Biomedical Engineering Reference
In-Depth Information
Z
∞
e
−
i
2π
ft
df
x
(
t
)=
X
(
f
)
(1.26)
−
∞
Z
∞
e
i
2π
ft
dt
X
(
f
)=
x
(
t
)
(1.27)
−
∞
1.4.3 Finite discrete signal
In practice we deal with discrete signals of finite duration. The Fourier transform
that operates on this kind of signal is called discrete Fourier transform (DFT) and the
algorithms that implement it are FFT (fast Fourier transform).
The DFT formula can be derived from (1.23). The signal to be transformed is
N
samples long
x
and the samples are taken every
T
s
seconds. It is assumed that the finite signal
x
is just one period of the infinite periodic
sequence with period
T
=
{
x
[
0
]
,...,
x
[
n
]
,...
x
[
N
−
1
]
}
=
N
·
T
s
. The process of sampling can be written as
x
[
n
]=
x
(
nT
s
)=
x
(
t
)
δ
(
t
−
nT
s
)
. Substituting this into (1.23) gives:
Z
T
N
1
n
=
0
x
[
n
]
e
i
2π
knT
s
−
N
1
n
=
0
x
[
n
]
e
i
2
N
−
1
T
1
NT
s
1
N
e
i
2π
t
k
dt
kn
X
[
k
]=
x
(
t
)
δ
(
t
−
nT
s
)
=
T
s
=
T
T
0
(1.28)
From the above formula it follows that
k
in the range
k
1 produces
different components in the sum. From the Euler formulas (1.21) it follows that for
a real signal a pair of conjunct complex exponentials is needed to represent one
frequency. Thus for real signals there are only
N
=
0
,...,
N
−
/
2 distinct frequency components.
The inverse discrete Fourier transform (IDFT) is given by
N
1
k
=
0
X
[
k
]
e
−
i
2
N
−
kn
x
[
n
]=
n
=
0
,...,
N
−
1
.
(1.29)
Note, that the signs of the exponents and the normalization factors by which the
DFT and IDFT are multiplied (here 1
N
and 1) are conventions, and may be written
differently by other authors. The only requirements of these conventions are that the
DFT and IDFT have opposite sign exponents and that the product of their normal-
ization factors be 1
/
/
N
.
1.4.4 Basic properties of Fourier transform
Given signals
x
(
t
)
,
y
(
t
)
,and
z
(
t
)
we denote their Fourier transforms by
X
(
f
)
,
Y
(
f
)
,
and
Z
, respectively. The Fourier transform has the following basic properties:
[Pinsky, 2002].
(
f
)
Linearity:
For any complex numbers
a
and
b
:
z
(
t
)=
ax
(
t
)+
by
(
t
)
⇒
Z
(
f
)=
a
·
X
(
f
)+
b
·
Y
(
f
)
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