Xampling. Analog-to-Digital at Sub-Nyquist Rates. Yonina Eldar

Transcription

1 Xampling Analog-to-Digital at Sub-Nyquist Rates Yonina Eldar Department of Electrical Engineering Technion Israel Institute of Technology Electrical Engineering and Statistics at Stanford Joint work with Moshe Mishali and Kfr Gedalyahu 1/20

2 Talk Outline Brief overview of standard sampling Classes of structured analog signals What is Xampling? Sub-Nyquist solutions Multiband communication Time delay estimation: Ultrasound Multipath medium identifcation: radar 2

3 Sampling: Analog Girl in a Digital World Judy Gorman 99 Digital world Analog world Sampling A2D Music Radar Image Reconstruction Very high sampling rates: D2A Signal processing Image denoising Analysis hardware excessive solutions High DSP rates (Interpolation) Main Idea: Exploit structure to reduce sampling and processing rates 3

4 Shannon-Nyquist Sampling Minimal Rate Signal Model Analog+Digital Implementation ADC DAC Digital Signal Processor Interpolation 4

5 Structured Analog Models (1) 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 is infeasible Question: How do we treat structured (non-subspace) models effciently? 5

6 Structured Analog Models (2) Medium identifcation: Similar problem arises in radar Channel Unknown delays non-subspace Can implement a digital match flter 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 effciently? 6

7 Ultrasound Imaging Tx pulse Ultrasonic probe Rx signal Unknowns Standard beamforming techniques require sampling at the Nyquist rate of Pulse shape is known more effcient sampling methods exist Goal: Laptop ultrasound (Work with General Electrics, Israel) 7

8 Proposed Framework Instead of a single subspace modeling use union of subspaces framework Adopt a new design methodology Xampling Results in simple hardware and low computational cost on the DSP + Modularity Union + Xampling = Practical Low Rate Sampling 8

9 Union Sampling where each (Lu and Do, Blumensath and Davies, E. and Mishali ) is a subspace Union of Subspaces Each is a subspace The union tells us more about the signal! Sum of Subspaces Includes: Vector sparsity / Block-sparsity: (fnite union of fnite-dim. subspaces) Many papers (E. and Mishali, E., Kuppinger and Bolcskei) 9

10 Union Types We Treat Finite union of SI subspaces, when only generators are active: (multiband model) Infnite union of SI subspaces: active generators out of uncountable possible number of generators (time delay estimation - multipath identifcation) Finite union of fnite subspaces (arbitrary not SI): where is selected from a given set 10

11 Why Not CS? CS is for fnite dimensional models (y=ax) Loss in resolution when discritizing Sensitivity to grid, analog bandwidth issues Is not able to exploit structure in analog signals Results in large computation on the digital side Samples do not typically interface with standard processing methods Possible coherence issues 11

12 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 12

13 Take-Home Message Compressed sensing uses fnite models Xampling works for analog signals Compression Sampling Must combine ideas from Sampling theory and algorithms from CS CS+Sampling = Xampling X prefx for compression, e.g. DivX 13

14 Signal Model ~ (Mishali and E , Mishali, E., Tropp 08) ~ 1. Each band has an uncountable number of non-zero elements 2. Band locations lie on the continuum 3. Band locations are unknown in advance no more than N bands, max width B, bandlimited to 14

15 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). 15

16 ~ ~ The Modulated Wideband Converter 16

17 ~ ~ Recovery From Xamples Cannot invert a fat matrix! 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 17

18 Reconstruction Approach Solve fnite problem Reconstruct 0 S = non-zero rows 1 2 CTF (Support recovery) 3 4 Continuous 5 Finite 6 The matrix V is any basis for the span of 18

19 Reconstruction High-level architecture CTF (Support recovery) Memory Detector DSP (Baseband) Analog Back-end (Realtime) Recover any desired spectrum slice at baseband 19

20 Reconstruction High-level architecture CTF (Support recovery) Memory Detector DSP (Baseband) Analog Back-end (Realtime) Can reconstruct: The original analog input exactly (without noise) Improve SNR for noisy inputs, due to rejection of out-of-band noise Any band of interest, modulated on any desired carrier frequency 20

21 A 2.4 GHz Prototype 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 Off-the-shelf devices, ~5k$, standard PCB production 21

22 Sub-Nyquist Aliasing AM input, MHz 5.7 MHz (+42 db gain) 22

23 Sub-Nyquist Aliasing FM input, MHz MHz (+39 db gain) 23

24 Sub-Nyquist Aliasing PAM input, MHz 6.73 MHz (+29.5 db gain) 24

25 Sub-Nyquist Reconstruction MHz + = MHz 910 MHz Overlayed sub-nyquist aliasing around MHz Signal reconstruction Magnitude Time x Carrier recovery Baseband (lowrate) Digital Processing Frequency (Hz) x MHz Time x MHz 25

