lecture 19 the wavelet transform - ncku
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The Wavelet Transform
Jean Baptiste Joseph Fourier (1768 – 1830)
1787: Train for priest (Left but Never married!!!).
1793: Involved in the local Revolutionary Committee.
1974: Jailed for the first time.
1797: Succeeded Lagrange as chair of analysis and
mechanics at É cole Polytechnique.
1798: Joined Napoleon's army in its invasion of Egypt.
1804-1807: Political Appointment. Work on Heat.
Expansion of functions as trigonometrical series.
Objections made by Lagrange and Laplace.
1817: Elected to the Académie des Sciences in and
served as secretary to the mathematical section.
Published his prize winning essay Théorie analytique de
la chaleur.
1824: Credited with the discovery that gases in the
atmosphere might increase the surface temperature of
the Earth (sur les températures du globe terrestre et des
espaces planétaires ). He established the concept of
planetary energy balance. Fourier called infrared
radiation "chaleur obscure" or "dark heat“.
MGP: Leibniz - Bernoulli - Bernoulli - Euler - Lagrange - Fourier – Dirichlet - ….
Windowed (Short-Time) Fourier Transform (1946)
James W. Cooley and John W. Tukey, "An algorithm
for the machine calculation of complex Fourier series,"
Math. Comput. 19, 297–301 (1965).
Independently re-invented an algorithm known to Carl
Friedrich Gauss around 1805
Fast Fourier Transform
Dennis Gabor
James W. Cooley and John W. Tukey
Winner of the 1971 Nobel Prize for contributions to the principles
underlying the science of holography, published his now-famous paper
“Theory of Communication.”2
C. F. Gauss
Stephane Mallat, Yves Meyer
Jean Morlet
Presented the concept of wavelets (ondelettes) in its present theoretical form
when he was working at the Marseille Theoretical Physics Center (France).
(Continuous Wavelet Transform)
(Discrete Wavelet Transform) The main algorithm dates
back to the work of Stephane Mallat in 1988. Then joined Y.
Meyer.
Some signals obviously have spectral
characteristics that vary with time
Motivation
STATIONARITY OF SIGNAL
• Stationary Signal
– Signals with frequency content unchanged in time
– All frequency components exist at all times
• Non-stationary Signal
– Frequency changes in time
– One example: the “Chirp Signal”
STATIONARITY OF SIGNAL
0 0.2 0.4 0.6 0.8 1-3
-2
-1
0
1
2
3
0 5 10 15 20 250
100
200
300
400
500
600
Time
Ma
gn
itu
d
e Ma
gn
itu
d
e
Frequency (Hz)
2 Hz + 10 Hz + 20Hz
Stationary
0 0.5 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 250
50
100
150
200
250
Time
Ma
gn
itu
d
e Ma
gn
itu
d
e
Frequency (Hz)
Non-
Stationary
0.0-0.4: 2 Hz +
0.4-0.7: 10 Hz +
0.7-1.0: 20Hz
CHIRP SIGNALS
0 0.5 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 250
50
100
150
Time
Ma
gn
itu
d
e
Ma
gn
itu
d
e
Frequency (Hz)
0 0.5 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 250
50
100
150
Time
Ma
gn
itu
d
e
Ma
gn
itu
d
e
Frequency (Hz)
Different in Time DomainFrequency: 2 Hz to 20 Hz Frequency: 20 Hz to 2 Hz
Same in Frequency Domain
At what time the frequency components occur? FT can not tell!
Fourier Analysis
Breaks down a signal into constituent
sinusoids of different frequencies
In other words: Transform the view of the
signal from time-base to frequency-base.
What’s wrong with Fourier?
By using Fourier Transform , we loose
the time information : WHEN did a
particular event take place ?
FT can not locate drift, trends, abrupt
changes, beginning and ends of events,
etc.
Calculating use complex numbers.
Short Time Fourier Analysis
In order to analyze small section of a
signal, Denis Gabor (1946), developed a
technique, based on the FT and using
windowing : STFT
STFT (or: Gabor Transform)
A compromise between time-based and
frequency-based views of a signal.
both time and frequency are
represented in limited precision.
The precision is determined by the size
of the window.
Once you choose a particular size for
the time window - it will be the same for
all frequencies.
What’s wrong with Gabor?
Many signals require a more flexible
approach - so we can vary the window
size to determine more accurately either
time or frequency.
What is Wavelet Analysis ?
And…what is a wavelet…?
A wavelet is a waveform of effectively
limited duration that has an average value
of zero.
Wavelet's properties
• Short time localized waves with zero
integral value.
• Possibility of time shifting.
