z変換の表(メモ)

\[
\begin{array}{|c|c|}
\hline f(t) & \mathcal{Z}[f](z) \\\hline
f(t) & \sum_{k=0}^\infty f(k)z^{-k} \\\hline
f(k)=1\;(k\geq 0) & \frac{z}{z-1}\;(|z|>1) \\\hline
a^k & \frac{z}{z-a}\;(|z|>|a|) \\\hline
f(k)=\begin{cases}
1& k=0\\
0& k\geq 1
\end{cases}
& 1 \\\hline
f(k)+g(k) &\mathcal{Z}[f](z)+\mathcal{Z}[g](z)\\\hline
\sum_{u=0}^k f(k-u)g(u) &\mathcal{Z}[f](z)\mathcal{Z}[g](z)\\\hline
g(k)=\begin{cases}
0& k=0 \\
f(k-{1})& k\geq 1
\end{cases}
& z^{-1}\mathcal{Z}[f](z) \\\hline
f(k)a^k\;(k\geq 0) &\mathcal{Z}[f](\frac{z}{a})\\\hline
f(k+1)\;(k\geq 0) &z\mathcal{Z}[f(k)](\frac{z}{a})-zf(0)\\\hline
\cos\omega k &\frac{z^2-z\cos\omega}{z^2-2z\cos\omega+1}\\\hline
\sin\omega k &\frac{z\sin\omega}{z^2-2z\cos\omega+1}\\\hline
k &\frac{z}{(z-1)^2}\\\hline
ka^k &\frac{az}{(z-a)^2}\\\hline
\end{array}
\]