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FrFT and Time-Frequency Distribution

DISP LAB. 2. Outline. IntroductionFrFT

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FrFT and Time-Frequency Distribution

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    1. FrFT and Time-Frequency Distribution ???????????? Guo-Cyuan Guo ??? ????:Jian Jiun Ding??? Institute of Communications Engineering National Taiwan University Feb., 2008

    2. DISP LAB 2 Outline Introduction FrFT & LCT Time-Frequency Distribution Applications DFrFT Acoustics & Music Signals Conclusions and Future works Reference

    3. DISP LAB 3 Outline Introduction FrFT & LCT Time-Frequency Distribution Applications DFrFT Acoustics & Music Signals Conclusions and Future works Reference

    4. DISP LAB 4 Introduction Fourier Transform(18-th century): Fractional Fourier Transform (FrFT): 1980 Victor Namias (Quantum mechanics) 1994 Almeida (Signal Processing) Ozaktas (Optics) LCT 1970 matrix optics— Fresnel transform Mathematics V. Namias eigenvalue – Hermite Gaussian…Green’s function, Stationary state Almeida ??? Portugal Ozaktas Turkey Fresnel transform Helmholtz equation,…Huygens-Fresnel principle ???? ?????? ?? FraunhoferV. Namias eigenvalue – Hermite Gaussian…Green’s function, Stationary state Almeida ??? Portugal Ozaktas Turkey Fresnel transform Helmholtz equation,…Huygens-Fresnel principle ???? ?????? ?? Fraunhofer

    5. DISP LAB 5 Introduction FT….alpha ? Like SVD matrix??? LCT more general math form of FrFT Since it came from T-F ???FT….alpha ? Like SVD matrix??? LCT more general math form of FrFT Since it came from T-F ???

    6. DISP LAB 6 Outline Introduction FrFT & LCT Time-Frequency Distribution Applications DFrFT Acoustics & Music Signals Conclusions and Future works Reference

    7. DISP LAB 7 Fractional Fourier Transform

    8. DISP LAB 8 FrFT & Linear Canonical Transform Definition:

    9. DISP LAB 9 FrFT (cont’)

    10. DISP LAB 10 Outline Introduction FrFT & LCT Time-Frequency Distribution Applications DFrFT Acoustics & Music Signals Conclusions and Future works Reference

    11. DISP LAB 11 Time-Frequency Distribution Short Time Fourier Transform(STFT) Gabor transform Wigner Distribution(WD) ????????????????? ????…?????? Cognitive Radio ????? TDMA,FDMA,CDMA….????????????????? ????…?????? Cognitive Radio ????? TDMA,FDMA,CDMA….

    12. DISP LAB 12 T-F Distribution(cont’) Input:

    13. DISP LAB 13 T-F Distribution(cont’)

    14. DISP LAB 14 Outline Introduction FrFT & LCT Time-Frequency Distribution Applications DFrFT Acoustics & Music Signals Conclusions and Future works Reference

    15. DISP LAB 15 Filter Design ??????….??????….

    16. DISP LAB 16 Filter Design(cont’) Chirp multiplication=freq ? Chirp conv = time change ??shift ????????…??Ding?paper… ???????lens???z-axis?Chirp multiplication=freq ? Chirp conv = time change ??shift ????????…??Ding?paper… ???????lens???z-axis?

    17. DISP LAB 17 Fourier Optics Rect -->sinc 2-D??????… Rect -->sinc 2-D??????…

    18. DISP LAB 18 Fourier Optics(cont’) Through free space:

    19. DISP LAB 19 Fourier Optics(cont’) Through thin lens

    20. DISP LAB 20 Fourier Optics(cont’) Through the gradient-index medium (GRIN) Reflection indexReflection index

    21. DISP LAB 21 Fourier Optics(cont’)

    22. DISP LAB 22 Outline Introduction FrFT & LCT Time-Frequency Distribution Applications DFrFT Acoustics & Music Signals Conclusions and Future works Reference

    23. DISP LAB 23 DFrFT Definition1: Definition2:

    24. DISP LAB 24 DFrFT Definition3:

    25. DISP LAB 25 DFrFT

    26. DISP LAB 26 Outline Introduction FrFT & LCT Time-Frequency Distribution Applications DFrFT Acoustics & Music Signals Conclusions and Future works Reference

    27. DISP LAB 27 Pronounce

    28. DISP LAB 28 Hearing

    29. DISP LAB 29 Masking Effect

    30. DISP LAB 30 MFCC

    31. DISP LAB 31 Music Sim.

    32. DISP LAB 32 Music Sim.

    33. DISP LAB 33 Problems The computation problem Real time Resolution Harmonics

    34. DISP LAB 34 Acoustics Signals ????

    35. DISP LAB 35 Problems Computation Resolution Frame decision Correlation

    36. DISP LAB 36 Outline Introduction FrFT & LCT Time-Frequency Distribution Applications DFrFT Acoustics & Music Signals Conclusions and Future works Reference

    37. DISP LAB 37 Conclusions and Future works FrFT & LCT &DFrFT Time-Frequency Distribution Applications Acoustics & Music Signals Fractional Fourier Series Discrete Time Fourier Transform Time-Frequency Resolution and Computation Music Autoscore

    38. DISP LAB 38 Reference [1] H.M. Ozaktas, Z. Zalevsky and M. A. Kutay, The fractional Fourier transform with Applications in Optics and Signal Processing, John Wiley & Sons, 2001. [2] J. J. Ding, Research of Fractional Fourier Transform and Linear Canonical Transform, Ph.D. thesis, National Taiwan University, Taipei, Taiwan, R.O.C, 2001. [3] S. Qian and D. Chen, Joint Time-Frequency Analysis: Methods and Applications, Prentice Hall, N.J., 1996. [4] R. L. Allen and D. W. Mills, Signal Analysis: Time, Frequency, Scale, and Structure, Wiley- Interscience, NJ, 2004. [5] S. C. Pei and J. J. Ding, “Relations between Gabor Transforms and Fractional Fourier Transforms and Their Applications for Signal Processing,” Revised Version: T-SP-04763- 2006.R1. [6] X. G. Xia, “On Bandlimited Signals with Fractional Fourier Transform,” IEEE Signal Processing Letters, Vol. 3, No. 3, March 1996. [7] P. Andres, W. D. Furlan and G. Saavedra, “Variable Fractional Fourier Processor: A Simple Implementation,” J. Opt. Soc. Am. A, Vol. 14, p.853-858, No. 4 , April 1997. [8] H. M. Ozaktas and D. Mendlovic, “Fractional Fourier Optics,” J. Opt. Soc. Am. A, Vol. 12, p.743-751, No. 4, April 1995. [9] D. Mendlovic, R. G. Dorsch, A. W. Lohmann, Z. Zalevsky, and C. Ferreira, “Optical Illustration of a Varied Fractional Fourier Transform Order and the Radon-Wigner Display,” Appl. Opt. Vol. 35, No. 20, 10, p.3925-3929, July 1996. [10] L. R. Rabiner and R. W. Schafer, Digital Processing of Speech Signals, Pren-tice-Hall, 1978. [11] ???, ??????, ????????????, Taipei, 2004. [12] A. Klapuri , “Signal Processing Methods for the Automatic Transcription of Mu-sic,” Ph. D thesis, Tampere University of Technology, Tampere, March 2004.

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