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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 [email protected] Dagstuhl Seminar December, 2008.

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Beyond nyquist compressed sensing of analog signals

Beyond Nyquist: Compressed Sensing of Analog Signals

Yonina Eldar

Technion – Israel Institute of Technology

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

[email protected]

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)


Compression
Compression

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

Donoho

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Outline
Outline

  • 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


Cs setup
CS Setup

  • K non-zero entries at least 2K measurements

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


Brief introduction to cs
Brief Introduction to CS

  • 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


Na ve extension to analog domain
Naïve Extension to Analog Domain

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


Na ve extension to analog domain1
Naïve Extension to Analog Domain

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


Goals
Goals

  • 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)


Multiband sensing
Multiband “Sensing”

(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


Rate requirements
Rate Requirements

  • 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


Sampling
Sampling

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


The sampler
The Sampler

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 ( )


Paradigm
Paradigm

Solve finiteproblem

Reconstruct

0

S = non-zero rows

1

2

3

4

5

6


Continuous to finite
Continuous to Finite

Solve finiteproblem

Reconstruct

CTF block

MMV

  • span a finite space

  • Any basis preserves the sparsity

Continuous

Finite


2 words on solving mmv
2-Words on Solving MMV

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.


Algorithm
Algorithm

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


Framework analog compressed sensing
Framework: Analog Compressed Sensing

(Eldar 2008)

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

Subspace

generators


Framework analog compressed sensing1
Framework: Analog Compressed Sensing

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


Framework analog compressed sensing2
Framework: Analog Compressed Sensing

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


Framework analog compressed sensing3
Framework: Analog Compressed Sensing

Low sampling rate = p/TPre-compression

System B

CTF

Theorem

Eldar (2008)


Simulations
Simulations

Reconstruction filter

Signal

Amplitude

Amplitude

Output

Time (nSecs)

Time (nSecs)


Simulations1

Minimal rate

Minimal rate

Simulations

Sampling rate

Sampling rate

Brute-Force

M-OMP


Simulations2
Simulations

0% Recovery

100% Recovery

0% Recovery

100% Recovery

Noise-free

Sampling rate

Sampling rate

SBR4

SBR2

Empirical recovery rate


Multi coset limitations
Multi-Coset Limitations

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


Efficient sampling
Efficient Sampling

(Mishali, Eldar, Tropp 2008)

Efficientimplementation

Use CTF


Hardware implementation
Hardware Implementation

A few first steps…


Pairs of bases
Pairs Of Bases

  • 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!


Uncertainty relations
Uncertainty Relations

  • How sparse can be in each basis?

  • Finite setting: vector in

    Elad and Brukstein 2002

Different bases

Uncertainty

relation


Analog uncertainty principle

Theorem

Eldar (2008)

Analog Uncertainty Principle

Theorem

Eldar (2008)


Bases with minimal coherence
Bases With Minimal Coherence

In the DFT domain

Fourier

Spikes

What are the analog counterparts ?

  • Constant magnitude

  • Modulation

  • “Single” component

  • Shifts


Analog setting bandlimited signals
Analog Setting: Bandlimited Signals

  • Minimal coherence:

  • Tightness:


Finding sparse representations
Finding Sparse Representations

  • 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


Conclusion
Conclusion

  • 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


Some things should remain at the nyquist rate
Some Things Should Remain At The Nyquist Rate

Thank you

Thank you

High-rate


References
References

  • 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.


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