Tabelle Zeitfunktion/Laplace-Transformation/z-Transformation

Tabelle

Nr.	Zeitfunktion $f(t)\;$	L-Transformierte $F(s)=L\left[f(t)\right]\;$	z-Transformierte $F_{z}(z)=Z\left[f(kT_{A})\right]\;$ mit$f(kT_{A})=f(t)\vert _{t=kT_{A}}\;$
1	Dirac-Impuls $\delta (t)\;$	$1\;$	$1\;$
2	Einheitssprung $\sigma (t)\;$	${\cfrac {1}{s}}$	${\cfrac {z}{z-1}}$
3	$t\;$	${\cfrac {1}{s^{2}}}$	${\cfrac {T_{A}z}{(z-1)^{2}}}$
4	$t^{2}\;$	${\cfrac {2}{s^{3}}}$	${\cfrac {T_{A}^{2}z(z+1)}{(z-1)^{3}}}$
5	$e^{-at}\;$	${\cfrac {1}{s+a}}$	${\cfrac {z}{(z-c)}};c=e^{-aT_{A}}$
6	$te^{-at}\;$	${\cfrac {1}{s+a}}^{2}$	${\cfrac {cT_{A}z}{(z-c)^{2}}};c=e^{-aT_{A}}$
7	$t^{2}e^{-at}\;$	${\cfrac {2}{s+a}}^{3}$	${\cfrac {cT_{A}^{2}z(z+c)}{(z-c)^{3}}};c=e^{-aT_{A}}$
8	$1-e^{-at}\;$	${\cfrac {a}{s(s+a)}}$	${\cfrac {(1-c)z}{(z-1)(z-c)}};c=e^{-aT_{A}}$
9	$sin\omega _{0}t\;$	${\cfrac {\omega _{0}}{s^{2}+\omega _{0}^{2}}}$	${\cfrac {zsin\omega _{0}T_{A}}{z^{2}-2zcos\omega _{0}T_{A}+1}}$
10	$cos\omega _{0}t\;$	${\cfrac {s}{s^{2}+\omega _{0}^{2}}}$	${\cfrac {z^{2}-zcos\omega _{0}T_{A}}{z^{2}-2zcos\omega _{0}T_{A}+1}}$
11	$1-(1+at)e^{-at}\;$	${\cfrac {a^{2}}{s(s+a)^{2}}}$	${\cfrac {z}{z-1}}-{\cfrac {z}{z-c}}-a{\cfrac {cT_{A}z}{(z-c)^{2}}};c=e^{-aT_{A}}$
12	$1-{\cfrac {be^{-at}-ae^{-at}}{a-b}}$	${\cfrac {ab}{s(s+a)(s+b)}}$	${\cfrac {z}{z-1}}+{\cfrac {bz}{(a-b)(z-c)}}-{\cfrac {az}{(a-b)(z-d)}};c=e^{-aT_{A}};d=e^{-bT_{A}}$
13	$e^{-at}sin\omega _{0}t\;$	${\cfrac {\omega _{0}}{s+a)^{2}+\omega _{0}^{2}}}$	${\cfrac {czsin\omega _{0}T_{A}}{z^{2}-2czcos\omega _{0}T_{A}+c^{2}}};c=e^{-aT_{A}}$
14	$e^{-at}cos\omega _{0}t\;$	${\cfrac {s+a}{s+a)^{2}+\omega _{0}^{2}}}$	${\cfrac {z^{2}-czcos\omega _{0}T_{A}}{z^{2}-2czcos\omega _{0}T_{A}+c^{2}}};c=e^{-aT_{A}}$
15	$a^{\cfrac {t}{T_{A}}}\;$	${\cfrac {1}{s-\left({\cfrac {1}{T_{A}}}\right)lna}}$	${\cfrac {z}{z-a}}$
16	$\delta (t-T_{A})\;$	$e^{-T_{A}s}\;$	$z^{-1}\;$
17	$\delta (t-kT_{A})\;$	$e^{-kT_{A}s}\;$	$z^{-k}\;$