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Chapter 11 Signal Processing with Wavelets

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Objectives Define and illustrate the difference between a stationary and non-stationary signal. Describe the relationship between wavelets and sub- band coding of a signal using quadrature mirror filters with the property of perfect reconstruction. Illustrate the multi-level decomposition of a signal into approximation and detail components using wavelet decomposition filters. Illustrate the application of wavelet analysis using MATLAB ® to noise suppression, signal compression, and the identification of transient features in a signal.

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Motivation for Wavelet Analysis Signals of practical interest are usually non- stationary, meaning that their time-domain and frequency-domain characteristics vary over short time intervals (i.e., music, seismic data, etc) Classical Fourier analysis (Fourier transforms) assumes a signal that is either infinite in extent or stationary within the analysis window. Non-stationary analysis requires a different approach: Wavelet Analysis Wavelet analysis also produces better solutions to important problems such as the transform compression of images (jpeg versus jpeg2000)

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Basic Theory of Wavelets Wavelet analysis can be understood as a form of sub-band coding with quadrature mirror filters The two basic wavelet processes are decomposition and reconstruction

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Wavelet Decomposition A single level decomposition puts a signal through 2 complementary low-pass and high-pass filters The output of the low-pass filter gives the approximation (A) coefficients, while the high pass filter gives the detail (D) coefficients Decomposition Filters for Daubechies-8 Wavelets

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Wavelet Reconstruction The A and D coefficients can be used to reconstruct the signal perfectly when run through the mirror reconstruction filters of the wavelet family

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Wavelet Families Wavelet families consist of a particular set of quadrature mirror filters with the property of perfect reconstruction. These families are completely determined by the impulse responses of the set of 4 filters.

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Example: Filter Set for the Daubechies-5 Wavelet Family % Set wavelet name. >> wname = 'db5'; % Compute the four filters associated with wavelet name given % by the input string wname. >> [Lo_D,Hi_D,Lo_R,Hi_R] = wfilters(wname); >> subplot(221); stem(Lo_D); >> title('Decomposition low-pass filter'); >> subplot(222); stem(Hi_D); >> title('Decomposition high-pass filter'); >> subplot(223); stem(Lo_R); >> title('Reconstruction low-pass filter'); >> subplot(224); stem(Hi_R); >> title('Reconstruction high-pass filter'); >> xlabel('The four filters for db5')

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Example: Filter Set for the Daubechies-5 Wavelet Family

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Decomposition FiltersReconstruction Filters >> fvtool(Lo_D,1,Hi_D,1) >> fvtool(Lo_R,1,Hi_R,1)

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Multi-level Decomposition of a Signal with Wavelets The decomposition tree can be schematically described as:

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Multi-level Decomposition of a Signal with Wavelets Frequency Domain (Sub-band Coding)

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Example: One-level Decomposition of a Noisy Signal >> x=analog(100,4,40,10000); % Construct a 100 Hz sinusoid of amplitude 4 >> xn=x+0.5*randn(size(x)); % Add Gaussian noise >> [cA,cD]=dwt(xn,'db8'); % Compute the first level decomposition with dwt % and the Daubechies-8 wavelet >> subplot(3,1,1),plot(xn),title('Original Signal') >> subplot(3,1,2),plot(cA),title('One Level Approximation') >> subplot(3,1,3),plot(cD),title('One Level Detail') Single level discrete wavelet decomposition with the Daubechies-8 wavelet family

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One-Level Decomposition of a Non-Stationary Signal >> fs=2500; >> len=100; >> [x1,t1]=analog(50,.5,len,fs); % The time vector t1 is in milliseconds >> [x2,t2]=analog(100,.25,len,fs); >> [x3,t3]=analog(200,1,len,fs); >> y1=cat(2,x1,x2,x3); % Concatenate the signals >> ty1=[t1,t2+len,t3+2*len]; %Concatenate the time vectors 1 to len, len to 2*len, etc. >> [A1,D1]=dwt(y1,'db8'); >> subplot(3,1,1),plot(y1),title('Original Signal') >> subplot(3,1,2),plot(A1),title('One Level Approximation') >> subplot(3,1,3),plot(D1),title('One Level Detail') The detail coefficients reveal the transitions in the non-stationary signal

