[PPT] - Sampled-data Control and Signal Processing Beyond the Shannon PowerPoint Presentation

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May 23, 2011 SontagFest 1

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

Yutaka Yamamoto

yy@i.kyoto-u.ac.jp www-ics.acs.i.kyoto-u.ac.jp Workshop in honor of Eduardo Sontag

n the occasion of his 60th birthday

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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 paradigm

Via Shannon ⇒ Upper limit in high frequencies

CAN BE SAVED via sampled-data control

theory

Some 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 Nyquist

freq. (= 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

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

This does not produce a sound

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

Old CD players

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

More recent players

Oversampling DA converter

Hold device is necessary

Simple 0-order hold

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

Questions

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Typical problem: Sampling → Aliasing

Intersample information can be lost If 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 Theorem

Band limiting hypothesis ⇒ unique

recovery

ω

π/h I deal Filter

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May 23, 2011 SontagFest 12

CD recording

Nyquist freq.

ω

Freq. domain

energy distribution Recorded signal Band-limiting filter Sampling frequency: 44.1kHz Nyquist frequency: 22.05kHz Alleged audible limit: 20kHz

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Digital Recording (CD): sharp anti-aliasing filter No signal components beyond 20kHz Very sharp anti-aliasing filter But you won’t be able to hear them anyway??

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Effect of a band-limiting filter

Big amount of ringing due to 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 filter Problem: mixture of continuous- and discrete-time

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Difficulties

Plant P(s) is continuous-time Controller K(z) is discrete-time The overall system is not time-

invariant

No transfer function No steady-state response No frequency response

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May 23, 2011 SontagFest 20

Response against a sinusoid

H

1 1

2 +

s

) ( 2 1

2 2 h h

e z e

− −

t t r ) 20 1 sin( ) ( π + =

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θ π θ ) 20 1 sin( ) ( + = v

Response

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May 23, 2011 SontagFest 22

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

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Lifting of Functions

f(t)

) (

0 θ

f ) (

1 θ

f ) (

2 θ

f ) (

3 θ

f

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Does this make a difference?

--Yes
ptimization

2

H

1 1

2

+ + s s ] [z K

h

S

h

H ) (t y ) (t w ) (t u ] [k yd ] [k ud

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

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Time Response

Discrete-time H2 design Sampled-data H2 design

a) Discrete-time H2 with no intersample consideration b) 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 . +

× 2 .

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Utilizing analog characteristic

Imaging Nyquist freq.

ω

1

ω

1 2

/ 2 ω π ω − = h

Freq. domain

energy distribution conventional new

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Imaging components

upsample Original frequency response Filtering

Interpolate with zeros

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Sampled-data Design Model

Problem:

) (s F M ↑

) (z K

M h/

H

h +

mhs

e−

w e

upsampler sampling

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

Proposed Square wave resp.

Virtually no ringing

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Response of the Johnston filter

Big amount of ringing due to 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

ptical Fiber

cable

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MDLP4(66kbps)

Example in MD(mini disk) players Example 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

ver 20 million units.

By the courtesy of SANYO Corporation

After “YY”

More natural high

freq. response

Faithful recover of high. Freq.

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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
0.5

0.0

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 compresed

music 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, WMA Bitrates: 64kbps, 96kbps, 128kbps Showing 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 noise Lack of sufficient bandwidth Mosquito noise – Gibbs phenomenon Can sampled-data filter help?

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Original ⇓ 2 downsample and hold YYa I nterpolation Via equiripple filter

4times upsample+ twice downsample via 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 model Error frequency response to be

minimized (doesn’t exist in the conventional approach)

⇐ sampled-data H∞ control

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