1. Paradoxes on Instantaneous Frequency
a la Leon Cohen
Time-Frequency Analysis, Prentice Hall, 1995
Chapter 2: Instantaneous Frequency, P. 40

2. The Five Paradoxes
1. Instantaneous frequency of a signal may not be one of the frequencies in the spectrum.
2. For a signal with a line spectrum consisting of only a few sharp frequencies, the instantaneous frequency may be continuous and range over an infinite number of values.
3. Although the spectrum of analytic signal is zero for negative frequencies, the instantaneous frequency may be negative
4. For the band limited signal the instantaneous frequency may be outside the band.
5. The value of the Instantaneous frequency should depend only on the present time, but the analytic signal, from which the instantaneous frequency is computed, depends on the signal values for the whole time space.

3. Observations I
By ‘spectrum’, Cohn is limiting the term to ‘Fourier spectrum’.
By ‘instantaneous Frequency’, Cohn is limiting the terms to be the IF obtained through Hilbert Transform. In fact, as we see, IF could be determined through many other methods. Hilbert transform is only an approximation.

4. Observations II
1. Paradoxes 1, 2 and 4 are essentially the same: Instantaneous Frequency values may be different from the frequency in the spectrum.
2. The negative frequency in analytic signal seems to violate Gabor’s construction.
3. The analytic function, or the Hilbert Transform, involves the functional values over the whole time domain; therefore, it is not local.

5. Resolution for paradoxes 1, 2 and 4
Two Examples

6. The First Example
Sin A + c*Sin B

7. Data: Sin (πt/360) + Sin (πt/320) : t=0:23040

8. Hilbert Spectrum X

9. Spectrogram X

10. Morlet Wavelet X

11. Instantaneous frequency X

12. Instantaneous frequency X : Details

13. Marginal Spectra X

14. Data: Sin (πt/360) + 0.8* Sin (πt/320) : t=0:23040

15. Hilbert Spectrum X08

16. Marginal Spectra X08

17. Two ways to view modulated wave

18. New developments
G. RILLING, P. FLANDRIN, 2008 :  "One or Two Frequencies? The Empirical Mode Decomposition Answers,“ IEEE Trans. on Signal Proc., Vol. 56, No. 1, pp
“….close tones are no longer perceived as such by the human ear but are rather considered as a whole, one can wonder whether a decomposition into tones is a good answer if the aim is to get a representation matched to physics (and/or perception) rather than to mathematics.”

19. Example

20. General case

21. Derivatives of HF and LF components
Af < 1
Af2 > 1

22. Numerical experiments

23. Numerical Experiments : C
In general, when frequency separation is more than a factor of 2, and also when the amplitude of low frequency is small.

24. Numerical Experiments : C

25. One or two-frequency?
Mathematically, if we select strict Fourier basis, it is two-frequency signal.
Physically, it is a modulated one frequency signal.
Using EMD, we could separate the signal, if the amplitude-frequency combination satisfies certain condition*, the condition coincides with physical perception.
*The condition: if frequency separation more than a factor of 2; and the amplitude of the low frequency is relatively small.

