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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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PostPosted: Thu Feb 19, 1998 9:01 am    
Subject: Answer: Modulus and phase of Morlet wavelet using FFT (WD 7.2 #19)
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#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 1-dimension set and cannot get modulus and phaze
>differently. May be it is impossible with completed FFT-IFFT
>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 anti-FFT. 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 anti-FFT 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
All times are GMT + 1 Hour
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