Nr.
Zeitfunktion
f
(
t
)
{\displaystyle f(t)\;}
L-Transformierte
F
(
s
)
=
L
[
f
(
t
)
]
{\displaystyle F(s)=L\left[f(t)\right]\;}
z-Transformierte
F
z
(
z
)
=
Z
[
f
(
k
T
A
)
]
{\displaystyle F_{z}(z)=Z\left[f(kT_{A})\right]\;}
mit
f
(
k
T
A
)
=
f
(
t
)
|
t
=
k
T
A
{\displaystyle f(kT_{A})=f(t)\vert _{t=kT_{A}}\;}
1
Dirac-Impuls
δ
(
t
)
{\displaystyle \delta (t)\;}
1
{\displaystyle 1\;}
1
{\displaystyle 1\;}
2
Einheitssprung
σ
(
t
)
{\displaystyle \sigma (t)\;}
1
s
{\displaystyle {\cfrac {1}{s}}}
z
z
−
1
{\displaystyle {\cfrac {z}{z-1}}}
3
t
{\displaystyle t\;}
1
s
2
{\displaystyle {\cfrac {1}{s^{2}}}}
T
A
z
(
z
−
1
)
2
{\displaystyle {\cfrac {T_{A}z}{(z-1)^{2}}}}
4
t
2
{\displaystyle t^{2}\;}
2
s
3
{\displaystyle {\cfrac {2}{s^{3}}}}
T
A
2
z
(
z
+
1
)
(
z
−
1
)
3
{\displaystyle {\cfrac {T_{A}^{2}z(z+1)}{(z-1)^{3}}}}
5
e
−
a
t
{\displaystyle e^{-at}\;}
1
s
+
a
{\displaystyle {\cfrac {1}{s+a}}}
z
(
z
−
c
)
;
c
=
e
−
a
T
A
{\displaystyle {\cfrac {z}{(z-c)}};c=e^{-aT_{A}}}
6
t
e
−
a
t
{\displaystyle te^{-at}\;}
1
s
+
a
2
{\displaystyle {\cfrac {1}{s+a}}^{2}}
c
T
A
z
(
z
−
c
)
2
;
c
=
e
−
a
T
A
{\displaystyle {\cfrac {cT_{A}z}{(z-c)^{2}}};c=e^{-aT_{A}}}
7
t
2
e
−
a
t
{\displaystyle t^{2}e^{-at}\;}
2
s
+
a
3
{\displaystyle {\cfrac {2}{s+a}}^{3}}
c
T
A
2
z
(
z
+
c
)
(
z
−
c
)
3
;
c
=
e
−
a
T
A
{\displaystyle {\cfrac {cT_{A}^{2}z(z+c)}{(z-c)^{3}}};c=e^{-aT_{A}}}
8
1
−
e
−
a
t
{\displaystyle 1-e^{-at}\;}
a
s
(
s
+
a
)
{\displaystyle {\cfrac {a}{s(s+a)}}}
(
1
−
c
)
z
(
z
−
1
)
(
z
−
c
)
;
c
=
e
−
a
T
A
{\displaystyle {\cfrac {(1-c)z}{(z-1)(z-c)}};c=e^{-aT_{A}}}
9
s
i
n
ω
0
t
{\displaystyle sin\omega _{0}t\;}
ω
0
s
2
+
ω
0
2
{\displaystyle {\cfrac {\omega _{0}}{s^{2}+\omega _{0}^{2}}}}
z
s
i
n
ω
0
T
A
z
2
−
2
z
c
o
s
ω
0
T
A
+
1
{\displaystyle {\cfrac {zsin\omega _{0}T_{A}}{z^{2}-2zcos\omega _{0}T_{A}+1}}}
10
c
o
s
ω
0
t
{\displaystyle cos\omega _{0}t\;}
s
s
2
+
ω
0
2
{\displaystyle {\cfrac {s}{s^{2}+\omega _{0}^{2}}}}
z
2
−
z
c
o
s
ω
0
T
A
z
2
−
2
z
c
o
s
ω
0
T
A
+
1
{\displaystyle {\cfrac {z^{2}-zcos\omega _{0}T_{A}}{z^{2}-2zcos\omega _{0}T_{A}+1}}}
