1. Sep.2008DISP Lab @MD5311 Time-Frequency Analysis 時頻分析  Speaker: Wen-Fu Wang 王文阜  Advisor: Jian-Jiun Ding 丁建均 教授  E-mail: r96942061@ntu.edu.tw  Graduate Institute of Communication Engineering  National Taiwan University, Taipei, Taiwan, ROC

2. Sep.2008DISP Lab @MD5312 Outline  Introduction  Short Time Fourier Transform  Gabor Transform  Wigner Distribution  Gabor-Wigner Transform  Cohen’s Class Time-Frequency Distribution

3. Sep.2008DISP Lab @MD5313 Outline  S Transform  Hilbert-Huang Transform  Applications of Time-Frequency Analysis  Conclusions  References

4. Sep.2008DISP Lab @MD5314 Introduction  Fourier Transform (FT)  Example : x(t) = cos( t) when t < 10, x(t) = cos(3 t) when 10  t < 20, x(t) = cos(2 t) when t  20

5. Sep.2008DISP Lab @MD5315 Introduction  Instantaneous Frequency If then the instantaneous frequency of f (t) are If order of >1, then instantaneous frequency varies with time

6. Sep.2008DISP Lab @MD5316 Introduction  Example : → chirp function Instantaneous frequency =

7. Sep.2008DISP Lab @MD5317 Introduction  Time-Frequency analysis  Example : t -axis f -axis amplitude

8. Sep.2008DISP Lab @MD5318 Short Time Fourier Transform  The earliest Time-Frequency representation was the short time Fourier transform (STFT)  This scheme divides the temporal signal into a series of small overlapping pieces.  The STFT of a function is defined by

9. Sep.2008DISP Lab @MD5319 Short Time Fourier Transform  is the window function.  The principle of STFT time s(t) h(t) FT time frequency STFT

10. Sep.2008DISP Lab @MD53110 Short Time Fourier Transform  Example :

11. Sep.2008DISP Lab @MD53111 Short Time Fourier Transform  Advantage: (1) Least computation time for digital implementation compared with other (2) Its ability to avoid cross-term problem

12. Sep.2008DISP Lab @MD53112 Gabor Transform  In fact, Gabor transform is a special case of STFT.  When the of STFT, it can be rewritten as  Why does it choose the Gaussian function as a window?

13. Sep.2008DISP Lab @MD53113 Gabor Transform  The principle of Gabor Transform time s(t) h(t) FT time frequency STFT

14. Sep.2008DISP Lab @MD53114 Gabor Transform  Example:

15. Sep.2008DISP Lab @MD53115 Gabor Transform  Advantage: (1) Its ability to avoid cross-term problem (2) The resolution is better than STFT

16. Sep.2008DISP Lab @MD53116 Wigner Distribution  The Wigner distribution is defined as  In terms of the spectrum, it is  is the Fourier transform of and * means the complex conjugate.

17. Sep.2008DISP Lab @MD53117 Wigner Distribution  WD has much better time-frequency resolution  WD is not a linear distribution  WD has more computation time  If the signal is composed by several time-frequency components, additional interference will be produced.

18. Sep.2008DISP Lab @MD53118 Wigner Distribution  Example:

19. Sep.2008DISP Lab @MD53119 Wigner Distribution  The inner interference is caused by interference between positive and negative frequency of the signal itself  In order to reduce to inner interference problem, Wigner Ville Distribution can be used

20. Sep.2008DISP Lab @MD53120 Wigner Distribution  The outer interference is caused by mutual interference of multi- component in signal  In order to reduce to outer interference problem, Modified Wigner Distribution can be used

21. Sep.2008DISP Lab @MD53121 Wigner Ville Distribution  Due the inner interference is caused by interference between positive and negative frequency of the signal itself  We can use analytic version signal to replace the original signal for filtering out negative frequency

22. Sep.2008DISP Lab @MD53122 Wigner Ville Distribution  We denote the analytic signal of real valued signal by  is Hilbert transform of  We can redefine WD by analytic signal

23. Sep.2008DISP Lab @MD53123 Modified Wigner Distribution  Due outer interference is caused by mutual interference of multi- component in signal  We can select suitable window function is a way to suppress the outer interference, but retain the sharpness of auto terms.

