Frequency and Instantaneous Frequency

1. Frequency and Instantaneous Frequency A Totally New View of Frequency

2. In search of frequency I found the trend and other information, e. g., quantification of nonlinearity Instantaneous Frequencies and Trends for Nonstationary Nonlinear Data IMA Hot Topic Conference 2011

3. 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

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

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

6. Equivalence : • The definition of frequency is equivalent to defining velocity as Velocity = Distance / Time • But velocity should be V = dS / dt .

7. 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

8. Definitions of Frequency : 1For any data from linear Processes

9. Jean-Baptiste-Joseph Fourier • “On the Propagation of Heat in Solid Bodies” • 1812Grand Prize of Paris Institute • “Théorie analytique de la chaleur” • ‘... the manner in which the author arrives at these equations is not exempt of difficulties and that his analysis to integrate them still leaves something to be desired on the score of generality and even rigor.’ • Elected to Académie des Sciences • Appointed as Secretary of Académie • paper published Fourier’s work is a great mathematical poem.Lord Kelvin

10. Fourier Spectrum

11. 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:

12. Problem with Fourier Frequency • Limited to linear stationary cases: same spectrum for white noise and delta function. • Fourier is essentially a mean over the whole domain; therefore, information on temporal (or spatial) variations is all lost. • Phase information lost in Fourier Power spectrum: many surrogate signals having the same spectrum.

13. The Importance of Phase

15. Random and Delta Functions

16. Fourier Components : Random Function

17. Fourier Components : Delta Function

18. Surrogate Signal I. Hello

19. The original data : Hello

20. The surrogate data : Hello

21. The Fourier Spectra : Hello

22. The IMF of Surrogate data : Hello

23. The Hilbert spectrum of Surrogate data : Hello

24. The IMF of original data : Hello

25. The Hilbert spectrum of original data : Hello

26. Surrogate Signal II. Duffing Wave

27. The original data : Duffing Pure Tone

28. Compare Duffing and Sine Duffing Sine

29. The surrogate data : Duffing

30. The Fourier Spectra : Duffing

31. The IMF of Surrogate data : Duffing

32. The Hilbert spectrum of Surrogate data : Duffing

33. The Hilbert spectrum of original data : Hello

34. Observations • The sound qualities of the original and the surrogate is totally different, yet they have the same Fourier spectrum. • The Hilbert spectra are totally different that reflect the different sound quality. • Therefore, the ear should not perceive sound based on Fourier based analysis with the linear and stationary assumption.

36. Problems with Integral methods • Frequency is not a function of time within the integral limit; therefore, the frequency variation could not be found in any differential equation, other than a constant. • The integral transform pairs suffer the limitation imposed by the uncertainty principle.

37. Definitions of Frequency : 2For Simple Dynamic System This is an system analysis but not a data analysis method.

38. Definitions of Frequency : 3Instantaneous Frequency for IMF only

39. Teager Energy Operator : the Idea H. M. Teager, 1980: Some observations on oral air flow during phonation, IEEE Trans. Acoustics, Speech, Signal Processing, ASSP-28-5, 599-601.

40. Generalized Zero-Crossing :By using intervals between all combinations of zero-crossings and extrema. T1 T2 T4

41. Generalized Zero-Crossing :Computing the weighted frequency.

42. Problems with TEO and GZC • TEO has super time resolution but it is strictly for linear processes. • GZC is robust but its resolution is still crude with resolution to ¼ wave length.

43. Definitions of Frequency : 4Instantaneous Frequency for IMF only

44. Instantaneous Frequency

45. Instantaneous Frequency is indispensable for nonlinear Processes x

46. 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. But how to find φ(x, t)?

47. Ideal case for Instantaneous Frequency • Obtain the analytic signal based on real valued function through Hilbert Transform. • Compute the Instantaneous frequency by taking derivative of the phase function from AS. • This is true only if the function is an IMF, and its imaginary part of the analytic signal is identical to the quadrature of the real part. Unfortunately, this is true only for very special and simple cases.

48. Hilbert Transform : Definition

49. Limitations for IF computed through Hilbert Transform • Data must be expressed in terms of Intrinsic Mode Function. (Note : Traditional applications using band-pass filter distorts the wave form; therefore, it can only be used for linear processes.) IMF is only necessary but not sufficient. • Bedrosian Theorem: Hilbert transform of a(t) cos θ(t) might not be exactly a(t) sin θ(t). Spectra of a(t) and cos θ(t) must be disjoint. • Nuttall Theorem: Hilbert transform of cos θ(t) might not be sin θ(t) for an arbitrary function of θ(t). Quadrature and Hilbert Transform of arbitrary real functions are not necessarily identical. • Therefore, a simple derivative of the phase of the analytic function for an arbitrary function may not work.

50. Data : Hello