11
1
−
(
1
+
a
t
)
e
−
a
t
{\displaystyle 1-(1+at)e^{-at}\;}
a
2
s
(
s
+
a
)
2
{\displaystyle {\cfrac {a^{2}}{s(s+a)^{2}}}}
z
z
−
1
−
z
z
−
c
−
a
c
T
A
z
(
z
−
c
)
2
;
c
=
e
−
a
T
A
{\displaystyle {\cfrac {z}{z-1}}-{\cfrac {z}{z-c}}-a{\cfrac {cT_{A}z}{(z-c)^{2}}};c=e^{-aT_{A}}}
12
1
−
b
e
−
a
t
−
a
e
−
a
t
a
−
b
{\displaystyle 1-{\cfrac {be^{-at}-ae^{-at}}{a-b}}}
a
b
s
(
s
+
a
)
(
s
+
b
)
{\displaystyle {\cfrac {ab}{s(s+a)(s+b)}}}
z
z
−
1
+
b
z
(
a
−
b
)
(
z
−
c
)
−
a
z
(
a
−
b
)
(
z
−
d
)
;
c
=
e
−
a
T
A
;
d
=
e
−
b
T
A
{\displaystyle {\cfrac {z}{z-1}}+{\cfrac {bz}{(a-b)(z-c)}}-{\cfrac {az}{(a-b)(z-d)}};c=e^{-aT_{A}};d=e^{-bT_{A}}}
13
e
−
a
t
s
i
n
ω
0
t
{\displaystyle e^{-at}sin\omega _{0}t\;}
ω
0
s
+
a
)
2
+
ω
0
2
{\displaystyle {\cfrac {\omega _{0}}{s+a)^{2}+\omega _{0}^{2}}}}
c
z
s
i
n
ω
0
T
A
z
2
−
2
c
z
c
o
s
ω
0
T
A
+
c
2
;
c
=
e
−
a
T
A
{\displaystyle {\cfrac {czsin\omega _{0}T_{A}}{z^{2}-2czcos\omega _{0}T_{A}+c^{2}}};c=e^{-aT_{A}}}
14
e
−
a
t
c
o
s
ω
0
t
{\displaystyle e^{-at}cos\omega _{0}t\;}
s
+
a
s
+
a
)
2
+
ω
0
2
{\displaystyle {\cfrac {s+a}{s+a)^{2}+\omega _{0}^{2}}}}
z
2
−
c
z
c
o
s
ω
0
T
A
z
2
−
2
c
z
c
o
s
ω
0
T
A
+
c
2
;
c
=
e
−
a
T
A
{\displaystyle {\cfrac {z^{2}-czcos\omega _{0}T_{A}}{z^{2}-2czcos\omega _{0}T_{A}+c^{2}}};c=e^{-aT_{A}}}
15
a
t
T
A
{\displaystyle a^{\cfrac {t}{T_{A}}}\;}
1
s
−
(
1
T
A
)
l
n
a
{\displaystyle {\cfrac {1}{s-\left({\cfrac {1}{T_{A}}}\right)lna}}}
z
z
−
a
{\displaystyle {\cfrac {z}{z-a}}}
16
δ
(
t
−
T
A
)
{\displaystyle \delta (t-T_{A})\;}
e
−
T
A
s
{\displaystyle e^{-T_{A}s}\;}
z
−
1
{\displaystyle z^{-1}\;}
17
δ
(
t
−
k
T
A
)
{\displaystyle \delta (t-kT_{A})\;}
e
−
k
T
A
s
{\displaystyle e^{-kT_{A}s}\;}
z
−
k
{\displaystyle z^{-k}\;}
![{\displaystyle F(s)=L\left[f(t)\right]\;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a550b4d74de0c2a7e80dc0c00127019a562b611)
![{\displaystyle F_{z}(z)=Z\left[f(kT_{A})\right]\;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e1a07b0b4915ff63c9a1438aae87f86b3e64c9a)
