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

j (n) = X k
C j (k) jk (n) (35)
is the detail signal at scale j.
Figure 14 shows as an example the multiresolution decomposition method applied
to an event-related potential and the corresponding reconstruction achieved by using
the inverse transform. The method starts by correlating the signal with shifted versions
(i.e. thus giving the time evolution) of a contracted wavelet function, the coecients
obtained thus giving the detail of the high frequency components. The remaining part
will be a coarser version of the original signal that can be obtained by correlating the
signal with the scaling function, which is orthogonal to the wavelet function. Then, the
wavelet function is dilated and from the coarser signal the procedure is repeated, thus
giving a decomposition of the signal in dierent scale levels, or what it is analog, in
dierent frequency bands. This method gives a decomposition of the signal that can be
implemented with very ecient algorithms due to the fact that it can be achieved just
by applying simple lters, as showed by Mallat (1989, see table 1 for an example). The
lower levels give the details corresponding to the high frequency components and the
higher levels the ones corresponding the the low frequencies. As pointed out in sec. x4.1,
the levels related with higher frequencies have more coecients that the lower ones, due
to the varying window size of the Wavelet Transform.
4.2.4 B-Splines wavelets
An important point to be discussed is howtochoose the mother functions to be compared
with the signal. In principle, the wavelet function should have a certain shape that we
would like to localize in the original signal. However, due to mathematical restrictions,
not every function can be used as a wavelet. Then, one criterion for choosing the wavelet
function is that \it looks similar" to the patterns of the original signal. In this respect,
B-Spline functions seem suitable for decomposing ERPs (see bottom part of gure 13).
B-Splines are piecewise polynomial functions of a certain degree that form a base in
L 2 (R) (see e.g. Chui, 1992). Filter coecients corresponding to quadratic B-Splines are
shown in table 1. We can remark the following properties that makes them very suitable
for the analysis of ERP (Unser et al., 1992, 1993 Unser, 1997 Chui, 1992 Strang and
Nguyen, 1996):
1. Smoothness: the smooth behavior of B-Splines is very important in order to avoid
border eects when making the correlation between the original signal and a
wavelet function with abrupt patterns.
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