When I perform the haar maximal overlap wavelet transform of a signal I get a series of coefficients and one approximation of the signal itself.
x=rand(1,1000);
swc=modwt(x,'haar',4);
The modwt haar has the summation property of the coefficients so the sum of the coefficients xsum=sum(swc) is equal to the original signal.
max(abs(x-xsum))
ans =
4.4409e-16
One of the properties of the modwt is the conservation of energy. So the sum of the squares of the signal Ex= sum(abs(x).^2) is equal to the sum of the squares of the coefficients Eswc= sum(sum(abs(swc).^2,2)).
Ex =
336.1986
Eswc=
336.1986
But since x=sum(swc) then it means that the sum of the squares of the coefficients is equal to the square of the sum of the same. This property is normally invalid and it isn't after the thresholding of the swc coefficients even if I correct the energy of the thresholded coefficients in order to obtain the same energy of swc. I have come to know that this particular property has to do with Parseval's therom and the conservation of the Fourier transform energy.
What are the properties that a data series must have to satisfy the conservation of energy and what transformations I have to do to get this result?
Thank you!