From Sampled-data Control to Signal Processing

Transcription

1 From Sampled-data Control to Signal Processing Beyond the Shannon Paradigm Yutaka Yamamoto www-ics.acs.i.kyoto-u.ac.jp Sept. 10, 2010 MSC2010 Yokohama 1

2 Thanks to Masaaki Nagahara, Kyoto U. Pramod Khargonekar, U. Florida Koji Fujiyama, SANYO Co. and of course, my mentors, many friends, colleagues who taught me many things. Sept. 10, 2010 MSC 2010 Yokohama 2

3 Message of this talk You can do better signal processing using sampled-data control theory Optimal recovery of freq. components beyond the Nyquist freq. (=1/2 of sampling freq.) Sept. 10, 2010 MSC 2010 Yokohama 3

4 Outline Problems arising from the band-limiting hypothesis in digital signal processing Sampled-data control theory; review Sound reconstruction beyond the Nyquist frequency Generalized sampling theorem Application to images Addendum: how your patent can materialize? - collaboration with industry Sept. 10, 2010 MSC 2010 Yokohama 4

5 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? Sept. 10, 2010 MSC 2010 Yokohama 5

6 Part I: Current digital signal processing Basics Sept. 10, 2010 MSC2010 Yokohama 6

7 Sampling continuous-time signals 0 h 2h 3h 4h 5h 6h 7h This does not produce a sound 0 h 2h 3h 4h 5h 6h 7h Sept. 10, 2010 MSC 2010 Yokohama 7

8 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

9 0 h 2h 3h 4h 5h 6h 7h 0 h 2h 3h 4h 5h 6h 7h

10 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 Sept. 10, 2010 MSC 2010 Yokohama 10

11 Sampling Theorem Band limiting hypothesis unique recovery Ideal Filter π/h Sept. 10, 2010 MSC 2010 Yokohama 11 ω

12 Problems Band limiting hypothesis unrealistic The formula not causal (you need infinitely many future sampled values) Very slow convergence Must approximate the ideal filter in practice Sharp cut-off characteristics High degree of filters Distortion due to the Gibbs phenomenon Sept. 10, 2010 MSC 2010 Yokohama 12

13 Real signals: how much high. freq. are there FFT of an Analog Record: very wide range 20kHz

14 In contrast: Digital Recording (CD): sharp anti-aliasing filter No signal beyond 20kHz But you won t be able to hear them anyway?? Very sharp anti-aliasing filter

15 CD recording Band-limiting filter Freq. domain energy distribution Recorded signal ω Sampling frequency: 44.1kHz Nyquist frequency: 22.05kHz Alleged audible limit: 20kHz Nyquist freq. Sept. 10, 2010 MSC 2010 Yokohama 15

16 Effect of a band-limiting filter Big amount of ringing due to the Gibbs phenomenon Very unnatural sound of CD Sept. 10, 2010 MSC 2010 Yokohama 16

17 Mosquito Noise Truncated freq. response モスキートノイズ Mosquito noise Sept. 10, 2010 MSC 2010 Yokohama 17

18 Part II: Review of Sampleddata Control Theory Sept. 10, 2010 MSC2010 Yokohama 18

19 Sampled-data Control Systems What are they? Continuous-time plant Discrete-time controller sample/hold devices K(z) H P(s) Optimal platform for digital signal processing Problem: mixture of continuous- and discrete-time P(s): signal generator; K(z): digital filter Sept. 10, 2010 MSC 2010 Yokohama 19

20 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 Sept. 10, 2010 MSC 2010 Yokohama 20

21 Response against a sinusoid r ( t) = sin( 1+ 20π ) t e 2( z 2h 1 2 e h ) H 1 s Sept. 10, 2010 MSC 2010 Yokohama 21

22 Response v( θ ) = sin(1 + 20π ) θ

23 What to do? & solutions A new technique: lifting that turns SD system to discrete-time LTI digital controller that makes cont.-time performance optimal Sept. 10, 2010 MSC 2010 Yokohama 23

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

25 Time Response Discrete-time H2 design Sampled-data H2 design a) Discrete-time H 2 with no intersample consideration b) sampled-data design Sept. 10, 2010 MSC 2010 Yokohama 25

26 Part III: How can sampled-data theory help signal processing? Sept. 10, 2010 MSC2010 Yokohama 26

27 With a little bit of a priori information Sept. 10, 2010 MSC 2010 Yokohama 27

28 Utilizing analog characteristic Freq. domain energy distribution conventional Imaging new ω 1 Nyquist freq. 2 π / ω ω = 2 h ω 1 Sept. 10, 2010 MSC 2010 Yokohama 28

29 Overall decay envelope Sept. 10, 2010 MSC 2010 Yokohama 29

30 Original frequency response upsample Interpolate with zeros Imaging components Filtering

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

32 Example of F(s) Sampling period h=1, delay step m=2 F ( s) = ( s 1 + 1)(7.0187s + 1) f [khz] -32 Sept. 10, 2010 MSC 2010 Yokohama 32

33 Interpolator via the proposed method Proposed Square wave resp. Virtually no ringing Sept. 10, 2010 MSC 2010 Yokohama 33

34 Response of the Johnston filter Big amount of ringing due to the Gibbs phenomenon Sept. 10, 2010 MSC 2010 Yokohama 34

35 Comparison New Conventional Signal model (SD) Perfect Band-limiting Explicit error model No explicit error model Clear performance index pass-band, stop-band characterization phase information in the design no phase distortion considered Robustness can be considered only implicit in the design Sept. 10, 2010 MSC 2010 Yokohama 35

