Dears, I'm working with signal processing and wavelet analysis but I have got problems with energy conservation using the function cwt in Matlab 2015. See one example in small code below. The outputs are differents, but Parseval's Theorem implies that is wrong!
CWT in Matlab don't conservate the energy in Parseval sense?
clear all;
t = linspace(0, 6*pi, 4096);
f = sin(t);
W = cwt(f, 1:1:500, 'mexh');
escalogram = abs(W).^2;
energy_domaintime = sum(abs(f).^2)
energy_domainfrequency = sum(sum(escalogram))
Thank's. Luciano
Quick guess: I reckon there are two problems that you might be facing here jointly:
1) There is a problem of normalization here since MATLAB's frequency domain operators are not orthonormalized. You can try confirming using fft command in MATLAB and it won't work. It will only work after manual orthonormalization.
2) Parseval's Theorem holds not EXACTLY similarly in DFT and DWT/ CWT. I remember that its implementation might be little different as well as conditional to tight wavelet frames (Check out any Wavelet book). Mexican hat is nearly tight and would give some error here.
I would recommend confirming both the above guesses to move forward.
Dear Luciano
As far as I know parseval theorem, it s applicable only when we consider the Fourier atoms, you can take a look at a wavelet tour of signal processing, the author makes all the study from fourier to wavelet (STFT is a particular case of wavelet).
Also I believe the parseval's theorem requires a division or multiplication by 2*pi.
There is also a way to transform from time-scale to time-frequency, I do not remember all the details but after transforming, parseval's should hold.
Regards
The conservation of energy property does not takes the same form for all atomic decompositions. There are specific equations and the functions under analysis and the wavelet used should satisfy some conditions for the equality to hold.
For the continuous wavelet transform, the specific equation can be found in the book "A wavelet tour of signal processing", in the second edition that I have you can found it in equation 4.38, page 82.
But even if you apply the right equation you will find another problem. That equation requires integration, and knowledge of al times from -infinite to infinite and ALL SCALES with continuity from 0 to + infinity. As you are working in matlab in a digital computer, you don't have calculated te CWT bu a subsampled version of it (sometimes it is called semicontinuous), time is discretized at sampling frequency, and SCALE is evaluated according to the options you entered, but NOT with continuity. Instead of that a finite set of scales was evaluated. Thus you cannot expect that a discretized version of the integral formula will produce exact results.
- Badges
- Science topic