# Documentation

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# modwpt

Maximal overlap discrete wavelet packet transform

## Syntax

wpt = modwpt(x)
wpt = modwpt(x,wname)
wpt = modwpt(x,lo,hi)
wpt = modwpt(___,lev)
[wpt,packetlevs] = modwpt(___)
[wpt,packetlevs,cfreq] = modwpt(___)
[wpt,packetlevs,cfreq,energy] = modwpt(___)
[wpt,packetlevs,cfreq,energy,relenergy] = modwpt(___)
[___] = modwpt(___,Name,Value)

## 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.

example

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 , where n = 0,...,15, and 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 =

16       12998

Obtain the MODWPT of an ECG waveform using the Fejer-Korovkin length 18 scaling and wavelet filters.

[lo,hi] = wfilters('fk18');
wpt = modwpt(wecg,lo,hi);

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 =

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.

sum(energy)-(bandpower(wecg)*length(wecg))
ans = 3.6120e-09

Plot the MODWPT energy by node.

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

sprintf('Total power in passband is %.2f', energy(1))
ans =
'Total power in passband is 200.84'

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');

sprintf('Percentage of signal power in passband is %.1f',(relenergy(1)*100))
ans =
'Percentage of signal power in passband is 67.3'

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 filter specified as a that corresponds to an orthogonal wavelet. If you specify the scaling (lo) and wavelet (hi) filters, modwpt ignores the wname input.

Valid orthogonal wavelet families begin with one of the following s, followed by an integer, N, for example, sym4. Note, however, that 'haar' is not followed by an integer.

• 'haar' — Haar wavelet, which is the same as Daubechies wavelet with one vanishing moment, 'db1'.

• 'dbN' — Daubechies wavelet with N vanishing moments

• 'symN' — Symlets wavelet with N vanishing moments

• 'coifN' — Coiflets wavelet with N vanishing moments

• 'fkN' — Fejer-Korovkin wavelet with N coefficients

To check if your wavelet is orthogonal, use wavemngr('type',wname) and verify that it returns 1 as the wavelet type. To determine valid values for N, use waveinfo, for example, waveinfo('fk').

Scaling filter, specified as an even-length real-valued vector. lo must satisfy the conditions necessary to generate an orthogonal scaling function. You cannot specify both the scaling-wavelet filters and the wname input.

Wavelet filter, specified as an even-length real-valued vector. hi must satisfy the conditions necessary to generate an orthogonal wavelet. You cannot specify both the scaling-wavelet filters and the wname input.

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

### Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

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

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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.