MATLAB: Cwt.m normalization

cwtnormalization

Hello, in cwt.m in section L1 Norm from cwt.m is the mention from: "To preserve the energy of the original signal, you must multiply the CWT by 1/sqrt(s)"
should I normalize the scalogram or is it already done from the cwt.m?
Thx

Best Answer

Hi Simon, that energy normalization should be interpreted here in the correct way. With the CWT, we don't preserve the energy in either case with the L1 or L2 normalization. That energy preservation is only in the integral form of the CWT which is not implemented numerically. The same is true of the spectrogram in the Signal Processing Toolbox. If you look at the integral forms for the CWT with the L2 normalization, then the energy is preserved. However, when you implement the CWT numerically, that is not the case. We will make that clear in the documentation.
Now in the case of the DWT with very specific conditions, i.e. when we implement the classic DWT with a power of two input and the signal length some power of two. You will see the energy preserved. For example:
dwtmode('per')
x = randn(1024,1);
norm(x,2)^2
[C,L] = wavedec(x,10,'sym4');
norm(C,2)^2
But that won't happen with the CWT (by design) and it has nothing to do with the L2 vs L1 normalization. In fact if you look at the legacy CWT, we didn't preserve signal energy there either even though the wavelets were normalized by 1/\sqrt{s}.
If you want a redundant wavelet or wavelet packet transform that does preserve the energy, then MODWT and MODWPT will do that. They are what are referred to as "tight wavelet (and wavelet packet) frames".
Again, the reason for the L1 normalization in the CWT was so that if you have equal amplitude oscillatory components in your data at different scales, they should have equal magnitude in the CWT and NOT be multiplied by a scale factor.
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