Tabelle Zeitfunktion/Laplace-Transformation/z-Transformation

Tabelle

Nr.	Zeitfunktion $f(t)$	L-Transformierte $F(s)=L[f(t)]$	z-Transformierte $F_z(z)=Z[f(k{T}_A)]$ mit$f(k{T}_A)=f(t){|}_{t=k{T}_A}$
1	Dirac-Impuls $\delta (t)$	$1$	$1$
2	Einheitssprung $\sigma (t)$	$\frac{1}{s}$	$\frac{z}{z-1}$
3	$t$	$\frac{1}{s^2}$	$\frac{T_{A}z}{(z-1{)}^2}$
4	$t^2$	$\frac{2}{s^3}$	$\frac{T_A^{2}z(z+1)}{(z-1{)}^3}$
5	$e^{-at}$	$\frac{1}{s+a}$	$\frac{z}{(z-c)};c=e^{-a{T}_A}$
6	$t{e}^{-at}$	${\frac{1}{s+a}}^2$	$\frac{c{T}_{A}z}{(z-c{)}^2};c=e^{-a{T}_A}$
7	$t^2{e}^{-at}$	${\frac{2}{s+a}}^3$	$\frac{c{T}_A^{2}z(z+c)}{(z-c{)}^3};c=e^{-a{T}_A}$
8	$1-e^{-at}$	$\frac{a}{s(s+a)}$	$\frac{(1-c)z}{(z-1)(z-c)};c=e^{-a{T}_A}$
9	$sin{\omega }_{0}t$	$\frac{{\omega }_0}{s^2+{\omega }_0^2}$	$\frac{zsin{\omega }_0{T}_A}{z^2-2zcos{\omega }_0{T}_A+1}$
10	$cos{\omega }_{0}t$	$\frac{s}{s^2+{\omega }_0^2}$	$\frac{z^2-zcos{\omega }_0{T}_A}{z^2-2zcos{\omega }_0{T}_A+1}$
11	$1-(1+at)e^{-at}$	$\frac{a^2}{s(s+a{)}^2}$	$\frac{z}{z-1}-\frac{z}{z-c}-a\frac{c{T}_{A}z}{(z-c{)}^2};c=e^{-a{T}_A}$
12	$1-\frac{b{e}^{-at}-a{e}^{-at}}{a-b}$	$\frac{ab}{s(s+a)(s+b)}$	$\frac{z}{z-1}+\frac{bz}{(a-b)(z-c)}-\frac{az}{(a-b)(z-d)};c=e^{-a{T}_A};d=e^{-b{T}_A}$
13	$e^{-at}sin{\omega }_{0}t$	$\frac{{\omega }_0}{s+a{)}^2+{\omega }_0^2}$	$\frac{czsin{\omega }_0{T}_A}{z^2-2czcos{\omega }_0{T}_A+c^2};c=e^{-a{T}_A}$
14	$e^{-at}cos{\omega }_{0}t$	$\frac{s+a}{s+a{)}^2+{\omega }_0^2}$	$\frac{z^2-czcos{\omega }_0{T}_A}{z^2-2czcos{\omega }_0{T}_A+c^2};c=e^{-a{T}_A}$
15	$a^{\frac{t}{T_A}}$	$\frac{1}{s-\left(\frac{1}{T_A}\right)lna}$	$\frac{z}{z-a}$
16	$\delta (t-T_A)$	$e^{-T_{A}s}$	$z^{-1}$
17	$\delta (t-k{T}_A)$	$e^{-k{T}_{A}s}$	$z^{-k}$