Digital Signal Processing Reference
In-Depth Information
Section 6.2. Section 6.3 computes the time-domain representation of a Laplace-
transformed signal, while Section 6.4 considers the properties of the Laplace
transform. Sections 6.5 to 6.9 propose several applications of the Laplace trans-
form, ranging from solving differential equations (Section 6.5), evaluating the
location of poles and zeros (Section 6.6), determining the causality and stability
of LTIC systems from their Laplace transfer functions (Sections 6.7 and 6.8),
and analyzing the outputs of LTIC systems (Section 6.9). Section 6.10 presents
the cascaded, parallel, and feedback configurations for interconnecting LTI
systems, and Section 6.11 concludes the chapter.
6.1 Analytical deve
lopment
CTFT
←−−→
X
(j
ω
), was defined as follows:
In Section 5.1, the CTFT pair,
x
(
t
)
∞
1
2
π
X
(j
ω
)e
j
ω
t
d
ω
;
CTFT synthesis equation
x
(
t
)
=
(6.1)
−∞
∞
−
j
ω
t
d
t
.
CTFT analysis equation
X
(j
ω
)
=
x
(
t
)e
(6.2)
−∞
In Eqs. (6.1) and (6.2), the CTFT of
x
(
t
) is expressed as
X
(j
ω
), instead of
the earlier notation
X
(
ω
), to emphasize that the CTFT is computed on the
imaginary j
ω
-axis in the complex s-plane. For a CT signal
x
(
t
), the expression
for the bilateral Laplace transform is derived by considering the CTFT of the
modified version,
x
(
t
)e
−σ
t
, of the signal. Based on Eq. (6.2), the CTFT of the
−σ
t
modified signal
x
(
t
)e
is given by
∞
−σ
t
=
−σ
t
e
−
j
ω
t
d
t
,
ℑ
x
(
t
)e
x
(
t
)e
(6.3)
−∞
which reduces to
∞
−σ
t
=
−
(
σ +
j
ω
)
t
d
t
ℑ
x
(
t
)e
x
(
t
)e
−∞
=
X
(
σ
+
j
ω
)
.
(6.4)
Substituting
s
= σ +
j
ω
in Eq. (6
.
4) leads to the following definition for the
bilateral Laplace transform:
†
∞
−σ
t
=
−
st
d
t
.
Laplace analysis equation
X
(
s
)
=ℑ
x
(
t
)e
x
(
t
)e
(6.5)
−∞
†
The Laplace transform was discovered originally by Leonhard Euler (1707-1783), a prolific
Swiss mathematician and physicist. However, it is named in honor of another mathematician
and astronomer, Pierre-Simon Laplace (1749-1827), who used the transform in his work on
probability theory.
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