This section takes you through the features of one-dimensional critically-sampled wavelet analysis using the Wavelet Toolbox™ software.
The toolbox provides these functions for one-dimensional signal analysis. For more information, see the reference pages.
Function Name | Purpose |
---|---|
Penalized threshold for wavelet 1-D or 2-D denoising | |
Thresholds for wavelet 1-D using Birgé-Massart strategy | |
Wavelet denoising and compression | |
Automatic wavelet denoising | |
Threshold settings manager |
In this section, you'll learn how to
Load a signal
Perform a single-level wavelet decomposition of a signal
Construct approximations and details from the coefficients
Display the approximation and detail
Regenerate a signal by inverse wavelet transform
Perform a multilevel wavelet decomposition of a signal
Extract approximation and detail coefficients
Reconstruct the level 3 approximation
Reconstruct the level 1, 2, and 3 details
Display the results of a multilevel decomposition
Reconstruct the original signal from the level 3 decomposition
Remove noise from a signal
Refine an analysis
Compress a signal
Show a signal's statistics and histograms
Since you can perform analyses either from the command line or using the graphical interface tools, this section has subsections covering each method.
The final subsection discusses how to exchange signal and coefficient information between the disk and the graphical tools.
This example involves a real-world signal — electrical consumption measured over the course of 3 days. This signal is particularly interesting because of noise introduced when a defect developed in the monitoring equipment as the measurements were being made. Wavelet analysis effectively removes the noise.
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))
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.
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 calculated in step
6. 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 graphical interface tools include an easy-to-use denoising feature that includes automatic thresholding.
More information on the denoising process can be found in the following sections:
Wavelet Denoising and Nonparametric Function Estimation in the Wavelet Toolbox User's Guide
One-Dimensional Adaptive Thresholding of Wavelet Coefficients
One-Dimensional Wavelet Variance Adaptive Thresholding in the Wavelet Toolbox User's Guide
In this section, we explore the same electrical consumption signal as in the previous section, but we use the graphical interface tools to analyze the signal.
Start the 1-D Wavelet Analysis Tool.
From the MATLAB prompt, type
wavemenu
The Wavelet Toolbox Main Menu appears.
Click the Wavelet 1-D menu item.
The discrete wavelet analysis tool for one-dimensional signal data appears.
Load a signal.
From the File menu, choose the Load > Signal option.
When the Load
Signal dialog box appears, select the MAT-file leleccum.mat
,
which is in the MATLAB folder toolbox/wavelet/wavedemo
.
Click the OK button.
The electrical consumption signal is loaded into the Wavelet 1-D tool.
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 One-Dimensional 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 graphical interface tools 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:
"Separate Mode," which shows the details and the approximations in separate columns.
"Superimpose Mode," which shows the details on a single plot superimposed in different colors. The approximations are plotted similarly.
"Tree Mode," which shows the decomposition tree, the original signal, and one additional component of your choice. Click on the decomposition tree to select the signal component you'd like to view.
"Show and Scroll Mode," which displays three windows. The first shows the original signal superimposed on an approximation you select. The second window shows a detail you select. The third window shows the wavelet coefficients.
"Show and Scroll Mode (Stem Cfs)" is very similar to the "Show and Scroll Mode" except that it displays, in the third window, the wavelet coefficients as stem plots instead of colored blocks.
You can change the default display mode on a per-session basis. Select the desired mode from the View > Default Display Mode submenu.
Note The Compression and Denoising windows opened from the Wavelet 1-D tool will inherit the current coefficient visualization attribute (stems or colored blocks). |
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 graphical interface tools feature 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.
Continue by clicking the denoise button.
The denoised signal appears superimposed on the original. The tool also plots the wavelet coefficients of both signals.
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.
Note If you want to experiment with manual thresholding, choose the By Level thresholding option from the menu located at the top right of the Wavelet 1-D Compression window. The sliders located below this menu then control the level-dependent thresholds, indicated by yellow dotted lines running horizontally through the graphs on the left of the window. The yellow dotted lines can also be dragged directly using the left mouse button. |
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.
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.
The Wavelet 1-D graphical interface tool lets you import information from and export information to disk and the MATLAB workspace.
