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Window Fourier and wavelet
transforms.Properties and applications of the
wavelets.
A.S. Yakovlev
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Contents1. Fourier Transform
2. Introduction To Wavelets
3. Wavelet Transform4. Types Of Wavelets
5. Applications
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Window Fourier TransformOrdinary Fourier Transform
Contains no information about time localizationWindow Fourier Transform
Whereg(t) - window functionIn discrete form
( )1
( ) ( )2
i tFf f t e dt
=
( )win ( , ) ( ) ( ) i tT f s f t g t s e dt =
( )win, 0( ) ( )i t
m nT f f t g t ns e dt =
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Window Fourier Transform
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Window Fourier Transform
Examples of window functionsHat function
Gauss function
Gabor function
( ) 00 22( )1
( ) exp ( ) exp22
t tg t i t t i
=
= 20
2 2
)(
exp2
1
)(
tt
tg
>==
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Window Fourier Transform
Examples of window functionsGabor function
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Fourier Transform
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Window Fourier Transform
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Window Fourier Transform
Disadvantage
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Multi Resolution AnalysisMRA is a sequence of spaces {Vj} with the
following properties:
1. 2.
3.
4.
If5. If
6. Set of functions wheredefines basis in Vj
1+
jj
VV
( )Zj j RLV = 2{ } Zj jV = 0
1)2()( + jj VtfVtf
jj VktfVtf )()(
{ }kj ,
)2(2 2/, ktjj
kj =
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Multi Resolution Analysis
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Multi Resolution Analysis
DefinitionsFather function basis in V
Wavelet function basis in W
Scaling equation
Dilation equation
Filter coefficients hi , gi
)2(2 2/, ktjj
kj =
( ) (2 )ii Z
x h x i
=
1
( ) 2 (2 )
( 1)
i
i Z
i
i L i
x g x i
g h
=
=
Zi
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Continuous Wavelet
Transform (CWT)
( )wave 1/ 2( , ) | | ( )t b
T f s a f t dt a
=
( )wave( ) ( , )t b
f t T f s d dsa
=
Direct transform
Inverse transform
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Discrete Wavelet
DecompositionFunctionf(x)
Decomposition
We want
In orthonormal case
2 1
, ,
0
( ) ( )
j
j k j k j
k
f t s t V
=
= 1 2 1 2 1
, , , ,
0 0
( ) ( ) ( )j LJ
j k j k L k L k
j L k k
f t w t s t
= = =
= +
, ,
, ,
( ) ( )
( ) ( )
j k j k
j k j k
s f t t dt
w f t t dt
=
=
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Discrete Wavelet
Decomposition
0321
0121
WWWW
VVVVV
nnn
nnn
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Fast Wavelet Transform
(FWT) Formalism
In the same way
, , 2 1,
2 , 2 1,
( ) ( ) ( ) ( )
( ) ( )
j k j k l k j l
l Z
l k j k l k j l l Z l Z
w f t t dt f t g t
g f t t dt g s
+
+
= = =
=
, 2 1,j k l k j l
l Z
s h s +
=
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Fast Wavelet Transform
(FWT)1,0 0,0
1,1 0,1
1,2 0,2 0,0 0,0
1,3 0,3 0,1 0,1
1,4 0,0 0,2 0,2
1,5 0,1 0,3 0,3
1,6 0,2
1,7 0,3
s s
s s
s s s ws s s w
Ts w s w
s w s w
s w
s w
+
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Fast Wavelet Transform
(FWT) Matrix notation0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
2 3 0 1
2
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
2 0 0 1
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 00 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
D
h h h h
h h h h
h h h h
h h h h
h h h hT
g g g g
g g g gg g g g
g g g g
g g g g
=
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Fast Wavelet Transform
(FWT) Matrix notation0 2 0 2
1 3 1 3
2 0 2 0
3 1 3 1
2 0 2 0
2 2
3 1 3 1
2 0 2 0
3 1 3 1
2 0 2 0
3 1 3 1
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 00 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
rev t
D D
h h g g
h h g g
h h g g
h h g g
h h g g T T
h h g g
h h g g h h g g
h h g g
h h g g
= =
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Fast Wavelet Transform
(FWT) NoteFWT is an orthogonal transform
It has linear complexity
1
*
rev t
rev
T T T
T T I
= =
=
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Conditions on wavelets1. Orthogonality:
2. Zero moments of father function andwavelet function:
2 ,k k l l k Z
h h l Z +
=
( ) 0,
( ) 0.
i
i
i
i
M t t dt
t t dt
= =
= =
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Conditions on wavelets3. Compact support:
Theorem: if wavelet has nonzero
coefficients with only indexes fromn to n+m the father functionsupport is [n,n+m].
4. Rational coefficients.
5. Symmetry of coefficients.
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Types Of Wavelets
Haar Wavelets1. Orthogonal inL2
2. Compact Support
3. Scaling function is symmetricWavelet function is antisymmetric
4. Infinite support in frequency domain
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Types Of Wavelets
Haar WaveletsSet of equation to calculate coefficients:
First equation corresponds to orthonormality in
L2, Second is required to satisfy dilation
equation.
