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 Contains no information about time localization Window Fourier Transform Where g(t) - window function In discrete form

Window Fourier Transform

Window Fourier Transform Examples of window functions Hat function Gauss function Gabor function

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 {V j } with the following properties: If 5. If 6. Set of functions where defines basis in V j

Multi Resolution Analysis

Multi Resolution Analysis Definitions Father function basis in V Wavelet function basis in W Scaling equation Dilation equation Filter coefficients h i, g i

Continuous Wavelet Transform (CWT) Direct transform Inverse transform

Discrete Wavelet Decomposition Function f(x) Decomposition We want In orthonormal case

Discrete Wavelet Decomposition

Fast Wavelet Transform (FWT) Formalism In the same way

Fast Wavelet Transform (FWT)

Fast Wavelet Transform (FWT) Matrix notation

Fast Wavelet Transform (FWT) Note FWT is an orthogonal transform It has linear complexity

Conditions on wavelets 1. Orthogonality: 2. Zero moments of father function and wavelet function:

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 L 2 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: First equation corresponds to orthonormality in L 2, Second is required to satisfy dilation equation. Obviously the solution is

Types Of Wavelets Haar 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

Types Of Wavelets Haar Wavelets And 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.

Types Of Wavelets Haar Wavelets Haar a)Father function and B)Wavelet function a) b)

Types Of Wavelets Shannon Wavelet Father function Wavelet function

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 L 2 2. Compact support 3. Zero moments of father-function

Types Of Wavelets Daubechies Wavelets First two equation correspond to orthonormality In L 2, 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 L 2 2. Compact support 3. Zero moments of father-function 4. Zero moments of wavelet function

Types Of Wavelets Coifmann Wavelets (Coiflets) Set of equations to calculate coefficients

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

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 8 th 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 9 th 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