1. Learning Wavelet Transform by MATLAB Toolbox Professor : R.J. Chang Student : Chung-Hsien Chao Date : 2011/12/02

2. Outline Wavelet evolution Wavelet transform toolbox in MATLAB Wavelet function in MATLAB

3. Wavelet evolution Fourier transform: Time-frequency tile for Fourier transform: Poor time-localization f t

4. Wavelet evolution(count.) Short-time Fourier transform: Time-frequency tile for STFT: w(t-τ) is the window function. f t f t

5. Wavelet evolution(count.) Wavelet transform: Time-frequency tile for wavelet transform: f t Translations and Scaling of a Wavelet

6. Wavelet evolution(count.) Compare these three methods: Fourier transformSTFTWavelet Resolution in time domain and frequency domain No resolution in time domain Low resolution in time domain. High resolution in time domain. Can analyze the non- stationary signal? NoYesGood

7. Wavelet transform toolbox in MATLAB Decomposition and reconstruction : This structure contains for J = 3 the terminal nodes of the following tree. Input signal Level 1 Level 2 Level 3

8. Wavelet transform toolbox in MATLAB(count.) Step 1: Type command “wavemenu” at command window and hit the Wavelet 1-D button. Step 2: Load signal from “*.mat files” or “workspace”. Step 3: Select the mother wavelet and levels, for example: db7, level 5. Step 4: Push “Analyze” button.

9. Wavelet transform toolbox in MATLAB(count.) The signals include approximations(a) and Details(d): s ＝ a5+d5+d4+d3+d2+d1 a5d5 d4 d3 d2 d1

10. Wavelet transform toolbox in MATLAB(count.) Step 6: Push “Statistics” button to show the statistics of signal of each levels.

11. Wavelet transform toolbox in MATLAB(count.) Step 7: Push “De-noise” button to remove the detail parts.

12. Wavelet transform toolbox in MATLAB(count.)

13. Wavelet function in MATLAB 1. Single-level discrete 1-D wavelet transform: [cA,cD] = dwt(X,'wname') Example: load noissin s = noissin(1:1000); [ca1,cd1] = dwt(s,'haar'); plot(s) subplot(121),plot(ca1) subplot(122),plot(cd1) ca1cd1 s

14. Wavelet function in MATLAB(count.) 2. Single-level inverse discrete 1-D wavelet transform: X = idwt(cA,cD,'wname') Example: load noissin s = noissin(1:1000); [ca1,cd1] = dwt(s,'haar'); X = idwt(ca1,cd1,'haar'); plot(X) s X ca1 cd1 +

15. Wavelet function in MATLAB(count.) 3. Multilevel 1-D wavelet decomposition: [C,L] = wavedec(X,N,'wname') Example: load sumsin s = sumsin; [c,l] = wavedec(s,3, 'db1');

16. Wavelet function in MATLAB(count.) 4. Reconstruct single branch from 1-D wavelet coefficients: X = wrcoef('type',C,L,'wname',N) Example: load sumsin s = sumsin; [c,l] = wavedec(s,5, 'sym4'); a5 = wrcoef('a',c,l, 'sym4',5);

17. Thank you for your attention.