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Learning Wavelet Transform by MATLAB Toolbox Professor : R.J. Chang Student : Che-Wei Chen Date :2013/12/13 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 1. Wavelet evolution Fourier transform: F ( ) i t f (t ) e dt Time-frequency tile for Fourier transform: f t Poor time-localization 1. Wavelet evolution(count.) Short-time Fourier transform: S ( , ) f (t ) w (t ) e i t dt Time-frequency tile for STFT: f f f t t w(t-τ) is the window function. t 1. Wavelet evolution(count.) Wavelet transform: a ,b ( t ) 1 ( tb ) a,b Z a a (a , b ) a , b ( t ), x ( t ) x ( t ) a ,b d t * Time-frequency tile for wavelet transform: f t Morlet 1. Wavelet evolution(count.) Compare these three methods: Fourier transform STFT Wavelet 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 nonstationary signal? No Yes Good 2. Wavelet transform toolbox in MATLAB Decomposition and reconstruction : 2 h[n ] ~ h [n] 2 x[n ] ~y [ n ] H y H [ n ] y H [n] g[n] ~y [ n ] L y L [ n ] y L [n] 2 2 ~ x [n] g~ [ n ] This structure contains for J = 3 the terminal nodes of the following tree. Input signal Level 1 Level 2 Level 3 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. 2. Wavelet transform toolbox in MATLAB(count.) The signals include approximations(a) and Details(d): s＝a5+d5+d4+d3+d2+d1 d1 d2 d3 d4 a5 d5 Wavelet transform toolbox in MATLAB(count.) Step 6: Push “Statistics” button to show the statistics of signal of each levels. Wavelet transform toolbox in MATLAB(count.) Step 7: Push “De-noise” button to remove the detail parts. Wavelet transform toolbox in MATLAB(count.) 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) 1.5 1 0.5 0 -0.5 -1 -1.5 0 100 200 300 400 500 s 2.5 0.8 2 0.6 1.5 600 700 800 900 1000 0.4 1 0.2 0.5 0 0 -0.2 -0.5 -0.4 -1 -0.6 -1.5 -2 0 50 100 150 200 250 300 ca1 350 400 450 -0.8 500 0 50 100 150 200 250 cd1 300 350 400 450 500 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) 2.5 2 1.5 1 0.5 0 -0.5 1.5 1.5 -1 -1.5 1 -2 1 0 50 100 150 0.5 200 250 300 ca1 + 350 400 450 500 0.5 0.8 0 0 0.6 0.4 -0.5 -0.5 0.2 0 -1 -1 -0.2 -0.4 -1.5 0 100 200 300 400 s 500 -1.5 -0.6 600 700 800 900 1000 -0.8 0 50 100 150 200 250 300 cd1 350 400 450 500 0 100 200 300 400 500 X 600 700 800 900 1000 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'); 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); 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. 4. Wavelet packet transform toolbox in MATLAB Select the mother wavelet, this step will influence your results. Suggest you choose haar, db, sym. You can chose different entropy for your wavelet packet. Some wavelet must be defined, you can compare results with different value at this button. This button decides wavelet packet levels. Tapping different nodes will show you results. 4. Wavelet packet transform toolbox in MATLAB 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) 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 30 20 10 0 -10 -20 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) -30 -40 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 6. Definition of SNR This definition in time-domain is the basic definition of SNR. It is defined to be the average power of signal divided by that of noise, and referred as Time Domain 1. N 2 1 SN R1 10 log 1 0 N 1 N 0 N 0 y S (t ) 2 y N (t ) This definition in time-domain is defined by Mitaim and Kosko, and referred as Time Domain 2. (Signal frequency=f0) SN R 2 10 log 10 2 Y [k0 ] L 1 k 0 k 0 f 0 L Ts 2 2 Y [k ] 2 Y [k0 ] 2 L 1 Y [k ] i yt e 2 k t L t0 This definition in frequency-domain is defined to be the power spectrum at signal frequency divided by the noise spectrum at signal frequency, and referred as Narrow Band 1.(Signal frequency=) lim SN R 10 log 1 0 0 S S ( ) d S N ( ) 6. Definition of SNR This definition in frequency-domain is defined to be the power spectrum of signal and noise at signal frequency divided by the noise spectrum, and referred as Narrow Band 2.(Signal frequency=) lim S N R 1 0 lo g 1 0 0 S ( ) d S N ( ) This definition in frequency-domain is defined to be the total signal spectrum divided by the total noise spectrum, and referred as Wide Band. .(Signal k frequency=) SN R w 10 log 10 k 1 lim 0 S S ( ) d k S 0 N ( ) d Thank you for your attention.