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

and the band maximum peak frequency (f (i)
M (t)) of the i-band, as the frequency value for
which I(f t) is maximum in the interval ( f (i) min f (i) max )ata time t, i.e.
I(f M t) >I(f t)
8 f 6= f M (f (i)
min f(i) max) (25)
Uncertainty Principle
One critical limitation appears when windowing the data due to the Uncertainty
Principle (Cohen, 1995 Chui, 1992 Qian and Chen, 1996 Kaiser, 1994). If the window
is too narrow, the frequency resolution will be poor, and if the window is to wide, the
time localization will be not so precise. Or in another words, sharp localizations in
time and frequency are mutually exclusive because a frequency can not be calculated
instantaneously. If we denote by t the time uncertainty (time duration) and by f the
uncertainty in the frequencies (frequency bandwidth), in the case of normalized signals
the Uncertainty Principle can be expressed as follows (see appendix xA for a proof):
t f 1
(26)
4
This limitation becomes important when the signal has transient components localized
in time as in the case of EEGs or ERPs. Gabor (1946) suggested a Gaussian
function as the smoothing function, owing to its good localization in time and frequency.
In fact, with a Gaussian function eq. 64 is an equality and furthermore, the equality also
holds for the mother function of the Gabor Transform, g D e i2ft (see appendix xA).
3.3 Application to intracranially recorded tonic-clonic seizures
3.3.1 Methods and Materials
Seizure EEG recordings were obtained from a 21 years old male patient. A nine hours
recording was performed with 12 depth electrodes ( each electrode having 5 to 15 contacts
) placed in the epileptogenic zone and propagating brain areas. Each signal was
amplied and ltered using a 1;40Hz band-pass lter. A 4 pole Butterworth lter was
used as low-pass lter and as an anti-aliasing scheme. After 10 bits A/D conversion the
EEG data was written continuously onto a hard drive with a sampling rate of 256Hz
per channel. Selected data sets of ictal and interictal activity were stored for subsequent
o-line analysis. We established the necessary frequency resolution in f = 0:25Hz
in order to have enough frequency values for calculating the mean and maximum band
frequencies and the time resolution was set to t = 0:25sec. This was done by using
a Gaussian window of D = 4sec width 2 with slide displacement steps of 0:25sec. The
2 from eq. 8, f = 1 = 1
= 1 Hz =0:25Hz
N 4256Hz(1=256Hz) 4
25