26 Hardware Verifcation After a long-day at the lab 26

27 Online Demonstrations GUI package of the MWC Video recording of sub-nyquist sampling + carrier recovery in lab 27

28 Streams of Pulses Applications: Communication Radar Bio-imaging Ghost imaging degrees of freedom per time unit Special case of Finite Rate of Innovation (FRI) signals (Vetterli et. al 2002) Minimal sampling rate the rate of innovation: (Dragotti, Vetterli & Blu) Previous work: (Kusuma & Goyal, Seelamantula & Unser) The rate of innovation is not achieved Pulse shape often limited to diracs Unstable for high model orders 28

29 Analog Sampling Stage Naïve attempt: direct sampling at low rate Most samples do not contain information!! Sampling rate reduction requires proper design of the analog front-end Each Fourier coeffcient satisfes: Spectral sstimation: sum of complex exponentials problem Solved using measurements Methods: annihilating flter, MUSIC, ESPRIT (Stoica & Moses) 29

30 Multichannel Scheme Proposed scheme: Mix & integrate Take linear combinations from which Fourier coeff. can be obtained Samples Fourier coeff. vector Gedalyahu, Tur & Eldar (2010) Supports general pulse shapes (time limited) Operates at the rate of innovation Stable in the presence of noise Practical implementation based on the MWC Single pulse generator can be used 30

31 Noise Robustness MSE of the delays estimation, versus integrators approach (Kusuma & Goyal ) MSE [db] MSE [db] L=10 pulses, 21 samples L=2 pulses, 5 samples proposed method integrators proposed method integrators SNR [db] SNR [db] The proposed scheme is stable even for high rates of innovation! 31 90

32 Noise Robustness 32

33 Application: Multipath Medium Identifcation (Gedalyahu and E ) LTV channel propagation paths pulses per period Medium identifcation: Recovery of the time delays Recovery of time-variant gain coeffcients The proposed method can recover the channel parameters from sub-nyquist samples 33

34 Application: Radar (1) (Bajwa, Gedalyahu, E. 2010) Each target is defned by: Range delay Velocity doppler Delay-Doppler spreading function: Assumption: highly underspread setting LTV system K targets Probing signal Received signal 34

35 Application: Radar (2) Main result: The system can be identifed with infnite resolution as long as the time-bandwidth product of the input signal satisfy Example: 4 targets, Matched fltering would require (empirically) true locations MF peaks max ] 0.3 doppler [ν doppler [νmax] original estimated delay [Tp] delay [T ] p 35

36 Comparison with CS Radar (Herman & Strohmer 09) Limited resolution to Analog signals and sampling process are not explicitly described A discrete version of the channel is being estimated Discretized channel doppler [νmax] doppler [νmax] Real channel delay [Tp] delay [Tp] Leakage effect fake targets 36

37 Ultrasound Experiment Real data acquired by GE Healthcare s Vivid-i imaging system Method applied on this noisy signal Excellent reconstruction from sub-nyquist samples 5 equally spaced scatterers 37

38 Conclusions Compressed sampling and processing of many analog signals sub-nyquist sampler in hardware Union of subspaces: broad and fexible model Many research opportunities: extensions, stability, hardware Compressed sensing can be extended practically to the infnite analog domain! 38

39 References Y. C. Eldar, Compressed sensing of analog signals in shift-invariant spaces, IEEE Trans. Signal Processing, vol. 57, no. 8, pp , August Y. C. Eldar, Uncertainty relations for analog signals, IEEE Trans. Inform. Theory, vol. 55, no. 12, pp , Dec M. Mishali and Y. C. Eldar, Blind multiband signal reconstruction: Compressed sensing for analog signals, IEEE Trans. Signal Processing, vol. 57, pp , Mar M. Mishali and Y. C. Eldar, From theory to practice: sub-nyquist sampling of sparse wideband analog signals, IEEE Journal of Selected Topics on Signal Processing, vol. 4, pp , April M. Mishali, Y. C. Eldar, O. Dounaevsky and E. Shoshan, " Xampling: Analog to Digital at SubNyquist Rates," CCIT Report #751 Dec-09, EE Pub No. 1708, EE Dept., Technion. arxiv M. Mishali and Y. C. Eldar, Reduce and boost: Recovering arbitrary sets of jointly sparse vectors, IEEE Trans. Signal Processing, vol. 56, no. 10, pp , Oct Y. C. Eldar and M. Mishali, Robust recovery of signals from a structured union of subspaces, IEEE Trans. Inform. Theory, vol. 55, no. 11, pp , November K. Gedalyahu and Y. C. Eldar, "Time-Delay Estimation From Low-Rate Samples: A Union of Subspaces Approach," IEEE Trans. Signal Processing, vol. 58, no. 6, pp , June R. Tur, Y. C. Eldar and Z. Friedman, "Low Rate Sampling of Pulse Streams with Application to Ultrasound Imaging," submitted to IEEE Transactions on Signal Processing; [Online] arxiv: K. Gedalyahu, R. Tur and Y. C. Eldar, "Multichannel Sampling of Pulse Streams at the Rate of Innovation," to IEEE Trans. on Signal Processing, April 2010.; [Online] arxiv: Y. C. Eldar, P. Kuppinger and H. Bolcskei, "Block-Sparse Signals: Uncertainty Relations and Effcient Recovery," IEEE Trans. Signal Processing, vol. 58, no. 6, pp , June

40 Thank you 40