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

Sampled-data Control and Signal Processing – Beyond the Shannon Paradigm Workshop in honor of Eduardo Sontag on the occasion of his 60 th birthday Yutaka Yamamoto yy@i.kyoto-u.ac.jp 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

7 produce a sound This does not Sampling continuous-time 7h 7h 6h 6h 5h 5h SontagFest 4h 4h 3h 3h 2h 2h signals h h 0 0 May 23, 2011

Hold device is necessary Simple 0-order hold 0 h 2h 3h 5h 6h 7h 4h Old CD players 0 h 2h 3h 5h 6h 7h 4h Oversampling DA converter More recent players

7h 7h 6h 6h 5h 5h 4h 4h 3h 3h 2h 2h h h Questions 0 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 I deal Filter ω π/ h May 23, 2011 SontagFest 11

Band-limiting filter CD recording Freq. domain energy distribution Recorded signal ω Nyquist freq. Sampling frequency: 44.1kHz Nyquist frequency: 22.05kHz Alleged audible limit: 20kHz May 23, 2011 SontagFest 12

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

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

16 What can we do? SontagFest May 23, 2011

Part II: Review of Sampled- data 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 time- invariant � No transfer function � No steady-state response � No frequency response May 23, 2011 SontagFest 19

Response against a sinusoid = + π r ( t ) sin( 1 20 ) t − − 2 h 1 e 1 H 2 + − − 2 h 2 ( z e ) s 1 May 23, 2011 SontagFest 20

θ ) π 20 + 1 sin( = ) θ ( v Response

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 May 23, 2011 SontagFest 22

Lifting of Functions f(t) 1 θ f ( ) 2 θ f ( ) 0 θ f ( ) 3 θ f ( )

Does this make a difference? ---Yes 2 w ( t ) H optimization y ( t ) u ( t ) 1 + s + 2 s 1 y d [ k ] u d [ k ] S K [ z ] H h h a) Discrete-time H 2 with no intersample consideration b) sampled-data design

Time Response Discrete-time H2 design Sampled-data H2 design a) Discrete-time H 2 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

28 With a little bit of a priori × SontagFest 2 . 0 information… + × 8 . 0 May 23, 2011

Utilizing analog characteristic conventional Freq. domain energy distribution Imaging new ω ω ω = π − ω 2 / h 1 2 1 Nyquist freq. May 23, 2011 SontagFest 29

Original frequency response upsample Interpolate with zeros Imaging components Filtering

Sampled-data Design Model Reconstruction Contrinuous-time error delay e − mhs Exogenous signals ∈ L 2 - h w e ↑ K ( z ) H M F ( s ) h / M sampling + upsampler Band-limiting filter (musical instruments) Signal reconstruction Problem: Find K[z] satisfying Sampled-data H ∞ control problem

Interpolator via the proposed method Proposed Virtually no ringing Square wave resp. May 23, 2011 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

36 SontagFest YYLab May 23, 2011

Analog Output DSP (TI C6713) Digital readout (44.1kHz) via optical Fiber cable May 23, 2011 SontagFest 37

Example in MD(mini disk) players 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 PEAQ値 Tested on 100 compresed � 0.0 -0.5 -1.0 -1.5 -2.0 -2.5 -3.0 music sources via PEAQ AAC, 128kbps+YY ( Perceptual Evaluation of -0.381 good WMA, 128kbps+YY -0.521 Audio Quality ) AAC, 128kbps -0.527 PEAQ values: � MP3, 128kbps+YY -0.753 0 … indistinguishable from AAC, 96kbps+YY -0.804 WMA, 128kbps CD -0.831 MP3, 128kbps -1.041 -1 … distinguishable but does WMA, 96kbps+YY -1.104 not bother the listener AAC, 96kbps -1.386 bad -2 … not disturbing MP3, 96kbps+YY -1.495 -3 … disturbing WMA, 96kbps -1.726 AAC, 64kbps+YY -1.847 -4 … very disturbing WMA, 64kbps+YY -1.960 MP3, 96kbps -2.191 Note how YY improves � AAC, 64kbps -2.700 the sound quality WMA, 64kbps -2.759 http://en.wikipedia.org/wiki/PEAQ Compression formats: MP3, AAC, WMA Bitrates: 64kbps, 96kbps, 128kbps By the courtesy of SANYO Showing average values corporation

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 I nterpolation and hold Via equiripple filter 4times upsample+ twice downsample via YY YYa May 23, 2011 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

47 SontagFest May 23, 2011