Documentation

Load the signal and select a portion for wavelet analysis.

load leleccum;
s = leleccum(1:3920);
l_s = length(s);

Perform a single-level wavelet decomposition of a signal.

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

[cA1,cD1] = dwt(s,'db1');

This generates the coefficients of the level 1 approximation (cA1) and detail (cD1).

Construct approximations and details from the coefficients.

To construct the level 1 approximation and detail (A1 and D1) from the coefficients cA1 and cD1, type

A1 = upcoef('a',cA1,'db1',1,l_s);
D1 = upcoef('d',cD1,'db1',1,l_s);

or

A1 = idwt(cA1,[],'db1',l_s);
D1 = idwt([],cD1,'db1',l_s);

Display the approximation and detail.

To display the results of the level-one decomposition, type

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

Regenerate a signal by using the Inverse Wavelet Transform.

To find the inverse transform, enter

A0 = idwt(cA1,cD1,'db1',l_s);
err = max(abs(s-A0))

Perform a multilevel wavelet decomposition of a signal.

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

[C,L] = wavedec(s,3,'db1');

The coefficients of all the components of a third-level decomposition (that is, the third-level approximation and the first three levels of detail) are returned concatenated into one vector, C. Vector L gives the lengths of each component.

Extract approximation and detail coefficients.

To extract the level 3 approximation coefficients from C, type

cA3 = appcoef(C,L,'db1',3);

To extract the levels 3, 2, and 1 detail coefficients from C, type

cD3 = detcoef(C,L,3);
cD2 = detcoef(C,L,2);
cD1 = detcoef(C,L,1);

or

[cD1,cD2,cD3] = detcoef(C,L,[1,2,3]);

Reconstruct the Level 3 approximation and the Level 1, 2, and 3 details.

To reconstruct the level 3 approximation from C, type

A3 = wrcoef('a',C,L,'db1',3);

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

D1 = wrcoef('d',C,L,'db1',1); 
D2 = wrcoef('d',C,L,'db1',2); 
D3 = wrcoef('d',C,L,'db1',3);

Display the results of a multilevel decomposition.

To display the results of the level 3 decomposition, type

subplot(2,2,1); plot(A3);  
title('Approximation A3') 
subplot(2,2,2); plot(D1);  
title('Detail D1') 
subplot(2,2,3); plot(D2);  
title('Detail D2') 
subplot(2,2,4); plot(D3);  
title('Detail D3')

Reconstruct the original signal from the Level 3 decomposition.

To reconstruct the original signal from the wavelet decomposition structure, type

A0 = waverec(C,L,'db1');  
err = max(abs(s-A0))  

Crude denoising of a signal.

Using wavelets to remove noise from a signal requires identifying which component or components contain the noise, and then reconstructing the signal without those components.

In this example, we note that successive approximations become less and less noisy as more and more high-frequency information is filtered out of the signal.

The level 3 approximation, A3, is quite clean as a comparison between it and the original signal.

To compare the approximation to the original signal, type

subplot(2,1,1);plot(s);title('Original'); axis off 
subplot(2,1,2);plot(A3);title('Level 3 Approximation'); 
axis off

Of course, in discarding all the high-frequency information, we've also lost many of the original signal's sharpest features.

Optimal denoising requires a more subtle approach called thresholding. This involves discarding only the portion of the details that exceeds a certain limit.

Remove noise by thresholding.

Let's look again at the details of our level 3 analysis.

To display the details D1, D2, and D3, type

subplot(3,1,1); plot(D1); title('Detail Level 1'); axis off 
subplot(3,1,2); plot(D2); title('Detail Level 2'); axis off 
subplot(3,1,3); plot(D3); title('Detail Level 3'); axis off

Most of the noise occurs in the latter part of the signal, where the details show their greatest activity. What if we limited the strength of the details by restricting their maximum values? This would have the effect of cutting back the noise while leaving the details unaffected through most of their durations. But there's a better way.

Note that cD1, cD2, and cD3 are just MATLAB^® vectors, so we could directly manipulate each vector, setting each element to some fraction of the vectors' peak or average value. Then we could reconstruct new detail signals D1, D2, and D3 from the thresholded coefficients.

To denoise the signal, use the ddencmp command to calculate the default parameters and the wdencmp command to perform the actual denoising, type

[thr,sorh,keepapp] = ddencmp('den','wv',s); 
clean = wdencmp('gbl',C,L,'db1',3,thr,sorh,keepapp);

Note that wdencmp uses the results of the decomposition (C and L) that we already calculated. We also specify that we used the db1 wavelet to perform the original analysis, and we specify the global thresholding option 'gbl'. See ddencmp and wdencmp in the reference pages for more information about the use of these commands.

To display both the original and denoised signals, type

subplot(2,1,1); plot(s(2000:3920)); title('Original') 
subplot(2,1,2); plot(clean(2000:3920)); title('denoised')

We've plotted here only the noisy latter part of the signal. Notice how we've removed the noise without compromising the sharp detail of the original signal. This is a strength of wavelet analysis.

While using command line functions to remove the noise from a signal can be cumbersome, the software's Wavelet Analyzer app includes an easy-to-use denoising feature that includes automatic thresholding.

More information on the denoising process can be found in the following sections:

Start the 1-D Wavelet Analysis Tool.

From the MATLAB prompt, type waveletAnalyzer.

The Wavelet Analyzer appears.

Click the Wavelet 1-D menu item.

The discrete wavelet analysis tool for 1-D signal data appears.

Load a signal.

At the MATLAB command prompt, type

load leleccum;

Perform a single-level wavelet decomposition.

