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Engineering Tables/Z Transform Table
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Engineering Tables
Here:
u
[
n
]
=
1
{\displaystyle u[n]=1}
for
n
>=
0
{\displaystyle n>=0}
,
u
[
n
]
=
0
{\displaystyle u[n]=0}
for
n
<
0
{\displaystyle n<0}
δ
[
n
]
=
1
{\displaystyle \delta [n]=1}
for
n
=
0
{\displaystyle n=0}
,
δ
[
n
]
=
0
{\displaystyle \delta [n]=0}
otherwise
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
−
u
[
−
n
−
1
]
{\displaystyle -u[-n-1]\,}
1
1
−
z
−
1
{\displaystyle {\frac {1}{1-z^{-1}}}}
|
z
|
<
1
{\displaystyle |z|<1\,}
5
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\,}
6
−
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\,}
7
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\,}
8
−
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\,}
9
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\,}
10
−
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\,}
11
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|\,}
12
−
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|\,}
13
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|\,}
14
−
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|\,}
15
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|\,}
16
−
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|\,}
17
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\,}
18
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\,}
19
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|\,}
20
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|\,}
![{\displaystyle x[n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/864cbbefbdcb55af4d9390911de1bf70167c4a3d)
![{\displaystyle \delta [n]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71bf83f188a8d5c544e4565e7690b4029baed424)
![{\displaystyle \delta [n-n_{0}]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37420bd43ae13a5c7f300b7e74cae6f08d38929a)
![{\displaystyle u[n]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e693a2911b29e6c8d440d97e46d27760559af7c5)
![{\displaystyle -u[-n-1]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ffe6af41d5e73ca723916ce158e69325ab37f0f)
![{\displaystyle nu[n]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/666189dd08672bcd9366962001fdeb012ef48db6)
![{\displaystyle -nu[-n-1]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9c7bfd00539cf805ba91e15a60b73576194dbd1)
![{\displaystyle n^{2}u[n]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/556ff0a974378872385484fbeadf8bee8da04e97)
![{\displaystyle -n^{2}u[-n-1]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bcc9d247970a92c7c6a69da9b5a272190dadcd24)
![{\displaystyle n^{3}u[n]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0458b9d3eaa2c032ea05c85f0b2dc59809e61815)
![{\displaystyle -n^{3}u[-n-1]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f66cd02f765990cd569d2cc47dd44cc75bfad0a3)
![{\displaystyle a^{n}u[n]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec08f4507612bf171a306727ec543e8b22909a8b)
![{\displaystyle -a^{n}u[-n-1]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf8f7594a9aa188a37767ad6cbbcd1c932f5408c)
![{\displaystyle na^{n}u[n]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b15d6e4309b253c58fdf105b270217869762ad)
![{\displaystyle -na^{n}u[-n-1]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a7df946d3646eb5c94313eb414edf4c9070d782)
![{\displaystyle n^{2}a^{n}u[n]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0dc9a83f632b9781ec66653e486b9f77fafe3bb0)
![{\displaystyle -n^{2}a^{n}u[-n-1]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d85083b09d1b7c73d32bdfcc120bd85c56504b66)
![{\displaystyle \cos(\omega _{0}n)u[n]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07538580111cb3bee33a686fa3970039105fee57)
![{\displaystyle \sin(\omega _{0}n)u[n]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0a4e25fc75ace79158f75ab06339a9b87ce3665)
![{\displaystyle a^{n}\cos(\omega _{0}n)u[n]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a451552ec134261501b282063fca173b531a0ba)
![{\displaystyle a^{n}\sin(\omega _{0}n)u[n]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee7ec057f4c8c7c307f3a58a9091e6e03d91f01d)
![{\displaystyle u[n]=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c3405c534c1af1bf479ff017d87682729722321)
![{\displaystyle u[n]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75f0ec7d52e8703689ee40dc9655fbecb74265b2)
![{\displaystyle \delta [n]=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4953a952218d3ccd27c2a5797e4b2d0c7eb08b39)
![{\displaystyle \delta [n]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2fb443c0e773350512b296fea00e317bbcf0d7d)