24. Sep.2008DISP Lab @MD53124 Modified Wigner Distribution  The Modified Wigner Distribution is defined as  When the window function, it also become Wigner distribution.

25. Sep.2008DISP Lab @MD53125 Modified Wigner Distribution  Example: WD MWD

26. Sep.2008DISP Lab @MD53126 Modified Wigner Ville Distribution  Combined form of the WVD and MWD  The advantage of WVD is filtering out inner interference  The advantage of MWD is suppressing the outer interference, but retain the sharpness of auto terms  The MWVD can avoid the inner and outer interference at the same time

27. Sep.2008DISP Lab @MD53127 Gabor-Wigner Transform  We have compared the properties of the WD and Gabor transform in previous section  The advantage of WD is its high clarity, and the disadvantage of WD is it’s the cross-term problem

28. Sep.2008DISP Lab @MD53128 Gabor-Wigner Transform  In contrast, the advantage of Gabor transform is its ability to avoid cross- term problem, but its clarity is not as good as that of the WDF  The Gabor-Wigner transform (GWT) can achieve the higher clarity and avoiding cross-term problem at the same time.

29. Sep.2008DISP Lab @MD53129 Gabor-Wigner Transform  Combined form of the WDF and the Gabor transform (a) (b) (c) (d)

30. Sep.2008DISP Lab @MD53130 Cohen’s Class Time-Frequency Distribution  The definition of the Cohen class is  is the ambiguity function  How does the Cohen’s class distribution avoid the cross term?

31. Sep.2008DISP Lab @MD53131  For the ambiguity function: (1) The auto term is always near to the origin (2) The cross-term is always far from the origin AF WD Cohen’s Class Time-Frequency Distribution

32. IFT f  FT t  IFT f  FT t  IFT f  FT t  Sep.2008DISP Lab @MD53132 Cohen’s Class Time-Frequency Distribution  Relationship between Wigner distribution and ambiguity function

33. Sep.2008DISP Lab @MD53133 Cohen’s Class Time-Frequency Distribution  How does the Cohen’s class distribution avoid the cross term?  Choi-Williams Distribution  Cone-Shape Distribution

34. Sep.2008DISP Lab @MD53134 Cohen’s Class Time-Frequency Distribution  Example: Ambiguity Function Choi-Williams Distribution

35. Sep.2008DISP Lab @MD53135 Cohen’s Class Time-Frequency Distribution  Example: Ambiguity Function Wigner Distribution

36. Sep.2008DISP Lab @MD53136 Cohen’s Class Time-Frequency Distribution  Some popular distributions and their kernels TFDKernelFormulation Page Levin Kirkwood Spectrogram Wigner-Ville 1 Choi-Williams Cone shape

37. Sep.2008DISP Lab @MD53137 Cohen’s Class Time-Frequency Distribution  Advantage: The Cohen’s class distribution may avoid the cross term and has higher clarity  Disadvantage: It requires more computation time and lacks of well mathematical properties

38. Sep.2008DISP Lab @MD53138 S Transform  Unlike STFT, the width of S transform’s window changes with frequency.  Closely related to the wavelet transform

39. Sep.2008DISP Lab @MD53139 S Transform Example:

40. Sep.2008DISP Lab @MD53140 Hilbert-Huang Transform  Traditional data analysis methods are all based on linear and stationary assumptions  The HHT consists of two parts: empirical mode decomposition (EMD) and Hilbert spectral analysis (HSA).

41. Sep.2008DISP Lab @MD53141 Hilbert-Huang Transform---EMD  Intrinsic Mode Functions (IMF)

42. Sep.2008DISP Lab @MD53142 Hilbert-Huang Transform--EMD  The Sifting Process 102030405060708090100110120 -2 0 1 2 IMF 1; iteration 0 102030405060708090100110120 -2 0 1 2 IMF 1; iteration 0

43. Sep.2008DISP Lab @MD53143 Hilbert-Huang Transform--EMD

44. Sep.2008DISP Lab @MD53144 Hilbert-Huang Transform--EMD 102030405060708090100110120 -1.5 -0.5 0 0.5 1 1.5 residue

45. Sep.2008DISP Lab @MD53145 Hilbert-Huang Transform--EMD

46. Sep.2008DISP Lab @MD53146 Hilbert-Huang Transform--HSA  We have obtained the intrinsic mode function components by EMD process method  Then we will do the Hilbert transform to each IMF component.

