1. Professor : R.J. Chang Student : Che-Wei Chen Date :2013/12/13 Learning Wavelet Transform by MATLAB Toolbox

2. Outline Wavelet evolution Wavelet transform toolbox in MATLAB Wavelet function in MATLAB Wavelet packet transform toolbox in MATLAB Wavelet packet function in MATLAB Definition of SNR

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

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

5. 1. Wavelet evolution(count.) Wavelet transform: Time-frequency tile for wavelet transform: f t Morlet

6. 1. 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. 2. 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. 2. 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. 2. 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. 3. 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. 3. 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. 3. 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. 3. 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. 4. Wavelet packet transform toolbox in MATLAB Step 1: Type command “wavemenu” at command window and hit the Wavelet Packet 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, shannon. Step 4: Push “Analyze” button.

18. 4. Wavelet packet transform toolbox in MATLAB Select the mother wavelet, this step will influence your results. Suggest you choose haar, db, sym. Some wavelet must be defined, you can compare results with different value at this button. This button decides wavelet packet levels. You can chose different entropy for your wavelet packet. Tapping different nodes will show you results.

19. 4. Wavelet packet transform toolbox in MATLAB

20. 5. Wavelet packet function in MATLAB 1. Single-level discrete 1-D wavelet packet transform: T = wpdec(X,N,'wname',E) Example: load noissin s = noissin(1:1000); wpt = wpdec(s,3,'haar','shannon'); plot(wpt)

21. 5. Wavelet packet function in MATLAB 2. Single-level denoise discrete 1-D wavelet packet transform: [XD,TREED,PERF0,PERFL2]=wpdencmp(X,SORH,N,'wname',CRIT,PAR,K EEPAPP) Example: load noissin s = noissin(1:1000); n = length(s); keepapp = 1 SORH=soft or hard entropy criterion= CRIT thr = sqrt(2*log(n*log(n)/log(2))); xwpd = wpdencmp(s,'s',3,'haar','sure',thr,1); plot(t,xx,'r',t,xwpd)

22. 6. Definition of SNR

24. Thank you for your attention.