# modwpt

Maximal overlap discrete wavelet packet transform

## Description

example

wpt = modwpt(x) returns the terminal nodes for the maximal overlap discrete wavelet packet transform (MODWPT) for the 1-D real-valued signal, x.

Note

The output of the MODWPT is time-delayed compared to the input signal. Most filters used to obtain the MODWPT have a nonlinear phase response, which makes compensating for the time delay difficult. This is true for all orthogonal scaling and wavelet filters, except the Haar wavelet. It is possible to time-align the coefficients with the signal features, but the result is an approximation, not an exact alignment with the original signal. The MODWPT partitions the energy among the wavelet packets at each level. The sum of the energy over all the packets equals the total energy of the input signal. The output of MODWPT is useful for applications where you want to analyze the energy levels in different packets.

The MODWPT details (modwptdetails) are the result of zero-phase filtering of the signal. The features in the MODWPT details align exactly with features in the input signal. For a given level, summing the details for each sample returns the exact original signal. The output of the MODWPT details is useful for applications that require time-alignment, such as nonparametric regression analysis.

example

wpt = modwpt(x,wname) returns the MODWPT using the orthogonal wavelet filter specified by the wname.

wpt = modwpt(x,lo,hi) returns the MODWPT using the orthogonal scaling filter, lo, and wavelet filter, hi.

wpt = modwpt(___,lev) returns the terminal nodes of the wavelet packet tree at positive integer level lev.

example

[wpt,packetlevs] = modwpt(___) returns a vector of transform levels corresponding to the rows of wpt.

[wpt,packetlevs,cfreq] = modwpt(___) returns the center frequencies of the approximate passbands corresponding to the rows of wpt.

example

[wpt,packetlevs,cfreq,energy] = modwpt(___) returns the energy (squared L2 norm) of the wavelet packet coefficients for the nodes in wpt.

example

[wpt,packetlevs,cfreq,energy,relenergy] = modwpt(___) returns the relative energy for the wavelet packets in wpt.

example

[___] = modwpt(___,Name,Value) returns the MODWPT with additional options specified by one or more Name,Value pair arguments.

## Examples

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Obtain the MODWPT of an electrocardiogram (ECG) signal using the default length 18 Fejer-Korovkin ('fk18') wavelet.

wpt = modwpt(wecg);

wpt is a 16-by-2048 matrix containing the sequency-ordered wavelet packet coefficients for the wavelet packet transform nodes. In this case, the nodes are at level 4. Each node corresponds to an approximate passband filtering of $\left[n{f}_{s}/{2}^{5},\left(n+1\right){f}_{s}/{2}^{5}\right)$, where n = 0,...,15, and ${f}_{s}$ is the sampling frequency. Plot the wavelet packet coefficients at node (4,2), which is level 4, node 2.

plot(wpt(3,:))
title('Node 4 Wavelet Packet Coefficients')

Obtain the MODWPT of Southern Oscillation Index data with the Daubechies extremal phase wavelet with two vanishing moments ('db2').

wsoi = modwpt(soi,'db2');

Verify that the size of the resulting transform contains 16 nodes. Each node is in a separate row.

size(wsoi)
ans = 1×2

16       12998

Obtain the MODWPT and full wavelet packet tree of an ECG waveform using the default length 18 Fejer-Korovkin ('fk18') wavelet. Extract and plot the node coefficients at level 3, node 2.

[wpt,packetlevels,cfreq] = modwpt(wecg,'FullTree',true);
p3 = wpt(packetlevels==3,:);
plot(p3(3,:))
title('Level 3, Node 2 Wavelet Coefficients')

Display the center frequencies at level 3.

cfreq(packetlevels==3,:)
ans = 8×1

0.0312
0.0938
0.1562
0.2188
0.2812
0.3438
0.4062
0.4688

Obtain and plot the MODWPT energy and relative energy of an ECG waveform.

[wpt,~,cfreq,energy,relenergy] = modwpt(wecg);

Show that the sum of the MODWPT energies is equal to the sum of the energy in the original signal. The difference between the total MODWPT energy and the signal energy is small enough to be considered insignificant.

disp(['Difference between MODWPT energy and signal energy: ',num2str(sum(energy)-sum(wecg.^2))])
Difference between MODWPT energy and signal energy: 3.6122e-09

Plot the MODWPT energy by node.

figure
bar(1:16,energy)
xlabel('Node')
ylabel('Energy')
title('Energy by Node')

disp(['Total power in passband: ',num2str(energy(1))])
Total power in passband: 200.8446

Plot the relative energy and show the percentage of signal energy in the first passband [0,5.6250].

figure
bar(1:16,relenergy*100)
xlabel('Node')
ylabel('Percent Energy')
title('Energy Relative to Signal Energy by Node')

disp(['Percentage of signal power in passband: ',num2str(relenergy(1)*100)])
Percentage of signal power in passband: 67.3352

Obtain the time-aligned MODWPT of two intermittent sine waves in noise. The sine wave frequencies are 150 Hz and 200 Hz. The data is sampled at 1000 Hz.

dt = 0.001;
t = 0:dt:1-dt;
x = cos(2*pi*150*t).*(t>=0.2 & t<0.4)+ sin(2*pi*200*t).*(t>0.6 & t<0.9);
y = x+0.05*randn(size(t));
[wpta,~,Falign] = modwpt(x,'TimeAlign',true);
[wptn,~,Fnon] = modwpt(x);

Compare the nonaligned and time-aligned time-frequency plots.

subplot(2,1,1);
contour(t,Fnon.*(1/dt),abs(wptn).^2);
grid on;
ylabel('Hz');
title('Time-Frequency Plot (Nonaligned)');
subplot(2,1,2)
contour(t,Falign.*(1/dt),abs(wpta).^2);
grid on;
xlabel('Time');
ylabel('Hz');
title('Time-Frequency Plot (Aligned)');

## Input Arguments

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Input signal, specified as a real-valued row or column vector. x must have at least two elements.

