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> Volume 7, Issue 3
Answer: Modulus and phase of Morlet wavelet using FFT (WD 7.2 #19)

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Posted: Thu Feb 19, 1998 9:01 am Subject: Answer: Modulus and phase of Morlet wavelet using FFT (WD 7.2 #19) 



#17 Answer: Modulus and phase of Morlet wavelet using FFT (WD 7.2 #19)
>Please, can anybody explain: Is it possible to divide modulus and
>phaze while calculating Morlet wavelet with use available FFT and
>IFFT procedures (for instance in MATHCAD)? I can calculate modulus
>and phaze throuh direct integrating but when I use these procedures I
>get a single 1dimension set and cannot get modulus and phaze
>differently. May be it is impossible with completed FFTIFFT
>procedures and it is necessary to use only algoritms?
Hello Anna, I have seen your question on the wavelet digest. I am
presently using the Morlet wavelet to analyse turbulence measurements
coming from a hot plasma, so I can perhaps try to help you. First of
all, I don't know how the FFT is implemented in Mathcad. If well
implemented, it should give a complex result. In general, the
procedure for calculating the wavelet transform of a time series x(t)
through FFT is to compute the FFT of x(t), to compute the FFT of the
wavelet w(t), to multiply them and to compute the antiFFT. Now, the
FFT of x(t), which I'll call X(f), is complex, with the property that
the negative frequency compnents are complex conjugates of the
positive ones, since x(t) is real, i.e. X*(f)=X(f) [I use the * to
indicate the complex conjugation]. The FFT of the Morlet wavelet
w(t), which I'll call W(f), is real, and does not have simmetry
between negative and positive frequencies, i.e. W(f) is not equal to
W(f). The product of the two, P(f)=X(f)W(f), is therefore complex,
and does not possess the same property as the FFT of x, i.e. IT IS NOT
TRUE that P*(f)=P(f). As a consequence, the antiFFT of P(f) is
complex, and you can therefore compute a modulus and a phase. A
warning: are you sure that the phase is so meaningful? I have been
thinking about this. The wavelet transform is localised in time, and
the phase simply gives a fine adjustment of this localisation, by an
amount less than the distance between 2 subsequent points of the
transform. In my applications, this is seldom important, but may be in
your case it is different. I hope to have been helpful. Contact me if
you need further clarifications. Yours.
Emilio
Emilio Martines
Consorzio RFX phone: +39 (49) 8295065
corso Stati Uniti, 4 fax: +39 (49) 8700718
35127 Padova
Italy 





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