Presents. The Story of Wavelets. Robi Polikar Dept. of Electrical & Computer Engineering Rowan University. The Story of Wavelets. Technical Overview But…We cannot do that with Fourier Transform…. Time  frequency representation and the STFT Continuous wavelet transform
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Presents
The Story of Wavelets
Robi PolikarDept. of Electrical & Computer EngineeringRowan University
Jean B. Joseph Fourier
(17681830)
“An arbitrary function, continuous or with discontinuities, defined in a finite interval by an arbitrarily capricious graph can always be expressed as a sum of sinusoids”
J.B.J. Fourier
F
F
F
F
Complex exponentials (sinusoids) as basis functions:
F
An ultrasonic Ascan using 1.5 MHz transducer, sampled at 10 MHz
Concatenation
X4(ω)
Perfect knowledge of what
frequencies exist, but no
information about where
these frequencies are
located in time
X5(ω)
Shortcomings of the FT
Frequency
parameter
Time
parameter
Signal to
be analyzed
FT Kernel
(basis function)
STFT of signal x(t):
Computed for each
window centered at t=t’
Windowing
function
Windowing function
centered at t=t’
t’=8t’=2
t’=4t’=8
Two extreme cases:
STFT then gives the time signal back, with a phase factor. Excellent time information (good time resolution), but no frequency information
Wide analysis window poor time resolution, good frequency resolution
Narrow analysis windowgood time resolution, poor frequency resolution
Once the window is chosen, the resolution is set for both time and frequency.
Frequency resolution: How well two spectral components can be separated from each other in the transform domain
Time resolution: How well two spikes in time can be separated from each other in the transform domain
Both time and frequency resolutions cannot be arbitrarily high!!! We cannot precisely know at what time instance a frequency component is located. We can only know what interval of frequencies are present in which time intervals
A normalization
constant
Translation parameter, measure of time
Scale parameter, measure of frequency
Signal to be analyzed
Continuous wavelet transform of the signal x(t) using the analysis wavelet (.)
The mother wavelet. All kernels are obtained by translating (shifting) and/or scaling the mother wavelet
Scale = 1/frequency
High frequency (small scale)
Low frequency (large scale)
H
x[n]
y[n]
x[n]
x[n]
~
G
2
2
2
2
2
2
2
2
2
2
~
~
G
H
G
G
+
+
~
H
H
H
Decomposition
Reconstruction
Downsampling
Upsampling
Half band high pass filter
Half band low pass filter
G
H
2level DWT decomposition. The decomposition can be continues as long as there are enough samples for downsampling.
H(jw)
w
/2
/2
2
2
2
2
2
Length: 512
B: 0 ~
g[n]
h[n]
Length: 256
B: 0 ~ /2 Hz
Length: 256
B: /2 ~ Hz
a1
G(jw)
d1: Level 1 DWT
Coeff.
g[n]
h[n]
Length: 128
B: 0 ~ /4 Hz
w
Length: 128
B: /4 ~ /2 Hz
/2
/2

a2
d2: Level 2 DWT
Coeff.
g[n]
h[n]
2
Length: 64
B: 0 ~ /8 Hz
Length: 64
B: /8 ~ /4 Hz
…a3….
d3: Level 3 DWT
Coeff.
Level 3 approximation
Coefficients
Choose wavelet
and number
of levels
Load signal
Hit Analyze
button
s=a5+d5+…+d1
Approx. coef.
at level 5
Level 1 coeff.
Highest freq.
(Wavedemo_signal1)
Applications of
Wavelets
First, analyze the signal with appropriate wavelets
Hit
Denoise
(Noisy Doppler)
Choose thresholding
method
Choose noise type
Choose thrsholds
Hit
Denoise
(microdisc.mat)
(microdisc.mat)
143 years
Ingrid Daubechies:
Stephane Mallat:
Ph.D. dissertation, 1988
Martin Vetterli & Jelena Kovacevic
The Story of Wavelets