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wavelet tutorial

wavelet tutorial

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    Wavelet Wavelet Presentation Transcript

    • Introduction to Wavelet Transform
    • TABLE OF CONTENT
      • Frequency analysis
      • Limitations of Fourier Transform
      • Wavelet Transform
      • Summary
    • Time Domain Signal 10 Hz 2 Hz 20 Hz 2 Hz + 10 Hz + 20Hz Time Time Time Time Magnitude Magnitude Magnitude Magnitude
    • 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
    • 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.
    • 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
      • 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
      • 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.)
    • Solutions = Wavelet Transform
    • WAVELET TRANSFORM Translation (The location of the window) Scale Window function
    • 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
    • S = 5
    • S = 20
    • 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.
    • 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.
    • End