Biomedical Engineering Reference
In-Depth Information
1
2
∞
(
{}
()
() ( )
−
1
∫
FX
ω
=
xt
=
X
ω
exp
j
ω
t d
ω
(3.2)
π
−∞
Because most EEG signal processing is carried out using computers, the signal
ω
1
S
x
(
t
) is sampled with a sampling frequency of
f
==
, where
T
s
is the sam-
S
2
π
T
S
pling time interval. The sampling process generates the sequence
x
(
n
) where
n
denotes the discrete sample time. The discrete time Fourier transform (DTFT) of a
discrete-time signal
x
(
n
) is defined as
∞
∑
(
{}
()
() ( )
j
ω
DTFT xn
=
Xe
=
xn
exp
−
j n
ω
(3.3)
n
=−∞
where
DTFT
{
x
(
n
)} is a continuous and periodic function of
ω
with period 2
π
.
is sampled on the unit circle, then we have the discrete Fourier transform of
an
N
-length sequence
x
(
n
):
If
ω
N
−
∑
1
2
π
⎛
⎜
⎞
⎟
(
{}
()
()
DFT
x n
=
X k
=
x n
exp
−
j
kn
(3.4)
N
n
=
0
2
π
where
ω
=
k
,
n
=
0, 1, ...,
N
−
1, and
k
=
0, 1, ...,
N
−
1.
N
is the number of spec-
k
N
tral samples in one period of the spectrum
X
(
e
j
ω
). Increasing the sequence length
N
will improve the frequency resolution of the spectrum by decreasing the discrete fre-
quency spacing of the spectrum. The inverse DFT, which transforms a signal from
the discrete frequency domain into the discrete time domain, is
1
N
−
∑
1
2
π
⎛
⎜
⎞
⎟
(
{}
()
()
IDFT
Xk
=
xn
=
Xk
exp
j
N
kn
(3.5)
N
k
=
0
where
n
= 0, 1, …,
N
- 1 and
k
= 0, 1, …,
N
- 1.
The fast Fourier transform (FFT) algorithm is used to compute the discrete Fou-
rier transform [38, 40]. The FFT algorithm utilizes some properties of the discrete
Fourier transform to perform fast calculations of the transform. The FFT reduces
the number of computations from
N
2
to
N
log(
N
). MATLAB provides several FFT
functions for computing spectra.
Y = fft(x)
for example, returns the complex discrete Fourier transform
Y
of a discrete time
vector
x
, computed with the FFT algorithm [41]. The magnitude and phase of the
spectrum are computed using
MY = abs(Y)
and
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