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Comprehensive Guide to Continuous Wavelet Transform (CWT)

Learn about the definition, properties, and applications of CWT with detailed explanations, theorems, and wavelet types like Haar and Morlet. Discover how CWT helps in time and frequency localization, resolution, and filtering for signal analysis. Explore correlations, transforms, and comparisons with Fourier analysis methods. Dive into CWT principles, oscillation, and localization techniques to enhance your understanding of wavelet transforms.

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Comprehensive Guide to Continuous Wavelet Transform (CWT)

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  1. Wavelet Transform

  2. Definition of The Continuous Wavelet Transform CWT The continuous-time wavelet transform (CWT) of f(x) with respect to a wavelet (x): L2(R)

  3. Mother WaveletDilation / Translation  Mother Wavelet a Dilation Scale b Translation

  4. Properties of a Basic Wavelet   L2(R) is called a Basic Wavelet if the following admissibility condition is satisfied: Admissibility condition. Necessary condition to obtain the inverse from the CWT by the basic Wavelet . Sufficient, but not a necessary condition to obtain the inverse by general Wavelet. Oscillation (Wave) 1. Finite energy (Let) fast decay 2. Oscillation + fast decay = Wave + let = Wavelet

  5. Haar Wavelet Dilation / Translation Haar 1 1 1 2-1/2 2-1/2 1 4 2 4 4 2 -1 -1 -1

  6. Morlet Wavelet Dilation / Translation Morlet

  7. Forward / Inverse Transform [1/5] Forward Inverse Admissibility condition.

  8. Forward / Inverse Transform [2/5] Theorem cwt_001 Proof

  9. Forward / Inverse Transform [3/5] Theorem cwt_002 Proof

  10. Forward / Inverse Transform [4/5] Theorem cwt_003 Proof

  11. Forward / Inverse Transform [5/5] Theorem cwt_004 Proof

  12. Wavelet TransformMorlet Wavelet - Stationary Signal

  13. Wavelet TransformMorlet Wavelet - Transient Signal

  14. Wavelet TransformMorlet Wavelet - Transient Signal

  15. Wavelet TransformMorlet Wavelet - Non-visible Oscillation [1/3]

  16. Wavelet TransformMorlet Wavelet - Non-visible Oscillation [2/3]

  17. Wavelet TransformMorlet Wavelet - Non-visible Oscillation [3/3]

  18. Wavelet TransformHaar Wavelet - Stationary Signal

  19. Wavelet TransformHaar Wavelet - Transient Signal

  20. Wavelet TransformMexican Hat - Stationary Signal

  21. Wavelet TransformMexican Hat - Transient Signal

  22. Wavelet TransformMorlet WaveletFourier/Wavelet Fourier Wavelet

  23. Wavelet TransformMorlet WaveletFourier/Wavelet Fourier Wavelet

  24. CWT - Correlation 1 Cross- correlation CWT CWT W(a,b) is the cross-correlation at lag (shift)  between f(x) and the wavelet dilated to scale factor a.

  25. CWT - Correlation 2 W(a,b) always exists The global maximum of |W(a,b)| occurs if there is a pair of values (a,b) for which ab(t) = f(t). Even if this equality does not exists, the global maximum of the real part of W2(a,b) provides a measure of the fit between f(t) and the corresponding ab(t) (se next page).

  26. CWT - Correlation 3 The global maximum of the real part of W2(a,b) provides a measure of the fit between f(x) and the corresponding ab(x) ab(x) closest to f(x) for that value of pair (a,b) for which Re[W(a,b)] is a maximum. -ab(x) closest to f(x) for that value of pair (a,b) for which Re[W(a,b)] is a minimum.

  27. CWT - Localization both in time and frequency The CWT offers position/time and frequency selectivity; that is, it is able to localize events both in position/time and in frequency. Time: The segment of f(x) that influences the value of W(a,b) for any (a,b) is that stretch of f(x) that coinsides with the interval over which ab(x) has the bulk of its energy. This windowing effect results in the position/time selectivity of the CWT. Frequency: The frequency selectivity of the CWT is explained using its interpretation as a collection of linear, time-invariant filters with impulse responses that are dilations of the mother wavelet reflected about the time axis (se next page).

  28. CWT - Frequency - Filter interpretation Convolution CWT CWT is the output of a filter with impulse response *ab(-b) and input f(b). We have a continuum of filters parameterized by the scale factor a.

  29. CWT - Time and frequency localization 1 Time Center of mother wavelet Frequency Center of the Fourier transform of mother wavelet

  30. CWT - Time and frequency localization 2 Time Frequency Time-bandwidth product is a constant

  31. CWT - Time and frequency localization 3 Time Frequency Small a: CWT resolve events closely spaced in time. Large a: CWT resolve events closely spaced in frequency. CWT provides better frequency resolution in the lower end of the frequency spectrum. Wavelet a natural tool in the analysis of signals in which rapidly varying high-frequency components are superimposed on slowly varying low-frequency components (seismic signals, music compositions, …).

  32. CWT - Time and frequency localization 4  a=1/2 a=1 a=2 t Time-frequency cells for a,b(t)

  33. Filtering / Compression Data compression Remove low W-values Highpass-filtering Lowpass-filtering Replace W-values by 0 for high a-values Replace W-values by 0 for low a-values

  34. CWT - DWT CWT DWT Binary dilation Dyadic translation Dyadic Wavelets

  35. Mexican Hat

  36. Rotation - Scaling2 dim Rotation Scaling

  37. Translation - Rotation - Scaling3 dim Translation Rotation Scaling

  38. Mexican Hat - 3 Dim

  39. End

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