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Frequency and Instantaneous Frequency A Totally New View of Frequency.

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Presentation on theme: "Frequency and Instantaneous Frequency A Totally New View of Frequency."— Presentation transcript:

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 on Instantaneous 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 : 1 For any data from linear Processes

9 Jean-Baptiste-Joseph Fourier 1807 “On the Propagation of Heat in Solid Bodies” 1812 Grand 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. ’ 1817 Elected to Académie des Sciences 1822 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

14

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.

35

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

38 Definitions of Frequency : 3 Instantaneous 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. T1T1 T2T2 T4T4

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 : 4 Instantaneous 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

51

52

53 Empirical Mode Decomposition Sifting to produce IMFs

54 Bedrosian Theorem Let f(x) and g(x) denotes generally complex functions in L 2 (-∞, ∞) of the real variable x. If (1) the Fourier transform F(ω) of f(x) vanished for │ω│> a and the Fourier transform G(ω) of g(x) vanishes for │ω│< a, where a is an arbitrary positive constant, or (2) f(x) and g(x) are analytic (i. e., their real and imaginary parts are Hilbert pairs), then the Hilbert transform of the product of f(x) and g(x) is given H { f(x) g(x) } = f(x) H { g(x) }. Bedrosian, E., 1963: A Product theorem for Hilbert Transform, Proceedings of the IEEE, 51, 868-869.

55 Nuttall Theorem For any function x(t), having a quadrature xq(t), and a Hilbert transform xh(t); then, where Fq(ω) is the spectrum of xq(t). Nuttall, A. H., 1966: On the quadrature approximation to the Hilbert Transform of modulated signal, Proc. IEEE, 54, 1458

56 Difficulties with the Existing Limitations Data are not necessarily IMFs. Even if we use EMD to decompose the data into IMFs. IMF is only necessary but not sufficient because of the following limitations: Bedrosian Theorem adds the requirement of not having strong amplitude modulations. Nuttall Theorem further points out the difference between analytic function and quadrature. The discrepancy, however, is given in term of the quadrature spectrum, which is an unknown quantity. Therefore, it cannot be evaluated. Nuttall Theorem provides a constant limit not a function of time; therefore, it is not very useful for non-stationary processes.

57 Analytic vs. Quadrature X(t) Y(t) Z(t) Analytic Hilbert Transform Q(t) Quadrature, not analytic No Known general method Analytic functions satisfy Cauchy-Reimann equation, but may be x 2 + y 2 ≠ 1. Then the arc-tangent would not recover the true phase function. Quadrature pairs are not analytic, but satisfy strict 90 o phase shift; therefore, x 2 + y 2 = 1, and the arc-tangent always gives the true phase function. For cosθ(t) with arbitrary function of θ(t) :

58 A Difficulty of Hilbert Transform Bedrosian Theorem

59 Data : Step-function with Carrier

60 Fourier Spectra for Step-function and Carrier

61 Hilbert Spectrum : Step-function with Carrier

62 Morlet Wavelet : Step-function with Carrier

63 Spectrogram : Step-function with Carrier

64 Data : Step-function with Carrier III

65 Hilbert Spectrum : Step-function with Carrier III

66 Problems with Hilbert Transform method If there is any amplitude change, the Fourier Spectrum for the envelope and carrier are not separable. Thus, we violated the limitations stated in the Bedrosian Theorem; drastic amplitude change produce drastic deteriorating results. Once we cannot separate the envelope and the carrier, the analytic signal through Hilbert Transform would not give the phase function of the carrier alone without the influence of the variation from the envelope. Therefore, the instantaneous frequency computed through the analytic signal ceases to have full physical meaning; it provides an approximation only.

67 Normalization To overcome the limitation imposed by Bedrosian Theorem

68 Why do we need Decomposition and Normalization : We need a method to reduce the data to Intrinsic Mode Functions; then we also need a method for AM FM decomposition to over come the difficulties stated in Bedrosian Theorem. An Example : Step-function with Carrier

69 Effects of Normalization Normalization method can give a true AM FM decomposition to over come the difficulties stated in Bedrosian Theorem, and also provide a sharper error index than Nuttall Theorem.

70 NHHT : Procedures Obtain IMF representation of the data from siftings. Find local maxima of the absolute value of IMF (to take advantage of using both upper and lower envelopes) and fix the end values as maxima to ameliorate the end effects. Construct a Spline Envelope (SE) through the maxima. When envelope goes under the data, straight line envelope will be used for that section of the SE. Normalize the data using SE : N-data = Data/SE. This steps can be repeated. Compute IF (FM) and Absolute Value (AV) from Hilbert Transform of N-data. Definition : Error Index = (AV-1) 2. Compute Instantaneous Frequency for SE (AM).

71 NHHT Procedures : 1. IMF from Data through siftings

72 NHHT Procedures : 2. Locate local maxima and fix the ends

73 NHHT Procedures : 3. Construct the Cubic Spline Envelope (CSE)

74 NHHT procedures:

75 NHHT Procedures : 4. Normalize the IMF through CSE

76 NHHT Procedures : 5. Compute IF through Hilbert Transform

77 NHHT Procedures : 6. Comparison of Hilbert Transforms of Data and Normalized data

78 NHHT Procedures : 7. Define the Error Index = (AV – 1) 2.

79 NHHT Procedures : 8. Define the IMF of Envelope

80 NHHT Procedures : 9. AM and FM of y=c3y(7001:8000,9)

81 NHHT : Procedures Obtain IMF representation of the data from siftings. Find local maxima of the absolute value of IMF (to take advantage of using both upper and lower envelopes) and fix the end values as maxima to ameliorate the end effects. Construct a Spline Envelope (SE) through the maxima. When envelope goes under the data, straight line envelope will be used for that section of the SE. Normalize the data using SE : N-data = Data/SE. This steps can be repeated. Compute IF (FM) and Absolute Value (AV) from Hilbert Transform of N-data. Definition : Error Index = (AV-1) 2. Compute Instantaneous Frequency for SE (AM).

