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Beyond Nyquist: Compressed Sensing of Analog Signals

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Beyond Nyquist: Compressed Sensing of Analog Signals

Yonina Eldar

Technion – Israel Institute of Technology

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

Dagstuhl Seminar

December, 2008

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

Digital world

Analog world

Sampling

A2D

Signal processing

Denoising

Image analysis…

Reconstruction

D2A

(Interpolation)

“Can we not just directly measure the part that will not end up being thrown away ?”

Donoho

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- Compressed sensing – background
- From discrete to analog
- Goals
- Part I : Blind multi-band reconstruction
- Part II : Analog CS framework

- Implementations
- Uncertainty relations

Can break the Shannon-Nyquist barrier

by exploiting signal structure

- K non-zero entries at least 2K measurements

- Recovery: brute-force, convex optimization, greedy algorithms, …

- Uniqueness:

is uniquely determined by

Donoho and Elad, 2003

with high probability

is random

Donoho, 2006 and Candès et. al., 2006

Recovery:

Convex and tractable

Donoho, 2006 and Candès et. al., 2006

Greedy algorithms: OMP, FOCUSS, etc.

NP-hard

Tropp, Elad, Cotter et. al,. Chen et. al., and many others

Standard CS

Discrete Framework

Analog Domain

Sparsity prior

what is a sparse analog signal ?

Generalized sampling

Continuoussignal

Operator

Infinite

sequence

Finite dimensional elements

Stability

Randomness Infinitely many

Random is stable w.h.p

Need structure for efficient implementation

Reconstruction

Finite program, well-studied

Undefined program over a continuous signal

Standard CS

Discrete Framework

Analog Domain

- Questions:
- What is the definition of analog sparsity ?
- How to select a sampling operator ?
- Can we introduce stucture in sampling and still preserve stability ?
- How to solve infinite dimensional recovery problems ?

Sparsity prior

what is a sparse analog signal ?

Generalized sampling

Continuoussignal

Operator

Infinite

sequence

Finite dimensional elements

Stability

Randomness Infinitely many

Random is stable w.h.p

Need structure for efficient implementation

Reconstruction

Finite program, well-studied

Undefined program over a continuous signal

- Concrete analog sparsity model
- Reduce sampling rate (to minimal)
- Simple recovery algorithms
- Practical implementation in hardware

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

- More generally

only sequences are non-zero (Eldar 2008)

Analog Compressed Sensing

What is the definition of analog sparsity ?

- A signal with a multiband structure in some basis

- Each band has an uncountable number of non-zero elements

- Band locations lie on an infinite grid

- Band locations are unknown in advance

(Mishali and Eldar 2007)

(Mishali and Eldar 2007)

bands

Sampling

Reconstruction

Analog

Infinite

Analog

Goal: Perfect reconstruction

We are interested in unknown spectral support (a union of subspace prior)

- Known band locations (subspace prior):
- Minimal-rate sampling and reconstruction (NB) with known band locations (Lin and Vaidyanathan 98)
- Half blind system (Herley and Wong 99, Venkataramani and Bresler 00)

- Next steps:
- What is the minimal rate requirement ?
- A fully-blind system design

- The minimal rate is doubled
- For , the rate requirement is samples/sec (on average)

Theorem (blind recovery)

Mishali and Eldar (2007)

Theorem (non-blind recovery)

Landau (1967)

Average sampling rate

Multi-Coset: Periodic Non-uniform on the Nyquist grid

In each block of samples, only are kept, as described by

2

Analog signal

0

Point-wise samples

0

3

3

2

0

3

2

in vector form

unknowns

Length .

known

matrix

known

Observation:

is sparse

DTFT

of sampling sequences

Constant

Problems:

- Undetermined system – non unique solution
- Continuous set of linear systems

is jointly sparse and unique under appropriate parameter selection ( )

Solve finiteproblem

Reconstruct

0

S = non-zero rows

1

2

3

4

5

6

Solve finiteproblem

Reconstruct

CTF block

MMV

- span a finite space
- Any basis preserves the sparsity

Continuous

Finite

Find a matrix U that has as few non-zero rows as possible

- Variety of methods based on optimizing mixed column-row norms
- We prove equivalence results by extending RIP and coherence to allow for structured sparsity (Mishali and Eldar, Eldar and Bolcskei)
- New approach: ReMBo – Reduce MMV and Boost
- Main idea: Merge columns of Vto obtain a single vector problem y=Aa
- Sparsity pattern of a is equal to that of U
- Can boost performance by repeating the merging with different coeff.

