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

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

2. Contents Fourier Transform Introduction To Wavelets Wavelet Transform Types Of Wavelets Applications

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

4. Window Fourier Transform

5. Window Fourier TransformExamples of window functions Hat function Gauss function Gabor function

6. Window Fourier TransformExamples of window functions Gabor function

7. Fourier Transform

8. Window Fourier Transform

9. Window Fourier TransformDisadvantage

10. Multi Resolution Analysis MRA is a sequence of spaces {Vj} with the following properties: If If Set of functions where defines basis in Vj

11. Multi Resolution Analysis

12. Multi Resolution Analysis Definitions Father function basis in V Wavelet function basis in W Scaling equation Dilation equation Filter coefficients hi , gi

13. Continuous Wavelet Transform (CWT) Direct transform Inverse transform

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

15. Discrete Wavelet Decomposition

16. Fast Wavelet Transform (FWT) Formalism In the same way

17. Fast Wavelet Transform (FWT)

18. Fast Wavelet Transform (FWT) Matrix notation

19. Fast Wavelet Transform (FWT) Matrix notation

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

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

22. Conditions on wavelets Compact support:Theorem: if wavelet has nonzero coefficients with only indexes from n to n+m the father function support is [n,n+m]. Rational coefficients. Symmetry of coefficients.

23. Types Of WaveletsHaar Wavelets Orthogonal in L2 Compact Support Scaling function is symmetricWavelet function is antisymmetric Infinite support in frequency domain

24. Types Of WaveletsHaar Wavelets Set of equation to calculate coefficients: First equation corresponds to orthonormality in L2, Second is required to satisfy dilation equation. Obviously the solution is

25. 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

26. Types Of WaveletsHaar 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.

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

28. Types Of WaveletsShannon Wavelet Father function Wavelet function

29. Types Of WaveletsShannon Wavelet Fourier transform of father function

30. Types Of WaveletsShannon Wavelet Orthogonal Localized in frequency domain Easy to calculate Infinite support and slow decay

31. Types Of WaveletsShannon Wavelet Shannon a)Father function and b)Wavelet function a) b)

32. Types Of WaveletsMeyer Wavelets Fourier transform of father function

33. Types Of WaveletsDaubishes Wavelets Orthogonal in L2 Compact support Zero moments of father-function

34. 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

35. Types Of WaveletsDaubechies Wavelets Note: Daubechhies D1 wavelet is Haar Wavelet

36. Types Of WaveletsDaubechies Wavelets Daubechhies D2 a)Father function and b)Wavelet function a) b)

37. Types Of WaveletsDaubechies Wavelets Daubechhies D3 a)Father function and b)Wavelet function a) b)

38. Types Of WaveletsDaubechhies Symmlets (for reference only) Symmlets are not symmetric! They are just more symmetric than ordinary Daubechhies wavelets

39. Types Of WaveletsDaubechies Symmlets Symmlet a)Father function and b)Wavelet function a) b)

40. Types Of WaveletsCoifmann Wavelets (Coiflets) Orthogonal in L2 Compact support Zero moments of father-function Zero moments of wavelet function

41. Types Of WaveletsCoifmann Wavelets (Coiflets) Set of equations to calculate coefficients

42. Types Of WaveletsCoifmann Wavelets (Coiflets) Coiflet K1 a)Father function and b)Wavelet function a) b)

43. Types Of WaveletsCoifmann Wavelets (Coiflets) Coiflet K2 a)Father function and b)Wavelet function a) b)

44. How to plot a function Using the equation

45. How to plot a function

46. Applications of the wavelets Data processing Data compression Solution of differential equations 

47. �Digital� signal Suppose we have a signal:

48. �Digital� signalFourier method Fourier spectrum Reconstruction

49. �Digital� signalWavelet Method 8th Level Coefficients Reconstruction

50. �Analog� signal Suppose we have a signal:

51. �Analog� signalFourier Method Fourier Spectrum

52. �Analog� signalFourier Method Reconstruction

53. �Analog� signalWavelet Method 9th level coefficients

54. �Analog� signalWavelet Method Reconstruction

55. Short living stateSignal

56. Short living stateGabor transform

57. Short living stateWavelet transform

58. Conclusion Stationary signal � Fourier analysis Stationary signal with singularities � Window Fourier analysis Nonstationary signal � Wavelet analysis

59. Acknowledgements Prof. Andrey Vladimirovich Tsiganov Prof. Serguei Yurievich Slavyanov