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Introduction : TimeFrequency Analysis

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Title: Introduction : TimeFrequency Analysis


1
Introduction Time-Frequency Analysis
  • HHT, Wigner-Ville and Wavelet

2
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?

3
Traditional Methodsfor Time Series Analysis
  • Various probability distributions
  • Spectral analysis and Spectrogram
  • Wavelet Analysis
  • Wigner-Ville Distributions
  • Empirical Orthogonal Functions aka Singular
    Spectral Analysis

4
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.

5
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.

6
Morlet Wavelet Spectrum
7
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 .
8
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.
9
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.
10
Addativity of Fourier Transforms (Spectra)
11
Non-addativity of Power Spectral Properties
Therefore, for Wigner-Ville Distribution, it is
impossible to have two events occur at different
time independently with different frequency to be
totally independent of each other. Both Wavelet
and Spectrogram can separate events. But, Sum of
Spectrogram is not the Fourier Spectrum.
12
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.
  • 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?

13
Non-addativity Example Data 2 Waves
14
Non-addativity Example Fourier Spectra
15
Non-addativity Example Hilbert Spectrum
16
Non-addativity Example Wavelet Spectrum
17
Non-addativity Example Wigner-Ville Spectrum
18
Non-addativity Example Wigner-Ville Spectrum
and Components
19
Non-addativity Example Fourier Components
20
Non-addativity Example Hilbert,Wigner-Ville
Wavelet Spectra
21
Non-addativity Example Marginal Hilbert and
Fourier Spectra
22
Non-addativity Example Marginal Hilbert and
Fourier Spectra Details
23
New Example Data LOD 1962-1972
24
New Example Spectrogram (730)
25
New Example Spectrogram Details
26
New Example Wigner-Ville
27
New Example Morlet wavelet
28
New Example Hilbert Spectrum
29
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 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.
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