Window Fourier and wavelet transforms.

Properties and applications of the wavelets.

A.S. Yakovlev

Contents

1. Fourier Transform2. Introduction To Wavelets3. Wavelet Transform4. Types Of Wavelets5. Applications

Window Fourier TransformOrdinary Fourier Transform

Contains no information about time localization

Window Fourier Transform

Where g(t) - window functionIn discrete form

1( ) ( )

2i 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

Window Fourier Transform

Window Fourier TransformExamples of window functions

Hat function

Gauss function

Gabor function 0

0 22

( )1( ) exp ( ) exp

22

t tg t i t t i

2

0

2 2)(

exp2

1)(

tt

tg

1,0

]1,0[,1

0,0

xg

xg

xg

Window Fourier TransformExamples of window functions

Gabor function

Fourier Transform

Window Fourier Transform

Window Fourier TransformDisadvantage

Multi Resolution Analysis MRA is a sequence of spaces {Vj} with

the following properties:1. 2. 3. 4. If 5. If 6. Set of functions where

defines basis in Vj

1 jj VV

Zj j RLV

2

Zj jV

0

1)2()( jj VtfVtf

jj VktfVtf )()(

kj ,)2(2 2/

, ktjjkj

Multi Resolution Analysis

Multi Resolution Analysis Definitions

Father function basis in V Wavelet function basis in WScaling equationDilation equationFilter coefficients hi , gi

)2(2 2/, ktjjkj

( ) (2 )ii Z

x h x i

1

( ) 2 (2 )

( 1)

ii Z

ii L i

x g x i

g h

Zi

Continuous Wavelet Transform (CWT)

wave 1/ 2( , ) | | ( )t b

T f s a f t dta

wave( ) ( , )t b

f t T f s d dsa

Direct transform

Inverse transform

Discrete Wavelet DecompositionFunction f(x)

Decomposition

We want

In orthonormal case

2 1

, ,0

( ) ( )j

j k j k jk

f t s t V

1 2 1 2 1

, , , ,0 0

( ) ( ) ( )j LJ

j k j k L k L kj 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

Discrete Wavelet Decomposition

0321

0121

WWWW

VVVVV

nnn

nnn

Fast Wavelet Transform (FWT) Formalism

In the same way

, , 2 1,

2 , 2 1,

( ) ( ) ( ) ( )

( ) ( )

j k j k l k j ll Z

l k j k l k j ll 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 ll Z

s h s

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 w

s s s wT

s w s w

s w s w

s w

s w

Fast Wavelet Transform (FWT) Matrix notation

0 1 2 3

0 1 2 3

0 1 2 3

0 1 2 3

2 3 0 12

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 0

0 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 g

g g g g

g g g g

g g g g

Fast Wavelet Transform (FWT) Matrix notation

0 2 0 2

1 3 1 3

2 0 2 0

3 1 3 1

2 0 2 02 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 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

rev tD D

h h g g

h h g g

h h g g

h h g g

h h g gT T

h h g g

h h g g

h h g g

h h g g

h h g g

Fast Wavelet Transform (FWT) Note

FWT is an orthogonal transform

It has linear complexity

1

*

rev t

rev

T T T

T T I

Conditions on wavelets

1. Orthogonality:

2. Zero moments of father function and wavelet function:

2 , k k l lk Z

h h l Z

( ) 0,

( ) 0.

ii

ii

M t t dt

t t dt

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 WaveletsHaar Wavelets

1. Orthogonal in L2

2. Compact Support3. Scaling function is symmetric

Wavelet function is antisymmetric

4. Infinite support in frequency domain

Types Of WaveletsHaar 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 20 1

0 1

1

2

h h

h h

0 1

1

2h h

Types Of WaveletsHaar Wavelets

Theorem: The only orthogonal basis with the symmetric, compactly supported father-function is the Haar basis.

