Wavelet Toolbox

One-Dimensional Analysis Using the Command Line

This example involves a noisy Doppler test signal.

1. Load a signal.

1. From the MATLAB prompt, type

   • load noisdopp
     

2. Set the variables. Type

• s = noisdopp; 
  

1. For the SWT, if a decomposition at level k is needed, 2^k must divide evenly into the length of the signal. If your original signal does not have the correct length, you can use the Signal Extension GUI tool or the  wextend function to extend it.

3. Perform a single-level Stationary Wavelet Decomposition.

1. Perform a single-level decomposition of the signal using the db1 wavelet. Type

   • [swa,swd] = swt(s,1,'db1');
     

   This generates the coefficients of the level 1 approximation (swa) and detail (swd). Both are of the same length as the signal. Type

   • whos 
     
     

     Name	Size	Bytes	Class
     noisdopp	1x1024	8192	double array
     s	1x1024	8192	double array
     swa	1x1024	8192	double array
     swd	1x1024	8192	double array

4. Display the coefficients of approximation and detail.

1. To display the coefficients of approximation and detail at level 1, type

   • subplot(1,2,1), plot(swa); title('Approximation cfs')
     subplot(1,2,2), plot(swd); title('Detail cfs')
     
     
     

5. Regenerate the signal by Inverse Stationary Wavelet Transform.

1. To find the inverse transform, type

   • A0 = iswt(swa,swd,'db1'); 
     

   To check the perfect reconstruction, type

   • err = norm(s-A0) 
     err =
      2.1450e-14 
     

6. Construct and display approximation and detail from the coefficients.

1. To construct the level 1 approximation and detail (A1 and D1) from the coefficients swa and swd, type

   • nulcfs = zeros(size(swa));
     A1 = iswt(swa,nulcfs,'db1'); 
     D1 = iswt(nulcfs,swd,'db1');
     

   To display the approximation and detail at level 1, type

   • subplot(1,2,1), plot(A1); title('Approximation A1');
     subplot(1,2,2), plot(D1); title('Detail D1');
     
     
     

7. Perform a multilevel Stationary Wavelet Decomposition.

1. To perform a decomposition at level 3 of the signal (again using the db1 wavelet), type

   • [swa,swd] = swt(s,3,'db1');
     

   This generates the coefficients of the approximations at levels 1, 2, and 3 (swa) and the coefficients of the details (swd). Observe that the rows of swa and swd are the same length as the signal length. Type

   • clear A0 A1 D1 err nulcfs
     whos
     

Name	Size	Bytes	Class
noisdopp	1x1024	8192	double array
s	1x1024	8192	double array
swa	3x1024	24576	double array
swd	3x1024	24576	double array

8. Display the coefficients of approximations and details.

1. To display the coefficients of approximations and details, type

   • kp = 0;
     for i = 1:3 
         subplot(3,2,kp+1), plot(swa(i,:)); 
         title(['Approx. cfs level ',num2str(i)])
         subplot(3,2,kp+2), plot(swd(i,:)); 
         title(['Detail cfs level ',num2str(i)])
         kp = kp + 2;
     end
     
     
     

9. Reconstruct approximation at Level 3 From coefficients.

1. To reconstruct the approximation at level 3, type

   • mzero = zeros(size(swd));
     A = mzero;
     A(3,:) = iswt(swa,mzero,'db1');
     

10. Reconstruct details from coefficients.

1. To reconstruct the details at levels 1, 2 and 3, type

   • D = mzero;
     for i = 1:3
         swcfs = mzero;
         swcfs(i,:) = swd(i,:);
         D(i,:) = iswt(mzero,swcfs,'db1');
     end
     

11. Reconstruct and display approximations at Levels 1 and 2 from approximation at Level 3 and details at Levels 2 and 3.

1. To reconstruct the approximations at levels 2 and 3, type

   • A(2,:) = A(3,:) + D(3,:);
     A(1,:) = A(2,:) + D(2,:);
     

   To display the approximations and details at levels 1, 2 and 3, type

   • kp = 0;
     for i = 1:3
         subplot(3,2,kp+1), plot(A(i,:));
         title(['Approx. level ',num2str(i)])
         subplot(3,2,kp+2), plot(D(i,:));
         title(['Detail level ',num2str(i)])
         kp = kp + 2;
     end
     
     
     

12. Remove noise by thresholding.

1. To de-noise the signal, use the ddencmp command to calculate a default global threshold. Use the wthresh command to perform the actual thresholding of the detail coefficients, and then use the iswt command to obtain the de-noised signal.

   • [thr,sorh] = ddencmp('den','wv',s);
     dswd = wthresh(swd,sorh,thr);
     clean = iswt(swa,dswd,'db1');
     

   To display both the original and de-noised signals, type

   • subplot(2,1,1), plot(s);
     title('Original signal')
     subplot(2,1,2), plot(clean);
     title('De-noised signal')
     
     
     

   The obtained signal remains a little bit noisy. The result can be improved by considering the decomposition of s at level 5 instead of level 3, and repeating steps 14 and 15. To improve the previous de-noising, type

   • [swa,swd] = swt(s,5,'db1');
     [thr,sorh] = ddencmp('den','wv',s);
     dswd = wthresh(swd,sorh,thr);
     clean = iswt(swa,dswd,'db1');
     subplot(2,1,1), plot(s); title('Original signal')
     subplot(2,1,2), plot(clean); title('De-noised signal')
     
     
     

   A second syntax can be used for the swt and iswt functions, giving the same results:

   • lev = 5;
     swc = swt(s,lev,'db1');
     swcden = swc;
     swcden(1:end-1,:) = wthresh(swcden(1:end-1,:),sorh,thr);
     clean = iswt(swcden,'db1');
     

   You can obtain the same plot by using the same plot commands as in step 16 above.

One-Dimensional Discrete Stationary Wavelet Analysis

One-Dimensional Analysis for De-Noising Using the Graphical Interface