47. Sep.2008DISP Lab @MD53147 Applications of Time-Frequency Analysis  (1) Finding the Instantaneous Frequency  (2) Sampling Theory  (3) Modulation and Multiplexing  (4) Filter Design  (5) Signal Representation  (6) Random Process Analysis

48. Sep.2008DISP Lab @MD53148 Applications of Time-Frequency Analysis  (7) Acoustics  (8) Data Compression  (9) Spread Spectrum Analysis  (10) Radar Signal Analysis  (11) Biomedical Engineering  (12) Economic Data Analysis

49. Sep.2008DISP Lab @MD53149 Conclusions AdvantageDisadvantage STFT and Gabor transform 1.Low computation 2.The range of the integration is limited 3.No cross term 4.Linear operation 1.Complex value 2.Low resolution Wigner distribution function 1.Real 2.High resolution 3.If the time/frequency limited, time/frequency of the WDF is limited with the same range 1.High computation 2.Cross term 3.Non-linear operation Cohen’s class distribution 1.Avoid the cross term 2.Higher clarity 1.High computation 2.Lack of well mathematical properties Gabor-Wigner distribution function 1.Combine the advantage of the WDF and the Gabor transform 2.Higher clarity 3.No cross-term 1.High computation

50. Sep.2008DISP Lab @MD53150 Conclusions  Compare with Fourier, wavelet and HHT analyses FourierWaveletHilbert Basis A priori Adaptive Frequency Convolution: globalConvolution: regionalDifferentiation: local Presentation Energy-frequencyEnergy-time-frequency Non-linear No Yes Non-stationary NoYes Uncertainty Yes No Harmonics Yes No Feature extraction No Discrete: no Continuous: yes Yes Theoretical base Theory complete Empirical

51. Sep.2008DISP Lab @MD53151 Conclusions  Advantage: The instantaneous frequency can be observed  Disadvantage: Higher complexity for computation  Which method is better?

52. Sep.2008DISP Lab @MD53152 References  N. E. Huang, Z. Shen and S. R. Long, et al., ＂ The Empirical Mode Decomposition and the Hilbert Spectrum for Nonlinear and Non- Stationary Time Series Analysis ＂, Proc. Royal Society, vol. 454, pp.903-995, London, 1998 ＂ The Empirical Mode Decomposition and the Hilbert Spectrum for Nonlinear and Non- Stationary Time Series Analysis ＂  N. E. Huang, S. Shen, ＂ Hilbert-Huang Transform and its Applications ＂, World scientific, Singapore, 2005. Hilbert-Huang Transform and its Applications

53. Sep.2008DISP Lab @MD53153 References  S. C. Pei and J. J. Ding, “Relations between the fractional operations and the Wigner distribution, ambiguity function,” IEEE Trans. Signal Processing, vol. 49, no. 8, pp. 1638- 1655, Aug. 2001.Relations between the fractional operations and the Wigner distribution, ambiguity function  L. Cohen, ＂ Time-Frequency distributions-A review” Proc. IEEE, Vol. 77, No. 7, pp. 941- 981, July 1989. Time-Frequency distributions-A review  C. H. Page, “Instantaneous Power Spectra,” National Bureau of Standards, Washington, D. C., 1951Instantaneous Power Spectra

54. Sep.2008DISP Lab @MD53154 References  S. C. Pei and J. J. Ding, “Relations between Gabor transforms and fractional Fourier transforms and their applications for signal processing,” IEEE Trans. Signal Processing, vol. 55, no. 10, pp. 4839-4850, Oct. 2007  R. G. Stockwell, L. Mansinha, and R. P. Lowe, “Localization of the complex spectrum: the S transform,” IEEE Trans. Signal Processing, vol. 44, no. 4, pp. 998–1001, Apr. 1996.