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

4 Wavelet Transform
4.1 Introduction
Fourier Transform consists in making a correlation between the signal to be analyzed
and complex sinusoids of dierent frequencies (see upper part of g. 13). As stated in
chapter x2, Fourier Transform gives no information about time and requires stationarity
of the signal. By \windowing" the complex sinusoidal mother functions of the Fourier
Transform, a time evolution of the frequencies can be obtained just sliding the windows
throughout the signal. This procedure, called the Gabor Transform, consists in correlating
the original signal with modulated complex sinusoids as showed in the middle part of
g. 13 (for details see section x3.2). Gabor Transform gives an optimal time-frequency
representation, but one critical limitation appears when windowing the data due to the
Uncertainty Principle (see x3.2 and xA Cohen, 1995 Strang and Nguyen, 1996 Chui,
1992). If the window is too narrow, the frequency resolution will be poor, and if the
window is too wide, the time localization will be not so precise. Data involving slow
processes will require wide windows and on the other hand, for data with fast transients
(high frequency components) a narrow window will be more suitable. Then, owing to
its xed window size, Gabor Transform is not suitable for analyzing signals involving
dierent range of frequencies.
Grossmann and Morlet (1984) introduced the Wavelet Transform in order to overcome
this problem. The main advantage of wavelets is that they have avarying window
size, being wide for slow frequencies and narrow for the fast ones (see lower part of
g. 13), thus leading to an optimal time-frequency resolution in all the frequency ranges
(Chui, 1992 Strang and Nguyen, 1996 Mallat, 1989). Furthermore, owing to the fact
that windows are adapted to the transients of each scale, wavelets lack of the requirement
of stationarity.
This chapter starts with a brief theoretical background of the Wavelet Transform.
Details of a straightforward implementation called the multiresolution analysis and an
alternative decomposition of time signals called the Wavelet Packets analysis will be
given. In section x4.4, I will show the results obtained by applying trigonometric wavelet
packets (Serrano, 1996) to the study of tonic-clonic seizures (Blanco et al., 1998a). In
sections x4.5 and x4.6, I will analyze two dierent types of event related potentials. By
using the multiresolution decomposition method, in the rst case I will study the alpha
band responses (Quian Quiroga and Schurmann, 1998, 1999) and in the second case the
ones corresponding to the gamma band (Sakowicz et al., 1999). Finally, I will discuss
advantages of Wavelets in the study of brain signals.
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