Quantitative analysis of EEG signals: Time-frequency methods and ...

(ml)
j ( n ) =
2 m X
d=1
and the corresponding coecients are easily computed as
where 1 d 2 m .
D (ml)
j ( d ) =
D (ml)
j ( d ) (ml)
jd
( n ) : (40)
l+2 m ;1
X
k=l
M (m)
dk
C j ( k ) (41)
The trigonometric wavelet packets (ml)
jd
(n) have zero mean, oscillate on the interval
2 ;j l n 2 ;j (l +2 m ) and decay with exponential ratio. Moreover, their wave-forms
resemble modulated sines or cosines. In fact, it can be demonstrated that each Fourier
(ml)
transform ^
jd
(!) is centered at the fundamental frequency ! mh , when d = 2h or
(ml)
d =2h +1. Moreover, ^ jd (!) =0on the other fundamental frequencies.
In other words, the coecients fD (ml)
j
spectrum
for the local signal r (ml)
j
of coecients fC j (k)D (ml)
j
(d)g can be considered as the discrete Fourier
(n). Summing up, we can resume in the double set
(d)g the time-scale-frequency information of the local signal
r (ml)
j (n).
Finally, to analyze the complete function r j (n), that is, the details at level j, we
choose some partition in local components r (m il i )
j (n), according the structure of the
signal,
r j ( n ) = X m i
r (m il i )
j ( n ) (42)
where the sequence of index l i veries l i+1 = l i +2 m i
. Then, we implement the above
refereed time-scale-frequency technique for each local signal.
4.3 Short review of wavelets applied to the study of EEG signals
Several works applied the Wavelet Tranform to the study of EEGs and ERPs (see a
review in Unser and Aldroubi, 1996 or in Samar et al., 1995). One rst line of applications
is for pattern recognition in the EEG. This is achieved by correlating dierent
transients of the EEG with wavelet coecients of dierent scales. Schi et al. (1994a)
used amultiresolution decomposition implemented with B-Splines mother functions for
extracting features of EEG seizure recordings. They showed a better performance of
wavelets in comparison with Gabor Transform, and a similar resolution of the multiresolution
decomposition compared with the continuous Wavelet Transform but with a
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