Sampled-data Control and Signal Processing – Beyond the Shannon Paradigm Workshop in honor of Eduardo Sontag on the occasion of his 60th birthday
Yutaka Yamamoto
[email protected] www-ics.acs.i.kyoto-u.ac.jp May 23, 2011
SontagFest
1
Thanks to My schoolmate, dear friend, colleague, and even a teacher
One of the very rare pictures of Eduardo with a Tie： at my (YY) fest
Outline Current signal processing paradigm Via Shannon ⇒ Upper limit in high frequencies
CAN BE SAVED via sampled-data control theory Some examples
May 23, 2011
SontagFest
3
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.)
May 23, 2011
SontagFest
4
Let’s first listen to a demo アプリケーション
Red： Original（up to 22kHz） Blue： downsampled to 11k, and then processed 4 times upsampled via YY filter Did you hear the difference? May 23, 2011
SontagFest
5
Part I: Current digital signal processing – Basics☺
May 23, 2011
SontagFest
6
Sampling continuous-time signals
0
h
2h
3h
4h
5h
6h
7h
This does not produce a sound
May 23, 2011
0
h
2h
3h
4h 5h SontagFest
6h
7h
7
Hold device is necessary Simple 0-order hold 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
0
h
2h
3h
4h
5h
6h
7h
h
2h
3h
4h
5h
6h
7h
Questions
0
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 May 23, 2011
SontagFest
10
Sampling Theorem Band limiting hypothesis ⇒ unique recovery
Ideal Filter
May 23, 2011
π/h
SontagFest
ω
11
Band-limiting filter
CD recording Freq. domain energy distribution
Recorded signal
ω Sampling frequency: 44.1kHz Nyquist frequency: 22.05kHz Alleged audible limit: 20kHz May 23, 2011
SontagFest
Nyquist freq.
12
Digital Recording (CD): sharp anti-aliasing filter No signal components beyond 20kHz
But you won’t be able to hear them anyway??
Very sharp anti-aliasing filter
Effect of a band-limiting filter Big amount of ringing due to the Gibbs phenomenon
Very unnatural sound of CD May 23, 2011
SontagFest
14
Mosquito Noise-another Gibbs phoenomenon
Truncated freq. response
モスキートノイズ Mosquito noise
May 23, 2011
SontagFest
15
What can we do?
May 23, 2011
SontagFest
16
Part II: Review of Sampleddata Control Theory
May 23, 2011
SontagFest
17
Sampled-data Control Systems – What are they? • Continuous-time plant Discrete-time controller • sample/hold devices H
K(z)
P(s)
Optimal platform for digital signal processing Problem: mixture of continuous- and discrete-time P(s): signal generator; K(z): digital filter May 23, 2011
SontagFest
18
Difficulties Plant P(s) is continuous-time Controller K(z) is discrete-time The overall system is not timeinvariant No transfer function No steady-state response No frequency response May 23, 2011
SontagFest
19
Response against a sinusoid r(t) = sin(1+ 20π )t
May 23, 2011
e −2 h − 1 2( z − e − 2 h )
SontagFest
H
1 s2 +1
20
Response
v(θ ) = sin(1 + 20π )θ
What to do? & solutions A new technique: lifting (1990) that turns SD system to discretetime LTI
∃ digital controller that makes cont.-time performance optimal May 23, 2011
SontagFest
22
Lifting of Functions f(t)
f 0 (θ )
f1 (θ )
f 2 (θ )
f 3 (θ )
Does this make a difference? ---Yes
H
2
w(t )
optimization
y (t )
u (t )
1 2 s + s +1
Sh
yd [k ]
K [z ]
u d [k ]
a) Discrete-time H2 with no intersample consideration b) sampled-data design
Hh
Time Response Discrete-time H2 design Sampled-data H2 design
a) Discrete-time H2 with no intersample consideration b) sampled-data design May 23, 2011
SontagFest
25
Can this be used for signal processing?
May 23, 2011
SontagFest
26
Part III: How can sampled-data theory help?
May 23, 2011
SontagFest
27
With a little bit of a priori information…
0 .8 ×
May 23, 2011
+ 0.2 ×
SontagFest
28
Utilizing analog characteristic conventional
Freq. domain energy distribution
Imaging new
ω1
ω
ω2 = 2π / h − ω1
Nyquist freq.
May 23, 2011
SontagFest
29
Original frequency response
upsample
Interpolate with zeros
Imaging components
Filtering
Sampled-data Design Model Contrinuous-time delay
Exogenous signals ∈ L2
w
F (s )
Band-limiting filter (musical instruments)
e
Reconstruction error
− mhs
-e
h sampling
↑M
K (z )
H h/
upsampler
Signal reconstruction
Problem: Find K[z] satisfying Sampled-data H∞ control problem
M
+
Interpolator via the proposed method Proposed Square wave resp.
May 23, 2011
Virtually no ringing
SontagFest
32
Response of the Johnston filter Big amount of ringing due to the Gibbs phenomenon
May 23, 2011
SontagFest
33
Part IV: Application to Sound Restoration
May 23, 2011
SontagFest
34
Sound restoration
アプリケーション
アプリケーション
May 23, 2011
SontagFest
35
YYLab
May 23, 2011
SontagFest
36
Analog Output DSP (TI C6713)
Digital readout (44.1kHz) via optical Fiber cable May 23, 2011
SontagFest
37
Example in MD(mini disk) players
MDLP4(66kbps)
Faithful recover of high. Freq.
After “YY”
More natural high freq. response By the courtesy of SANYO Corporation
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.
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
PEAQ値
0.0
AAC, 128kbps+YY
http://en.wikipedia.org/wiki/PEAQ By the courtesy of SANYO corporation
-1.0
-0.521
AAC, 128kbps
-0.527 -0.753
AAC, 96kbps+YY
-0.804
WMA, 128kbps
-0.831
MP3, 128kbps
-1.041
WMA, 96kbps+YY
-1.104
MP3, 96kbps+YY WMA, 96kbps AAC, 64kbps+YY MP3, 96kbps
-2.0
-2.5
good
MP3, 128kbps+YY
AAC, 96kbps
-1.5
-0.381
WMA, 128kbps+YY
WMA, 64kbps+YY
Note how YY improves the sound quality
-0.5
-1.386
bad
-1.495 -1.726 -1.847 -1.960 -2.191
AAC, 64kbps
-2.700
WMA, 64kbps
-2.759
Compression formats: MP3, AAC, WMA Bitrates: 64kbps, 96kbps, 128kbps Showing average values
-3.0
Part V: Application to Images
May 23, 2011
SontagFest
40
Same Problems as Sounds Block and Mosquito noise Lack of sufficient bandwidth Mosquito noise – Gibbs phenomenon Can sampled-data filter help?
May 23, 2011
SontagFest
41
Original
⇓ 2 downsample and hold
4times upsample+ twice downsample via YY
YYa May 23, 2011
Interpolation Via equiripple filter
SontagFest
42
Another application: How can we zoom “digitally”?
May 23, 2011
SontagFest
43
Interpolation via bicubic filter
May 23, 2011
SontagFest
44
Interpolation via sampled-data filter
May 23, 2011
SontagFest
45
Summarizing Analog signal generator model Error frequency response to be minimized (doesn’t exist in the conventional approach) ⇐ sampled-data H∞ control
May 23, 2011
SontagFest
46
May 23, 2011
SontagFest
47
