# Window Fourier and wavelet transforms. Properties and applications

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```					  Window Fourier and wavelet
transforms.
Properties and applications of the
wavelets.

A.S. Yakovlev
Contents
1.    Fourier Transform
2.    Introduction To Wavelets
3.    Wavelet Transform
4.    Types Of Wavelets
5.    Applications
Window Fourier Transform
Ordinary Fourier Transform
1
 Ff  ( )             f (t )e  it dt
2
Contains no information about time localization
Window Fourier Transform
 T win f  (, s)   f (t ) g (t  s)eit dt
Where g(t) - window function
In discrete form
Tm,n  f    f (t ) g (t  ns0 )eit dt
win
Window Fourier Transform
Window Fourier Transform
Examples of window functions
Hat function
 g  0, x  0

 g  1, x  [0,1]
 g  0, x  1


Gauss function
1         (t  t0 ) 
g (t )           exp       2 
2 2
 2 
Gabor function
1                                 (t  t0 ) 
g (t )          exp  i(t  t0 )  i  exp       2 
2 2
 2 
Window Fourier Transform
Examples of window functions
Gabor function
Fourier Transform
Window Fourier Transform
Window Fourier Transform
Disadvantage
Multi Resolution Analysis
MRA is a sequence of spaces {Vj} with the
following properties:
1.   V j  V j 1
2.      jZ
V j  L2 R 
3.      jZ
V j  0
4.       If f (t ) V j  f (2t ) V j 1
5.       If f (t )  V j  f (t  k )  V j
6.       Set of functions  j ,k  where
 j , k  2 j / 2  (2 j t  k )     defines basis in Vj
Multi Resolution Analysis
Multi Resolution Analysis
Definitions
Father function  basis in V
Wavelet function  basis in W
Scaling equation     j , k  2 j / 2  (2 j t  k )

Dilation equation  ( x)   hi (2 x  i)
iZ

Filter coefficients hi , gi i  Z
 ( x)  2  gi (2 x  i )
iZ

gi  (1)i hL i 1
Continuous Wavelet
Transform (CWT)
Direct transform
 t b 
T wave f  (s, ) | a |1/ 2      f (t ) 
 a 
dt
Inverse transform
 t b 
f (t )   T   wave
f  ( s, )       d ds
 a 
Discrete Wavelet
Decomposition
Function f(x)
2 j 1

Decomposition f (t )   s j ,k j ,k (t ) V j
k 0
J 1 2 j 1                      2L 1

We want f (t )   w j ,k j ,k (t )   sL,kL,k (t )
j  L k 0                       k 0


In orthonormal case s j ,k                    

f (t ) j ,k (t )dt


w j ,k     

f (t ) j ,k (t )dt
Discrete Wavelet
Decomposition
Vn      Vn 1     Vn 2       V1    V0
                                
Wn 1      Wn 2      Wn 3         W0
Fast Wavelet Transform (FWT)
Formalism
                             
w j ,k         f (t ) j ,k (t )dt        f (t ) gl 2 k j 1,l (t ) 
                                    lZ


g
lZ
l 2k       f (t ) j ,k (t )dt   gl 2 k s j 1,l
lZ


In the same way
s j ,k   hl  2 k s j 1,l
lZ
Fast Wavelet Transform (FWT)
 s1,0         s0,0 
                   
 s1,1         s0,1 
 s1,2         s0,2         s0,0         w0,0 
                                             
 s1,3 
T              s0,3 
    s0,1 
    w0,1 
s1,4          w0,0         s0,2         w0,2 
                          
s           w 
      
 s1,5         w0,1         0,3          0,3 
s            w 
 1,6          0,2 
s            w 
 1,7          0,3 
Fast Wavelet Transform (FWT)
Matrix notation
 h0   h1   h2   h3   0    0    0    0    0    0
                                                 
0     0    h0   h1   h2   h3   0    0    0    0
0     0    0    0    h0   h1   h2   h3   0    0
                                                 
0     0    0    0    0    0    h0   h1   h2   h3 
 h2   h3   0    0    0    0    0    0    h0   h1 
TD 2                                                    
 g0   g1   g2   g3   0    0    0    0    0    0
0     0    g0   g1   g2   g3   0    0    0    0
                                                 
0     0    0    0    g0   g1   g2   g3   0    0
0     0    0    0    0    0    g0   g1   g2   g3 
                                                 
g                                             g1 
 2    g0   0    0    0    0    0    0    g0      
Fast Wavelet Transform (FWT)
Matrix notation
 h0   0    0    0    h2   g0   0    0    0    g2 
                                                 
 h1   0    0    0    h3   g1   0    0    0    g3 
 h2   h0   0    0    0    g2   g0   0    0    0
                                                 
 h3   h1   0    0    0    g3   g1   0    0    0
0     h2   h0   0    0    0    g2   g0   0    0
TD 2  TD 2
rev    t
                                                 
0     h3   h1   0    0    0    g3   g1   0    0
0     0    h2   h0   0    0    0    g2   g0   0
                                                 
0     0    h3   h1   0    0    0    g3   g1   0
0     0    0    h2   h0   0    0    0    g2   g0 
                                                 
0                                             g1 
      0    0    h3   h1   0    0    0    g3      
Fast Wavelet Transform (FWT)
Note
FWT is an orthogonal transform
T rev  T t  T 1
T *T rev  I

It has linear complexity
Conditions on wavelets
1.      Orthogonality:
h h
k Z
k k  2l    l , l  Z

2.      Zero moments of father function and
wavelet function:
M i   t i (t )dt  0,
i   t i (t )dt  0.
Conditions on wavelets
3.    Compact support:
Theorem: if wavelet has nonzero
coefficients with only indexes from
n to n+m the father function
support is [n,n+m].

