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

A
Time-frequency resolution and the Uncertainty
Principle
In this section after the introduction of some basic concepts from signal analysis, I will
show the proof of the Uncertainty Principle. Then, I will describe its application to the
Fourier, Gabor and Wavelet Transform, stressing the advantages of each method related
with time-frequency localization properties.
A.1 Preliminary concepts
Lets consider a normalized signal x(t) (i.e. x2 (t)dt = 1) with a corresponding
Fourier Transform X(!) 9 . The energy density can be written as jx(t)j 2 in the time
domain or as jX(!)j 2 in the frequency domain. Since the total energy or intensity should
be the same in both domains, we have the following relation (Parceval's theorem)
I =
Z 1
;1
jx(t)j 2 dt = 1
2
R 1
;1
Z 1
;1
jX(!)j 2 d! (56)
Considering the energy densities per time and frequency, the average time can be
dened as:
and the average frequency as:
=
Z 1
;1
t jx(t)j 2 dt (57)
=
Z 1
;1
! jX(!)j 2 d! (58)
Furthermore, the mean frequency can be calculated without the previous computation
of the Fourier spectrum, just by using the following relation (see demonstration
in Cohen, 1995 pp:11)
=
Z 1
;1
! jX(!)j 2 d! =
Z 1
;1
x (t) 1 i
d
x(t) dt (59)
dt
were denotes complex conjugation. From the energy densities we can also dene the
second order moments as:
=
Z 1
;1
t 2 jx(t)j 2 dt (60)
9 for simplicity I will use in this section the angular frequency ! =2f
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