You can save synthesized signals, coefficients, and decompositions from the Wavelet 1-D tool to the disk, where the information can be manipulated and later reimported into the graphical tool.
Saving Synthesized Signals. You can process a signal in the Wavelet
1-D tool and then save the processed signal to a MAT-file
(with extension mat
or other).
For example, load the example analysis: File > Example Analysis > Basic Signals > with db3 at level 5 → Sum of sines, and perform a compression or denoising operation on the original signal. When you close the Denoising or Compression window, update the synthesized signal by clicking Yes in the dialog box.
Then, from the Wavelet 1-D tool, select the File > Save > Signal menu option.
A dialog box appears allowing you to select a folder and filename
for the MAT-file. For this example, choose the name synthsig
.
To load the signal into your workspace, simply type
load synthsig;
When the synthesized signal is obtained using any thresholding method except a global one, the saved structure is
whos
Name | Size | Bytes | Class |
---|---|---|---|
synthsig | 1x1000 | 8000 | double array |
thrParams | 1x5 | 580 | cell array |
wname | 1x3 | 6 | char array |
The synthesized signal is given by the variable synthsig
.
In addition, the parameters of the denoising or compression process
are given by the wavelet name (wname
) and the level
dependent thresholds contained in the thrParams
variable,
which is a cell array of length 5 (same as the level of the decomposition).
For i
from 1 to 5, thrParams{i}
contains
the lower and upper bounds of the thresholding interval and the threshold
value (since interval dependent thresholds are allowed, see One-Dimensional Adaptive Thresholding of Wavelet Coefficients).
For example, for level 1,
thrParams{1} ans = 1.0e+03 * 0.0010 1.0000 0.0014
When the synthesized signal is obtained using a global thresholding method, the saved structure is
Name | Size | Bytes | Class |
---|---|---|---|
synthsig | 1x1000 | 8000 | double array |
valTHR | 1x1 | 8 | double array |
wname | 1x3 | 6 | char array |
where the variable valTHR
contains the global
threshold:
valTHR valTHR = 1.2922
Saving Discrete Wavelet Transform Coefficients. The Wavelet 1-D tool lets you save the coefficients of a discrete wavelet transform (DWT) to disk. The toolbox creates a MAT-file in the current folder with a name you choose.
To save the DWT coefficients from the present analysis, use the menu option File > Save > Coefficients.
A dialog box appears that lets you specify a folder and filename for storing the coefficients.
Consider the example analysis:
File > Example Analysis > Basic Signals > with db1 at level 5 → Cantor curve.
After saving the wavelet coefficients to the file cantor.mat
,
load the variables in the workspace:
load cantor whos
Name | Size | Bytes | Class |
---|---|---|---|
coefs | 1x2190 | 17520 | double array |
longs | 1x7 | 56 | double array |
thrParams | 0x0 | 0 | double array |
wname | 1x3 | 6 | char array |
Variable coefs
contains the discrete wavelet
coefficients. More precisely, in the above example coefs
is
a 1-by-2190 vector of concatenated coefficients, and longs
is
a vector giving the lengths of each component of coefs
.
Variable wname
contains the wavelet name
and thrParams
is empty since the synthesized signal
does not exist.
Saving Decompositions. The Wavelet 1-D tool lets you
save the entire set of data from a discrete wavelet analysis to disk.
The toolbox creates a MAT-file in the current folder with a name you
choose, followed by the extension wa1
(wavelet
analysis 1-D).
Open the Wavelet 1-D tool and load the example analysis:
File > Example Analysis > Basic Signals > with db3 at level 5 → Sum of sines
To save the data from this analysis, use the menu option File > Save > Decomposition.
A dialog box appears that lets you specify a folder and filename
for storing the decomposition data. Type the name wdecex1d
.
