# modwt

Maximal overlap discrete wavelet transform

## Description

example

w = modwt(x) returns the maximal overlap discrete wavelet transform (MODWT) of x. x can be a real- or complex-valued vector or matrix. If x is a matrix, modwt operates on the columns of x. modwt computes the wavelet transform down to level floor(log2(length(x))) if x is a vector and floor(log2(size(x,1))) if x is a matrix. By default, modwt uses the Daubechies least-asymmetric wavelet with four vanishing moments ('sym4') and periodic boundary handling.

example

w = modwt(x,wname) uses the orthogonal wavelet, wname, for the MODWT.

example

w = modwt(x,Lo,Hi) uses the scaling filter, Lo, and wavelet filter, Hi, to compute the MODWT. These filters must satisfy the conditions for an orthogonal wavelet. You cannot specify both wname and a filter pair, Lo and Hi.

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w = modwt(___,lev) computes the MODWT down to the specified level, lev, using any of the arguments from previous syntaxes.

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w = modwt(___,'reflection') computes the MODWT using reflection boundary handling. Other inputs can be any of the arguments from previous syntaxes. Before computing the wavelet transform, modwt extends the signal symmetrically at the terminal end to twice the signal length. The number of wavelet and scaling coefficients that modwt returns is equal to twice the length of the input signal. By default, the signal is extended periodically.

You must enter the entire character vector 'reflection'. If you added a wavelet named 'reflection' using the wavelet manager, you must rename that wavelet prior to using this option. 'reflection' may be placed in any position in the input argument list after x.

## Examples

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Obtain the MODWT of an electrocardiogram (ECG) signal using the default sym4 wavelet down to the maximum level. The data are taken from Percival & Walden (2000), p.125 (data originally provided by William Constantine and Per Reinhall, University of Washington).

wtecg = modwt(wecg);
whos wtecg
Name        Size               Bytes  Class     Attributes

wtecg      12x2048            196608  double

The first eleven rows of wtecg are the wavelet coefficients for scales ${2}^{1}$ to ${2}^{11}$. The final row contains the scaling coefficients at scale ${2}^{11}$. Plot the detail (wavelet) coefficients for scale ${2}^{3}$.

plot(wtecg(3,:))
title('Level 3 Wavelet Coefficients')

Obtain the MODWT of Southern Oscillation Index data with the db2 wavelet down to the maximum level.

wsoi = modwt(soi,'db2');

Obtain the MODWT of the Deutsche Mark - U.S. Dollar exchange rate data using the Fejer-Korovkin length 8 scaling and wavelet filters.

[Lo,Hi] = wfilters('fk8');
wdm = modwt(DM_USD,Lo,Hi);

Obtain the MODWT of an ECG signal down to scale ${2}^{4}$, which corresponds to level four. Use the default sym4 wavelet. The data are taken from Percival & Walden (2000), p.125 (data originally provided by William Constantine and Per Reinhall, University of Washington).

wtecg = modwt(wecg,4);
whos wecg wtecg
Name          Size              Bytes  Class     Attributes

wecg       2048x1               16384  double
wtecg         5x2048            81920  double

The row size of wtecg is L+1, where, in this case, the level (L) is 4. The column size matches the number of input samples.

Obtain the MODWT of an ECG signal using reflection boundary handling. Use the default sym4 wavelet and obtain the transform down to level 4. The data are taken from Percival & Walden (2000), p.125 (data originally provided by William Constantine and Per Reinhall, University of Washington).

wtecg = modwt(wecg,4,'reflection');
whos wecg wtecg
Name          Size               Bytes  Class     Attributes

wecg       2048x1                16384  double
wtecg         5x4096            163840  double

wtecg has 4096 columns, which is twice the length of the input signal, wecg.

Load the 23 channel EEG data Espiga3 [3]. The channels are arranged column-wise. The data is sampled at 200 Hz.

Compute the maximal overlap discrete wavelet transform down to the maximum level.

wt = modwt(Espiga3);

Obtain the squared signal energies and compare them against the squared energies obtained from summing the wavelet coefficients over all levels. Use the log-squared energy due to the disproportionately large energy in one component.

sigN2 = vecnorm(Espiga3).^2;
wtN2 = sum(squeeze(vecnorm(wt,2,2).^2));
bar(1:23,log(sigN2))
hold on
scatter(1:23,log(wtN2),'filled','SizeData',100)
alpha(0.75)
legend('Signal Energy','Energy in Wavelet Coefficients', ...
'Location','NorthWest')
xlabel('Channel')
ylabel('ln(squared energy)')

This example demonstrates the differences between the functions MODWT and MODWTMRA. The MODWT partitions a signal's energy across detail coefficients and scaling coefficients. The MODWTMRA projects a signal onto wavelet subspaces and a scaling subspace.

