Sampled-data Control and Signal Processing
Transcript of Sampled-data Control and Signal Processing
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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
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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 paradigmVia Shannon⇒ Upper limit in high frequencies
CAN BE SAVED via sampled-data control theorySome examples
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May 23, 2011 SontagFest 4
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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May 23, 2011 SontagFest 6
Part I: Current digital signal processing – Basics☺
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May 23, 2011 SontagFest 7
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
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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
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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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May 23, 2011 SontagFest 12
CD recording
Nyquist freq.
ω
Freq. domain energy distribution
Recorded signal
Band-limiting filter
Sampling frequency: 44.1kHzNyquist frequency: 22.05kHzAlleged audible limit: 20kHz
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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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May 23, 2011 SontagFest 17
Part II: Review of Sampled-data Control Theory
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May 23, 2011 SontagFest 18
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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May 23, 2011 SontagFest 20
Response against a sinusoid
H 11
2 +s)(212
2
h
h
eze
−
−
−−
ttr )201sin()( π+=
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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
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Lifting of Functions
f(t)
)(0 θf)(1 θf )(2 θf
)(3 θf
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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
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Imaging components
upsample
Original frequency response
Filtering
Interpolate with zeros
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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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May 23, 2011 SontagFest 36
YYLab
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DSP(TI
C6713)
Analog Output
Digital readout (44.1kHz) via optical Fiber
cable
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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.
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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
-3.0-2.5-2.0-1.5-1.0-0.50.0
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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May 23, 2011 SontagFest 41
Same Problems as Sounds
Block and Mosquito noiseLack of sufficient bandwidthMosquito noise – Gibbs phenomenonCan sampled-data filter help?
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May 23, 2011 SontagFest 42
Original ⇓ 2 downsampleand hold
YYa
InterpolationVia equiripplefilter
4times upsample+ twice downsamplevia YY
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May 23, 2011 SontagFest 43
Another application:How can we zoom “digitally”?
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May 23, 2011 SontagFest 44
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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