Digital Signal Processing Reference
In-Depth Information
Because of this, the
LT
enables us to examine the transient response or the reaction
of frequency dependent linear systems to turn-on processes in detail (in Illustration
308, the common oscillating circuit was calculated solely in steady state). This is of
great importance for circuits or systems working with feedback/ negative feedback.
Here, the risk of instability exists, ie unexpectedly strong oscillations may appear
that do not match the planned performance. In practice, the system then moves to
the determined limits and creates non- linear distortions (see Illustration 124).
Note that in measurement technologies, you can reexamine such transients with the aid of
the “trick” using an
FT
, as shown in Illustration 115.
Mathematical interpretation
Any information about the characteristics and advantages of the
FT
is also valid for
the
LT
. You can transfer linear differential equations that are of great importance
for physics and, even more, engineering, into more simple algebraic equations. In
contrast to the
FT
, the
LT
does not provide direct physically interpretable results.
In fact it represents a pure formal schema and is of little interest to metrology.
In some cases though, the
FT
- integral cannot be integrated, that is, it does not have
a finite value. For instance, the step function is not integrable from 0 to
but any
function
x(t)
for a capital t can be led towards zero with the help of the “decay func-
tion” (as indicated above), thus making it integrable.
e
-σ
t
therefore is called the
con-
vergence factor
.
μ,
Switching operations start at
t = 0
sec, therefore the integration area of the
LT
only
leads from 0 to
. Otherwise, for
t < 0
e
-s
t
would become a divergence factor
The convolution theorems are of the same importance for
LT
and
FT
, but we will
not deal with them further at this point.
∝
Deriving the inverse LAPLACE- transformation ILT
The
LT
is defined as
∞
∞
(
)
LT
³
−
st
³
−+
σω
jt
x t
()
⎯⎯→=
X s
( )
x t e
()
dt
=
x t e
()
dt
0
0
Comparison to the accordant
IFT
of
x(t)
shows that the LAPLACE- transformed
X(s)
represents the FOURIER- transformed of the function
x(t)e
-
σ
t
.
If you now multiply these equations by the factor
e
σ
t
, the result is
∞
∞
∞
1
1
1
³
σω
t
j
t
³
(
σω
+
j
)
t
³
t
x t
()
=
X s e
()
e
d
ω
=
X s e
()
d
ω
=
X s e d
()
ω
2
π
2
π
2
π
−∞
−∞
−∞