• Flexibility.
12/9/2020 16
Fourier vs. Wavelet
• FFT, basis functions: sinusoids
• Wavelet transforms: small waves, called wavelet
• FFT can only offer frequency information
• Wavelet: frequency + temporal information
• Fourier analysis doesn’t work well on discontinuous, “bursty” data
– music, video, power, earthquakes,…
The Continuous Wavelet
Transform (CWT) A mathematical representation of the
Fourier transform:
Meaning: the sum over all time of the
signal f(t) multiplied by a complex
exponential, and the result is the Fourier
coefficients F() .
dtetfwF iwt)()(
18
What is wavelet transform?
• Provides time-frequency representation
• Wavelet transform decomposes a signal into a set of basis functions (wavelets)
• Wavelets are obtained from a single prototype wavelet Ψ(t) called mother wavelet by dilationsand shifting:
• where a is the scaling parameter and b is the shifting parameter
)(1
)(,a
bt
atba
Wavelet Transform (Cont’d)
Those coefficients, when multiplied by a
sinusoid of appropriate frequency ,
yield the constituent sinusoidal
component of the original signal:
Wavelet Transform
And the result of the CWT are Wavelet
coefficients .
Multiplying each coefficient by the
appropriately scaled and shifted wavelet
yields the constituent wavelet of the
original signal:
Scaling Wavelet analysis produces a time-scale
view of the signal.
Scaling means stretching or
compressing of the signal.
scale factor (a) for sine waves:
f t a
f t a
f t a
t
t
t
( )
( )
( )
sin( )
sin( )
sin( )
;
;
;
1
2 12
4 14
Scaling (Cont’d)
Scale factor works exactly the same
with wavelets:
f t a
f t a
f t a
t
t
t
( )
( )
( )
( )
( )
( )
;
;
;
1
2 12
4 14
Wavelet function
a
by
abx
abba
yxyxyx
,1, ,,
• b – shift
coefficient
• a – scale
coefficient
• 2D function
a
bxxba
a
1,
12/9/2020 24
Five Easy Steps to a Continuous Wavelet
Transform
1. Take a wavelet and compare it to a section at the start of
the original signal.
2. Calculate a correlation coefficient c
12/9/2020 25
Five Easy Steps to a Continuous Wavelet
Transform3. Shift the wavelet to the right and repeat steps 1 and 2 until you've
covered the whole signal.
4. Scale (stretch) the wavelet and repeat steps 1 through 3.
5. Repeat steps 1 through 4 for all scales.
12/9/2020 26
Coefficient Plots
Wavelets examplesDyadic transform
• For easier calculation we can
to discrete continuous signal.
• We have a grid of discrete
values that called dyadic grid .
• Important that wavelet
functions compact (e.g. no
overcalculatings) .
j
j
kb
a
2
2
Haar transform
Wavelet functions examples
• Haar function
• Daubechies function
Mallat* Filter Scheme
Mallat was the first to implement this
scheme, using a well known filter design
called “two channel sub band coder”,
yielding a ‘Fast Wavelet Transform’
12/9/2020 31
Discrete Wavelet transform
signal
filters
Approximation
(a)Details
(d)
lowpass highpass
Approximations and Details:
Approximations: High-scale, low-
frequency components of the signal
Details: low-scale, high-frequency
components
Input Signal
LPF
HPF
Decimation
The former process produces twice the
data it began with: N input samples
produce N approximations coefficients and
N detail coefficients.
To correct this, we Down sample (or:
Decimate) the filter output by two, by simply
throwing away every second coefficient.
Decimation (cont’d)
Input
Signal
LPF
HPF
A*
D*
So, a complete one stage block looks like:
Multi-level Decomposition
Iterating the decomposition process,
breaks the input signal into many lower-
resolution components: Wavelet
decomposition tree:
Orthogonality
• For 2 vectors
• For 2 functions
0*, nnwvwvn
0*, dttgtftgtf
b
a
Wavelet Transform
Inverse Wavelet Transform
All wavelet derived from mother wavelet
Inverse Wavelet Transform
wavelet with
scale, s and time, t
time-series
coefficients
of wavelets
build up a time-series as sum of wavelets of different
scales, s, and positions, t
An example:
40
Subband Coding Revisited
41
Subband Coding
Split the signal spectrum with a bank of filters as:
42
Wavelet Transform
• Continuous Wavelet Transform (CWT)
• Discrete Wavelet Transform (DWT)
43
CWT
• Continuous wavelet transform (CWT) of
1D signal is defined as
• The a,b is computed from the mother
wavelet by translation and dilation
dxxxfbfW baa )()()( ,
a
bx
axba
1)(,
44
Separates the high and low-frequency portions of a
signal through the use of filters
One level of transform:
Signal is passed through G & H filters.
