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

where k g D k= R 1
;1 jg D(t 0 )j 2 dt 0 . Eq. 18 is the inverse Gabor Transform and implies that
the original signal can be completely reconstructed from the coecients G D (f t).
Gabor Transform is highly redundant because it gives a time-frequency map from
every time value of the original signal. In order to decrease redundancy, asampled Gabor
Transform can be dened by taking discrete values of time and frequency, i.e.:
G D (f t) ! G D (mF nT ) (19)
where F and T represent the frequency and time sampling steps. Small F steps are
obtained by using large windows, and small T steps are obtained by using high overlapping
between successive windows. Depending on the resolution required, a proper
choice of F and T will decrease the redundancy and save computational time, but the
price to pay is that the reconstruction will be no longer straightforward as with eq. 18
(Qian and Chen, 1996).
In the following, for convenience I will keep the notation of the continuous Gabor
Transform dened in eq. 14. The spectrum can be dened as
I(f t) =jG D (f t)j 2 = G D(f t) G D (f t) (20)
and a time-frequency representation of the signal can be obtained by using a recursive
algorithm that slides the time window and plots the energy (I) as a function of the
frequency and time. These plots, called spectrograms (see g. 8), give an elegant visual
description of the time evolution of the dierent frequencies, but it is dicult to extract
from them any quantitative measure.
In order to quantify this information, I will dene the band power spectral intensity
for each one of the EEG traditional frequency bands i (i = :::) as:
I (i) (t) =
Z f
(i) max
f (i) min
I(f t) df i = ::: (21)
where ( f (i) min f (i) max ) are the frequency limits for the band i. The division and grouping
of the spectrum in frequency bands is not arbitrary, since as showed in section x1.1.1
and section x2.3.3, EEG bands are correlated with dierent sources and functions of the
brain.
Obviously the total power spectral intensity will be
I T (t) = X i
I (i) (t) i = ::: (22)
and we can dene the band relative intensity ratio (RIR) foreach frequency band i as
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