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wavelet tutorial

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  1. 1. Introduction to Wavelet Transform
  2. 2. TABLE OF CONTENT <ul><li>Frequency analysis </li></ul><ul><li>Limitations of Fourier Transform </li></ul><ul><li>Wavelet Transform </li></ul><ul><li>Summary </li></ul>
  3. 3. Time Domain Signal 10 Hz 2 Hz 20 Hz 2 Hz + 10 Hz + 20Hz Time Time Time Time Magnitude Magnitude Magnitude Magnitude
  4. 4. FREQUENCY ANALYSIS <ul><li>Frequency Spectrum </li></ul><ul><ul><li>the frequency components (spectral components) of that signal </li></ul></ul><ul><ul><li>Show what frequencies exists in the signal </li></ul></ul><ul><li>Fourier Transform (FT) </li></ul><ul><ul><li>Most popular way to find the frequency content </li></ul></ul><ul><ul><li>Tells how much of each frequency exists in a signal </li></ul></ul>
  5. 5. Limitations of FT <ul><li>Stationary Signal </li></ul><ul><ul><li>Signals with frequency content unchanged in time </li></ul></ul><ul><ul><li>All frequency components exist at all times </li></ul></ul><ul><ul><li>Fourier transform is good at stationary signals. </li></ul></ul><ul><li>Non-stationary Signal </li></ul><ul><ul><li>Frequency changes in time </li></ul></ul><ul><ul><li>Fourier transform is not good at representing non-stationary signals. </li></ul></ul>
  6. 6. Limitations of FT Occur at all times Do not appear at all times Time Magnitude Magnitude Frequency (Hz) 2 Hz + 10 Hz + 20Hz Stationary Time Magnitude Magnitude Frequency (Hz) Non-Stationary 0.0-0.4: 2 Hz + 0.4-0.7: 10 Hz + 0.7-1.0: 20Hz
  7. 7. <ul><li>Frequency: 2 Hz to 20 Hz </li></ul>Limitations of FT Same in Frequency Domain <ul><li>Frequency: 20 Hz to 2 Hz </li></ul>At what time the frequency components occur? FT can not tell! Time Magnitude Magnitude Frequency (Hz) Time Magnitude Magnitude Frequency (Hz) Different in Time Domain
  8. 8. <ul><li>FT Only Gives what Frequency Components Exist in the Signal </li></ul><ul><li>Time-frequency Representation of the Signal is Needed </li></ul>Limitations of FT Many Signals in our daily life are Non-stationary. (We need to know whether and also when an incident was happened.)
  9. 9. Solutions = Wavelet Transform
  10. 10. WAVELET TRANSFORM Translation (The location of the window) Scale Window function
  11. 11. SCALE and Translation <ul><li>Large scale: </li></ul><ul><li>Large window size, low frequency components. </li></ul><ul><li>Small scale: </li></ul><ul><li>Small window size, high frequency components. </li></ul><ul><li>Translation: </li></ul><ul><li>Shift the window to different location of the signal </li></ul>
  12. 12. S = 5
  13. 13. S = 20
  14. 14. COMPUTATION OF CWT <ul><li>Step 1: The wavelet is placed at the beginning of the signal, and set s=1 (the most compressed wavelet); </li></ul><ul><li>Step 2: The wavelet function at scale “1” is multiplied by the signal, and integrated over all times; then multiplied by ; </li></ul><ul><li>Step 3: Shift the wavelet to t = , and get the transform value at t = and s =1; </li></ul><ul><li>Step 4: Repeat the procedure until the wavelet reaches the end of the signal; </li></ul><ul><li>Step 5: Scale s is increased by a sufficiently small value, the above procedure is repeated for all s; </li></ul><ul><li>Step 6: Each computation for a given s fills the single row of the time-scale plane; </li></ul><ul><li>Step 7: CWT is obtained if all s are calculated. </li></ul>
  15. 15. Summary <ul><li>Frequency analysis can obtain frequency content of the signal. </li></ul><ul><li>Fourier transform cannot deal with non-stationary signal. </li></ul><ul><li>Wavelet transform can give a good time-frequency representation of the non-stationary signal. </li></ul>
  16. 16. End