Signal, x [ n ] {\displaystyle x[n]}
Z-transform, X ( z ) {\displaystyle X(z)}
ROC
1
δ [ n ] {\displaystyle \delta [n]\,}
1 {\displaystyle 1\,}
all z {\displaystyle {\mbox{all }}z\,}
2
δ [ n − n 0 ] {\displaystyle \delta [n-n_{0}]\,}
z − n 0 {\displaystyle z^{-n_{0}}\,}
z ≠ 0 {\displaystyle z\neq 0\,}
3
u [ n ] {\displaystyle u[n]\,}
1 1 − z − 1 {\displaystyle {\frac {1}{1-z^{-1}}}}
| z | > 1 {\displaystyle |z|>1\,}
4
e − α n u [ n ] {\displaystyle \,e^{-\alpha n}u[n]}
1 1 − e − α z − 1 {\displaystyle 1 \over 1-e^{-\alpha }z^{-1}}
| z | > | e − α | {\displaystyle |z|>|e^{-\alpha }|\,}
5
− u [ − n − 1 ] {\displaystyle -u[-n-1]\,}
1 1 − z − 1 {\displaystyle {\frac {1}{1-z^{-1}}}}
| z | < 1 {\displaystyle |z|<1\,}
6
n u [ n ] {\displaystyle nu[n]\,}
z − 1 ( 1 − z − 1 ) 2 {\displaystyle {\frac {z^{-1}}{(1-z^{-1})^{2}}}}
| z | > 1 {\displaystyle |z|>1\,}
7
− n u [ − n − 1 ] {\displaystyle -nu[-n-1]\,}
z − 1 ( 1 − z − 1 ) 2 {\displaystyle {\frac {z^{-1}}{(1-z^{-1})^{2}}}}
| z | < 1 {\displaystyle |z|<1\,}
8
n 2 u [ n ] {\displaystyle n^{2}u[n]\,}
z − 1 ( 1 + z − 1 ) ( 1 − z − 1 ) 3 {\displaystyle {\frac {z^{-1}(1+z^{-1})}{(1-z^{-1})^{3}}}}
| z | > 1 {\displaystyle |z|>1\,}
9
− n 2 u [ − n − 1 ] {\displaystyle -n^{2}u[-n-1]\,}
z − 1 ( 1 + z − 1 ) ( 1 − z − 1 ) 3 {\displaystyle {\frac {z^{-1}(1+z^{-1})}{(1-z^{-1})^{3}}}}
| z | < 1 {\displaystyle |z|<1\,}
10
n 3 u [ n ] {\displaystyle n^{3}u[n]\,}
z − 1 ( 1 + 4 z − 1 + z − 2 ) ( 1 − z − 1 ) 4 {\displaystyle {\frac {z^{-1}(1+4z^{-1}+z^{-2})}{(1-z^{-1})^{4}}}}
| z | > 1 {\displaystyle |z|>1\,}
11
− n 3 u [ − n − 1 ] {\displaystyle -n^{3}u[-n-1]\,}
z − 1 ( 1 + 4 z − 1 + z − 2 ) ( 1 − z − 1 ) 4 {\displaystyle {\frac {z^{-1}(1+4z^{-1}+z^{-2})}{(1-z^{-1})^{4}}}}
| z | < 1 {\displaystyle |z|<1\,}
12
a n u [ n ] {\displaystyle a^{n}u[n]\,}
1 1 − a z − 1 {\displaystyle {\frac {1}{1-az^{-1}}}}
| z | > | a | {\displaystyle |z|>|a|\,}
13
− a n u [ − n − 1 ] {\displaystyle -a^{n}u[-n-1]\,}
1 1 − a z − 1 {\displaystyle {\frac {1}{1-az^{-1}}}}
| z | < | a | {\displaystyle |z|<|a|\,}
14
n a n u [ n ] {\displaystyle na^{n}u[n]\,}
a z − 1 ( 1 − a z − 1 ) 2 {\displaystyle {\frac {az^{-1}}{(1-az^{-1})^{2}}}}
| z | > | a | {\displaystyle |z|>|a|\,}
15
− n a n u [ − n − 1 ] {\displaystyle -na^{n}u[-n-1]\,}
a z − 1 ( 1 − a z − 1 ) 2 {\displaystyle {\frac {az^{-1}}{(1-az^{-1})^{2}}}}
| z | < | a | {\displaystyle |z|<|a|\,}
16
n 2 a n u [ n ] {\displaystyle n^{2}a^{n}u[n]\,}
a z − 1 ( 1 + a z − 1 ) ( 1 − a z − 1 ) 3 {\displaystyle {\frac {az^{-1}(1+az^{-1})}{(1-az^{-1})^{3}}}}
| z | > | a | {\displaystyle |z|>|a|\,}
17
− n 2 a n u [ − n − 1 ] {\displaystyle -n^{2}a^{n}u[-n-1]\,}
a z − 1 ( 1 + a z − 1 ) ( 1 − a z − 1 ) 3 {\displaystyle {\frac {az^{-1}(1+az^{-1})}{(1-az^{-1})^{3}}}}
| z | < | a | {\displaystyle |z|<|a|\,}
18
cos ( ω 0 n ) u [ n ] {\displaystyle \cos(\omega _{0}n)u[n]\,}
1 − z − 1 cos ( ω 0 ) 1 − 2 z − 1 cos ( ω 0 ) + z − 2 {\displaystyle {\frac {1-z^{-1}\cos(\omega _{0})}{1-2z^{-1}\cos(\omega _{0})+z^{-2}}}}
| z | > 1 {\displaystyle |z|>1\,}
19
sin ( ω 0 n ) u [ n ] {\displaystyle \sin(\omega _{0}n)u[n]\,}
z − 1 sin ( ω 0 ) 1 − 2 z − 1 cos ( ω 0 ) + z − 2 {\displaystyle {\frac {z^{-1}\sin(\omega _{0})}{1-2z^{-1}\cos(\omega _{0})+z^{-2}}}}
| z | > 1 {\displaystyle |z|>1\,}
20
a n cos ( ω 0 n ) u [ n ] {\displaystyle a^{n}\cos(\omega _{0}n)u[n]\,}
1 − a z − 1 cos ( ω 0 ) 1 − 2 a z − 1 cos ( ω 0 ) + a 2 z − 2 {\displaystyle {\frac {1-az^{-1}\cos(\omega _{0})}{1-2az^{-1}\cos(\omega _{0})+a^{2}z^{-2}}}}
| z | > | a | {\displaystyle |z|>|a|\,}
21
a n sin ( ω 0 n ) u [ n ] {\displaystyle a^{n}\sin(\omega _{0}n)u[n]\,}
a z − 1 sin ( ω 0 ) 1 − 2 a z − 1 cos ( ω 0 ) + a 2 z − 2 {\displaystyle {\frac {az^{-1}\sin(\omega _{0})}{1-2az^{-1}\cos(\omega _{0})+a^{2}z^{-2}}}}
| z | > | a | {\displaystyle |z|>|a|\,}