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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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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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Examples of window functionsGabor function


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Fourier Transform


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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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(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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(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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(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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(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

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