Spectral Analysis and Time Series - max planck research school ...
Spectral Analysis and Time Series - max planck research school ...
Spectral Analysis and Time Series - max planck research school ...
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A. Lagg – <strong>Spectral</strong> <strong>Analysis</strong><br />
Wavelets: Mathematical Approach<br />
WLtransform:<br />
CWT x = x , s = 1<br />
∣s∣ ∫ t<br />
*<br />
t−<br />
<br />
xt<br />
s<br />
dt<br />
Mexican Hat wavelet:<br />
t =<br />
−t 2<br />
1<br />
s e <br />
2 2<br />
3<br />
t2<br />
2 −1 <br />
Morlet wavelet:<br />
t = e i a t e − t2<br />
2<br />
inverse WLtransform:<br />
xt =<br />
1 2 ∫∫ x<br />
, s 1<br />
c s s t−<br />
2 s<br />
d ds<br />
admissibility<br />
condition:<br />
c <br />
={2∫<br />
−∞∞<br />
∣∣ 2<br />
∣∣<br />
1/2<br />
d } ∞ with<br />
FT<br />
⇔<br />
t
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