To start our analysis, let's perform a single-level decomposition using the db1 wavelet, just as we did using the command-line functions in 1-D Analysis Using the Command Line.

In the upper right portion of the Wavelet 1-D tool, select the db1 wavelet and single-level decomposition.

Click the Analyze button.

After a pause for computation, the tool displays the decomposition.

Zoom in on relevant detail.

One advantage of using the Wavelet Analyzer app is that you can zoom in easily on any part of the signal and examine it in greater detail.

Drag a rubber band box (by holding down the left mouse button) over the portion of the signal you want to magnify. Here, we've selected the noisy part of the original signal.

Click the X+ button (located at the bottom of the screen) to zoom horizontally.

The Wavelet 1-D tool zooms all the displayed signals.

The other zoom controls do more or less what you'd expect them to. The X- button, for example, zooms out horizontally. The history function keeps track of all your views of the signal. Return to a previous zoom level by clicking the left arrow button.

Perform a multilevel decomposition.

Again, we'll use the graphical tools to emulate what we did in the previous section using command line functions. To perform a level 3 decomposition of the signal using the db1 wavelet:

Select 3 from the Level menu at the upper right, and then click the Analyze button again.

After the decomposition is performed, you'll see a new analysis appear in the Wavelet 1-D tool.

Selecting Different Views of the Decomposition

The Display mode menu (middle right) lets you choose different views of the wavelet decomposition.

The default display mode is called “Full Decomposition Mode.” Other alternatives include:

You can change the default display mode on a per-session basis. Select the desired mode from the View > Default Display Mode submenu.

Depending on which display mode you select, you may have access to additional display options through the More Display Options button.

These options include the ability to suppress the display of various components, and to choose whether or not to display the original signal along with the details and approximations.

Remove noise from a signal.

The Wavelet Analyzer app features a denoising option with a predefined thresholding strategy. This makes it very easy to remove noise from a signal.

Bring up the denoising tool: click the denoise button, located in the middle right of the window, underneath the Analyze button.

The Wavelet 1-D Denoising window appears.

While a number of options are available for fine-tuning the denoising algorithm, we'll accept the defaults of soft fixed form thresholding and unscaled white noise. The Unscaled white noise option corresponds to setting the multiplicative threshold input argument SCAL of wden to 'one'. Choosing Scaled white noise corresponds to 'sln', and Non-white noise corresponds to 'mln'. For more information, see wden.

Continue by clicking the denoise button.

The denoised signal appears superimposed on the original. The tool also plots the wavelet coefficients of both signals. The original detail coefficients appear on the left side of the display. In order to time align decomposition levels across all scales, wavelet coefficients are replicated at each scale to account for the missing time points. Therefore, as the scale becomes coarser, the coefficients assume a staircase-like appearance.

Zoom in on the plot of the original and denoised signals for a closer look.

Drag a rubber band box around the pertinent area, and then click the XY+ button.

The denoise window magnifies your view. By default, the original signal is shown in red, and the denoised signal in yellow.

Dismiss the Wavelet 1-D Denoising window: click the Close button.

You cannot have the denoise and Compression windows open simultaneously, so close the Wavelet 1-D Denoising window to continue. When the Update Synthesized Signal dialog box appears, click No. If you click Yes, the Synthesized Signal is then available in the Wavelet 1-D main window.

Refine the analysis.

The graphical tools make it easy to refine an analysis any time you want to. Up to now, we've looked at a level 3 analysis using db1. Let's refine our analysis of the electrical consumption signal using the db3 wavelet at level 5.

Select 5 from the Level menu at the upper right, and select the db3 from the Wavelet menu. Click the Analyze button.

Compress the signal.

The graphical interface tools feature a compression option with automatic or manual thresholding.

Bring up the Compression window: click the Compress button, located in the middle right of the window, underneath the Analyze button.

The Compression window appears.

While you always have the option of choosing by level thresholding, here we'll take advantage of the global thresholding feature for quick and easy compression.

Click the Compress button, located at the center right.

After a pause for computation, the electrical consumption signal is redisplayed in red with the compressed version superimposed in yellow. Below, we've zoomed in to get a closer look at the noisy part of the signal.

You can see that the compression process removed most of the noise, but preserved 99.99% of the energy of the signal.

Show the residuals.

From the Wavelet 1-D Compression tool, click the Residuals button. The More on Residuals for Wavelet 1-D Compression window appears.

Displayed statistics include measures of tendency (mean, mode, median) and dispersion (range, standard deviation). In addition, the tool provides frequency-distribution diagrams (histograms and cumulative histograms), as well as time-series diagrams: autocorrelation function and spectrum. The same feature exists for the Wavelet 1-D Denoising tool.

Dismiss the Wavelet 1-D Compression window: click the Close button. When the Update Synthesized Signal dialog box appears, click No.

Show statistics.

You can view a variety of statistics about your signal and its components.

From the Wavelet 1-D tool, click the Statistics button.

The Wavelet 1-D Statistics window appears displaying by default statistics on the original signal.

Select the synthesized signal or signal component whose statistics you want to examine. Click the appropriate option button, and then click the Show Statistics button. Here, we've chosen to examine the synthesized signal using 100 bins instead of 30, which is the default:

Displayed statistics include measures of tendency (mean, mode, median) and dispersion (range, standard deviation).

In addition, the tool provides frequency-distribution diagrams (histograms and cumulative histograms). You can plot these histograms separately using the Histograms button from the Wavelets 1-D window.

Click the Approximation option button. A menu appears from which you choose the level of the approximation you want to examine.

Select Level 1 and again click the Show Statistics button. Statistics appear for the level 1 approximation.