Proof:Orthogonality:For l=2n this is For l=2n-2 this is

1 0 0 1[..., ,..., , , , ,..., ,...]n nh a a a a a a

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

Types Of WaveletsHaar WaveletsAnd so on. The only possible sequences are:

Among these possibilities only the Haar filterleads to convergence in the solution of

dilationequation.End of proof.

1 1[...,0,0, ,0,0,0,0,0,0, ,0,0,...]

2 2

Types Of WaveletsHaar WaveletsHaar a)Father function and B)Wavelet

function

a) b)

Types Of WaveletsShannon Wavelet

Father function

Wavelet functionx

xxx

)sin(

)(sinc)(

xxx

)sin()2sin(

Types Of WaveletsShannon Wavelet

Fourier transform of father function

Types Of WaveletsShannon Wavelet

1. Orthogonal2. Localized in frequency domain3. Easy to calculate4. Infinite support and slow decay

Types Of WaveletsShannon Wavelet

Shannon a)Father function and b)Wavelet function

a) b)

Types Of WaveletsMeyer Wavelets

Fourier transform of father function

Types Of WaveletsDaubishes Wavelets

1. Orthogonal in L2

2. Compact support3. Zero moments of father-function

( ) 0iiM x x dx

Types Of WaveletsDaubechies Wavelets

First two equation correspond to orthonormality

In L2, Third equation to satisfy dilation

equation, Fourth one – moment of the father-function

2 2 2 20 1 2 3

0 2 1 3

0 1 2 3

1 2 3

1

0

2

2 3 0

h h h h

h h h h

h h h h

h h h

Types Of WaveletsDaubechies Wavelets

Note: Daubechhies D1 wavelet is Haar Wavelet

Types Of WaveletsDaubechies Wavelets

Daubechhies D2 a)Father function and b)Wavelet function

a) b)

Types Of WaveletsDaubechies Wavelets

Daubechhies D3 a)Father function and b)Wavelet function

a) b)

Types Of WaveletsDaubechhies Symmlets

(for reference only)Symmlets are not symmetric!They are just more symmetric than

ordinary Daubechhies wavelets

Types Of WaveletsDaubechies Symmlets

Symmlet a)Father function and b)Wavelet function

a) b)

Types Of WaveletsCoifmann Wavelets (Coiflets)

1. Orthogonal in L2

2. Compact support

3. Zero moments of father-function

4. Zero moments of wavelet function

( ) 0iiM x x dx

( ) 0ii x x dx

Types Of WaveletsCoifmann Wavelets (Coiflets)

Set of equations to calculate coefficients2 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

0

0

2

2 2 3 0

2 2 3 0

h h h h h h

h h h h h h h h

h h h h

h h h h h h

h h h h h

h h h h h

Types Of WaveletsCoifmann Wavelets (Coiflets)

Coiflet K1 a)Father function and b)Wavelet function

a) b)

Types Of WaveletsCoifmann Wavelets (Coiflets)

Coiflet K2 a)Father function and b)Wavelet function

a) b)

How to plot a function

Using the equation ( ) (2 )ii Z

x h x i

How to plot a function

Applications of the wavelets

1. Data processing2. Data compression3. Solution of differential equations

“Digital” signal

Suppose we have a signal:

“Digital” signalFourier method

Fourier spectrum Reconstruction

“Digital” signalWavelet Method

8th Level Coefficients Reconstruction

“Analog” signal

Suppose we have a signal:

“Analog” signalFourier Method

Fourier Spectrum

“Analog” signalFourier Method

Reconstruction

“Analog” signalWavelet Method

9th level coefficients

“Analog” signalWavelet Method

Reconstruction

Short living stateSignal

Short living stateGabor transform

Short living stateWavelet transform

Conclusion

Stationary signal – Fourier analysisStationary signal with singularities –

Window Fourier analysisNonstationary signal – Wavelet

analysis

Acknowledgements

1. Prof. Andrey Vladimirovich Tsiganov

2. Prof. Serguei Yurievich Slavyanov