How to reconstruct original signal from continuous wavelet coefficients

Dear all, After using CWT (signal,3:10,'haar'), i got one matrix with CWT coefficients with scale from 3 to 8. I still do not understand 2 bellow problems: 1. How can i reconstruct signal from each scale? 2. in CWT (signal,3:10,'haar'), haar at scale 3 is [1/sqrt(2^3 1/sqrt(2^3 -1/sqrt(2^3 -1/sqrt(2^3] ? Thank you so much

Answer 1

[Wayne King]

Dear Hai, I would first ask. Are you sure you need the CWT here and not the MODWT? The MODWT is limited to dyadic scales (that is a limitation as opposed to CWT), but it has the property that it allows for perfect reconstruction (perfect inversion) and it allows you to obtain a reconstruction (projection) of the signal onto each scale with MODWTMRA.

The MODWT and associated functionality work with orthogonal wavelets -- the Haar is one such wavelet.

For example:

load wecg;  % load ecg signal
 dt = 1/180; % data sampled at 180 Hz
 t = 0:dt:(length(wecg)*dt)-dt;
 wtecg = modwt(wecg,3,'haar');  % obtain the MODWT at three levels of resolution -- scales, 2,4,8
 ecgmra = modwtmra(wtecg,'haar');
 subplot(5,1,1);
 plot(t,wecg); title('Original Data');
 for kk = 1:size(ecgmra,1)
   subplot(5,1,kk+1)
   plot(t,ecgmra(kk,:));
end

The top plot is the original signal, the first plot is the projection onto the wavelet (detail) subspace corresponding to a scale of 2 (the highest resolution or shortest scale), the next is the projection onto the wavelet subspace of scale 4, and finally the last row is the projection onto the scaling (lowpass) space at scale 2^3=8

If you sum the rows of ecgmra (summing the MODWT MRA at each time point), you will get back the original signal.

figure;
 ts = sum(ecgmra,1);
 plot(t,[wecg ts'])
 grid on;

[Hai Tran]

Dear Dr.Wayne King, First, I am very much appreciated your answer. Because, now my analysis has some problem with WAVEDEC, and MODWT but I still do not know exactly where is it wrong? So, I just try to use CWT, that is a reason why i asked you. Okay, I will consider to your advice and check my code again. Regards, Hai