FrFT and Time-Frequency Distribution 分數傅立葉轉換與時頻分析 Guo-Cyuan Guo 郭國銓 指導教授 :Jian Jiun Ding 丁建均 Institute of Communications Engineering National Taiwan University Feb., 2008

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

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

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

DISP LAB 5 Introduction FTFrFTLCT

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

DISP LAB 7 Fractional Fourier Transform FT 0.1 FT ?

DISP LAB 8 FrFT & Linear Canonical Transform Definition:

DISP LAB 9 FrFT (cont’)

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

DISP LAB 11 Time-Frequency Distribution Short Time Fourier Transform(STFT) Gabor transform Wigner Distribution(WD)

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

DISP LAB 13 T-F Distribution(cont’) GaborWDFGabor-Wigner

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

DISP LAB 15 Filter Design

DISP LAB 16 Filter Design(cont’) u v u v u v u v u v u v

DISP LAB 17 Fourier Optics output planeinput plane output planeinput plane

DISP LAB 18 Fourier Optics(cont’) Through free space: output plane input plane

DISP LAB 19 Fourier Optics(cont’) Through thin lens output plane input plane

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

DISP LAB 21 Fourier Optics(cont’) output planeinput plane

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

DISP LAB 23 DFrFT Definition1: Definition2:

DISP LAB 24 DFrFT Definition3:

DISP LAB 25 DFrFT

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

DISP LAB 27 Pronounce Pulmonary alveolus Resonant cavityvoice Random sequence generator voiced Periodic pulse train generator unvoiced x[n] Vocal Tract Model G

DISP LAB 28 Hearing Frequency …… Weighting Bark Scale

DISP LAB 29 Masking Effect Sound Pressure Level Frequency Masking signal Masked signals Unmasked signal Hearing threshold Masking threshold

DISP LAB 30 MFCC Speech signal x(n) Pre-emphasis Window DFT Mel filter bank DCT Energy Derivatives MFCC

DISP LAB 31 Music Sim.

DISP LAB 32 Music Sim.

DISP LAB 33 Problems The computation problem Real time Resolution Harmonics

DISP LAB 34 Acoustics Signals ㄞㄟㄠㄡ

DISP LAB 35 Problems Computation Resolution Frame decision Correlation

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

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

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, [2] J. J. Ding, Research of Fractional Fourier Transform and Linear Canonical Transform, Ph.D. thesis, National Taiwan University, Taipei, Taiwan, R.O.C, [3] S. Qian and D. Chen, Joint Time-Frequency Analysis: Methods and Applications, Prentice Hall, N.J., [4] R. L. Allen and D. W. Mills, Signal Analysis: Time, Frequency, Scale, and Structure, Wiley- Interscience, NJ, [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 R1. [6] X. G. Xia, “On Bandlimited Signals with Fractional Fourier Transform,” IEEE Signal Processing Letters, Vol. 3, No. 3, March [7] P. Andres, W. D. Furlan and G. Saavedra, “Variable Fractional Fourier Processor: A Simple Implementation,” J. Opt. Soc. Am. A, Vol. 14, p , No. 4, April [8] H. M. Ozaktas and D. Mendlovic, “Fractional Fourier Optics,” J. Opt. Soc. Am. A, Vol. 12, p , No. 4, April [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 , July [10] L. R. Rabiner and R. W. Schafer, Digital Processing of Speech Signals, Pren-tice-Hall, [11] 王小川, 語音訊號處理, 全華科技圖書股份有限公司, Taipei, [12] A. Klapuri, “Signal Processing Methods for the Automatic Transcription of Mu-sic,” Ph. D thesis, Tampere University of Technology, Tampere, March 2004.