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1-D Decimated Wavelet Transforms

This section takes you through the features of 1-D critically-sampled wavelet analysis using the Wavelet Toolbox™ software.

The toolbox provides these functions for 1-D signal analysis. For more information, see the reference pages.

Analysis-Decomposition Functions

Function Name

Purpose

dwt

Single-level decomposition

wavedec

Decomposition

wmaxlev

Maximum wavelet decomposition level

Synthesis-Reconstruction Functions

Function Name

Purpose

idwt

Single-level reconstruction

waverec

Full reconstruction

wrcoef

Selective reconstruction

upcoef

Single reconstruction

Decomposition Structure Utilities

Function Name

Purpose

detcoef

Extraction of detail coefficients

appcoef

Extraction of approximation coefficients

upwlev

Recomposition of decomposition structure

Denoising and Compression

Function Name

Purpose

wdenoise

Automatic wavelet signal denoising (recommended)

ddencmp

Provide default values for denoising and compression

wbmpen

Penalized threshold for wavelet 1-D or 2-D denoising

wdcbm

Thresholds for wavelet 1-D using Birgé-Massart strategy

wdencmp

Wavelet denoising and compression

wthrmngr

Threshold settings manager

Apps

App

Purpose

Signal Multiresolution AnalyzerDecompose signals into time-aligned components
Wavelet Signal AnalyzerAnalyze and compress signals using wavelets
Wavelet Signal DenoiserVisualize and denoise time series data

1-D Decimated Wavelet Analysis

This example involves a real-world signal — electrical consumption measured over the course of three 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.

Decomposition and Reconstruction

Load the signal and select a portion for wavelet analysis.

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

Use the dwt function to perform a single-level decomposition of the signal using the db1 wavelet. This generates the coefficients of the level 1 approximation (cA1) and detail (cD1).

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

Construct the level 1 approximation and detail (A1 and D1) from the coefficients cA1 and cD1. You can use either the idwt function or the upcoef function.

useIDWT = true;
if useIDWT
    A1 = idwt(cA1,[],"db1",l_s);
    D1 = idwt([],cD1,"db1",l_s);
else
    A1 = upcoef("a",cA1,"db1",1,l_s);
    D1 = upcoef("d",cD1,"db1",1,l_s);
end

Display the results of the level-one decomposition.

tiledlayout(1,2)
nexttile
plot(A1)
title("Approximation A1")
nexttile
plot(D1)
title("Detail D1")

Figure contains 2 axes objects. Axes object 1 with title Approximation A1 contains an object of type line. Axes object 2 with title Detail D1 contains an object of type line.

Reconstruct a signal by using the idwt function. Compare the reconstruction with the original signal.

A0 = idwt(cA1,cD1,"db1",l_s);
err = max(abs(s-A0))
err = 
2.2737e-13

Use the wavedec function to perform a level 3 decomposition of the signal.

[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.

ch02_usewv91.png

Use the appcoef function to extract the level 3 approximation coefficients from C.

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

Use the detcoef function to extract the levels 3, 2, and 1 detail coefficients from C. The detcoef function gives you the option to extract the coefficients at the different levels one at a time.

extractTogether = true;
if extractTogether
    [cD1,cD2,cD3] = detcoef(C,L,[1,2,3]);
else
    cD3 = detcoef(C,L,3);
    cD2 = detcoef(C,L,2);
    cD1 = detcoef(C,L,1);
end

Use the wrcoef function to reconstruct the Level 3 approximation and the Level 1, 2, and 3 details.

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

Reconstruct the details at levels 1, 2, and 3.

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 the level 3 decomposition.

figure
tiledlayout(2,2)
nexttile
plot(A3)  
title("Approximation A3") 
nexttile
plot(D1)  
title("Detail D1") 
nexttile
plot(D2)  
title("Detail D2") 
nexttile
plot(D3) 
title("Detail D3")

Figure contains 4 axes objects. Axes object 1 with title Approximation A3 contains an object of type line. Axes object 2 with title Detail D1 contains an object of type line. Axes object 3 with title Detail D2 contains an object of type line. Axes object 4 with title Detail D3 contains an object of type line.

Reconstruct the original signal from the Level 3 decomposition.

A0 = waverec(C,L,"db1");  
err = max(abs(s-A0))  
err = 
4.5475e-13

Crude Denoising

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.

Compare the approximation to the original signal.

figure
tiledlayout(2,1)
nexttile
plot(s)
title("Original")
axis off 
nexttile
plot(A3)
title("Level 3 Approximation") 
axis off

Figure contains 2 axes objects. Hidden axes object 1 with title Original contains an object of type line. Hidden axes object 2 with title Level 3 Approximation contains an object of type line.

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.

Denoise by Thresholding

Let's look again at the details of our level 3 analysis. Display the details D1, D2, and D3.

figure
tiledlayout(3,1)
nexttile
plot(D1)
title("Detail Level 1")
axis off 
nexttile
plot(D2)
title("Detail Level 2")
axis off 
nexttile
plot(D3)
title("Detail Level 3")
axis off

Figure contains 3 axes objects. Hidden axes object 1 with title Detail Level 1 contains an object of type line. Hidden axes object 2 with title Detail Level 2 contains an object of type line. Hidden axes object 3 with title Detail Level 3 contains an object of type line.

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 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 a signal, use the wdenoise function. The wdenoise function is recommended for denoising 1-D signals. The function provides a simple interface to a variety of denoising methods. Default parameters are provided are automatically provided for quick and easy use.

Instead of using all the default function values, denoise the signal using the db1 wavelet and a wavelet decomposition down to level 3.

clean = wdenoise(s,3,Wavelet="db1");

Display the original and denoised signals. Notice how we have removed the noise without compromising the sharp detail of the original signal. This is a strength of wavelet analysis.

figure
tiledlayout(2,1)
nexttile
plot(s(2000:3920))
title("Original") 
nexttile
plot(clean(2000:3920))
title("Denoised")

Figure contains 2 axes objects. Axes object 1 with title Original contains an object of type line. Axes object 2 with title Denoised contains an object of type line.

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

You can also use the Wavelet Signal Denoiser app to denoise 1-D signals.

See Also

Apps

Related Topics