Obtain the MODWPT of an electrocardiogram (ECG) signal using the default length 18 Fejer-Korovkin ('fk18') wavelet and the default level, 4.

load wecg;
wptdetails = modwptdetails(wecg);

Demonstrate that summing the MODWPT details over each sample reconstructs the signal. The largest absolute difference between the original signal and the reconstruction is on the order of $$10^{-11}$$, which demonstrates perfect reconstruction.

xrec = sum(wptdetails);
max(abs(wecg-xrec'))

ans = 1.7903e-11

Obtain the MODWPT details for a signal containing 100 Hz and 450 Hz sine waves. Each row of the modwptdetails output corresponds to a separate frequency band.

dt = 0.001;
fs = 1/dt;
t = 0:dt:1;
x = (sin(2*pi*100*t)+sin(2*pi*450*t));
[lo,hi] = wfilters('fk22');
wptdetails = modwptdetails(x,lo,hi);

Use modwpt to obtain the energy and center frequencies of the signal. Plot the energy in the wavelet packets. The fourth and fifteenth frequency bands contain most of the energy. Other frequency bands have significantly less energy. The frequency ranges of fourth and fifteenth bands are approximately 94-125 Hz and 438-469 Hz, respectively.

[wpt,~,cfreqs,energy] = modwpt(x,lo,hi);
figure
bar(1:16,energy);
xlabel('Packet')
ylabel('Packet Energy')
title('Energy by Wavelet Packet')

Plot the power spectral density of the input signal.

pwelch(x,[],[],[],fs,'onesided');
title('Power Spectral Density of Input Signal')

Show that the MODWPT details have zero-phase shift from the 100 Hz input sine.

p4 = wptdetails(4,:);
plot(t,sin(2*pi*100*t).*(t>0.3 & t<0.7))
hold on
plot(t,p4.*(t>0.3 & t<0.7),'r')
legend('Sine Wave','MODWPT Details')

Obtain the MODWPT details for a 100 Hz time-localized sine wave in noise. The sampling rate is 1000 Hz. Obtain the MODWPT at level 4 using the length 22 Fejer-Korovkin ('fk22') wavelet.

dt = 0.001;     
t = 0:dt:1;
x = cos(2*pi*100*t).*(t>0.3 & t<0.7)+0.25*randn(size(t));
wptdetails = modwptdetails(x,'fk22');
p4 = wptdetails(4,:);

Plot the MODWPT details for level 4, packet number 4. The MODWPT details represent zero-phase filtering of the input signal with an approximate passband of $$[3Fs/2^5,4Fs/2^5)$$, where $$Fs$$ is the sampling frequency.

plot(t,cos(2*pi*100*t).*(t>0.3 & t<0.7));
hold on
plot(t,p4,'r')
legend('Sine Wave','MODWPT Details')

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

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

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

load wecg;     
[w,packetlevels] = modwptdetails(wecg,'FullTree',true);
p3 = w(packetlevels==3,:);
plot(p3(3,:))
title('Level 3, Node 2 MODWPT Details')