What is Frequency?

2. What is Frequency? Norden E. Huang Research Center for Adaptive Data Analysis Center for Dynamical Biomarkers and translational Medicine National Central University Zhongli, Taiwan, ROC The 15th Wu Chien Shiung Science Camp, 2012

3. Definition of Frequency Given the period of a wave as T ; the frequency is defined as

4. T This definition is easy for regular sine wave, but not very practical for complicated oscillations.

5. Impossible for complicated waves Need Decomposition

6. Jean-Baptiste-Joseph Fourier Fourier’s work is a great mathematical poem.Lord Kelvin • “On the Propagation of Heat in Solid Bodies” 1812Grand Prize of Paris Institute • Elected to Académie des Sciences • Appointed as Secretary of Math Section • paper published Ever since Fourier’s ground breaking work, people always think of any changes in terms of waves. Fourier’s work is a great mathematical poem.Lord Kelvin

7. Definitions of Fourier Type Frequency : For any data from linear Processes

8. Fourier Analysis Fourier proved that the Fourier expansion in terms of trigonometric function is complete, convergent, orthogonal and unique. Therefore, every signal could be think as combination of sinusoidal waves each with constant amplitude and frequency. Are those frequency meaningful?

9. Definition of Power Spectral Density Since a signal with nonzero average power is not square integrable, the Fourier transforms do not exist in this case. Fortunately, the Wiener-Khinchin Theroem provides a simple alternative. The PSD is the Fourier transform of the auto-correlation function, R(τ), of the signal if the signal is treated as a wide-sense stationary random process:

10. Fourier Spectrum

11. Surrogate Signal I. Hello

12. The original data : Hello

13. The surrogate data : Hello

14. The Fourier Spectra : Hello

15. Surrogate Signal II. delta function and white noise Non-causality: Event involves both past and future

16. Random and Delta Functions

17. Fourier Components : Random Function

18. Fourier Components : Delta Function

19. The Importance of Phase

21. Can we just use phase to explore physical processes? Yes, to some extent.

22. Then:In search of frequency,I found the methods to quantify nonlinearity anddetermine the trend. After more than 15 years of searching, I found the key to nonlinear and nonstationary data analysis is the proper definition for frequency.

23. Hot Topic Conference University of Minnesota,Institute for Mathematics and Its Applications, 2011 Instantaneous Frequencies and Trends for Nonstationary Nonlinear Data

24. How to define frequency? It seems to be trivial. But frequency is an important parameter for us to understand many physical phenomena.

25. Traditional Definition of Frequency • frequency = 1/period. • Definition too crude • Only work for simple sinusoidal waves • Does not apply to nonstationary processes • Does not work for nonlinear processes • Does not satisfy the need for wave equations

26. Instantaneous Frequency

27. Prevailing Views onInstantaneous Frequency The term, Instantaneous Frequency, should be banished forever from the dictionary of the communication engineer. J. Shekel, 1953 The uncertainty principle makes the concept of an Instantaneous Frequency impossible. K. Gröchennig, 2001

28. The Idea and the need of Instantaneous Frequency According to the classic wave theory, the wave conservation law is based on a gradually changing φ(x,t) such that Therefore, both wave number and frequency must have instantaneous values and differentiable. This is the true definition of frequency.

29. Hilbert Transform : Definition

30. The Traditional use of the Hilbert Transform for Data Analysisfailed miserablyand gave IF a bad break.

31. Traditional Viewa la Hahn (1995) : Data LOD

32. Traditional Viewa la Hahn (1995) : Hilbert

33. Traditional Approacha la Hahn (1995) : Phase Angle

34. Traditional Approacha la Hahn (1995) : Phase Angle Details

35. Traditional Approacha la Hahn (1995) : Frequency

36. Why the traditional approach does not work?

37. Hilbert Transform a cos  + b : Data

38. Hilbert Transform a cos  + b : Phase Diagram

39. Hilbert Transform a cos  + b : Phase Angle Details

40. Hilbert Transform a cos  + b : Frequency

41. The Empirical Mode Decomposition Method and Hilbert Spectral AnalysisSifting

42. Empirical Mode Decomposition: Methodology : Test Data

43. Empirical Mode Decomposition: Methodology : data and m1

44. Empirical Mode Decomposition: Methodology : data & h1

45. Empirical Mode Decomposition: Methodology : h1 & m2

46. Empirical Mode Decomposition: Methodology : h3 & m4

47. Empirical Mode Decomposition: Methodology : h4 & m5

48. Empirical Mode DecompositionSifting : to get one IMF component

49. The Stoppage Criteria The Cauchy type criterion: when SD is small than a pre-set value, where Or, simply pre-determine the number of iterations.

50. Empirical Mode Decomposition: Methodology : IMF c1