After saving the decomposition data to the file wdecex1d.wa1
,
load the variables into your workspace:
load wdecex1d.wa1 -mat whos
Name | Size | Bytes | Class |
---|---|---|---|
coefs | 1x1023 | 8184 | double array |
data_name | 1x6 | 12 | char array |
longs | 1x7 | 56 | double array |
thrParams | 0x0 | 0 | double array |
wave_name | 1x3 | 6 | char array |
Note Save options are also available when performing denoising or compression inside the Wavelet 1-D tool. In the Wavelet 1-D Denoising window, you can save denoised signal and decomposition. The same holds true for the Wavelet 1-D Compression window. This way, you can save many different trials from inside the Denoising and Compression windows without going back to the main Wavelet 1-D window during a fine-tuning process. |
Note
When saving a synthesized signal, a decomposition or coefficients
to a MAT-file, the |
The Wavelet 1-D tool allows you to export your 1-D wavelet analysis to the MATLAB workspace in a number of formats.
For example, load the example analysis for the freqbrk
signal.
After the wavelet 1-D analysis loads, select File
—> Export to Workspace
.
You have the option to
Export Signal
—
This option exports the synthesized signal vector.
Export Coefficients
—
This option exports the vector of wavelet and scaling coefficients,
the bookkeeping vector, and the analyzing wavelet in a structure array.
The wavelet and scaling coefficient and bookkeeping vectors are identical
to the output of wavedec
.
Export Decomposition
—
This option is identical to Export Coefficients
except
that it also contains the name of the analyzed signal.
Export All Approximations
—
This option exports a L-by-N matrix where L is the value of Level and
N is the length of the input signal. Each row of the matrix is the
projection onto the approximation space at the corresponding level.
For example, the first row of the matrix is the projection onto the
approximation space at level 1.
Export All Details
—
This option exports a L-by-N matrix where L is the value of Level and
N is the length of the input signal. Each row of the matrix is the
projection onto the detail (wavelet) space at the corresponding level.
For example, the first row of the matrix is the projection onto the
detail space at level 1.
You can load signals, coefficients, or decompositions into the graphical interface. The information you load may have been previously exported from the graphical interface, and then manipulated in the workspace, or it may have been information you generated initially from the command line.
In either case, you must observe the strict file formats and data structures used by the Wavelet 1-D tool, or else errors will result when you try to load information.
Loading Signals. To load a signal you've constructed in your MATLAB workspace
into the Wavelet 1-D tool, save the
signal in a MAT-file (with extension mat
or other).
For instance, suppose you've designed a signal called warma
and
want to analyze it in the Wavelet 1-D tool.
save warma warma
The workspace variable warma must be a vector.
sizwarma = size(warma) sizwarma = 1 1000
To load this signal into the Wavelet 1-D tool, use the menu option File > Load > Signal.
A dialog box appears that lets you select the appropriate MAT-file to be loaded.
Note The first one-dimensional variable encountered in the file is considered the signal. Variables are inspected in alphabetical order. |
Loading Discrete Wavelet Transform Coefficients. To load discrete wavelet transform coefficients into the Wavelet 1-D tool, you must first save the appropriate
data in a MAT-file, which must contain at least the two variables coefs
and longs
.
Variable coefs
must be a vector of DWT coefficients
(concatenated for the various levels), and variable longs
a
vector specifying the length of each component of coefs
,
as well as the length of the original signal.
After constructing or editing the appropriate data in your workspace, type
save myfile coefs longs
Use the File > Load > Coefficients menu option from the Wavelet 1-D tool to load the data into the graphical tool.
A dialog box appears, allowing you to choose the folder and file in which your data reside.
Loading Decompositions. To load discrete wavelet transform decomposition data into the Wavelet 1-D graphical interface, you must first
save the appropriate data in a MAT-file (with extension wa1
or
other).
The MAT-file contains the following variables.
Variable | Status | Description |
---|---|---|
coefs | Required | Vector of concatenated DWT coefficients |
longs | Required | Vector specifying lengths of components of |
wave_name | Required | String specifying name of wavelet used for decomposition
(e.g., |
data_name | Optional | String specifying name of decomposition |
After constructing or editing the appropriate data in your workspace, type
save myfile coefs longs wave_name
Use the File > Load > Decomposition menu option from the Wavelet 1-D tool to load the decomposition data into the graphical tool.
A dialog box appears, allowing you to choose the folder and file in which your data reside.
Note
When loading a signal, a decomposition or coefficients from
a MAT-file, the extension of this file is free. The |