Dr Rajkumar L. Biradar, Prof, GNITS
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Wavelet Transformation
Shri Bhagyavanthi Krupa
Dr. Rajkumar L. Biradar
Prof., ETE Dept
GNITS, HYD
Table of content
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Introduction
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Most of the signals in practice, are TIME-DOMAIN(Raw Signal) signals in their raw format. It means that measured signal is a function of time.
To obtain a further information from the signal that is not readily available in the raw signal.
EX: In many cases, the most distinguished information is hidden in the frequency content of the signal.
Why do we need the frequency information?
Stationary signal
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20Hz
80Hz
120Hz
Transformation
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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:
Every transformation technique has its own area of
application, with advantages and disadvantages.
Fourier Transformation
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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(2πft)
5Hz
10Hz
Non-Stationary signal
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Magnitude
20 Hz
80 Hz
120 Hz
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Frequency
(X) Amplitude
FT
Frequency: 2 Hz to 20 Hz
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Time
Magnitude
Magnitude
Frequency (Hz)
Time
Magnitude
Magnitude
Frequency (Hz)
Different in Time Domain
Frequency: 20 Hz to 2 Hz
Same in Frequency Domain
At what time the frequency components occur? FT can not tell!
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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.
Once again please note that, the FT gives what frequency components (spectral components) exist in the signal. Nothing more, nothing less.
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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).
ECG
EEG
EMG
Short-Time Fourier Transformation
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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 step
Frequency
Amplitude
time-frequency representation (TFR)
Window width = 0.05
Time step = 100 milisec
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FT
STFT
ω
ω
ω
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Time step
Frequency
Amplitude
Amplitude
Time step
Frequency
Amplitude
Narrow windows give good time resolution, but poor frequency resolution.
Window width = 0.02
Time step = 10 milisec
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Time step
Frequency
Amplitude
Amplitude
Time step
Frequency
Amplitude
Wide windows give good frequency resolution, but poor time resolution;
Window width = 0.1
Time step = 10 milisec
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What kind of a window to use
?
The answer, of course, is application dependent:
Overcomes the shortcoming of STFT by using variable length windows
.
The Wavelet transform (WT) solves the dilemma of resolution to a certain extent, as we will see.
Multi Resolution Analysis
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MRA, as implied by its name, analyzes the signal at different frequencies with different time resolutions. Every spectral component is not resolved equally as was the case in the STFT.
MRA is designed to give good time resolution (uses narrow window) and poor frequency resolution at high frequencies (bcoz time is very less) and good frequency resolution (uses wider window) 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.
What are Wavelets?
(1) varying frequency,
(2) limited duration, and
(3) an average value of zero.
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Sinusoid Wavelet
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Continuous Wavelet Transformation
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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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X
t = 0
Scale = 1
Ψ(t)
f(t)
×
Inner product
Like sines and cosines in FT, wavelets are used as basis functions Ψ(t)
in representing other functions f(t):
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X
t = 50 (position translation/shifting)
Scale = 1
Ψ(s,t)
f(t)
×
Inner product
Dr Rajkumar L. Biradar, Prof, GNITS
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X
t = 100 (position translation/shifting)
Scale = 1
Ψ(s,t)
f(t)
×
Inner product
Dr Rajkumar L. Biradar, Prof, GNITS
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X
t = 150 (position translation/shifting)
Scale = 1
Ψ(s,t)
f(t)
×
Inner product
Dr Rajkumar L. Biradar, Prof, GNITS
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X
t = 200 (position translation/shifting)
Scale = 1
Ψ(s,t)
f(t)
×
Inner product
Dr Rajkumar L. Biradar, Prof, GNITS
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X
t = 200 (position translation/shifting)
Scale = 1
Ψ(s,t)
f(t)
×
Inner product
0
Dr Rajkumar L. Biradar, Prof, GNITS
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X
t = 0
Scale = 10
Ψ(s,t)
f(t)
×
Inner product
Dr Rajkumar L. Biradar, Prof, GNITS
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X
t = 50 (position translation/shifting)
Scale = 10
Ψ(s,t)
f(t)
×
Inner product
Dr Rajkumar L. Biradar, Prof, GNITS
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X
t = 100 (position translation/shifting)
Scale = 10
Ψ(s,t)
f(t)
×
Inner product
Dr Rajkumar L. Biradar, Prof, GNITS
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X
t = 150 (position translation/shifting)
Scale = 10
Ψ(s,t)
f(t)
×
Inner product
Dr Rajkumar L. Biradar, Prof, GNITS
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X
t = 200 (position translation/shifting)
Scale = 10
Ψ(s,t)
f(t)
×
Inner product
Dr Rajkumar L. Biradar, Prof, GNITS
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X
Scale = 10
Ψ(s,t)
f(t)
×
Inner product
0
Dr Rajkumar L. Biradar, Prof, GNITS
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X
t=0
Scale = 20
Ψ(s,t)
f(t)
×
Inner product
Dr Rajkumar L. Biradar, Prof, GNITS
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X
t=0
Scale = 30
Ψ(s,t)
f(t)
×
Inner product
Dr Rajkumar L. Biradar, Prof, GNITS
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X
t=0
Scale = 40
Ψ(s,t)
f(t)
×
Inner product
Dr Rajkumar L. Biradar, Prof, GNITS
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X
t=0
Scale = 50
Ψ(s,t)
f(t)
×
Inner product
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The continuous wavelet transform (CWT)
∙ Fourier Transform
FT is the sum over all the time of signal f(t) multiplied by a complex exponential.
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This equation shows how a function ƒ(t) is decomposed into a set of basis functions , called the wavelets.
The variables s and τ are the new dimensions, scale and translation (position), after the wavelet transform.
∙ Similarly, the Continuous Wavelet Transform (CWT) is defined as the sum over all time of the signal multiplied by scale , shifted version of the wavelet function :
Dr Rajkumar L. Biradar, Prof, GNITS
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the so-called mother wavelet, by scaling and translation:
s is the scale factor, τ is the translation factor and the factor s-1/2 is for energy normalization across the different scales.
Dr Rajkumar L. Biradar, Prof, GNITS
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∙Scale
∙ Scaling a wavelet simply means stretching (or compressing) it.
Dr Rajkumar L. Biradar, Prof, GNITS
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∙Translation (shift)
∙ Translating a wavelet simply means delaying (advancing) its onset.
∙Scale and Frequency
∙ Low scale s
Compressed wavelet
Rapidly changing details
High frequency
∙ High scale s
stretched wavelet
slowly changing details
low frequency
Continuous Wavelet Transform (CWT)
Dr Rajkumar L. Biradar, Prof, GNITS
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As seen in the above equation , the transformed signal is a function of two variables, τ and s , the translation and scale parameters, respectively. Ψ(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.
Dr Rajkumar L. Biradar, Prof, GNITS
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Magnitude
20 Hz
50 Hz
120 Hz
Translation increment=50 milisecond
Scale inc.=0.5
What are Wavelets? (cont’d)
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Morlet
Haar
Daubechies
�� Shannon Wavelet �Ψ(t) = 2 sinc(2t) – sinc(t)
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mother wavelet
τ=5, s=2
time
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