freqanalysis_wltconvol

[Jason Sherfey]

Dear FieldTrip users,

I have been using freqanalysis_wltconvol to compute the wavelet power
spectrum for several months and recently began trying to understand
the code.  Reading past discussions on this listserv has been very
helpful, but I can't figure out why it does the particular
normalization that it does for the wavelet power calculation.  This
calculation can be found on line 299 of the 20081001 release.

According to the comments at the beginning of the function, jansch
"fixed [the] improper normalization of powdum" in Revision 1.17 on
2007/06/14.

It seems to me that "powdum" is the wavelet power spectrum (P) and
that "autspctrmacttap" is the signal's wavelet transform (T).  If that
is true, then the correction made on 2007/06/14 was the following:
P = (2*|T| / fs)^2   became   P = 2*|T|^2 / fs
where fs = sampling frequency

My current understanding (based on pp. 31-32 in Addison's text "The
illustrated wavelet transform handbook") is that the wavelet energy
density is given by E = |T|^2 and that P = E / (time period of the
signal).  Since the time period of the signal = (# of samples) /
(sampling frequency), that would imply P = fs*|T|^2 / (# of samples).

Thus, I thought |T|^2 should be multiplied by fs/(# of samples), but
in the code it's instead multiplied by 2/fs.  I am fairly new to
wavelet analysis and suspect there's a small detail I don't
understand.

Does anyone know why the wavelet energy is normalized by 2/fs instead
of fs/(# samples)?

Thanks in advance,
Jason

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