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Introduction In previous sessions we studied
Wavelet Transform
Meaning of Transform
Types of transforms
Algorithms
Applications
Among which Fourier transform is probably by far the mostpopular
BUT , is it necessary to have both the time and the frequencyinformation at the same time?!
TIME FREQUENCY
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Introduction For the given signals
Wavelet Transform
When in time these frequencies happened ??!!
0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0
- 6
- 4
- 2
0
2
4
6
8
0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0
- 8
- 6
- 4
- 2
0
2
4
6
8
FFT
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 0
2 0 0
4 0 0
6 0 0
8 0 0
1 0 0 0
1 2 0 0
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 0
2 0 0
4 0 0
6 0 0
8 0 0
1 0 0 0
1 2 0 0
FFT
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Introduction
Wavelet Transform
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Introduction
Time information is not required when the signal is so-calledstationary
Most of practical signals are non-stationary as in speech,radar, sonar, seismic and even two dimensional images arenon-stationary.
FT can be used for non-stationary signals, if we are onlyinterested in what spectral components exist, but not interestedwhere these occur.
ONE EARLIER SOLUTION: SHORT-TIME FOURIERTRANSFORM (STFT)
Wavelet Transform
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Short Time Fourier Transform
Windowed F.T. or Short Time F.T. (STFT) : Segmenting the signal
into narrow time intervals, narrow enough to be considered
stationary, and then take the Fourier transform of each segment,Gabor 1946.
Followed by other Time Frequency Representations (TFRs), which
differed from each other by the selection of the windowing
function
Wavelet Transform
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Short Time Fourier Transform
Wavelet Transform
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Short Time Fourier Transform
1. Choose a window function of finite length
2. Place the window on top of the signal at t=0
3. Truncate the signal using this window
4. Compute the FT of the truncated signal, save.5. Incrementally slide the window to the right
6. Go to step 3, until window reaches the end of the signal
For each time location where the window is centered, weobtain a different FT
Hence, each FT provides the spectral information of a separatetime-slice of the signal, providing simultaneous time andfrequency information
Wavelet Transform
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FTX
Short Time Fourier Transform
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FTX
Short Time Fourier Transform
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FTX
Short Time Fourier Transform
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FTX
Short Time Fourier Transform
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FTX
Short Time Fourier Transform
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FTX
Short Time Fourier Transform
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FTX
Short Time Fourier Transform
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FTX
Short Time Fourier Transform
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FTX
Short Time Fourier Transform
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FTX
Short Time Fourier Transform
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FTX
Short Time Fourier Transform
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? A d!dt
tjx dtettWtxtS
[[ [ )()(),(
STFT of signal x(t):
Computed for each
window centered at t=t
Time
parameter
Frequency
parameterSignal to
beanalyzed
Windowing
function
Windowing function
centered at t=t
FT Kernel
(basis function)
Short Time Fourier Transform
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The window size determines what we callRESOLUTION
We are concerned in Time & Frequency resolution. Resolution may also vary from window type to another. Once you choose a particular size for the time window - it will
be the same for all frequencies. For a window of infinite length we get FT which gives perfect
frequency resolution but no time resolution. The narrower the window the better time resolution but poorer
frequency resolution.
Wavelet Transform
Short Time Fourier Transform
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Wavelet Transform
The problem with STFT is the fact whose roots go back towhat is known as the HeisenbergUncertainty Principle .
Simply, this principle states that one cannot know the exacttime-frequency representation of a signal
Many signals require a more flexible approach - so we canvary the window size to determine more accurately eithertime or frequency.
2/1. u!(( constt [
Short Time Fourier Transform
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Short Time Fourier Transform
Wavelet Transform
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Short Time Fourier Transform
Wavelet Transform
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Short Time Fourier Transform
Wavelet Transform
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Short Time Fourier Transform
Wavelet Transform
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Short Time Fourier Transform
Wavelet Transform
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Short Time Fourier Transform
Time and frequency resolution problems are results of aphysical phenomenon and exist regardless of the transformused.
Multi-resolution analysis (MRA)is alternative approach toanalyze the signal at different frequencies with differentresolutions. Every spectral component is not resolved equallyas was the case in the STFT.
MRA is designed to give good time resolution and poorfrequency resolution at high frequencies and good frequencyresolution and poor time resolution at low frequencies.
This approach makes sense as signals in practical applicationshave high frequency components for short durations and lowfrequency components for long durations.
Wavelet Transform
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Continuous Wavelet Transform (CWT)
Wavelet Transform
Time
Frequency Better timeresolution;
Poor frequency
resolution
Better frequency
resolution
Poor time
resolution
Each box represents a equal portionResolution in STFT is selected once for entire analysis
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Continuous Wavelet Transform (CWT)
Wavelet analysis is done in a similar way to the STFTanalysis.
Wavelet Transform
FT of the windowed signals are not taken. Width of the window is changed as the transform iscomputed for every spectral component.
There are two main differences between the STFT and theCWT:
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Continuous Wavelet Transform (CWT)
Wavelet Transform
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Continuous Wavelet Transform (CWT)
dtsttxsss xx
X]y!X=!X ]]
*1,,WT
Wavelet Transform
Wavelet
Small wave
Means the window function is of finite length.
MotherWavelet
A prototype for generating the other window functions All the used windows are its dilated or compressed and
shifted versions
Translation
(The location of
the window)
Scale
Mother Wavelet
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Wavelet Transform
Wavelet analysis produces a time-scale view of thesignal.
Scalingmeans stretching or compressing of the signal
Continuous Wavelet Transform (CWT)
f t a
f t a
f t a
t
t
t
( )
( )
( )
sin( )
sin( )
sin( )
! !
! !
! !
;
;
;
scale factor (a) for sine waves:
f t a
f t a
f t a
t
t
t
( )
( )
( )
( )
( )
( )
! !
! !
! !
=
=
=
;
;
;
1
2 1
24 1
4
Scale factor works exactly the same with wavelets:
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Continuous Wavelet Transform (CWT)
Step 1: The wavelet is placed at the beginning of the signal,and set s=1 (the most compressed wavelet)
Wavelet Transform
Computation of the CWT
signal
Step 2: The wavelet function at scale 1 is multiplied by thesignal, and integrated over all times; then multiplied by 1/s;
Step 3: Shift the wavelet to t=, and get the transform value att= and s=1;
Step 4:Repeat the procedure until the wavelet reaches the endof the signal
Step 5: Scale s is increased by a sufficiently small value, theprevious steps are repeated for all s
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Continuous Wavelet Transform (CWT)
Wavelet Transform
Time
Frequency Better timeresolution;
Poor frequency
resolution
Better frequency
resolution
Poor time
resolution
Each box represents a equal portionResolution in STFT is selected once for entire analysis
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Continuous Wavelet Transform (CWT)
Wavelet Transform
Perfect timeresolution
No frequency
resolution
No time resolutionPerfect frequency
resolution
Constant time &
frequency resolution
Variable time &
frequency resolution