Continuous Wavelet Transform

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Institute for Advanced Studies in Basic Sciences
Zanjan
Continuous Wavelet Transformation
Mahdi Vasighi
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Table of content

• Introduction
• Fourier Transformation
• Short-Time Fourier Transformation
• Continuous Wavelet Transformation
• Applications of CWT

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Introduction
Most of the signals in practice, are TIME-DOMAIN
signals in their raw format. It means that
measured signal is a function of time.
Why do we need the frequency information?
In many cases, the most distinguished information
is hidden in the frequency content of the signal.
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Stationary signal

• Frequency content of stationary signals do not
  change in time.
• All frequency components exist at all times

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Transformation

• Fourier Transformation (FT) is probably the most
  popular transform being used (especially in
  electrical engineering and signal processing),
  There are many other transforms that are used
  quite often by engineers and mathematicians
• Hilbert transform
• Short-Time Fourier transform (STFT)
• Radon Transform,
• Wavelet transform, (WT)
• Every transformation technique has its own area
  of
• application, with advantages and disadvantages.

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Fourier Transformation
In 19th century, the French mathematician J.
Fourier, showed that any periodic function can be
expressed as an infinite sum of periodic complex
exponential functions.
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Raw Signal (time domain)
x(t)
cos(2?ft)
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1 Hz
? x(t).cos(2?ft) -8.8e-15
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2 Hz
? x(t).cos(2?ft) -5.7e-15
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3 Hz
? x(t).cos(2?ft) -4.6e-14
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4 Hz
? x(t).cos(2?ft) -2.2e-14
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4.8 Hz
? x(t).cos(2?ft) 74.5
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5 Hz
? x(t).cos(2?ft) 100
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5.2 Hz
? x(t).cos(2?ft) 77.5
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6 Hz
? x(t).cos(2?ft) 1.0e-14
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20, 80, 120 Hz
FT
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Non-Stationary signal

• Frequency content of stationary signals change in
  time.

20 Hz
80 Hz
120 Hz
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FT
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So, how come the spectrums of two entirely
different signals look very much alike? Recall
that the FT gives the spectral content of the
signal, but it gives no information regarding
where in time those spectral components appear.
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Almost all biological signals are non-stationary.
Some of the most famous ones are ECG (electrical
activity of the heart , electrocardiograph), EEG
(electrical activity of the brain,
electroencephalogram), and EMG (electrical
activity of the muscles, electromyogram).
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Short-Time Fourier Transformation
Can we assume that , some portion of a
non-stationary signal is stationary?
?
The answer is yes.
In STFT, the signal is divided into small
enough segments, where these segments (portions)
of the signal can be assumed to be stationary.
For this purpose, a window function "w" is chosen.
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FT
X
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FT
X
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FT
X
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FT
X
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FT
X
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FT
X
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FT
X
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FT
X
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FT
X
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FT
X
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FT
X
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time-frequency representation (TFR)
Window width 0.05 Time step 100 milisec
Amplitude
Time step
Frequency
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FT
STFT
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Narrow windows give good time resolution, but
poor frequency resolution.
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Wide windows give good frequency resolution, but
poor time resolution
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?
What kind of a window to use
The answer, of course, is application
dependent If the frequency components are well
separated from each other in the original signal,
than we may sacrifice some frequency resolution
and go for good time resolution, since the
spectral components are already well separated
from each other.
The Wavelet transform (WT) solves the dilemma of
resolution to a certain extent, as we will see.
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Multi Resolution Analysis
MRA, as implied by its name, analyzes the signal
at different frequencies with different
resolutions. Every spectral component is not
resolved equally as was the case in the STFT.
MRA is designed to give good time resolution and
poor frequency resolution at high frequencies and
good frequency resolution and poor time
resolution at low frequencies.
This approach makes sense especially when the
signal at hand has high frequency components for
short durations and low frequency components for
long durations.
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Continuous Wavelet Transformation
The continuous wavelet transform was developed
as an alternative approach to the short time
Fourier transform to overcome the resolution
problem. The wavelet analysis is done in a
similar way to the STFT analysis, in the sense
that the signal is multiplied with a function,
wavelet, similar to the window function in the
STFT, and the transform is computed separately
for different segments of the time domain signal.
wavelet
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Mexican hat
Morlet
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t 0 Scale 1

