1/20

Analog-to-Digital at Sub-Nyquist RatesAnalog-to-Digital at Sub-Nyquist Rates

– – Xampling – Xampling –

Yonina Eldar

Department of Electrical EngineeringTechnion – Israel Institute of Technology

Electrical Engineering and Statistics at Stanford

Joint work with Moshe Mishali and Kfr Gedalyahu

http://www.ee.technion.ac.il/people/YoninaEldar yonina@ee.technion.ac.il

2

Brief overview of standard sampling

Classes of structured analog signals

What is Xampling?

Sub-Nyquist solutionsMultiband communication

Time delay estimation: Ultrasound

Multipath medium identifcation: radar

Talk Outline

3

Sampling: “Analog Girl in a Digital World…” Judy Gorman 99

Digital worldAnalog world

ReconstructionD2A

SamplingA2D

Signal processing Image denoising Analysis…

(Interpolation)

Music Radar Image…

Very high sampling rates: hardware excessive solutions

High DSP rates

Main Idea:Exploit structure to reduce sampling and processing rates

4

Shannon-Nyquist Sampling

Signal Model Minimal Rate

Analog+DigitalImplementation

ADCDigitalSignal

Processor Interpolation

DAC

5

Structured Analog Models (1)

Can be viewed as bandlimited (subspace) But sampling at rate is a waste of resources For wideband applications Nyquist sampling is infeasible

Multiband communication:

Question:How do we treat structured (non-subspace) models effciently?

Unknown carriers – non-subspace

6

Structured Analog Models (2)

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 !

Medium identifcation:

Unknown delays – non-subspace

Channel

Question (same):How do we treat structured (non-subspace) models effciently?

Similar problem arises in radar

7

Ultrasound Imaging

Tx pulse Ultrasonic probe Rx signal

Standard beamforming techniques require sampling at the Nyquist rate of

Pulse shape is known – more effcient sampling methods exist

Goal: Laptop ultrasound

Unknowns

(Work with General Electrics, Israel)

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

9

Union Sampling

Many papers Vector sparsity / Block-sparsity: (fnite union of fnite-dim. subspaces)

Includes:

Each is a subspace

Sum of Subspaces

Union of Subspaces

The union tells us more about the signal!

where each is a subspace(Lu and Do, Blumensath and Davies, E. and Mishali …)

(E. and Mishali, E., Kuppinger and Bolcskei)

10

Union Types We Treat

Infnite union of SI subspaces:

Finite union of fnite subspaces (arbitrary – not SI):

Finite union of SI subspaces, when only generators are active:

active generators out of uncountable possible number of generators

where is selected from a given set

(multiband model)

(time delay estimation - multipath identifcation)

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

12

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

Identify subspaces involved

Recover using standard sampling results

Xampling: Main Idea

Hardware design ideas

DSP algorithms

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

14

no more than N bands, max width B, bandlimited to

(Mishali and E. 07-09, 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

Signal Model

~ ~~~

15

Rate Requirement

Average sampling rate

Theorem (Single multiband subspace)

(Landau 1967)

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).

16

The Modulated Wideband Converter

~ ~~~

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

18

Reconstruction Approach

Solve fniteproblem

Reconstruct

����������������������

0

1

2

3

4

5

6

S = non-zero rows

CTF(Support recovery)

Continuous Finite

The matrix V is any basis for the span of

19

Reconstruction

Memory

CTF(Support recovery) DSP

(Baseband)

AnalogBack-end

(Realtime)

High-level architecture

Detector

Recover any desired spectrum slice at baseband

20

Reconstruction

Memory

CTF(Support recovery) DSP

(Baseband)

AnalogBack-end

(Realtime)

High-level architecture

Detector

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

21

A 2.4 GHz Prototype

2.3 GHz Nyquist-rate, 120 MHz occupancy280 MHz sampling rateWideband receiver mode:

49 dB dynamic rangeSNDR > 30 dB over all input range

ADC mode:1.2 volt peak-to-peak full-scale42 dB SNDR = 6.7 ENOB

Off-the-shelf devices, ~5k$, standard PCB production

22

Sub-Nyquist Aliasing

AM input, 340.12 MHz 5.7 MHz (+42 dB gain)

23

Sub-Nyquist Aliasing

FM input, 629.2 MHz 4.816 MHz (+39 dB gain)

24

Sub-Nyquist Aliasing

PAM input, 1011.54 MHz 6.73 MHz (+29.5 dB gain)

