1. On the Marginal Hilbert Spectrum

2. Outline Definitions of the Hilbert Spectrum (HS) and the Marginal Hilbert Spectrum (MHS). Computation of MHS The relation between MHS and Fourier Spectrum MHS with different frequency resolutions Examples

3. Hilbert Spectrum

4. Definition of Hilbert Spectra

5. can be amplitude or the square of amplitude (energy). d ω d t Schematic of Hilbert Spectrum

6. Hilbert Spectra

7. Definition of the Marginal Hilbert Spectrum

8. Computing Hilbert Spectrum

9. Marginal Spectrum

10. Hilbert and Marginal Spectra

11. Some Properties

12. MHS and Fourier Spectra 1/T

13. MHS with different Resolutions

14. Observations Note that N/T is actually the sampling rate, so the conservation from Fourier to Hilbert is simply twice the sampling rate, if we use the full frequency range to the Nyquist limit. If we use any zoom, the additional factor is an additional

15. Some Properties The spectral density depends on the bin size that is on both temporal and frequency resolutions. For marginal Frequency spectrum, the temporal resolution is implicit. For instantaneous energy density, the frequency resolution is implicit. Frequency assumes instantaneous value, not mean; it is not limited by the Nyquist. We can zoom the spectrum to any temporal and frequency location.

16. Fourier vs. Hilbert Spectra Adaptive basis, Data Driven Time-frequency spectrum Physical meaningful frequency at both the high and low frequency ranges Resolution of the frequency adjustable Zoom capability Marginal spectra for frequency and time.

17. Example Uniformly distributed white noise

18. STD = 0.2 Data : White Noise STD = 0.2

19. Fourier Power Spectrum

20. IMF

21. Hilbert Marginal and Fourier Spectra Non-zero mean data : DC components

22. Factor = 1 Effects on Frequency Resolution MHS

23. Normalized MHS

24. [ 10 20 50 100 300 500 600 800]/1000

25. Effect Frequency Resolution : bin size

26. Normalized

27. Data

28. Data : IMF

29. Fourier Spectra

31. Hilbert Spectra : Various F-Resolutions

32. Hilbert Amplitude Spectra : Various F-Resolutions

33. Example Earthquake data

34. Earthquake data E921

35. IMF EEMD2(3,0.2,100)

36. IMF EEMD2(3,0.1,10)

37. IMF EEMD2(3,0,1)

38. Different Frequency Resolutions VS Fourier and Normalization

39. MHS and Fourier at full resolutions

40. MHS and Fourier Normalized

41. MHS Smoothed and Normalized

42. MHS Different Frequency Resolutions

43. MHS Different Resolutions Normalized

44. MHS EMD and EEMD

45. Zoom

46. MHS 100 Ensemble

48. MH Amplitude Spectrum

49. 10 Ensemble Poor normalization

50. Fourier and Hilbert Marginal Spectra

51. Normalized

52. Effect of Filter size : Fourier

53. Hilbert Spectrum

57. Marginal Spectra

58. Normalized

59. Zoom Effects

60. Normalized

61. Effect of bin size

62. Normalized

63. Effects of bin size and zoom

64. Normalized

65. Example Delta-Function

67. Influence of the resolution of frequency on the Hilbert-Huang spectrum [ 10 20 50 100 300 500 600 800]/1000

68. Effects of Frequency Resolution

69. Fourier Energy Spectrum

70. Summary Hilbert spectra are time-frequency presentations. The marginal spectra could have various resolutions and zoom capability. Hilbert marginal spectra could be smoothed without losing resolution drastically. Another marginal Hilbert quantity is the time- energy distribution.

71. Summary For long time, the Hilbert Marginal Spectrum was not defined absolutely. The energy and amplitude spectra were not clearly compared; they are totally different spectra. Clear conversion factor are given to make comparisons between MHS and Fourier easily. Conversion factor also was provided for MHS with different Frequency resolutions. In most cases the MHS in energy is very similar to Fourier in case the data are from stationary and linear processes, for the temporal has been integrated out.