X
?(s,t)
x(t)
Inner product
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t 50 Scale 1

X
?(s,t)
x(t)
Inner product
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t 100 Scale 1

X
?(s,t)
x(t)
Inner product
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t 150 Scale 1

X
?(s,t)
x(t)
Inner product
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t 200 Scale 1

X
?(s,t)
x(t)
Inner product
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t 200 Scale 1
0

X
?(s,t)
x(t)
Inner product
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t 0 Scale 10

X
?(s,t)
x(t)
Inner product
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t 50 Scale 10

X
?(s,t)
x(t)
Inner product
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t 100 Scale 10

X
?(s,t)
x(t)
Inner product
50
t 150 Scale 10

X
?(s,t)
x(t)
Inner product
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t 200 Scale 10

X
?(s,t)
x(t)
Inner product
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Scale 10
0

X
?(s,t)
x(t)
Inner product
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Scale 20

X
?(s,t)
x(t)
Inner product
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Scale 30

X
?(s,t)
x(t)
Inner product
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Scale 40

X
?(s,t)
x(t)
Inner product
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Scale 50

X
?(s,t)
x(t)
Inner product
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As seen in the above equation , the transformed
signal is a function of two variables, ? and s ,
the translation and scale parameters,
respectively. ?(t) is the transforming function,
and it is called the mother wavelet.
If the signal has a spectral component that
corresponds to the value of s, the product of the
wavelet with the signal at the location where
this spectral component exists gives a relatively
large value.
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Magnitude
20 Hz
50 Hz
120 Hz
Translation increment50 milisecond Scale
inc.0.5
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10 Hz
20 Hz
60 Hz
120 Hz
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CWT Applications

• Identifying time-scale (time-frequency) scheme
• Frequency filtering (Noise filtering)

Wavelet Synthesis Reconstructing signal using
selected range of scales
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CWT result for non-stationary signal (10 20 Hz )
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CWT Applications

• Solving peak overlapping problem in different
  analytical techniques (simultaneous determination)

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S20
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S50
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S100
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S150
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S200
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S250
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Unknown mixture spectrum
Calibration model
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The combination of both continuous wavelet and
chemometrics techniques
Canadian Journal of Analytical Sciences and
Spectroscopy 49 (2004) 218
Spectrophotometric Multicomponent Determination
of Tetramethrin, Propoxur and Piperonyl Butoxide
in Insecticide Formulation by Principal Component
Regression and Partial Least Squares Techniques
with Continuous Wavelet Transform
A continuous wavelet transform (CWT) followed by
a principal component regression (PCR) and
partial least squares (PLS) were applied for the
quantitative determination of tetramethrin (TRM),
propoxur (PPS) and piperonil butoxide (PPR) in
their formulations. A CWT was applied to the
absorbance data. The resulting CWT-coefficients
(xblock) and concentration set (y-block) were
used for the construction of CWT-PCR and CWT-PLS
calibrations. The combination of both continuous
wavelet and chemometrics techniques indicates
good results for the determination of insecticide
in synthetic mixtures and commercial formulation.
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References
Mathworks, Inc. Wavelet Toolbox Help Robi
Polikar, The Wavelet Tutorial Multiresolution
Wavelet Analysis of Event Related Potentials for
the Detection of Alzheimer's Disease, Iowa State
University, 06/06/1995 Robi Polikar An
Introduction to Wavelets, IEEE Computational
Sciences and Engineering, Vol. 2, No 2, Summer
1995, pp 50-61. Continuous wavelet and
derivative transforms for the simultaneous
quantitative analysis and dissolution test of
levodopabenserazide tablets, Journal of
Pharmaceutical and Biomedical Analysis (2007) In
press. Determination of bismuth and copper using
adsorptive stripping voltammetry couple with
continuous wavelet transform, Talanta 71 (2007)
324332 Canadian Journal of Analytical Sciences
and Spectroscopy 49 (2004) 218
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Thanks for your attention