals Xampling: Compression + sampling Sub-Nyquist solutions Multiband communication Time delay estimation: Ultrasound, radar, multipath medium identification Applications to digital processing 40
- 41. Streams of Pulses (Gedalyahu, Tur, Eldar 10, Tur, Freidman, Eldar 10) is replaced by an integrator Can equivalently be implemented as a single channel with Application to radar, ultrasound and general localization problems such as GPS 41
- 42. Unknown Pulses (Matusiak and Eldar, 11) Output corresponds to aliased version of Gabor coefficients Recovery by solving 2-step CS problem Row-sparse Gabor Coeff. 42
- 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 -20 MSE [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. Application: Multipath Medium Identification (Gedalyahu and Eldar 09-10) LTV channel propagation paths pulses per period Medium identification (collaboration with National Instruments): Recovery of the time delays Recovery of time-variant gain coefficients The proposed method can recover the channel parameters from sub-Nyquist samples 44
- 45. Application: Radar Each target is defined by: (Bajwa, Gedalyahu and Eldar, 10) Range – delay Velocity – doppler Targets can be identified with infinite resolution as long as the time-bandwidth product satisfies Previous results required infinite time- bandwidth product 45
- 46. Xampling in Ultrasound Imaging Wagner, Eldar, and Friedman, ’11 A scheme which enables reconstruction of a two dimensional image, from samples obtained at a rate 10-15 times below Nyquist The resulting image depicts strong perturbations in the tissue Obtained by beamforming in the compressed domain Standard Imaging Xampled beamforming 1662 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. Nonlinear Sampling Michaeli & Eldar, ’12 Results can be extended to include many classes of nonlinear sampling Example: Nonlinear Sampling In particular we have extended these ideas to phase retrieval problems where we recover signals from samples of the Fourier transform 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. 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. Source Separation Cont. Texture Separation: 49
- 50. Subspace Learning Prior work has focused primarily on learning a single subspace (Vidal et. al., Ma et. al., Elhamifar …) We developed methods for multiple subspace learning from training data (Rosenblum, Zelnik-Manor and Eldar, 10) Subspace learning from reduced-dimensional measured data: Blind compressed sensing (Gleichman and Eldar 10) Current work: Extending these ideas to more practical scenarios (Carin, Silva, Chen, Sapiro) 50
- 51. 50% Missing Pixels Interpolation by learning the basis from the corrupted image 51/20
- 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 + digital More 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 http://webee.technion.ac.il/Sites/People/YoninaEldar/books.html 52
- 53. Xampling Website webee.technion.ac.il/people/YoninaEldar/xampling_top.html Y. C. Eldar and G. Kutyniok, "Compressed Sensing: Theory and Applications", Cambridge University Press, to appear in 2012 53
- 54. Thank you 54