Choose the sym6 wavelet. Load and plot an electrocardiogram (ECG) signal. The sampling frequency for the ECG signal is 180 hertz. The data are taken from Percival and Walden (2000), p.125 (data originally provided by William Constantine and Per Reinhall, University of Washington).

t = (0:numel(wecg)-1)/180;
wv = 'sym6';
plot(t,wecg)
grid on
title(['Signal Length = ',num2str(numel(wecg))])
xlabel('Time (s)')
ylabel('Amplitude')

Take the MODWT of the signal.

wtecg = modwt(wecg,wv);

The input data are samples of a function $f\left(x\right)$ evaluated at $N$-many time points. The function can be expressed as a linear combination of the scaling function $\varphi \left(x\right)$ and wavelet $\psi \left(x\right)$at varying scales and translations: $f\left(x\right)=\sum _{k=0}^{N-1}{c}_{k}\phantom{\rule{0.16666666666666666em}{0ex}}{2}^{-{J}_{0}/2}\varphi \left({2}^{-{J}_{0}}\phantom{\rule{0.16666666666666666em}{0ex}}x-k\right)+\sum _{j=1}^{{J}_{0}}{f}_{j}\left(x\right)$ where ${f}_{j}\left(x\right)=\sum _{k=0}^{N-1}{d}_{j,k}\phantom{\rule{0.16666666666666666em}{0ex}}{2}^{-j/2}\phantom{\rule{0.16666666666666666em}{0ex}}\psi \left({2}^{-j}x-k\right)$ and ${J}_{0}$ is the number of levels of wavelet decomposition. The first sum is the coarse scale approximation of the signal, and the ${f}_{j}\left(x\right)$ are the details at successive scales. MODWT returns the $N$-many coefficients $\left\{{c}_{k}\right\}$and the $\left({J}_{0}×N\right)$-many detail coefficients $\left\{{d}_{j,k}\right\}$ of the expansion. Each row in wtecg contains the coefficients at a different scale.

When taking the MODWT of a signal of length $N$, there are $\text{floor}\left({\mathrm{log}}_{2}\left(N\right)\right)$-many levels of decomposition (by default). Detail coefficients are produced at each level. Scaling coefficients are returned only for the final level. In this example, since $N=2048$, ${J}_{0}=\text{floor}\left(\mathrm{log}2\left(2048\right)\right)=11$ and the number of rows in wtecg is ${J}_{0}+1=11+1=12$.

The MODWT partitions the energy across the various scales and scaling coefficients: ${||X||}^{2}=\sum _{j=1}^{{J}_{0}}{||{W}_{j}||}^{2}+{||{V}_{{J}_{0}}||}^{2}$ where $X$ is the input data, ${W}_{j}$ are the detail coefficients at scale $j$, and ${V}_{{J}_{0}}$ are the final-level scaling coefficients.

Compute the energy at each scale, and evaluate their sum.

energy_by_scales = sum(wtecg.^2,2);
Levels = {'D1';'D2';'D3';'D4';'D5';'D6';'D7';'D8';'D9';'D10';'D11';'A11'};
energy_table = table(Levels,energy_by_scales);
disp(energy_table)
Levels     energy_by_scales
_______    ________________

{'D1' }         14.063
{'D2' }         20.612
{'D3' }         37.716
{'D4' }         25.123
{'D5' }         17.437
{'D6' }         8.9852
{'D7' }         1.2906
{'D8' }         4.7278
{'D9' }         12.205
{'D10'}         76.428
{'D11'}         76.268
{'A11'}         3.4192
energy_total = varfun(@sum,energy_table(:,2))
energy_total=table
sum_energy_by_scales
____________________

298.28

Confirm the MODWT is energy-preserving by computing the energy of the signal and comparing it with the sum of the energies over all scales.

energy_ecg = sum(wecg.^2);
max(abs(energy_total.sum_energy_by_scales-energy_ecg))
ans = 7.4402e-10

Take the MODWTMRA of the signal.

mraecg = modwtmra(wtecg,wv);

MODWTMRA returns the projections of the function $f\left(x\right)$ onto the various wavelet subspaces and final scaling space. That is, MODWTMRA returns $\sum _{k=0}^{N-1}{c}_{k}\phantom{\rule{0.16666666666666666em}{0ex}}{2}^{-{J}_{0}/2}\varphi \left({2}^{-{J}_{0}}\phantom{\rule{0.16666666666666666em}{0ex}}x-k\right)$and the ${J}_{0}$-many $\left\{{f}_{j}\left(x\right)\right\}$evaluated at $N$-many time points. Each row in mraecg is a projection of $f\left(x\right)$ onto a different subspace. This means the original signal can be recovered by adding all the projections. This is not true in the case of the MODWT. Adding the coefficients in wtecg will not recover the original signal.