36 Comparison: Generalized sampling theorem (Unser et al.) Unser and co-workers (since around 1990) Using generalized hold and acquisition filters Generalized sampling theorem Survey: Unser 2000 (Trans. IEEE) Think analog, act digital (SP, 2005) Sept. 10, 2010 MSC 2010 Yokohama 36

37 General idea x (t) c ( ) c ( ) y t) = c ( k) ϕ ( t ) ϕ Q(z) ϕ2 1 1 k 2 k ( 2 2 k prefilter sampling filtering hold ϕ 1 : prefilter (our F(s)) Objective: reconstruct x(t) ill-posed Confine ourselves to the subsp. M 2 gen. by Idea similar to Wavelets Solution: Orthogonal (or oblique) Projection - generalized inverse Sept. 10, 2010 MSC 2010 Yokohama 37 ϕ 2

38 Does it work? M 2 is a very small subspace of L 2 Not necessary good for signals not in M 2 In contrast, our model: All signals are compactified to the image F* L 2 H sampled-data Big difference: sampled-data vs. discrete-time generalized inverse designed filter is often unstable Sept. 10, 2010 MSC 2010 Yokohama 38

39 A comparison (exp. Spline) Sept. 10, 2010 MSC 2010 Yokohama 39

40 Sampled-data H Sept. 10, 2010 MSC 2010 Yokohama 40

41 Part IV: Application to Sound Restoration Sept. 10, 2010 MSC2010 Yokohama 41

42 Sound restoration アプリケーション アプリケーション Sept. 10, 2010 MSC 2010 Yokohama 42

43 Lab Experiments ( ) Optical fiber digital readout DSP implementation of YY Sept. 10, 2010 MSC 2010 Yokohama 43

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46 Since 2006 Sept. 10, 2010 MSC 2010 Yokohama 46

47 Analog Output DSP (TI C6713 ) Digital readout (44.1kHz) via optical Fiber cable Sept. 10, 2010 MSC 2010 Yokohama 47

48 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 16 million units.

49 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 By the courtesy of SANYO corporation 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 good bad Compression formats: MP3, AAC, WMA Bitrates: 64kbps, 96kbps, 128kbps Showing average values -3.0

50 Part V: Application to Images Sept. 10, 2010 MSC2010 Yokohama 50

51 Same Problems as Sounds Block and Mosquito noise Lack of sufficient bandwidth Mosquito noise Gibbs phenomenon Can sampled-data filter help? Sept. 10, 2010 MSC 2010 Yokohama 51

52 Original 2 downsample and hold Interpolation Via equiripple filter YYa 4times upsample+ twice downsample via YY Sept. 10, 2010 MSC 2010 Yokohama 52

53 Another application: How can we zoom digitally? Sept. 10, 2010 MSC 2010 Yokohama 53

54 Interpolation via bicubic filter Sept. 10, 2010 MSC 2010 Yokohama 54

55 Interpolation via sampled-data filter Sept. 10, 2010 MSC 2010 Yokohama 55

56 Moving image application アプリケーション Reduction of so-called block noise Without blurring the boundaries of the image Sept. 10, 2010 MSC 2010 Yokohama 56

57 Final tips: How to commercialize your invention? Sept. 10, 2010 MSC2010 Yokohama 57

58 Invention and Practice Quite tricky subject I ll let you in on my secret?? The story goes: I ve invented a new algorithm, circuit, (you name it) Boy, what a genius I am!!! This should make me a billionaire! But Sept. 10, 2010 MSC 2010 Yokohama 58

59 Fact is: This usually doesn t happen. How come? Two cases: Your invention is not known too bad. Let s work more on publicity E.g. nice web pages people look at them There is someone who is willing to work with you, but Sept. 10, 2010 MSC 2010 Yokohama 59

60 Real Issues (problems) Often expected that inventors know everything (incl. implementation) A blind implementation is expected We (inventors) on the contrary, expect them to do everything for us This is B A D Sept. 10, 2010 MSC 2010 Yokohama 60

61 What to do? Best Scenario You (inventor) do what you know something about The industry side understands and needs to study the T H E O R Y In short, no free lunch (dinner?) In general, even if you find a competent person who is eager and willing to work, he/she is often surrounded by critical colleagues this is depressing They also need encouragement. Sept. 10, 2010 MSC 2010 Yokohama 61

62 My failure In the beginning, had no hardware to show Just a virtual demo no good Limit in human imagination Well, Prof. Yamamoto, what are other people saying about this? degrading Sad human nature not everyone is eager to be innovative Sept. 10, 2010 MSC 2010 Yokohama 62

63 My own experience with SANYO Very enthusiastic and competent collaborator on the SANYO side I knew something about the audio sound processing I set up a hardware demo An objective test (PEAQ) came around by chance These changed the scene completely Sept. 10, 2010 MSC 2010 Yokohama 63

64 Well, Opportunities abound. Control and signal processing are ubiquitous. With lots of preparation, and a little bit of luck, We can succeed. You can succeed. Be confident. Yes, you can! Sept. 10, 2010 MSC 2010 Yokohama 64

65 In summary Sampled-data theory (with analog signal generator model) allows us to recover intersample high-freq. signals Many problems can be handled by control theory Some knack in commercialization Sept. 10, 2010 MSC 2010 Yokohama 65

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