Wavelet Transform
Pavla Bromová
FIT BUT
November 18, 2012
Introduction
Fourier Transform
• frequency representation
• time information lost
• for stationary signals
4 □ MS
Introduction
Short-Time Fourier Transform
• windowing (static size)
window
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Ti rne
Introduction
Wavelet Transform
Pavla Bromová  (FIT BUT)
Wavelet Transform
4 □ ►   4 &
Wavelet Analysis
• wavelet - a waveform with a zero average
Haar Shannon or Sine Daubechies4 Daubechiss 20
• breaking a signal into shifted and scaled versions of the wavelet
Signal Constituent wavelets ot different scales and positions
inuous Wavelet Transform
/oo f'(t)tp(scale, position, t)dt -oo
Continuous Wavelet Transform
/oo f(t)tp(scale, position, t)dt -oo
« scalogram:
Large
Coefficients
Small
9960    4000    .4060    4100    4150    4200    4250 Coefficients Time
• continuous: scales, shifting
1
high-pass downsampling
filter
AAA
detail
D   High Frequency!
-jjonnwrcmiticiRnts
approximation
A  Low Frequency
low-pass downsampling fill or ~500 D WT coeilKJents
Bromová  (FIT BUT)
Wavelet Transform
November 18, 2012        8 / 16
uItipie-Level Decomposition
Pavla Bromová  (FIT BUT)
Wavelet Transform
4 □ ►   4 (5? ► 4
November 18, 2012
Applications
I Op      200      300      4t>0      500      QOO      TOO      BOO 900
Applications
Detecting long-term evolution
Signal and ftpprDxirr-H'io'rs;
jHii#WM'
*4
4
. i......-■" '.■—r
a1
100        2qo        300        400        500        «00        700 boo
Applications
Identifying pure frequencies
		■Ha
a5	VW\A	
a4		
a3 -■		
: a2	ill   ill''  lll/l k j'll	mm*
a1 o|
200 400 600 MO
d5
d4
d2
dl
200 400 600 BOO
Applications
De-noising
s o
Signal and Approximations
a5
a4
a3
a2
o|
200 400  SCO 000 1000
Signal and Deiail(s)
d5
d3
d1
200 400 SOU BOO 1000
Wavelet Power Spectrum
Example:
Simulated spectrum
140
1 2        3        4        5 1 2        3 4