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

Z 1
;1
tx(t) d dt x(t) dt = 1 2
Z 1
;1
t d dt x2 (t) dt = tx2 (t)
2
+1
;1
; 1 2
Z 1
;1
x 2 (t) dt = ; 1 2
replacing in eq. 68 and then in eq. 67, the uncertainty principle is proved.
A.3 Time-frequency resolution of the Fourier, Gabor and Wavelet
Transform
Fourier Transform is based in making a correlation between the original signal and
complex sinusoidal functions (see eq. 3). Since the Fourier Transform of these functions
is:
x(t) =e i! 0t
! X(!) =2 (! ; ! 0 ) (69)
then, the sinusoidal \mother functions" of the Fourier Transform are perfectly localized
in the frequency domain. However, they are not localized in time.
As stated in sec. x3.2, Gabor Transform consist in making a correlation between
the signal and amplitude modulated complex sinusoids, by using a windowing function.
Gabor (1946) suggested the use of a Gaussian function as window due to its simultaneous
localization in time and frequency domains. In fact, the Fourier Transform of a Gaussian
function is also a Gaussian:
r
x(t) =g (t) =e ;t2 ! X(!) =
e
; !2
4
(70)
Moreover, for a Gaussian function the inequality 68isanequality and ! t = 1 2 ,thus
having Gaussian functions the best time-frequency resolution allowed by the uncertainty
principle. In the case of the mother functions of the Gabor Transform, i.e. g e ;i!t
where g is the Gaussian windowing function, the equality in eq. 68 also holds, thus
having an optimal time-frequency resolution.
The frequency resolution (bandwidth) of a normalized Gaussian function can be
calculated by using eq. 63:
2 ! = 2
(71)
and in the case of the functions of the Gabor Transform, g e ;i!t , by applying eq. 59
and eq. 63 we obtain for one of these mother functions (! = ! 0 ) the same bandwidth
but localized in the frequency ! 0 , i.e.:
113