Down sample by a factor of two
Multiple levels (scales) are made by repeating the
filtering and decimation process on lowpass outputs
1D Discrete Wavelet Transform
45
Haar Wavelet Transform
• Find the average of each pair of samples
• Find the difference between the average and sample
• Fill the first half with averages
• Fill the second half with differences
• Repeat the process on the first half
• Step 1:[3 5 4 8 13 7 5 3]
[4 6 10 4 -1 -2 3 1]
Averaging
Differencing
46
Haar Wavelet Transform
• Step 2[4 6 10 4 -1 -2 3 1]
[5 7 -1 3 -1 -2 3 1]
ex. (4 + 6)/2 = 54 - 5 = -1
Averaging Differencing
47
Haar Wavelet Transform
• Step 3[5 7 -1 3 -1 -2 3 1]
[6 -1 -1 3 -1 -2 3 1]
ex. (5 + 7)/2 = 65 - 6 = -1
Averaging Differencing
row average
48
Discrete Wavelet Transform
LL2 HL2
LH2 HH2HL1
LH1 HH1
Mexican hat wavelet
2
2
2
223
11
2
tt
t e
Fig. 7 The Mexican hat wavelet[5]
Also called the second derivative of
the Gaussian function
49
Morlet wavelet
21 4 2imt tt e e
2
21 4ˆ mU e
U(ω): step function
Fig. 8 Morlet wavelet with m equals to 3[4]
50
Shannon wavelet
sinc 2 cos 3 2t t t
1 0.5 1
ˆ0
ff
otherwise
Fig. 9 The Shannon wavelet in time and frequency domains[5]
51
Comparison of resolution
• Fourier Transform
Fig. 17 the result using Fourier
Transform
52
Comparison of resolution
• Windowed Fourier Transform
Fig. 18 the result using Windowed Fourier Transform
53
Comparison of resolution
• Discrete Wavelet Transform
Fig. 19 the result using Discrete Wavelet Transform
54
55
STFT and Wavelets
Motivation….
Earthquake
Fourier Transform
1
0
/21ˆN
k
Nink
in efN
f
2,.....,1
2
NNn
dfdttf22 ˆ
dtetff ti 2ˆ
deftf ti2ˆ
Fourier Transform
Inverse Fourier Transform
Parseval Theorem
Discrete Fourier Transform
Phase!!!
Limitations???
Non-Stationary Signals…
Fourier does not provide information about when different periods(frequencies)
where important: No localization in time
has the same support for every
and , but the number of cycles varies
with frequency.
dttgtfuGf u,,
Windowed (Short-Time) Fourier Transform
ti
u eutgtg
2
,
tg u, u
Estimates locally around , the amplitude of
a sinusoidal wave of frequency
241 2ueug D. Gabor
u ug Function with local support.
Limitations??
Fixed resolution.
Related to the Heisenberg uncertainty principle. The product of the standard deviation
in time and frequency is limited.
The width of the windowing function relates to the how the signal is represented — it
determines whether there is good frequency resolution (frequency components close
together can be separated) or good time resolution (the time at which frequencies change).
Selection of determines and . ug gg
ggt ˆ00 Localization:
Example….
x(t) = cos(2π10t) for
x(t) = cos(2π25t) for
x(t) = cos(2π50t) for
x(t) = cos(2π100t) for
Wavelet Transform
dtttfuW u, , 0
uttu
1,
Gives good time resolution for high frequency events, and good frequency resolution for low
frequency events, which is the type of analysis best suited for many real signals.
0dtt
dtt
12dttMother wavelet
properties
0
2
,
0
2
,
0ˆ
ˆ
ˆ
,
d
d
t
t
t
0,10,1
0,1
ˆ0ˆ
0t
t,
t,ˆ
t,ˆ
0,1ˆ
0
ˆ0
ˆ
0,1
,
t
.
Wavelet Transform
Some Continuous Wavelets
241
2
0
ee
i
Morlet
uttu
1,
ti
u eutgtg
2
,
241 2ueug Gabor
Torrence and Compo (1998)
Continuous Wavelet
Transform
For Discrete Data
Time series
Wavelet
Defined as the convolution with a scaled
and translated version of
DFT (FFT) of the time series
N times for each s: Slow!
Using the convolution theorem,
the wavelet transform is the
inverse Fourier transform
Mallat's multiresolution framework
Design method of most of the
practically relevant discrete
wavelet transforms (DWT)
Doppler Signal
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