25

Sub-Nyquist Reconstruction

+ +FM @ 360 MHz AM @ 869.2 MHz Sine @ 910 MHz Overlayed sub-Nyquist

aliasing around 6.715 MHz

3.59 3.592 3.594 3.596 3.598 3.6 3.602 3.604 3.606 3.608 3.61

x 108

0.5

1

1.5

2

2.5

3

3.5

4

4.5

x 10-4

Frequency (Hz)

Tim

e

=

-1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1

x 10-4

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Time

Mag

nitu

de

Baseband (lowrate)Digital ProcessingCarrier recovery

Signal reconstruction

FM @ 360 MHz AM @ 869.2 MHz

26

Hardware Verifcation

After a long-day at the lab

27

Online Demonstrations GUI package of the MWC

Video recording of sub-Nyquist sampling + carrier recovery in lab

28

Applications:CommunicationRadarBio-imagingGhost imaging

Special case of Finite Rate of Innovation (FRI) signals (Vetterli et. al 2002)

Minimal sampling rate – the rate of innovation:

Previous work:The rate of innovation is not achievedPulse shape often limited to diracsUnstable for high model orders

Streams of Pulses

degrees of freedom per time unit

(Kusuma & Goyal, Seelamantula & Unser)

(Dragotti, Vetterli & Blu)

29

Each Fourier coeffcient satisfes:

Spectral sstimation: sum of complex exponentials problemSolved using measurements

Methods: annihilating flter, MUSIC, ESPRIT (Stoica & Moses)

Naïve attempt: direct sampling at low rateMost samples do not contain information!!

Analog Sampling Stage

Sampling rate reduction requires proper design of the analog front-end

30

Multichannel Scheme

Gedalyahu, Tur & Eldar (2010)

Proposed scheme:Mix & integrateTake linear combinations

from which Fourier coeff. can be obtained

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

Fourier coeff. vector

Samples

31

Noise Robustness

0 10 20 30 40 50 60 70 80 90

-100

-80

-60

-40

-20

0

20

40

SNR [dB]

MS

E [

dB]

proposed method

integrators

0 10 20 30 40 50 60 70 80 90

-100

-80

-60

-40

-20

0

20

40

SNR [dB]

MS

E [

dB]

proposed method

integrators

L=2 pulses, 5 samples L=10 pulses, 21 samples

MSE of the delays estimation, versus integrators approach (Kusuma & Goyal )

The proposed scheme is stable even for high rates of innovation!

32

Noise Robustness

33

Application: Multipath Medium Identifcation

LTV channel propagation paths

Medium identifcation:Recovery of the time delaysRecovery of time-variant gain coeffcients

pulses per period

The proposed method can recover the channel parameters from sub-Nyquist samples

(Gedalyahu and E. 09-10)

34

Each target is defned by:Range – delayVelocity – doppler

Delay-Doppler spreading function:

Assumption: highly underspread setting

Application: Radar (1)

LTV system K targets

Probing signal Received signal

(Bajwa, Gedalyahu, E. 2010)

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)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

delay [Tp]

dopp

ler

[ νm

ax]

original

estimated

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

delay [Tp]

dopp

ler

[ νm

ax]

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

true locations

MF peaks

36

Comparison with CS Radar

Limited resolution to Analog signals and sampling process are not explicitly describedA discrete version of the channel is being estimated

Leakage effect fake targets

Real channel Discretized channel

delay [Tp]

dopp

ler

[ νm

ax]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

delay [Tp]

dopp

ler

[ νm

ax]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

(Herman & Strohmer 09)

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

38

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 …

Conclusions

Compressed sensing can be extended practically to the infnite analog domain!

39

Y. C. Eldar, “Compressed sensing of analog signals in shift-invariant spaces”, IEEE Trans. Signal Processing, vol. 57, no. 8, pp. 2986-2997, August 2009.

Y. C. Eldar, “Uncertainty relations for analog signals,” IEEE Trans. Inform. Theory, vol. 55, no. 12, pp. 5742 - 5757, Dec. 2009.

M. Mishali and Y. C. Eldar, “Blind multiband signal reconstruction: Compressed sensing for analog signals,” IEEE Trans. Signal Processing, vol. 57, pp. 993–1009, Mar. 2009.

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. 375-391, April 2010.

M. Mishali, Y. C. Eldar, O. Dounaevsky and E. Shoshan, " Xampling: Analog to Digital at Sub-Nyquist Rates," CCIT Report #751 Dec-09, EE Pub No. 1708, EE Dept., Technion. arXiv 0912.2495.

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. 4692–4702, Oct. 2008.

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. 5302-5316, November 2009.

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. 3017–3031, June 2010.

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:1003.2822.

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:1004.5070.

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. 3042–3054, June 2010.

References

40

Thank you