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
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 May 23, 2011 SontagFest 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
Questions 0 h 2h 3h 4h 5h 6h 7h 0 h 2h 3h 4h 5h 6h 7h
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 π/h May 23, 2011 SontagFest 11 ω
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. May 23, 2011 SontagFest 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 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 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 e 2( z 2h 1 2 e h ) H 1 s 2 + 1 May 23, 2011 SontagFest 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 ) f 1( θ ) f ( ) 2 θ f 3 ( θ )
Does this make a difference? ---Yes H 2 optimization w(t) y(t) s 2 1 + 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
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
With a little bit of a priori information 0.8 + 0.2 May 23, 2011 SontagFest 28
Utilizing analog characteristic Freq. domain energy distribution conventional Imaging new ω 1 Nyquist freq. 2 π / ω ω = 2 h ω 1 May 23, 2011 SontagFest 29
Original frequency response upsample Interpolate with zeros Imaging components Filtering
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
Interpolator via the proposed method Proposed Square wave resp. Virtually no ringing 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
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 0.0-0.5-1.0 (Perceptual Evaluation of Audio Quality) -0.381-0.521 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 http://en.wikipedia.org/wiki/peaq 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 -0.527-0.753-0.804-0.831-1.041-1.104-1.386-1.495-1.726-1.847-1.960-2.191 PEAQ 値 -1.5-2.700-2.759-2.0-2.5 good bad 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 Interpolation Via equiripple filter YYa 4times upsample+ twice downsample via YY 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
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