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### Compressive Sampling(of Analog Signals)

Moshe Mishali Yonina C. Eldar

Technion – Israel Institute of Technology

http://www.technion.ac.il/~moshikomoshiko@tx.technion.ac.il

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

Advanced topics in sampling (Course 049029)

Seminar talk – November 2008

Compression

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

Donoho

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Outline

- Mathematical background
- From discrete to analog
- Uncertainty principles for analog signals
- Discussion

References

- 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.
- M. Mishali and Y. C. Eldar, "Blind Multi-Band Signal Reconstruction: Compressed Sensing for Analog Signals," CCIT Report #639, Sep. 2007, EE Dept., Technion.
- Y. C. Eldar, "Compressed Sensing of Analog Signals", submitted to IEEE Trans. on Signal Processing, June 2008.
- Y. C. Eldar and M. Mishali, "Robust Recovery of Signals From a Union of Subspaces", arXiv.org 0807.4581, submitted to IEEE Trans. Inform. Theory, July 2008. #
- Y. C. Eldar, "Uncertainty Relations for Analog Signals", submitted to IEEE Trans. Inform. Theory, Sept. 2008.

Mathematical background

- Basic ideas of compressed sensing
- Single measurement model (SMV)
- Multiple- and Infinite- measurement models (MMV, IMV)
- The “Continuous to finite” block (CTF)

Compressed Sensing

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

Donoho

“sensing … as a way of extracting information about an object from a small number of randomly selected observations”

Candès et. al.

AnalogAudioSignal

Nyquist rateSampling

CompressedSensing

Compression(e.g. MP3)

High-rate

Low-rate

Concept

Goal: Identify the bucket with fake coins.

Weigh a coinfrom each bucket

Compression

Nyquist:

Bucket #

numbers

1 number

Weigh a linear combinationof coins from all buckets

Compressed Sensing:

Bucket #

1 number

Mathematical Tools

non-zero entries at least measurements

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

CS theory – on 2 slides

Compressed sensing (2003/4 and on) – Main results

is uniquely determined by

Donoho and Elad, 2003

Maximal cardinality of linearly independent column subsets

Hard to compute !

CS theory – on 2 slides

Compressed sensing (2003/4 and on) – Main results

is uniquely determined by

Donoho and Elad, 2003

with high probability

is random

Donoho, 2006 and Candès et. al., 2006

Convex and tractable

Donoho, 2006 and Candès et. al., 2006

Greedy algorithms: OMP, FOCUSS, etc.

NP-hard

Tropp, Cotter et. al. Chen et. al. and many other

Sparsity models

unknowns

measurements

MMV

Joint sparsity

SMV

IMV = Infinite Measurement Vectors (countable or uncountable) with joint sparsity prior

How can be found ?

Infinite many variables

Infinite many constraints

Exploit prior Reduce problem dimensions

Reduction Framework

Find a frame

for

Solve MMV

Theorem

Mishali and Eldar (2008)

Deterministicreduction

IMV

MMV

Infinite structure allows

CS for analog signals

From discrete to analog

- Naïve extension
- The basic ingredients of sampling theorem
- Sparse multiband model
- Rate requirements
- Multicoset sampling and unique representation
- Practical recovery with the CTF block
- Sparse union of shift-invariant model
- Design of sampling operator
- Reconstruction algorithm

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

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

Whittaker

1915

Nyquist

1928

Kotelnikov

1933

Shannon

1949

A step backwardEvery bandlimited signal ( Hertz)

can be perfectly reconstructed from uniform sampling

if the sampling rate is greater than

A step backward

Every bandlimited signal ( Hertz)

can be perfectly reconstructed from uniform sampling

if the sampling rate is greater than

Fundamental ingredients of a sampling theorm

- A signal model
- A minimal rate requirement
- Explicit sampling and reconstruction stages

Discrete Compressed Sensing

Analog Compressive Sampling

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

- More generally

only sequences are non-zero

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)

(Eldar 2008)

Multi-Band Sensing: Goals

bands

Sampling

Reconstruction

Analog

Infinite

Analog

Goal: Perfect reconstruction

Constraints:

- Minimal sampling rate
- Fully blind system

What is the minimal rate ?

What is the sensing mechanism ?

How to reconstruct from infinite sequences ?

Rate Requirement

- 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

- Subspace scenarios:
- 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)

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

Bresler et. al. (96,98,00,01)

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 ( )

Continuous to Finite

Solve finiteproblem

Reconstruct

CTF block

MMV

- span a finite space
- Any basis preserves the sparsity

Continuous

Finite

Algorithm

Perfect reconstruction at minimal rate

Blind system: band locations are unkown

Can be applied to CS of general analog signals

Works with other sampling techniques

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

CTF

Blind reconstruction flow

Multi-coset with

Universal

SBR4

Yes

CTF

No

SBR2

No

Bi-section

CTF

Yes

Uniform at

Ideal low-pass filter

Spectrum-blind

Sampling

Spectrum-blind

Reconstruction

Final reconstruction (non-blind)

Bresler et. al. (96,00)

Framework: Analog Compressed Sensing

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

Subspace

generators

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 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 Sensing

Low sampling rate = p/TPre-compression

System B

CTF

Theorem

Eldar (2008)

Simulations (2)

0% Recovery

100% Recovery

0% Recovery

100% Recovery

Noise-free

Sampling rate

Sampling rate

SBR4

SBR2

Empirical recovery rate

Break

(10 min. please)

Uncertainty principles

- Coherence and the discrete uncertainty principle
- Analog coherence and principles
- Achieving the lower coherence bound
- Uncertainty principles and sparse representations

The discrete uncertainty principle

Uncertainty principle

Discrete coherence

Which bases achieve the lowest coherence ?

Discrete to analog

- Shift invariant spaces
- Sparse representations

- Questions:
- What is the analog uncertainty principle ?
- Which bases has the lowest coherence ?
- Which signal achieves the lower uncertainty bound ?

Bases with minimal coherence

In the DFT domain

Fourier

Spikes

What are the analog counterparts ?

- Constant magnitude
- Modulation

- “Single” component
- Shifts

Bases with minimal coherence

In the frequency domain

Sparse representations

- In discrete setting

Sparse representations

- Analog counterparts

Undefined program !

But, can be transformed into an IMV model

Discussion

- IMV model as a fundamental tool for treating sparse analog signals
- Should quantify the DSP complexity of the CTF block
- Compare approach with the “analog” model
- Building blocks of analog CS framework.

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