modwtmra

Obtain the MODWTMRA of a simple time-series signal and demonstrate perfect reconstruction.

Create a time-series signal

t = 1:10;
x = sin(2*pi*200*t);

Obtain the MODWT and the MODWTMRA and sum the MODWTMRA rows.

m = modwt(x);
mra = modwtmra(m);
xrec = sum(mra);

Use the maximum of the absolute values to show that the difference between the original signal and the reconstruction is extremely small. The largest absolute value is on the order of $$10^{-25}$$, which demonstrates perfect reconstruction.

max(abs(x-xrec))

ans = 
5.5738e-25

Construct an MRA of an ECG signal down to level four using the db2 wavelet. The data are taken from Percival & Walden (2000), p.125 (data originally provided by William Constantine and Per Reinhall, University of Washington). The sampling frequency for the ECG signal is 180 hertz.

load wecg;
lev = 4;
wtecg = modwt(wecg,'db2',lev);
mra = modwtmra(wtecg,'db2');

Plot the ECG waveform and the MRA.

t = (0:numel(wecg)-1)/180;
subplot(6,1,1)
plot(t,wecg)
for kk = 2:lev+2
    subplot(6,1,kk)
    plot(t,mra(kk-1,:))
end
xlabel('Time (s)')
set(gcf,'Position',[0 0 500 700])

Construct a multiresolution analysis for the Southern Oscillation Index data. The sampling period is one day. Plot the level eight details corresponding to a scale of $$2^8$$ days. The details at this scale capture oscillations on a scale of approximately one year.

load soi
wtsoi = modwt(soi);
mrasoi = modwtmra(wtsoi);
plot(mrasoi(8,:))
title('Level 8 Details')

Obtain the MRA for the Deutsch Mark - U.S. Dollar exchange rate data using the minimum bandwidth scaling and wavelet filters with four coefficients.

load DM_USD;
Lo = [0.4801755, 0.8372545, 0.2269312, -0.1301477];
Hi = qmf(Lo);
wdm = modwt(DM_USD,Lo,Hi);
mra = modwtmra(wdm,Lo,Hi);

Load the ECG data.

load wecg

Obtain the MODWT of the signal using the filters associated with the 8-coefficient Fejér-Korovkin filters.

[~,~,Lo,Hi] = wfilters("fk8");
wtecg = modwt(wecg,Lo,Hi);

Obtain the MRA of the signal using the filters.

mra = modwtmra(wtecg,Lo,Hi);

Obtain a second MRA of the signal using the wavelet name. Confirm the multiresolution analyses are equal.

mra2 = modwtmra(wtecg,"fk8");
max(abs(mra(:)-mra2(:)))

ans = 
0

Obtain the MRA for an ECG signal using 'reflection' boundary handling. The data are taken from Percival & Walden (2000), p.125 (data originally provided by William Constantine and Per Reinhall, University of Washington).

load wecg;
wtecg = modwt(wecg,'reflection');
mra = modwtmra(wtecg,'reflection');

Show that the number of columns in the MRA is equal to the number of elements in the original signal.

isequal(size(mra,2),numel(wecg))

ans = logical
   1

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

load Espiga3

Obtain the MRA of the multisignal.

w = modwt(Espiga3);
mra = modwtmra(w);