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Fig. 9.9
Representation of
the data matrix. There are
three dimensions: one for the
participants, one for the time
(46,000 points corresponding
to the number of points in
each 3 min EEG signals
recorded using a sampling
frequency of 256 Hz), and
one for the electrodes
9.4
Data Preprocessing
The data is speci
ed in 3 dimensions (time, electrodes, and participants). The
proposed approach is to extract a feature in 2 dimensions to implement common
classi
cation tools. To do this,
the signal energy, obtained by the wavelet
decomposition, is considered.
9.4.1 Wavelet Decomposition
Wavelet decomposition (Daubechies
1992
; Mallat
2008
) is a tool widely used in
signal processing. Its main advantage is that it can be used to analyze the evolution
of the frequency content of a signal in time. It is therefore more suitable than the
Fourier transform for analyzing non-stationary signals.
A wavelet is a function
such that
R
w
2
L
2
ð
R
Þ
R
w
ð
t
Þ
dt
¼
0
. The continuous
wavelet transform of a signal
X
can be written as:
Z
1
dt
X
ð
a
;
b
Þ¼
1
a
t
b
a
p
X
ð
t
Þ
w
ð
9
:
1
Þ
1
where
a
is called the scale factor that represents the inverse of the signal frequency,
b
is a time-translation term and function
is called the mother wavelet. The mother
wavelet is usually a continuous and differentiable function with compact support.
Several families of wavelet mother exist such as Daubechies wavelets or Coi
ψ
ets.
Some wavelets are given in Fig.
9.10
.
It is also possible to de
ne the discrete wavelet transform, starting from the
previous formula and discretizing parameters
a
and
b
. Then, let
a
=
a
j
, where
a
0
is
the resolution parameter such as
a
0
> 1 and
and let
b
=
kb
0
a
j
, where
j
2
N
k
2
N
and
b
0
> 0. It is very common to consider the
“
dyadic
”
wavelet transform which