Choose a time point, add the projections of $f\left(x\right)$ evaluated at that time point and compare with the original signal.

time_point = 1000;
abs(sum(mraecg(:,time_point))-wecg(time_point))
ans = 3.0846e-13

Confirm that, unlike MODWT, MODWTMRA is not an energy-preserving transform.

energy_ecg = sum(wecg.^2);
energy_mra_scales = sum(mraecg.^2,2);
energy_mra = sum(energy_mra_scales);
max(abs(energy_mra-energy_ecg))
ans = 115.7053

The MODWTMRA is a zero-phase filtering of the signal. Features will be time-aligned. Demonstrate this by plotting the original signal and one of its projections. To better illustrate the alignment, zoom in.

plot(t,wecg,'b')
hold on
plot(t,mraecg(4,:),'-')
hold off
grid on
xlim([4 8])
legend('Signal','Projection','Location','northwest')
xlabel('Time (s)')
ylabel('Amplitude')

Make a similar plot using the MODWT coefficients at the same scale. Note that features will not be time-aligned. The MODWT is not a zero-phase filtering of the input.

plot(t,wecg,'b')
hold on
plot(t,wtecg(4,:),'-')
hold off
grid on
xlim([4 8])
legend('Signal','Coefficients','Location','northwest')
xlabel('Time (s)')
ylabel('Amplitude')

## Input Arguments

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Input signal, specified as a vector or matrix. If x is a vector, x must have at least two elements. If x is a matrix, the row dimension of x must be at least 2.

Data Types: single | double
Complex Number Support: Yes

Analyzing wavelet, specified as one of the following:

• 'haar' — Haar wavelet

• 'dbN' — Extremal phase Daubechies wavelet with N vanishing moments, where N is a positive integer from 1 to 45.

• 'symN' — Symlets wavelet with N vanishing moments, where N is a positive integer from 2 to 45.

• 'coifN' — Coiflets wavelet with N vanishing moments, where N is a positive integer from 1 to 5.

• 'fkN' — Fejér-Korovkin wavelet with N coefficients, where N = 4, 6, 8, 14, 18 and 22.

Filters, specified as a pair of even-length real-valued vectors. Lo is the scaling filter, and Hi is the wavelet filter. The filters must satisfy the conditions for an orthogonal wavelet. The lengths of Lo and Hi must be equal. See wfilters for additional information. You cannot specify both a wavelet wname and filter pair Lo,Hi.

Transform level, specified as a positive integer less than or equal to floor(log2(N)), where N = length(x) if x is a vector, or N = size(x,1) if x is a matrix. If unspecified, lev defaults to floor(log2(N)).

## Output Arguments

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MODWT transform of x. w contains the wavelet coefficients and final-level scaling coefficients of x. If x is a vector, w is a lev+1-by-N matrix. If x is a matrix, w is a lev+1-by-N-by-NC array, where NC is the number of columns in x. N is equal to the input signal length unless you specify 'reflection' boundary handling, in which case N is twice the length of the input signal. The kth row of the array, w, contains the wavelet coefficients for scale 2k (wavelet scale 2(k-1)). The final, (lev+1)th, row contains the scaling coefficients for scale 2lev.

## Algorithms

The standard algorithm for the MODWT implements the circular convolution directly in the time domain. This implementation of the MODWT performs the circular convolution in the Fourier domain. The wavelet and scaling filter coefficients at level j are computed by taking the inverse discrete Fourier transform (DFT) of a product of DFTs. The DFTs in the product are the signal’s DFT and the DFT of the jth level wavelet or scaling filter.

Let Hk and Gk denote the length N DFTs of the MODWT wavelet and scaling filters, respectively. Let j denote the level and N denote the sample size.

The jth level wavelet filter is defined by

$\frac{1}{N}\sum _{k=0}^{N-1}{H}_{j,k}{e}^{i2\pi nk/N}$

where

${H}_{j,k}={H}_{{2}^{j-1}k\text{mod}N}\prod _{m=0}^{j-2}{G}_{{2}^{m}k\text{mod}N}$

The jth level scaling filter is

$\frac{1}{N}\sum _{k=0}^{N-1}{G}_{j,k}{e}^{i2\pi nk/N}$

where

${G}_{j,k}=\prod _{m=0}^{j-1}{G}_{{2}^{m}k\text{mod}N}$

## References

[1] Percival, Donald B., and Andrew T. Walden. Wavelet Methods for Time Series Analysis. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge ; New York: Cambridge University Press, 2000.

[2] Percival, Donald B., and Harold O. Mofjeld. “Analysis of Subtidal Coastal Sea Level Fluctuations Using Wavelets.” Journal of the American Statistical Association 92, no. 439 (September 1997): 868–80. https://doi.org/10.1080/01621459.1997.10474042.

[3] Mesa, Hector. “Adapted Wavelets for Pattern Detection.” In Progress in Pattern Recognition, Image Analysis and Applications, edited by Alberto Sanfeliu and Manuel Lazo Cortés, 3773:933–44. Berlin, Heidelberg: Springer Berlin Heidelberg, 2005. https://doi.org/10.1007/11578079_96.