Obviously the solution is
2 2
0 1
0 1
1
2
h h
h h
+ =
+ =
0 1
1
2h h= =
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Types Of Wavelets
Haar WaveletsTheorem: The only orthogonal basis with the
symmetric, compactly supported father-
function is the Haar basis.Proof:
Orthogonality:
For l=2n this isFor l=2n-2 this is
1 0 0 1[..., ,..., , , , ,..., ,...]n nh a a a a a a=r
2 0, if 0.k k lk Z
h h l+
=
1 1 0,n n n na a a a + =
3 1 2 2 1 3 0.n n n n n n n na a a a a a a a + + + =
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Types Of Wavelets
Haar WaveletsAnd so on.
The only possible sequences are:
Among these possibilities only the Haar filter
leads to convergence in the solution of dilation
equation.
End of proof.
1 1[..., 0, 0, , 0, 0, 0, 0,0,0, , 0, 0,...]
2 2
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Types Of Wavelets
Haar WaveletsHaar a)Father function and B)Wavelet function
a) b)
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Types Of Wavelets
Shannon WaveletFather function
Wavelet functionx
xxx
)sin()(sinc)( ==
x
xx
)sin()2sin( =
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Types Of Wavelets
Shannon WaveletFourier transform of father function
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Types Of Wavelets
Shannon Wavelet1. Orthogonal
2. Localized in frequency domain
3. Easy to calculate
4. Infinite support and slow decay
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Types Of Wavelets
Shannon WaveletShannon a)Father function and b)Wavelet function
a) b)
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Types Of Wavelets
Meyer WaveletsFourier transform of father function
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Types Of Wavelets
Daubishes Wavelets1. Orthogonal inL2
2. Compact support
3. Zero moments of father-function( ) 0iiM x x dx= =
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Types Of Wavelets
Daubechies Wavelets
First two equation correspond to orthonormality
InL2, Third equation to satisfy dilation
equation, Fourth one moment of the father-
function
2 2 2 2
0 1 2 3
0 2 1 3
0 1 2 3
1 2 3
1
0
22 3 0
h h h h
h h h h
h h h hh h h
+ + + =
+ =
+ + + = + + =
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Types Of Wavelets
Daubechies WaveletsNote: Daubechhies D1 wavelet is Haar Wavelet
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Types Of Wavelets
Daubechies WaveletsDaubechhies D2 a)Father function and b)Wavelet
function
a) b)
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Types Of Wavelets
Daubechies WaveletsDaubechhies D3 a)Father function and b)Wavelet
function
a) b)
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Types Of Wavelets
Daubechhies Symmlets(for reference only)
Symmlets are not symmetric!
They are just more symmetric thanordinary Daubechhies wavelets
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Types Of Wavelets
Daubechies SymmletsSymmlet a)Father function and b)Wavelet function
a) b)
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Types Of Wavelets
Coifmann Wavelets (Coiflets)1. Orthogonal inL2
2. Compact support
3. Zero moments of father-function
4. Zero moments of wavelet function
( ) 0iiM x x dx= =
( ) 0ii
x x dx = =
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Types Of Wavelets
Coifmann Wavelets (Coiflets)Set of equations to calculate coefficients
2 2 2 2
2 1 0 1 2 3
2 0 1 1 0 2 1 3
2 2 1 3
2 1 0 1 2 3
2 1 1 2 3
2 1 1 2 3
1
00
2
2 2 3 0
2 2 3 0
h h h h h h
h h h h h h h hh h h h
h h h h h h
h h h h h
h h h h h
+ + + + + =
+ + + = + =
+ + + + + =
+ + + = + + =
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Types Of Wavelets
Coifmann Wavelets (Coiflets)Coiflet K1 a)Father function and b)Wavelet function
a) b)
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Types Of Wavelets
Coifmann Wavelets (Coiflets)Coiflet K2 a)Father function and b)Wavelet function
a) b)
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How to plot a functionUsing the equation ( ) (2 )i
i Z
x h x i
=
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How to plot a function
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Applications of the wavelets1. Data processing
2. Data compression
3. Solution of differential equations
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Digital signalSuppose we have a signal:
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Digital signal
Fourier method Fourier spectrum Reconstruction
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Digital signal
Wavelet Method8th Level Coefficients Reconstruction
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Analog signalSuppose we have a signal:
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Analog signal
Fourier MethodFourier Spectrum
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Analog signal
Fourier MethodReconstruction
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Analog signal
Wavelet Method9th level coefficients
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Analog signal
Wavelet MethodReconstruction
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Short living state
Signal
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Short living state
Gabor transform
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Short living stateWavelet transform
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Conclusion
Stationary signal Fourier analysis
Stationary signal with singularities
Window Fourier analysis
Nonstationary signal Wavelet analysis
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Acknowledgements
1. Prof. Andrey Vladimirovich Tsiganov
2. Prof. Serguei Yurievich Slavyanov