26. Example 2
Duffing’s Pendulum

27. Duffing Pendulum x

28. Duffing Type Wave : Data: x = cos(wt+0.3 sin2wt)

29. Duffing Type Wave : Perturbation Expansion

30. Duffing Type Wave :Wavelet Spectrum

31. Duffing Type Wave : Hilbert Spectrum

32. Duffing Type Wave : Marginal Spectra

33. Duffing Equation

34. Duffing Equation : Data

35. Duffing Equation : IMFs

36. Duffing Equation : IMFs

37. Duffing Equation : Hilbert Spectrum

38. Duffing Equation : Detailed Hilbert Spectrum

39. Duffing Equation : Wavelet Spectrum

40. Duffing Equation : Hilbert & Wavelet Spectra

41. Duffing Equation : Marginal Hilbert Spectrum

42. Rössler Equation

43. Rössler Equation : Data

44. Rössler Equation : 3D Phase

45. Rössler Equation : 2D Phase

46. Rössler Equation : IMF Strips

47. Rössler Equation : IMF

48. Rössler Equation : Hilbert Spectrum

49. Rössler Equation : Hilbert Spectrum & Data Details

50. Rössler Equation : Wavelet Spectrum

51. Rössler Equation : Hilbert & Wavelet Spectra

52. Rössler Equation : Marginal Spectra

53. Rössler Equation : Marginal Spectra

54. Resolution for Paradox 3
Negative Frequency

55. Complex Representation of real functions

56. Complex Representation of real functions

57. Hilbert Transform

58. Examples of Negative Frequency 1
Different references

59. Hilbert Transform a cos  + b : Data

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

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

62. Hilbert Transform a cos  + b : Frequency

63. Negative Frequency
The frequency values in Fourier and Hilbert analyses have totally different meanings: one is the mean the other is the instantaneous value.
Although the instantaneous value could be negative, the mean should be still the same.

64. The Empirical Mode Decomposition Method and Hilbert Spectral Analysis Sifting

65. Examples of Negative Frequency 2
FM and AM Frequencies
a sin ω t + b sin φ t

66. sin ω t sin 4 ω t

67. Hilbert : sin ω t sin 4 ω t

68. sin ω t + sin 4 ω t

69. Hilbert : sin ω t + sin 4 ω t

70. a sin ωt + b sin φt The data need to be sifted first.
Whenever Hilbert Transform has a loop away from the original (negative maximum or positive minimum), there will be negative frequency.
Whenever the Hilbert pass through the original (both real and imaginary parts are zero), there will be a frequency singularity.
Hilbert Transform is local to a degree of 1/t.

71. IMF : sin ω t sin 4 ω

72. IMF : sin ω t sin 4 ω

73. IMF : sin ω t + sin 4 ω

74. Negative Frequency
Negative instantaneous frequency values are mostly due to riding waves.
IMF is a necessary (but not a sufficient) condition for having non-negative frequency.
There are occasion when abrupt amplitude change in an IMF (but no riding waves) can also generate negative frequency. The amplitude induced problem is covered by Bedrosian theorem; normalized HHT will take care of it.
Physically, the abrupt amplitude change also shows the non-local characteristics of the Hilbert Transform.

75. Resolution for Paradox 5
Non-local influence does exist, they may come from Gibbs Phenomenon, end effects, and the limitation of the 1/t window in the Hilbert Transform. But most of the problems can be rectified through the Normalized HHT.
In fact, the non-local property of Hilbert transform is fully resolved by Quadrature method, though the solution is no longer a ‘Hilbert Spectrum’.

76. Data with magnitude jump : Signal

77. Data with magnitude jump : Signal

78. Hilbert Spectrum

79. Spectrogram

80. Morlet Wavelet

81. Data with magnitude jump

82. Data with magnitude jump : Details

83. Normalized Hilbert Spectrum

84. Amplitude Effects on Marginal Hilbert & Fourier Spectra

85. Instantaneous frequency

86. Instantaneous frequency : Details

87. Resolution for Paradox 5
Hilbert Transform is Non-local; therefore, the instantaneous frequency is not local.

88. Instantaneous Frequency
Hilbert transform might not be local, but it is very close to being so, for the window is 1/t. Therefore, the instantaneous frequency through Hilbert Transform is only nearly local.
We can use the Empirical AM/FM decomposition, normalization and quadrature to compute the instantaneous frequency. Then, it is perfectly local.

89. Summary: The so called paradoxes are really not problems, once some misconceptions are clarified
Instantaneous Frequency (IF) has very different meaning than the Fourier frequency.
IF for special mono-component functions only: IMFs; a necessary but not a sufficient condition.
Even for IMFs, there are still problems associated with IF through Hilbert Transform (Bedrosian and Nuttall Theorems). We can rectify most of them with the Normalized HHT.
The better solution is through quadrature.