Data Types: double

Analyzing wavelet, specified as a character vector or string scalar. The wavelet must be orthogonal. Orthogonal wavelets are designated as type 1 wavelets in the wavelet manager, wavemngr.

Valid built-in orthogonal wavelet families are: Best-localized Daubechies ("bl"), Beylkin ("beyl"), Coiflets ("coif"), Daubechies ("db"), Fejér-Korovkin ("fk"), Haar ("haar"), Han linear-phase moments ("han"), Morris minimum-bandwidth ("mb"), Symlets ("sym"), and Vaidyanathan ("vaid").

For a list of wavelets in each family, see wfilters. You can also use waveinfo with the wavelet family short name. For example, waveinfo("db"). Use wavemngr("type",wname) to determine if wname is orthogonal (returns 1). For example, wavemngr("type","db6") returns 1.

Filters, specified as a pair of even-length real-valued vectors. lo is the orthogonal scaling filter and hi is the orthogonal wavelet filter. The filters must satisfy the conditions for an orthogonal wavelet. For more information, see wfilters and isorthwfb. You cannot specify both wname and a filter pair lo,hi.

Note

By default, the wfilters function returns two pairs of filters associated with an orthogonal or biorthogonal wavelet you specify. To agree with the usual convention in the implementation of MODWPT in numerical packages, when you specify an orthogonal wavelet wname, the modwpt function internally uses the second pair of filters returned by wfilters. For example,

wpt = modwpt(x,"db2");

is equivalent to

[~,~,lo,hi] = wfilters("db2"); wpt = modwpt(x,lo,hi);

This convention is different from the one followed by most Wavelet Toolbox™ discrete wavelet transform functions when decomposing a signal. Most functions internally use the first pair of filters.

Transform level, specified as a positive integer less than or equal to floor(log2(numel(x))).

### Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: 'Fulltree',true returns the full wavelet packet tree

Option to return the full wavelet packet tree, specified as the comma-separated pair consisting of 'FullTree' and either false or true. If you specify false, then modwpt returns only the terminal (final-level) wavelet packet nodes. If you specify true, then modwpt returns the full wavelet packet tree down to the specified level.

Example: 'Fulltree',true

Option to time align wavelet packet coefficients with signal features, specified as the comma-separated pair consisting of 'TimeAlign' and either true to time align or false to not align.

The scaling and wavelet filters have a time delay. Circularly shifting the wavelet packet coefficients in all nodes aligns the signal and wavelet coefficients in time. If you want to reconstruct the signal, such as by using imodwpt, do not shift the coefficients because time alignment is done during the inversion process.

Example: 'TimeAlign',true

## Output Arguments

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Wavelet packet tree, returned as a matrix with each row containing the sequency-ordered wavelet packet coefficients. By default, wpt contains only the terminal level for the MODWPT. The default terminal level is either level 4 or floor(log2(numel(x))), whichever is smaller. At level 4, wpt is a 16-by-numel(x) matrix. For the full tree, at level j, wpt is a 2j+2-2-by-numel(x) matrix, with each row containing the packet coefficients by level and index. The approximate passband for the nth row of wpt at level j is $\left[\frac{n-1}{{2}^{\left(j+1\right)}},\frac{n}{{2}^{\left(j+1\right)}}\right)$ cycles/sample, where n = 1,2,...2j.

Transform levels, returned as a vector. The levels correspond to the rows of wpt. If wpt contains only the terminal level coefficients, packetlevs is a vector of constants equal to the terminal level. If wpt contains the full wavelet packet table, packetlevs is a vector with 2j elements for each level, j. To select all the wavelet packet nodes at a particular level, use packetlevs with logical indexing.

Center frequencies of the approximate passbands in the wpt rows, returned as a vector. The center frequencies are in cycles/sample. To convert the units to cycles/unit time, multiply cfreq by the sampling frequency.

Energy of the wavelet packet coefficients for the wpt nodes, returned as a vector. The sum of the energies (squared L2 norms) for the wavelet packets at each level equals the energy in the signal.

Relative energy for each level, returned as a vector. The relative energy is the proportion of energy in each wavelet packet by level, relative to the total energy of that level. The sum of relative energies in all packets at each level equals 1.

## Algorithms

The modwpt performs a discrete wavelet packet transform and produces a sequency-ordered wavelet packet tree. Compare the sequency-ordered and normal (Paley)-ordered trees.

## References

[1] Percival, D. B., and A. T. Walden. Wavelet Methods for Time Series Analysis. Cambridge, UK: Cambridge University Press, 2000.

[2] Walden, A.T., and A. Contreras Cristan. “The phase-corrected undecimated discrete wavelet packet transform and its application to interpreting the timing of events.” Proceedings of the Royal Society of London A. Vol. 454, Issue 1976, 1998, pp. 2243-2266.

## Version History

Introduced in R2016a