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

4.2.5 Wavelet Packets
In the approach described before, only the successive signals x j (n) are decomposed, but
in many cases, it is also interesting to decompose the details r j (n). If we decompose
both x j (n) and r j (n), then the original signal can be represented in dierent ways as
combinations of x j (n) and r j (n) ofdierent levels j.
In this work, a decomposition called trigonometric wavelet packets was used (Serrano,
1996). The main idea is to decompose the components r j (n) in portions.
We dene any portion or local signal as
r (ml)
j ( n ) =
l+2 m ;1
X
k=l
C j (k) jk ( n ) (36)
where the parameters m and l are chosen for r (ml)
j (n) to cover the full time interval
2 ;j l n 2 ;j (l +2 m ), which is a relative long interval of length 2 m;j . Note that we
dened the local wavelet packet with 2 m basic functions jk (n) fork = l l+2 m ; 1.
Now, we dene the set of fundamental frequencies
! mh = +2h=2 m (37)
with 0 h 2 m;1 and associated Fourier matrix M (m) given by
M (m)
dk
= 2 ;m=2 8><
>:
sin[ (k +1=2) ]
if d = 1
2 1=2 cos[ ! mh (k +1=2) ] if d is even
2 1=2 sin[ ! mh (k +1=2) ] if d is odd
cos[ 2(k +1=2) ] if d = 2 m
with 1 d 2 m , 0 k < 2 m and h = [d=2], where [ ] denotes the integer part. It
can be demonstrated that M (m) is a 2 m 2 m dimensional orthogonal matrix (Serrano,
1996).
Then, we can dene the new set of elemental functions in order to expand r (ml)
j (n)
as a 2 m dimensional vector obtained from
for 1 d 2 m .
(ml)
jd
( n ) =
l+2 m ;1
X
k=l
(38)
M (m)
dk jk ( n ) (39)
Clearly, these functions constitute a new local orthonormal basis covering the interval
under analysis 2 ;j l n 2 ;j (l +2 m ). Therefore we can give a second description of
the local signal as
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