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Window Fourier and wavelet transforms. Properties and applications of the wavelets. A.S. Yakovlev

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Contents 1. Fourier Transform 2. Introduction To Wavelets 3. Wavelet Transform 4. Types Of Wavelets 5. Applications

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

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Window Fourier Transform

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

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Window Fourier Transform Examples of window functions Gabor 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 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

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Multi Resolution Analysis

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

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Continuous Wavelet Transform (CWT) Direct transform Inverse transform

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

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Discrete Wavelet Decomposition

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Fast Wavelet Transform (FWT) Formalism In the same way

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Fast Wavelet Transform (FWT)

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Fast Wavelet Transform (FWT) Matrix notation

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

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

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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.

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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

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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

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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

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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.

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

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Types Of Wavelets Shannon Wavelet Father function Wavelet function

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Types Of Wavelets Shannon Wavelet Fourier transform of father function

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Types Of Wavelets Shannon Wavelet 1. Orthogonal 2. Localized in frequency domain 3. Easy to calculate 4. Infinite support and slow decay

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Types Of Wavelets Shannon Wavelet Shannon a)Father function and b)Wavelet function a) b)

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Types Of Wavelets Meyer Wavelets Fourier transform of father function

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Types Of Wavelets Daubishes Wavelets 1. Orthogonal in L 2 2. Compact support 3. Zero moments of father-function

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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

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Types Of Wavelets Daubechies Wavelets Note: Daubechhies D1 wavelet is Haar Wavelet

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Types Of Wavelets Daubechies Wavelets Daubechhies D2 a)Father function and b)Wavelet function a) b)

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Types Of Wavelets Daubechies Wavelets Daubechhies 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 than ordinary Daubechhies wavelets

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Types Of Wavelets Daubechies Symmlets Symmlet a)Father function and b)Wavelet function a) b)

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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

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

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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 function Using the equation

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How to plot a function

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Applications of the wavelets 1. Data processing 2. Data compression 3. Solution of differential equations

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Digital signal Suppose we have a signal:

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Digital signal Fourier method Fourier spectrum Reconstruction

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Digital signal Wavelet Method 8 th Level Coefficients Reconstruction

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Analog signal Suppose we have a signal:

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Analog signal Fourier Method Fourier Spectrum

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Analog signal Fourier Method Reconstruction

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Analog signal Wavelet Method 9 th level coefficients

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Analog signal Wavelet Method Reconstruction

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Short living state Signal

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Short living state Gabor transform

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Short living state Wavelet 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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