4.   Rational coefficients.
5.   Symmetry of coefficients.
Types Of Wavelets
Haar Wavelets
1.   Orthogonal in L2
2.   Compact Support
3.   Scaling function is symmetric
Wavelet function is antisymmetric
4.   Infinite support in frequency domain
Types Of Wavelets
Haar Wavelets
Set of equation to calculate coefficients:
 h02  h12  1


 h0  h1  2


First equation corresponds to orthonormality in
L2, Second is required to satisfy dilation
equation.
1
Obviously the solution is   h0  h1 
2
Types Of Wavelets
Haar Wavelets
Theorem: The only orthogonal basis with the
symmetric, compactly supported father-
function is the Haar basis.
Proof: h  [..., an ,..., a1, a0 , a0 , a1,..., an ,...]
Orthogonality: hk hk  2l  0, if l  0.
k Z

For l=2n this is an an 1  an 1an  0,
For l=2n-2 this is
an an 3  an 1an  2  an  2 an 1  an 3an  0.
Types Of Wavelets
Haar Wavelets
And so on.
The only possible sequences are:
1                      1
[..., 0, 0,    , 0, 0, 0, 0, 0, 0,    , 0, 0,...]
2                      2
Among these possibilities only the Haar filter
leads to convergence in the solution of dilation
equation.
End of proof.
Types Of Wavelets
Haar Wavelets
Haar a)Father function and B)Wavelet function

a)                b)
Types Of Wavelets
Shannon Wavelet
Father function
sin(x)
 ( x)  sinc( x) 
x
Wavelet function
sin(2x)  sin(x)

x
Types Of Wavelets
Shannon Wavelet
Fourier transform of father function
Types Of Wavelets
Shannon Wavelet
1.   Orthogonal
2.   Localized in frequency domain
3.   Easy to calculate
4.   Infinite support and slow decay
Types Of Wavelets
Shannon Wavelet
Shannon a)Father function and b)Wavelet function

a)              b)
Types Of Wavelets
Meyer Wavelets
Fourier transform of father function
Types Of Wavelets
Daubishes Wavelets
1.   Orthogonal in L2
2.   Compact support
3.   Zero moments of father-function
M i   xi ( x)dx  0
Types Of Wavelets
Daubechies Wavelets
h02  h12  h2  h32  1
2


h0 h2  h1h3  0

h0  h1  h2  h3  2
h  2h  3h  0
 1       2     3

First two equation correspond to orthonormality
In L2, Third equation to satisfy dilation
equation, Fourth one – moment of the father-
function
Types Of Wavelets
Daubechies Wavelets
Note: Daubechhies D1 wavelet is Haar Wavelet
Types Of Wavelets
Daubechies Wavelets
Daubechhies D2 a)Father function and b)Wavelet
function

a)               b)
Types Of Wavelets
Daubechies Wavelets
Daubechhies D3 a)Father function and b)Wavelet
function

a)               b)
Types Of Wavelets
Daubechhies Symmlets
(for reference only)
Symmlets are not symmetric!
They are just more symmetric than
ordinary Daubechhies wavelets
Types Of Wavelets
Daubechies Symmlets
Symmlet a)Father function and b)Wavelet function

a)               b)
Types Of Wavelets
Coifmann Wavelets (Coiflets)
1.   Orthogonal in L2
2.   Compact support
3.   Zero moments of father-function
M i   xi ( x)dx  0
4.   Zero moments of wavelet function
i   xi ( x)dx  0
Types Of Wavelets
Coifmann Wavelets (Coiflets)
Set of equations to calculate coefficients
h2  h1  h02  h12  h22  h32  1

h2 h0  h1h1  h0 h2  h1h3  0
h2 h2  h1h3  0


h2  h1  h0  h1  h2  h3  2
2h  h  h  2h  3h  0
     2    1    1      2      3

2h2  h1  h1  2h2  3h3  0

Types Of Wavelets
Coifmann Wavelets (Coiflets)
Coiflet K1 a)Father function and b)Wavelet function

a)               b)
Types Of Wavelets
Coifmann Wavelets (Coiflets)
Coiflet K2 a)Father function and b)Wavelet function

a)               b)
How to plot a function
Using the equation  ( x)   hi (2 x  i)
iZ
How to plot a function
Applications of the wavelets
1.   Data processing
2.   Data compression
3.   Solution of differential equations
“Digital” signal
Suppose we have a signal:
“Digital” signal
Fourier method
Fourier spectrum   Reconstruction
“Digital” signal
Wavelet Method
8th Level Coefficients   Reconstruction
“Analog” signal
Suppose we have a signal:
“Analog” signal
Fourier Method
Fourier Spectrum
“Analog” signal
Fourier Method
Reconstruction
“Analog” signal
Wavelet Method
9th level coefficients
“Analog” signal
Wavelet Method
Reconstruction
Short living state
Signal
Short living state
Gabor transform
Short living state
Wavelet transform
Conclusion
Stationary signal – Fourier analysis
Stationary signal with singularities –
Window Fourier analysis
Nonstationary signal – Wavelet analysis
Acknowledgements
1.   Prof. Andrey Vladimirovich Tsiganov
2.   Prof. Serguei Yurievich Slavyanov

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