Wavelet
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wavelet tutorial

wavelet tutorial

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Wavelet Presentation Transcript

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