Digital Signal Processing Reference
In-Depth Information
Formal aspects
Comparison of
FOURIER - transformation and LAPLACE
LAPLACE- transformation
-transformation
∞
∞
(
)
∞
−+
σω
jt
−
st
LT
³
³
x t
()
⎯⎯→=
X s
( )
x t e
()
dt
=
x t e
()
dt
−
jt
ω
³
x t
()
⎯⎯→=
FT
X
( )
ω
x t e
()
dt
0
0
−∞
σ
+∞
1
0
∞
1
st
ILT
³
Xs
()
⎯⎯→
xt
()
=
Xse ds
()
jt
ω
IFT
³
X
()
ω
⎯⎯→
x t
()
=
X
()
ω
e
d
ω
2
π
j
2
π
σ
−∞
0
−∞
•
A function
f(t)
is called
causal
, if for all
t < 0 f(t) = 0
•
s =
σ
+ j
ω
is a complex variable
The LAPLACE - integral converges if the function
x
σ
(t) = x(t)e
-st
is absolutely integrable
•
•
The variable
s =
σ
+ j
ω
has the dimension of an angular frequency , that is sec
-1
e
-st
•
has no dimension
•
The dimension of the
LT
results from the dimension of
x(t)
and the dimension of the differential
dt.
If x(t)
equals a voltage
u(t),
the dimension of the
LT
is
Vs
,with a current
i(t) As.
Examples fot the LT
(1)
1 für
tt
tt
<
>
j
ω
-
°
0
Step function
sf t
( )
=
®
°
¯
0 für
0
∞
−
st
∞
ª º
e
1
{}
−
st
³
LT
sf t
()
=
e
dt
=
=
σ
« »
−
s
s
¬ ¼
0
0
ω
Convergence
domain
−
st
−
σ
t
−
jt
Convergence criterion: lim
e
=
lim
e
e
=
0
t
→∞
t
→∞
()
s
real part of
=>
σ
0
1
1,0
0,9
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0,0
sft
()
LT
Correspondence
sf t
( )
⎯⎯→
s
a
e ft
−
( ) with
a
=
1, 2
(2)
0
50
150
250
350
450
550
650
750
850
950
Y/t-Grafik 0
ms
−
at
Exponentially decaying function ( )
xt
=
e
sf t
( )
j
ω
∞
∞
{}
0
−
at
−
st
−
at
−+
(
ast
)
³
³
LT
x t
()
=
e
sf t e
()
dt
=
e
sf t e
()
dt
=
0
∞
−
st
ª º
e
sa
1
=
« »
+
-a
σ
sa
+
¬ ¼
0
−+
(
ast
)
Convergence criterion: lim
e
=
0
Convergence
domain
t
→∞
real part of (
sa a
+>
)
respectively real p
art of ( )
s
>−
a
1
Correspondence
a
e ft
-
( )
⎯⎯→
LT
s
+
a
Illustration 319:
Comparison of structure of FT and LT and two examples of the LT
The step function sf(t) is the standard function of turn- on processes. In both examples, the region of
convergence (ROC) results of the (different) convergence criteria. In the second example, the region of
convergence is larger, because e
-at
has done some “ground work” for the limit 0.
In almost any technical
textbook, there are table summaries of the LAPLACE- transforms for almost any meaningful excitation
function x(t
)
. The derivation of the ILT can be found in the following.
