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# Wavelet

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wavelet tutorial

wavelet tutorial

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### 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