Time-Frequency Analysis

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Time-Frequency Analysis

• HHT, Wigner-Ville and Wavelet

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Motivations

• The frequency and energy level of data from real
  world phenomena are seldom constant. For example
  our speech, music, weather and climate are highly
  variable.
• Traditional frequency analysis is inadequate.
• To describe such phenomena and understand the
  underlying mechanisms we need the detailed time
  frequency analysis.
• What is Time-Frequency Analysis?

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Traditional Methodsfor Time Series Analysis

• Various probability distributions
• Spectral analysis and Spectrogram
• Wavelet Analysis
• Wigner-Ville Distributions
• Empirical Orthogonal Functions aka Singular
  Spectral Analysis

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Time-Frequency Analysis

• All time-frequency-energy representations should
  be classified as time-frequency analysis thus,
  wavelet, Wigner-Ville Distribution and
  spectrogram should all be included.
• Almost by default, the term, time-frequency
  analysis, was monopolized by the Wagner-Ville
  distribution.

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Conditions for Time-Frequency Analysis

• To have a valid time-frequency representation, we
  have to have frequency and energy functions
  varying with time.
• Therefore, the frequency and energy functions
  should have instantaneous values.
• Ideally, separated event should not influence
  each other and be treated independently.

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Morlet Wavelet Spectrum
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Wigner-Ville Distribution
Wigner-Ville Distribution, W(?, t), is defined as
WV Distribution has to be identical to the
Fourier Power spectrum therefore, the mean of
Wigner-Ville Spectrum is the same as the Fourier
spectrum, S(?) 2 .
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VW Instantaneous Frequency
Therefore, at any given time, there is only one
instantaneous frequency value. What if there
are two independent components? In this case, VW
gives the weighted mean.
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Spectrogram Short-Time-Fourier Transform
Spectrogram is defined as
Note 1. G(t, ?t) is a window with zero value
outside the duration of ?t. Note 2. The
spectrogram represents power density.
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Addativity of Fourier Transforms (Spectra)
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Non-addativity of Power Spectral Properties
Therefore, for Wigner-Ville Distribution, it is
impossible to have two events occurring at
different time with different frequencies to be
totally independent of each other. Both Wavelet
and Spectrogram can separate events. But, Sum of
Spectrogram is not the Fourier Spectrum.
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Marginal Requirement

• Discrete Wavelet analysis with orthogonal basis
  should satisfy this requirement Continuous
  Wavelet with redundancy and leakage would not
  satisfy this requirement.
• As the Wigner-Ville distributions have the
  marginal distribution identical to that of Power
  Spectral Density, there is the extra requirement
  that the marginal spectrum has to be PSD.
• A genuine instantaneous frequency distribution
  will also not satisfy this requirement (i.e. sum
  equals PSD). But the energy is conserved.
• Spectrogram does not satisfy this requirement,
  for it suffers the poor frequency resolution due
  to the limitation of the window length.
• This is not a very reasonable requirement. If
  PSD is inadequate to begin with, why should it be
  used as a standard?

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Non-addativity Example Data 2 Waves
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Non-addativity Example Fourier Spectra
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Non-addativity Example Hilbert Spectrum
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Non-addativity Example Wavelet Spectrum
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Non-addativity Example Wigner-Ville Spectrum
and Components
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Non-addativity Example Wigner-Ville Spectrum
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Non-addativity Example Fourier Components
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Non-addativity Example Hilbert,Wigner-Ville
Wavelet Spectra
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Non-addativity Example Marginal Hilbert and
Fourier Spectra
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Non-addativity Example Marginal Hilbert and
Fourier Spectra Details
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New Example Data LOD 1962-1972
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New Example Spectrogram (730)
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New Example Spectrogram Details
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New Example Wigner-Ville
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New Example Morlet wavelet
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New Example Hilbert Spectrum
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Summary

• Wavelet, Spectrogram and HHT can all separate
  simultaneous events with different degrees of
  fidelity, but WV cannot.
• The instantaneous frequency defined by moments in
  WV is crude and illogical it gives only one
  weighted mean IF value at any given time.
• Though WV satisfies the marginal energy
  requirement, it does not give WV any advantage in
  time-frequency analysis or even as an analysis
  tool, for the marginal requirement is illogic.