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

2 Fourier Transform
2.1 Introduction
Time signals can be represented in many dierent ways depending on the interest in
visualizing certain given characteristics. Among these, the frequency representation is
the most powerful and standard one. The main advantage over the time representation
is that it allows a clear visualization of the periodicities of the signal, in many cases
helping to understand underlying physical phenomena. Frequency analysis, developed
by Jean Baptiste Fourier (1768-1830), reached innumerable applications in mathematics,
physics and natural sciences. Furthermore, the Fourier Transform is computationally
very attractive since it can be calculated by using an extremely ecient algorithm called
the Fast Fourier Transform (FFT Cooley and Tukey, 1965).
This chapter starts with a brief mathematical background of Fourier Transform and
then, applications of Fourier analysis to EEG signals will be reviewed. Four main
applications will be described: analysis of frequency bands, topographical mapping,
analysis of evoked responses and coherence.
2.2 Theoretical background
Under mild conditions, the Fourier Transform describes a signal x(t) (I will consider
only real signals) as a linear superposition of sines and cosines characterized by their
frequency f.
where
x(t) =
X(f) =
Z +1
;1
Z +1
;1
X(f) e i2ft df (1)
x(t) e ;i2ft dt (2)
are complex valued coecients that give the relative contribution of each frequency f.
Equation 2 is the continuous Fourier Transform of the signal x(t). It can be seen as an
inner product of the signal x(t) with the complex sinusoidal mother functions e ;i2ft ,
i.e.
X(f) = (3)
Its inverse transform is given by eq. 1 and since the mother functions e ;i2ft are
orthogonal, the Fourier Transform is nonredundant and unique.
Let us consider in the following that the signal consists of N discrete values, sampled
every time .
13