Biomedical Engineering Reference
In-Depth Information
The Fourier transform is a generalization of the Fourier series. It breaks up any
time-varying signal into its frequency components of varying magnitude and is
defined in (6.1).
∞
()
()
∫
−
2 π
ikx
Fk
=
fte
dx
(6.1)
−∞
Due to Euler's formula, this can also be written as shown in (6.2) for any com-
plex function
f
(
t
) where
k
is the
k
th harmonic frequency:
∞
∞
()
() (
)
() (
)
F k
=
∫
f t
cos
−
2
π
kx dx
+
∫
f t i
sin
−
2
π
kx dx
(6.2)
−∞
−∞
We can represent any time-varying signal as a summation of sine and cosine
waves of varying magnitudes and frequencies [17]. The Fourier transform is repre-
sented with the power spectrum. The power spectrum has a value for each harmonic
frequency, which indicates how strong that frequency is in the given signal. The
magnitude of this value is calculated by taking the modulus of the complex number
that is calculated from the Fourier transform for a given frequency (
|F
(
k
)
|
).
Stationarity is an issue that needs to be considered when using the Fourier trans-
form. A stationary signal is one that is constant in its statistical parameters over
time, and is assumed by the Fourier transform to be present. A signal that is made up
of different frequencies at different times will yield the same transform as a signal
that is made up of those same frequencies for the entire time period considered. As
an example, consider two functions
f
1
and
f
2
over the domain 0
≤
t
≤
T
, for any two
frequencies
ω
1
and
ω
2
shown in (6.3) and (6.4):
()
(
)
(
)
ft
=
sin
2
πω
t
+
cos
2
πω
t
if
0
≤ <
t T
(6.3)
1
1
2
and
(
)
⎨
⎩
sin
cos
2
πω
πω
t
if
if
0
≤<
≤<
t
T
2
()
1
ft
=
(6.4)
(
)
2
2
t
T t
2
T
2
When using the short-term Fourier transform, the assumption is made that the
signal is stationary for some small period of time,
T
s
. The Fourier transform is then
calculated for segments of the signal of length
T
s
. The short-term Fourier transform
at time
t
gives the Fourier transform calculated over the segment of the signal lasting
from (
t T
s
)to
t
. The length of
T
s
determines the resolution of the analysis. There is
a trade-off between time and frequency resolution. A short
T
s
yields better time res-
olution, but it limits the frequency resolution. The opposite of this is also true; a
long
T
s
increases frequency resolution while decreasing the time resolution of the
output. Wavelet analysis overcomes this limitation, and offers a tool that can main-
tain both time and frequency resolution. An example of Fourier transform calcu-
lated prior to, during, and following an epileptic seizure is given in Figure 6.2.
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