Digital Signal Processing Reference
In-Depth Information
Frequency-Domain Representation of Continuous-Time Signals.
The Fourier transformof a continuous-time signal
x
(
t
)
and its inversion for-
mula are defined as
(3)
l
g
r
,
y
i
d
.
,
©
,
L
s
∞
e
−j
Ω
t
dt
X
(
j
Ω)=
x
(
t
)
(9.4)
−∞
∞
1
2
e
j
Ω
t
d
x
(
t
)=
X
(
j
Ω)
Ω
(9.5)
π
−∞
The convergence of the above integrals is assured for functions which sat-
isfy the so-called Dirichlet conditions. In particular, the FT is always well
defined for square integrable (finite energy), continuous-time signals. The
Fourier transform in continuous time is a linear operator; for a list of its
properties, which mirror those that we saw for the DTFT, we refer to the bib-
liography. It suffices here to recall the conservation of energy, also known as
Parseval's theorem:
∞
∞
x
X
1
2
2
2
(
t
)
dt
=
(
j
Ω)
d
Ω
π
−∞
−∞
The FT representation can be formally extended to signals which are not
square summable by means of the Dirac delta notation as we saw in Sec-
tion 4.4.2. In particular we have
e
j
Ω
0
t
FT
{
}
=
δ
(Ω
−
Ω
)
(9.6)
0
from which the Fourier transforms of sine, cosine, and constant functions
can easily be derived. Please note that, in continuous-time, the FT of a com-
plex sinusoid is
not
a train of impulses but just a single impulse.
The Convolution Theorem.
The convolution theorem for continuous-
time signal exactly mirrors the theorem in Section 5.4.2; it states that if
h
(
t
)=(
f
∗
g
)(
t
)
then the Fourier transforms of the three signals are related
by
H
(
j
Ω)=
F
(
j
Ω)
G
(
j
Ω)
. In particular we can use the convolution theorem
to compute
∞
1
2
F
(
j
Ω)
G
(
j
Ω)
e
j
Ω
t
d
Ω
(
f ∗g
)(
t
)=
(9.7)
π
−∞
(3)
The notation
X
(
j
Ω)
mirrors the specialized notation that we used for the DTFT; in this
case, by writing
X
(
j
Ω)
we indicate that the Fourier transform is just the (two-sided)
x
e
−st
dt
computed on the imaginary axis.
Laplace transform
X
(
s
)=
(
t
)
