Obtain the MODWT of an ECG signal and demonstrate perfect reconstruction.

Load the ECG signal data and obtain the MODWT.

load wecg;

Obtain the MODWT and the Inverse MODWT.

w = modwt(wecg);
xrec = imodwt(w);

Use the L-infinity norm to show that the difference between the original signal and the reconstruction is extremely small. The largest absolute difference between the original signal and the reconstruction is on the order of $$10^{-12}$$, which demonstrates perfect reconstruction.

norm(abs(xrec'-wecg),Inf)

ans = 2.3255e-12

Obtain the MODWT of Deutsche Mark-U.S. Dollar exchange rate data and demonstrate perfect reconstruction.

Load the Deutsche Mark-U.S. Dollar exchange rate data.

load DM_USD;

Obtain the MODWT and the Inverse MODWT using the 'db2' wavelet.

wdm = modwt(DM_USD,'db2');
xrec = imodwt(wdm,'db2');

Use the L-infinity norm to show that the difference between the original signal and the reconstruction is extremely small. The largest absolute difference between the original signal and the reconstruction is on the order of $$10^{-13}$$, which demonstrates perfect reconstruction.

norm(abs(xrec'-DM_USD),Inf)

ans = 1.6370e-13

Obtain the MODWT of an ECG signal using the Fejer-Korovkin filters.

Load the ECG data.

load wecg;

Create the 8-coefficient Fejer-Korovkin filters.

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

Obtain the MODWT and Inverse MODWT.

wtecg = modwt(wecg,Lo,Hi);
xrec = imodwt(wtecg,Lo,Hi);

Plot the original data and the reconstruction.

subplot(2,1,1)
plot(wecg)
title('ECG Signal');
subplot(2,1,2)
plot(xrec)
title('Reconstruction')

Obtain the MODWT of an ECG signal down to the maximum level and obtain the projection of the ECG signal onto the scaling space at level 3.

Load the ECG data.

load wecg;

Obtain the MODWT.

wtecg = modwt(wecg);

Obtain the projection of the ECG signal onto $$V_3$$, the scaling space at level three by using the imodwt function.

v3proj = imodwt(wtecg,3);

Plot the original signal and the projection.

subplot(2,1,1)
plot(wecg)
title('Original Signal')
subplot(2,1,2)
plot(v3proj)
title('Projection onto V3')

Note that the spikes characteristic of the R waves in the ECG are missing in the $$V_3$$ approximation. You can see the missing details by examining the wavelet coefficients at level three.

Plot the level-three wavelet coefficients.

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

Obtain the inverse MODWT using reflection boundary handling for Southern Oscillation Index data. The sampling period is one day. imodwt with the 'reflection' option assumes that the input matrix, which is the modwt output, is twice the length of the original signal length. imodwt reflection boundary handling reduces the number of wavelet and scaling coefficients at each scale by half.

load soi;
wsoi = modwt(soi,4,'reflection');
xrecsoi = imodwt(wsoi,'reflection');

Use the L-infinity norm to show that the difference between the original signal and the reconstruction is extremely small. The largest absolute difference between the original signal and the reconstruction is on the order of $$10^{-11}$$, which demonstrates perfect reconstruction.

norm(abs(xrecsoi'-soi),Inf)

ans = 1.6421e-11