May 23, 2011 SontagFest 1
Sampled-data Control and Signal Processing – Beyond the Shannon Paradigm
Yutaka [email protected]
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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