May 23, 2011 SontagFest 1

Sampled-data Control and Signal Processing – Beyond the Shannon Paradigm

Yutaka Yamamotoyy@i.kyoto-u.ac.jp

www-ics.acs.i.kyoto-u.ac.jp

Workshop in honor of Eduardo Sontag on the occasion of his 60th birthday

Thanks to

One of the very rare pictures of Eduardo with a Tie：at my (YY) fest

My schoolmate, dear friend,colleague, and even a teacher

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Outline

Current signal processing paradigmVia Shannon⇒ Upper limit in high frequencies

CAN BE SAVED via sampled-data control theorySome examples

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Message of this talk

We can do better in signal processing using sampled-data control theory

⇒　Optimal recovery of freq. components beyond the Nyquistfreq. (=1/2 of sampling freq.)

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Let’s first listen to a demo

Red：　Original（up to 22kHz）Blue：　downsampled to 11k, and then processed 4 times upsampled via YY filter

アプリケーション

Did you hear the difference?

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Part I: Current digital signal processing – Basics☺

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Sampling continuous-time signals

0 h 2h 3h 4h 5h 6h 7h

0 h 2h 3h 4h 5h 6h 7h

This does not produce a sound

0 h 2h 3h 4h 5h 6h 7h

Old CD players

0 h 2h 3h 4h 5h 6h 7h

More recent players

Oversampling DA converter

Hold device is necessary

Simple 0-orderhold

0 h 2h 3h 4h 5h 6h 7h

0 h 2h 3h 4h 5h 6h 7h

Questions

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Typical problem: Sampling → Aliasing

Intersample information can be lostIf no high-freq. components beyond the Nyquist frequency (=1/2 of sampling freq.) →unique restoration　 → Whittaker-Shannon-Someya sampling theorem

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Sampling TheoremBand limiting hypothesis ⇒ unique recovery

ωπ/h

Ideal Filter

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CD recording

Nyquist freq.

ω

Freq. domain energy distribution

Recorded signal

Band-limiting filter

Sampling frequency: 44.1kHzNyquist frequency: 22.05kHzAlleged audible limit: 20kHz

Digital Recording (CD): sharp anti-aliasing filterNo signal components beyond 20kHz

Very sharp anti-aliasing filter

But you won’t be able tohear them anyway??

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Effect of a band-limiting filterBig amount of ringing dueto the Gibbs phenomenon

Very unnatural sound of CD

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Mosquito Noise-another Gibbs phoenomenon

Truncated freq. response

モスキートノイズ

Mosquito noise

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What can we do?

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Part II: Review of Sampled-data Control Theory

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Sampled-data Control Systems – What are they?

Discrete-time controller

P(s)K(z)

• Continuous-time plant

• sample/hold devices

H

Optimal platform for digital signal processing

P(s): signal generator; K(z): digital filterProblem: mixture of continuous- and discrete-time

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DifficultiesPlant P(s) is continuous-timeController K(z) is discrete-timeThe overall system is not time-invariant

No transfer functionNo steady-state responseNo frequency response

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Response against a sinusoid

H 11

2 +s)(212

2

h

h

eze

−

−

−−

ttr )201sin()( π+=

θπθ )201sin()( +=vResponse

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What to do? & solutions

A new technique: lifting (1990)

that turns SD system to discrete-

time LTI

∃ digital controller that makes

cont.-time performance optimal

Lifting of Functions

f(t)

)(0 θf)(1 θf )(2 θf

)(3 θf

Does this make a difference? ---Yes

optimization2H

11

2 ++ ss

][zKhS hH

)(ty

)(tw

)(tu

][kyd ][kud

a) Discrete-time H2 with no intersample considerationb) sampled-data design

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Time ResponseDiscrete-time H2 designSampled-data H2 design

a) Discrete-time H2 with no intersample considerationb) sampled-data design

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Can this be used for signal processing?

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Part III: How can sampled-data theory help?

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With a little bit of a priori information…

×8.0 + ×2.0

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Utilizing analog characteristic

Imaging

Nyquist freq.

ω1ω

12 /2 ωπω −= h

Freq. domain energy distribution

conventional

new

Imaging components

upsample

Original frequency response

Filtering

Interpolate with zeros

Sampled-data Design Model

Problem:

)(sF M↑ )(zK Mh /Hh

+

-

mhse−

w e

upsamplersampling

Exogenous signals ∈ L2

Band-limiting filter (musical instruments)

Contrinuous-time delay

Signal reconstruction

Reconstruction error

Sampled-data H∞ control problem

Find K[z] satisfying

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Interpolator via the proposed method

ProposedSquare wave resp.

Virtually no ringing

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Response of the Johnston filter

Big amount of ringing dueto the Gibbs phenomenon

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Part IV: Application to Sound Restoration

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アプリケーション

Sound restoration

アプリケーション

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YYLab

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DSP(TI

C6713)

Analog Output

Digital readout (44.1kHz) via optical Fiber

cable

MDLP4(66kbps)

Example in MD(mini disk) playersExample in MD(mini disk) players

This “YY filter” is implemented in custom LSI sound chips by SANYO Coop., and being used in MP 3 players, mobile phones, voice recorders. The cumulative sale has reached over 20 million units.

By the courtesy ofSANYO Corporation

After “YY” More natural high freq. response

Faithful recover of high. Freq.

-0.381

-0.521

-0.527

-0.753

-0.804

-0.831

-1.041

-1.104

-1.386

-1.495

-1.726

-1.847

-1.960

-2.191

-2.700

-2.759

-3.0-2.5-2.0-1.5-1.0-0.50.0

AAC, 128kbps+YY

WMA, 128kbps+YY

AAC, 128kbps

MP3, 128kbps+YY

AAC, 96kbps+YY

WMA, 128kbps

MP3, 128kbps

WMA, 96kbps+YY

AAC, 96kbps

MP3, 96kbps+YY

WMA, 96kbps

AAC, 64kbps+YY

WMA, 64kbps+YY

MP3, 96kbps

AAC, 64kbps

WMA, 64kbps

PEAQ値

Effect evaluation on compressed audio via PEAQ program

Tested on 100 compresedmusic sources via PEAQ(Perceptual Evaluation of Audio Quality)PEAQ values: 0…indistinguishable from

CD-1…distinguishable but does

not bother the listener-2…not disturbing-3…disturbing-4…very disturbing

Note how YY improves the sound quality

Compression formats: MP3, AAC, WMABitrates: 64kbps, 96kbps, 128kbpsShowing average values

good

bad

By the courtesy of SANYO corporation

http://en.wikipedia.org/wiki/PEAQ

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Part V: Application to Images

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Same Problems as Sounds

Block and Mosquito noiseLack of sufficient bandwidthMosquito noise – Gibbs phenomenonCan sampled-data filter help?

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Original ⇓ 2 downsampleand hold

YYa

InterpolationVia equiripplefilter

4times upsample+ twice downsamplevia YY

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Another application:How can we zoom “digitally”?

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Interpolation via bicubic filter

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Interpolation via sampled-data filter

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Summarizing

Analog signal generator modelError frequency response to be minimized (doesn’t exist in the conventional approach)⇐ sampled-data H∞ control

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