Transcript Introduction : Time-Frequency Analysis HHT, Wigner-Ville and Wavelet

Introduction :
Time-Frequency Analysis
HHT, Wigner-Ville and Wavelet
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?
Traditional Methods
for Time Series Analysis
•
•
•
•
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Various probability distributions
Spectral analysis and Spectrogram
Wavelet Analysis
Wigner-Ville Distributions
Empirical Orthogonal Functions aka Singular Spectral
Analysis
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.
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.
Morlet Wavelet Spectrum
1
W ( a ,b; x ,  ) 
a
1/ 2

 t b
 x( t )   a  dt .

where
 (  ) is the Basic Wavelet function; for
Morlet Wavelet , it is the Gaussian modulated
sine and cosine functions .
a is the dilation factor , 1 / a is the frequency.
b is the translation factor , b is the time.
Wigner-Ville Distribution
Wigner-Ville Distribution, W(ω, t), is defined as
W (  ,t ) 
1
2

 x ( t 

2
) x(t 

2
)  e  i d
where
S(  )
2
1

2

 x  ( t ) x( t   ) e  i d
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(ω) |2 .
VW Instantaneous Frequency
1
W (  ,t ) 
2
( t )



 W (  ,t )d






 x ( t  ) x ( t  )  e  i d
2
2


W (  , t ) d
.
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.
Spectrogram :
Short-Time-Fourier Transform
Spectrogram is defined as
S(  : t , t )
2
1

2

 x  ( t ) x( t   ) G( t , t )e  i d
Note 1. G(t, Δt) is a window with zero value outside the
duration of Δt.
Note 2. The spectrogram represents power density.
Addativity of Fourier Transforms (Spectra)
If x( t )  x1 ( t )  x 2 ( t ) then
S(  )  S1 (  )  S 2 (  ) ,
where
S(  ) 
1
2

t
x( t ) e  i t d t .
Non-addativity of Power Spectral
Properties
2
2
S  S1  S 2  S1  S 2
2
2
2


 2  S1 S 2 
2
 S1  S 2 .
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.
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?
Non-addativity Example : Data 2 Waves
Non-addativity Example : Fourier Spectra
Non-addativity Example : Hilbert Spectrum
Non-addativity Example : Wavelet Spectrum
Non-addativity Example : Wigner-Ville Spectrum
Non-addativity Example :
Wigner-Ville Spectrum and Components
Non-addativity Example : Fourier Components
Non-addativity Example :
Hilbert,Wigner-Ville & Wavelet Spectra
Non-addativity Example :
Marginal Hilbert and Fourier Spectra
Non-addativity Example :
Marginal Hilbert and Fourier Spectra Details
New Example : Data LOD 1962-1972
New Example : Spectrogram (730)
New Example : Spectrogram Details
New Example : Wigner-Ville
New Example : Morlet wavelet
New Example : Hilbert Spectrum
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.