Rücktransformation der Z-Transformation

Tabelle

Sequence	Transform	ROC
1. $\delta[n]$	1	All $z$
2. $u[n]$	$\frac{1}{1-z^{-1}}$	$\\|z\\|>1$
3. $-u[-n-1]$	$\frac{1}{1-z^{-1}}$	$\\|z\\|<1$
4. $\delta[n-m]$	$z^{-m}$	All $z$ except 0 (if $m>0$ ) $\quad$ or $\infty$ (if $m<0$ )
5. $a^n u[n]$	$\frac{1}{1-a z^{-1}}$	$\\|z\\|>\\|a\\|$
6. $-a^n u[-n-1]$	$\frac{1}{1-a z^{-1}}$	$\\|z\\|<\\|a\\|$
7. $n a^n u[n]$	$\frac{a z^{-1}}{\left(1-a z^{-1}\right)^2}$	$\\|z\\|>\\|a\\|$
8. $-n a^n u[-n-1]$	$\frac{a z^{-1}}{\left(1-a z^{-1}\right)^2}$	$\\|z\\|<\\|a\\|$
9. $\left[\cos \omega_0 n\right] u[n]$	$\frac{1-\left[\cos \omega_0\right] z^{-1}}{1-\left[2 \cos \omega_0\right] z^{-1}+z^{-2}}$	$\\|z\\|>1$
10. $\left[\sin \omega_0 n\right] u[n]$	$\frac{\left[\sin \omega_0\right] z^{-1}}{1-\left[2 \cos \omega_0\right] z^{-1}+z^{-2}}$	$\\|z\\|>1$
11. $\left[r^n \cos \omega_0 n\right] u[n]$	$\frac{1-\left[r \cos \omega_0\right] z^{-1}}{1-\left[2 r \cos \omega_0\right] z^{-1}+r^2 z^{-2}}$	$\\|z\\|>r$
12. $\left[r^n \sin \omega_0 n\right] u[n]$	$\frac{\left[r \sin \omega_0\right] z^{-1}}{1-\left[2 r \cos \omega_0\right] z^{-1}+r^2 z^{-2}}$	$\\|z\\|>r$
13. $\begin{cases}a^n, & 0 \leq n \leq N-1, \\0, & \text { otherwise }\end{cases}$	$\frac{1-a^N z^{-N}}{1-a z^{-1}}$	$\\|z\\|>0$