82 Example : Exponentially Decaying Cubic Chirp Model function

83 Exponentially decaying cubic chirp : Equation

84 Exponentially decaying cubic chirp : Data

85 Exponentially decaying cubic chirp : Normalizing function

86 Exponentially decaying cubic chirp : Normalized carrier

87 Exponentially decaying cubic chirp : Phase Diagram

88 Exponentially decaying cubic chirp : Instantaneous Frequency

89 Exponentially decaying cubic chirp : Error Indices

90 Another difficulty of Hilbert Transform Nuttal Theorem

91 Nuttall Theorem For any function x(t), having a quadrature xq(t), and a Hilbert transform xh(t); then, where Fq(ω) is the spectrum of xq(t). Nuttall, A. H., 1966: On the quadrature approximation to the Hilbert Transform of modulated signal, Proc. IEEE, 54, 1458

92 Why do we need Quadrature : To over come the difficulties stated in Nuttall Theorem for complicate phase functions. An Example : Duffing Pendulum

93 Duffing : Model Equation

94 Duffing : Expansions of the Model Equation

95 Duffing : Data

96 Duffing : Data, Quadrature & Hilbert

97 Duffing : Amplitude

98 Duffing : Phase

99 Duffing : Frequency truth is given by quadrature

100 Quadrature To circumvent the limitation imposed by Nuttall Theorem

101 Quadrature : Procedures Normalize the IMFs as in the NHHT method. Compute IF (FM) from Quadrature of N-data as follows:

102 Validation of NHHT and Quadrature Methods Through examples using NHHT HHT GZC TEO Quadrature

103 Example : Duffing Equation Model function

104 Damped Chirp Duffing Model

105

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112 Example : Speech Signal ‘ Hello ’ Real Data

113 Data : Hello

114 Data : Hello IMF

115 Hello : Data c3y(8)

116 Hello : Check Bedrosian Theorem

117

118 Hello : Instantaneous Frequency & data c3y(8)

119 Hello : Instantaneous Frequency & data Details c3y(8)

120 Some alternatives for Quadrature Different implement for IF

121 Hou’s Approach Let our data be x(t). Using Taylor’s expansion, we can write Therefore, we have

122 Hou’s Approach Thus, we should have the instantaneous frequency as the derivative of the phase function given by: This approach requires no normalization.. Hou, T. Y., M. P. Yan and Z. Wu, 2009: A variant of the EMD method for multi-scale data. Advances in Adaptive Data Analysis, 1, 483-516.

123 Wu’s Approach In this method, we do not have to compute arc- cosine. After normalization of the IMF, we have therefore, we can also find

124 Wu’s Approach

125 Summary Instantaneous frequency could be computed accurately. Our implementation is basically according to Wu’s algorithm. In case of EEMD, the component might not be bona fide IMF

126 A Physical Example : Water Surface Waves Real Laboratory Data

127 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.

128 The Idea and the need of Instantaneous Frequency According to the classic wave theory, there are other more important wave conservation laws for Energy and Action: Therefore, if frequency is a function of time, it has to satisfy certain condition for both laws to be valid.

129 Data

130 Governing Equations I:

131 Governing Equations II:

132 Governing Equations III:

133 Governing Equations IV: The 4 th order Nonlinear Schrodinger Equation

134 Dysthe, K. B., 1979: Note on a modification to the nonlinear Schrodinger equation for application to deep water waves. Proc. R. Soc. Lond., 369, 105-114. Equation by perturbation up to 4 th order. But ω = constant.

135 Data and IF : Station #1

136 Data and IF : Station #3

137 Data and IF : Station #5

138 Phase Averaged Data and IF : Station #1

139 Phase Averaged Data and IF : Station #2

140 Phase Averaged Data and IF : Station #3

141 Phase Averaged Data and IF : Station #4

142 Summary Instantaneous frequency could be highly variable with high gradient. The assumption used in the classic wave theory might not be totally attainable. Coupled with the fusion of waves, we might need a new paradigm for water wave studies.

143 Comparisons of Different Methods TEO extremely local but for linear data only. GZC most stable but offers only smoothed frequency over ¼ wave period at most. HHT elegant and detailed, but suffers the limitations of Bedrosian and Nuttall Theorems. NHHT, with Normalized data, overcomes Bedrosian limitation, offers local, stable and detailed Instantaneous frequency and Error Index for nonlinear and nonstationary data. Quadrature is the best, but the sampling rate has to be sufficiently high.

144 Conclusions Instantaneous Frequency could be calculated routinely from the normalized IMFs through quadrature (for high data density) or Hilbert Transform (for low data density). For any signal, there might be more than one IF value at any given time. For data from nonlinear processes, there has to be intra-wave frequency modulations; therefore, the Instantaneous Frequency could be highly variable. This variations is essential for quantifying nonlinearity.


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