Biomedical Engineering Reference
In-Depth Information
representations of the signal are equivalent. That is we can transform without any
loss of information signals from time to frequency representation and vice versa.
The frequency can be expressed in radians per second—in this case we shall de-
note it as ω or in cycles per second—in this case we shall denote it as
f
. Both quan-
tities are related:
f
2πω.
Depending on the signal, different kinds of the Fourier transform are used. They
will be described below.
=
1.4.1 Continuous periodic signal
Let us first consider the simplest case: the signal
x
(
t
)
is periodic with period
T
.
Such a signal can be expressed as a series:
∞
∑
c
n
e
−
i
2π
t
n
x
(
t
)=
(1.22)
T
n
=
−
∞
where:
Z
T
1
T
e
i
2π
t
n
dt
c
n
=
x
(
t
)
(1.23)
T
0
This fact can be easily checked by substitution:
Z
T
e
i
2π
T
k
dt
x
(
t
)
=
0
Z
T
∞
∑
c
n
e
−
i
2π
t
n
e
i
2π
t
k
dt
=
T
T
0
=
−
n
∞
Z
T
∞
∑
c
n
e
−
i
2π
t
(
k
−
n
)
dt
=
T
(1.24)
0
n
=
−
∞
Z
T
Z
T
c
n
e
−
i
2π
t
=
n
=
k
+
n
=
k
(
k
−
n
)
dt
c
n
dt
T
0
0
=
0
=
Tc
n
n
T
We can think of
c
n
in expression (1.23) as the contribution of the frequency
f
n
=
to the signal
x
(
t
)
X
n
T
c
n
=
(1.25)
Hence a periodic signal can be expressed by the linear combination of complex ex-
ponentials with a discrete set of frequencies.
1.4.2 Infinite continuous signal
We can extend the formula (1.23) for aperiodic signals. The trick is that we con-
sider the whole infinite aperiodic signal domain as a single period of an infinite peri-
odic signal. In the limit
T
→
∞ we obtain:
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