Perfect reconstruction at minimal rate

Blind system: band locations are unknown

Can be applied to CS of general analog signals

Works with other sampling techniques

Continuous-to-finite block: Compressed sensing for analog signals

CTF

(Eldar 2008)

Sampling signals from a union of shift-invariant spaces (SI)

Subspace

generators

There is no prior knowledge on the exact indices in the sum

What happen if only K<<N sequences are not zero ?

Not a subspace !

Only k sequences are non-zero

Step 1: Compress the sampling sequences

Step 2: “Push” all operators to analog domain

CTF

System A

High sampling rate = m/TPost-compression

Only k sequences are non-zero

Low sampling rate = p/TPre-compression

System B

CTF

Theorem

Eldar (2008)

Reconstruction filter

Signal

Amplitude

Amplitude

Output

Time (nSecs)

Time (nSecs)

Minimal rate

Minimal rate

Sampling rate

Sampling rate

Brute-Force

M-OMP

0% Recovery

100% Recovery

0% Recovery

100% Recovery

Noise-free

Sampling rate

Sampling rate

SBR4

SBR2

Empirical recovery rate

Analog signal

2

Point-wise samples

0

0

3

3

2

0

3

2

Delay

ADC

@ rate

- Impossible to match rate for wideband RF signals(Nyquist rate > 200 MHz)
- Resource waste for IF signals

3. Requires accuratetime delays

(Mishali, Eldar, Tropp 2008)

Efficientimplementation

Use CTF

A few first steps…

- Until now: sparsity in a single basis
- Can we have a sparse representation in two bases?
- Motivation: A combination of bases can sometimes better represent the signal
Both and are small!

- How sparse can be in each basis?
- Finite setting: vector in
Elad and Brukstein 2002

Different bases

Uncertainty

relation

Theorem

Eldar (2008)

Theorem

Eldar (2008)

In the DFT domain

Fourier

Spikes

What are the analog counterparts ?

- Constant magnitude
- Modulation

- “Single” component
- Shifts

- Minimal coherence:

- Tightness:

- Given a dictionary ,
expand using as few elements as possible:

minimize

- Solution is possible using CTF if is small enough
- Basic idea:

- Sample with basis
- Obtain an IMV model:
maximal value

- Apply CTF to recover
- Can establish equivalence with as long as is small enough

- Extend the basic results of CS to the analog setting - CTF
- Sample analog signals at rates much lower than Nyquist
- Can find a sparse analog representation
- Can be implemented efficiently in hardware
Questions:

Other models of analog sparsity?

Other sampling devices?

Compressed Sensing of Analog Signals

Thank you

Thank you

High-rate

- M. Mishali and Y. C. Eldar, "Blind Multi-Band Signal Reconstruction: Compressed Sensing for Analog Signals,“ to appear in IEEE Trans. on Signal Processing.
- M. Mishali and Y. C. Eldar, "Reduce and Boost: Recovering Arbitrary Sets of Jointly Sparse Vectors", IEEE Trans. on Signal Processing, vol. 56, no. 10, pp. 4692-4702, Oct. 2008.
- Y. C. Eldar , "Compressed Sensing of Analog Signals", submitted to IEEE Trans. on Signal Processing.
- Y. C. Eldar and M. Mishali, "Robust Recovery of Signals from a Union of Subspaces’’, submitted to IEEE Trans. on Inform. Theory.
- Y. C. Eldar, "Uncertainty Relations for Analog Signals", submitted to IEEE Trans. Inform. Theory.
- Y. C. Eldar and T. Michaeli, "Beyond Bandlimited Sampling: Nonlinearities, Smoothness and Sparsity", to appear in IEEE Signal Proc. Magazine.