# Engineering Tables/Z Transform Table

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Here:

• ${\displaystyle u[n]=1}$ for ${\displaystyle n>=0}$, ${\displaystyle u[n]=0}$ for ${\displaystyle n<0}$
• ${\displaystyle \delta [n]=1}$ for ${\displaystyle n=0}$, ${\displaystyle \delta [n]=0}$ otherwise
Signal, ${\displaystyle x[n]}$ Z-transform, ${\displaystyle X(z)}$ ROC
1 ${\displaystyle \delta [n]\,}$ ${\displaystyle 1\,}$ ${\displaystyle {\mbox{all }}z\,}$
2 ${\displaystyle \delta [n-n_{0}]\,}$ ${\displaystyle z^{-n_{0}}\,}$ ${\displaystyle z\neq 0\,}$
3 ${\displaystyle u[n]\,}$ ${\displaystyle {\frac {1}{1-z^{-1}}}}$ ${\displaystyle |z|>1\,}$
4 ${\displaystyle -u[-n-1]\,}$ ${\displaystyle {\frac {1}{1-z^{-1}}}}$ ${\displaystyle |z|<1\,}$
5 ${\displaystyle nu[n]\,}$ ${\displaystyle {\frac {z^{-1}}{(1-z^{-1})^{2}}}}$ ${\displaystyle |z|>1\,}$
6 ${\displaystyle -nu[-n-1]\,}$ ${\displaystyle {\frac {z^{-1}}{(1-z^{-1})^{2}}}}$ ${\displaystyle |z|<1\,}$
7 ${\displaystyle n^{2}u[n]\,}$ ${\displaystyle {\frac {z^{-1}(1+z^{-1})}{(1-z^{-1})^{3}}}}$ ${\displaystyle |z|>1\,}$
8 ${\displaystyle -n^{2}u[-n-1]\,}$ ${\displaystyle {\frac {z^{-1}(1+z^{-1})}{(1-z^{-1})^{3}}}}$ ${\displaystyle |z|<1\,}$
9 ${\displaystyle n^{3}u[n]\,}$ ${\displaystyle {\frac {z^{-1}(1+4z^{-1}+z^{-2})}{(1-z^{-1})^{4}}}}$ ${\displaystyle |z|>1\,}$
10 ${\displaystyle -n^{3}u[-n-1]\,}$ ${\displaystyle {\frac {z^{-1}(1+4z^{-1}+z^{-2})}{(1-z^{-1})^{4}}}}$ ${\displaystyle |z|<1\,}$
11 ${\displaystyle a^{n}u[n]\,}$ ${\displaystyle {\frac {1}{1-az^{-1}}}}$ ${\displaystyle |z|>|a|\,}$
12 ${\displaystyle -a^{n}u[-n-1]\,}$ ${\displaystyle {\frac {1}{1-az^{-1}}}}$ ${\displaystyle |z|<|a|\,}$
13 ${\displaystyle na^{n}u[n]\,}$ ${\displaystyle {\frac {az^{-1}}{(1-az^{-1})^{2}}}}$ ${\displaystyle |z|>|a|\,}$
14 ${\displaystyle -na^{n}u[-n-1]\,}$ ${\displaystyle {\frac {az^{-1}}{(1-az^{-1})^{2}}}}$ ${\displaystyle |z|<|a|\,}$
15 ${\displaystyle n^{2}a^{n}u[n]\,}$ ${\displaystyle {\frac {az^{-1}(1+az^{-1})}{(1-az^{-1})^{3}}}}$ ${\displaystyle |z|>|a|\,}$
16 ${\displaystyle -n^{2}a^{n}u[-n-1]\,}$ ${\displaystyle {\frac {az^{-1}(1+az^{-1})}{(1-az^{-1})^{3}}}}$ ${\displaystyle |z|<|a|\,}$
17 ${\displaystyle \cos(\omega _{0}n)u[n]\,}$ ${\displaystyle {\frac {1-z^{-1}\cos(\omega _{0})}{1-2z^{-1}\cos(\omega _{0})+z^{-2}}}}$ ${\displaystyle |z|>1\,}$
18 ${\displaystyle \sin(\omega _{0}n)u[n]\,}$ ${\displaystyle {\frac {z^{-1}\sin(\omega _{0})}{1-2z^{-1}\cos(\omega _{0})+z^{-2}}}}$ ${\displaystyle |z|>1\,}$
19 ${\displaystyle a^{n}\cos(\omega _{0}n)u[n]\,}$ ${\displaystyle {\frac {1-az^{-1}\cos(\omega _{0})}{1-2az^{-1}\cos(\omega _{0})+a^{2}z^{-2}}}}$ ${\displaystyle |z|>|a|\,}$
20 ${\displaystyle a^{n}\sin(\omega _{0}n)u[n]\,}$ ${\displaystyle {\frac {az^{-1}\sin(\omega _{0})}{1-2az^{-1}\cos(\omega _{0})+a^{2}z^{-2}}}}$ ${\displaystyle |z|>|a|\,}$