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De-Noising a Signal with Multilevel Wavelet Decomposition >> x=analog(100,4,40,10000); >> xn=x+0.5*randn(size(x)); >> [C,L] = wavedec(xn,4,'db8'); % Do a multi-level analysis to four levels with the % Daubechies-8 wavelet >> A1 = wrcoef('a',C,L,'db8',1); % Reconstruct the approximations at various levels >> A2 = wrcoef('a',C,L,'db8',2); >> A3 = wrcoef('a',C,L,'db8',3); >> A4 = wrcoef('a',C,L,'db8',4); >> subplot(5,1,1),plot(xn),title('Original Signal') >> subplot(5,1,2),plot(A1),title('Reconstructed Approximation - Level 1') >> subplot(5,1,3),plot(A2),title(' Reconstructed Approximation - Level 2') >> subplot(5,1,4),plot(A3),title(' Reconstructed Approximation - Level 3') >> subplot(5,1,5),plot(A4),title(' Reconstructed Approximation - Level 4') Significant de-noising occurs with the level-4 approximation coefficients (Daubechies-8 wavelets)

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Finding Signal Discontinuities >> x=analog(100,4,40,10000); >> x(302:305)=.25; >> [A,D]=dwt(x,'db8'); >> subplot(3,1,1),plot(x),title('Original Signal') >> subplot(3,1,2),plot(A),title('First Level Approximation') >> subplot(3,1,3),plot(D),title('First Level Detail') 3 sample discontinuity at sample 302 Discontinuity response in the 1 st level detail coefficients (sample 151 because of the 2X down-sampling) Daubechies-8 wavelets

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Simple Signal Compression Using a Wavelet Approximation >> load leleccum >> x=leleccum; >> w = 'db3'; >> [C,L] = wavedec(x,4,w); >> A4 = wrcoef('a',C,L,'db3',4); >> A3 = wrcoef('a',C,L,'db3',3); >> A2 = wrcoef('a',C,L,'db3',2); >> A1 = wrcoef('a',C,L,'db3',1); >> a3 = appcoef(C,L,w,3); >> subplot(2,1,1),plot(x),axis([0,4000,100,600]) >> title('Original Signal') >> subplot(2,1,2),plot(A3),axis([0,4000,100,600]) >> title('Approximation Reconstruction at Level 3 Using the Daubechies-3 Wavelet') >> (length(a3)/length(x))*100 ans = The wavelet approximation at level-3 contains only 13 % of the original signal values because of the wavelet down-sampling, but still retains the important signal characteristics.

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Compression by Thresholding >> load leleccum >> x=leleccum; >> w = 'db3'; % Specify the Daubechies-4 wavelet >> [C,L] = wavedec(x,4,w); % Multi-level decomposition to 4 levels. >> a3 = appcoef(C,L,w,3); % Extract the level 3 approximation coefficients >> d3 = detcoef(C,L,3); % Extract the level 3 detail coefficients. >> subplot(2,1,1), plot(a3),title('Approximation Coefficients at Level 3') >> subplot(2,1,2), plot(d3),title('Detail Coefficients at Level 3') These are the A3 and D3 coefficients for the signal. Many of the D3 coefficients could be “zeroed” without losing much signal information or power

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Compression by Thresholding >> load leleccum >> x=leleccum; % Uncompressed signal >> w = 'db3'; % Set wavelet family >> n=3; % Set decomposition level >> [C,L] = wavedec(x,n,w); % Find the decomposition structure of x to level n using w. >> thr = 10; % Set the threshold value >> keepapp = 1; %Logical parameter = do not threshold approximation coefficients >> sorh='h'; % Use hard thresholding >> [xd,cxd,lxd, perf0,perfl2] =wdencmp('gbl',C,L,w,n,thr,sorh,keepapp); >> subplot(2,1,1), plot(x),title('Original Signal') >> subplot(2,1,2),plot(xd),title('Compressed Signal (Detail Thresholding)') >> perf0 % Percent of coefficients set to zero >> perfl2 % Percent retained energy in the compressed signal perf0 = perfl2 = In this compression 83% of the coefficients were set to zero, but 99% of the energy in the signal was retained.

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Compression by Thresholding >> D1 = wrcoef('d',C,L,w,1); >> D2 = wrcoef('d',C,L,w,2); >> D3 = wrcoef('d',C,L,w,3); >> d1 = wrcoef('d',cxd,lxd,w,1); >> d2 = wrcoef('d',cxd,lxd,w,2); >> d3 = wrcoef('d',cxd,lxd,w,3); >> subplot(3,2,1),plot(D3),title('Original Detail - Levels 3 to 1') >> subplot(3,2,2),plot(d3),title('Thresholded Detail - Levels 3 to 1') >> subplot(3,2,3),plot(D2) >> subplot(3,2,4),plot(d2) >> subplot(3,2,5),plot(D1) >> subplot(3,2,6),plot(d1) Zeroing of coefficients by thresholding results in effective signal compression

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Summary Wavelet processing is based on the idea of sub- band decomposition and coding. Wavelet “families” are characterized by the low- pass and high-pass filters used for decomposition and perfect reconstruction of signals. Typical applications of wavelet processing include elimination of noise, signal compression, and the identification of transient signal features.

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