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|\,}
![{\displaystyle X(z)={\mathcal {Z}}\{x[n]\}=\sum _{n=-\infty }^{\infty }x[n]z^{-n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12f6e27003f8c3271124b8af3ea0092c2906ae3e)
![{\displaystyle X(z)={\mathcal {Z}}\{x[n]\}=\sum _{n=0}^{\infty }x[n]z^{-n}.\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a1d65caf4abfd4be45ca9941565d7db69e31e79)
![{\displaystyle x[n]={\mathcal {Z}}^{-1}\{X(z)\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05e642979d4bbea30a164bd3c3c0478dd4f42c2d)
![{\displaystyle X(z)={\mathcal {Z}}\{x[n]\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3aefa942e18926dd24f0a75ca1f495002704e35f)
![{\displaystyle a_{1}x_{1}[n]+a_{2}x_{2}[n]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdb707cd1570b5004afa6306d8e349d36efffb58)
![{\displaystyle x_{(k)}[n]={\begin{cases}x[r],&n=rk\\0,&n\not =rk\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d9f9952a352bd13efc6e0310e925f44aeced1b5)
![{\displaystyle x[n-k]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cd4e132f4bd95a617de0e6eb00ba09e2801e1e0)
![{\displaystyle {\begin{array}{lcl}Z\{x[n-k]\}=&\\\sum _{n=0}^{\infty }x[n-k]z^{-n}\&\\{\text{ , let }}j=n-k&\\=\sum _{j=-k}^{\infty }x[j]z^{-(j+k)}&\\=\sum _{j=-k}^{\infty }x[j]z^{-j}z^{-k}&\\=z^{-k}\sum _{j=-k}^{\infty }x[j]z^{-j}&\\=z^{-k}\sum _{j=0}^{\infty }x[j]z^{-j}&\\{\text{ , since }}x[\beta ]=0{\text{ if }}\beta <0&\\=z^{-k}X(z)&\\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ce367dc21940488c169d25bdb2adf22af59ba67)
![{\displaystyle a^{n}x[n]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/654e28509ebf86a5253c1e79e7a6b12fa1087466)
![{\displaystyle {\begin{array}{lcl}Z\{a^{n}x[n]\}=&\\\sum _{n=-\infty }^{\infty }a^{n}x(n)z^{-n}&\\=\sum _{n=-\infty }^{\infty }x(n)(a^{-1}z)^{-n}&\\=X(a^{-1}z)&\\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b968c04b8900e9e16c42ef0333062c3d385d4523)
![{\displaystyle x[-n]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/71ca1d46f1a6942599a856f413f141deb2ab36cf)
![{\displaystyle x^{*}[n]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/836731dd6ac0987320b601f404492523187f4dec)
![{\displaystyle {\begin{array}{lcl}Z\{x^{*}(n)\}=&\\\sum _{n=-\infty }^{\infty }x^{*}(n)z^{-n}\ &\\=\sum _{n=-\infty }^{\infty }[x(n)(z^{*})^{-n}]^{*}\ &\\=[\sum _{n=-\infty }^{\infty }x(n)(z^{*})^{-n}\ ]^{*}&\\=X^{*}(z^{*})&\\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c974b68d4c86dd8624f0f5ef6fd71634e789cfac)
![{\displaystyle \operatorname {Re} \{x[n]\}\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3c4d81639a0f761dbc7855d7de10d03c2a42d6b)
![{\displaystyle {\frac {1}{2}}\left[X(z)+X^{*}(z^{*})\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cbea203386496bdc11a51c6f14798cf0b7373ef7)
![{\displaystyle \operatorname {Im} \{x[n]\}\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6aaa2a8950fdcb31a68819605c826960e6c48c5c)
![{\displaystyle {\frac {1}{2j}}\left[X(z)-X^{*}(z^{*})\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75e00a599cca7e1c1a78f75d7f7710b7474c12da)
![{\displaystyle nx[n]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b9d129545d7bb8f225e3d3d6cf29828055b2aa0)
![{\displaystyle x_{1}[n]*x_{2}[n]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a88f085e632f498b5ad7cae9fc45379ab931ff7)
![{\displaystyle {\begin{array}{lcl}{\mathcal {Z}}\{x_{1}(n)*x_{2}(n)\}=&\\{\mathcal {Z}}\{\sum _{l=-\infty }^{\infty }x_{1}(l)x_{2}(n-l)\}\ &\\=\sum _{n=-\infty }^{\infty }[\sum _{l=-\infty }^{\infty }x_{1}(l)x_{2}(n-l)]z^{-n}\ &\\=\sum _{l=-\infty }^{\infty }x_{1}(l)\sum _{n=-\infty }^{\infty }x_{2}(n-l)z^{-n}]\ &\\=[\sum _{l=-\infty }^{\infty }x_{1}(l)z^{-l}][\sum _{n=-\infty }^{\infty }x_{2}(n)z^{-n}]\ &\\=X_{1}(z)X_{2}(z)&\\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48ac821be34871813f81e8a3db121d9082cf83b0)
![{\displaystyle r_{x_{1},x_{2}}=x_{1}^{*}[-n]*x_{2}[n]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a3137b2b225055345f857b5fa1262472fef95e7)
![{\displaystyle x[n]-x[n-1]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/64d1f6e2e3298702699bdb047de5c5241347dca2)
![{\displaystyle \sum _{k=-\infty }^{n}x[k]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5cfdbdd32d7420715b393ab382646f76dc6e3dc2)
![{\displaystyle {\begin{array}{lcl}\sum _{n=-\infty }^{\infty }\sum _{k=-\infty }^{n}x[k]\cdot z^{-n}\\=\sum _{n=-\infty }^{\infty }(x[n]+x[n-1]+\\x[n-2]\cdots x[-\infty ])z^{-n}\\=X[z](1+z^{-1}+z^{-2}+z^{-3}\cdots )\\=X[z]\sum _{j=0}^{\infty }z^{-j}\\=X[z]{\frac {1}{1-z^{-1}}}\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29e2696cbdd13918cdaad3b71ec2802b54a2c7fb)
![{\displaystyle x_{1}[n]x_{2}[n]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b8ee2f74b87c0783a1e479136a290c947ff945)
![{\displaystyle \sum _{n=-\infty }^{\infty }x_{1}[n]x_{2}^{*}[n]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/14b55a573a79d115fd4d071bd6f6cb3bbf5172a1)
![{\displaystyle x[0]=\lim _{z\rightarrow \infty }X(z)\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/34e74d32b1013c9db6578a43b03066f7dd31202c)
![{\displaystyle x[n]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b575dce543a2cf10a5a3e108204b928c2c9aaa54)
![{\displaystyle x[\infty ]=\lim _{z\rightarrow 1}(z-1)X(z)\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/71d6d17cc3b7ef5c69a58caf4ebc6d2acc2c8ee3)
![{\displaystyle u[n]={\begin{cases}1,&n\geq 0\\0,&n<0\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/421a23ee8e78e1a635a55f77b90157e36ace10ce)
![{\displaystyle \delta [n]={\begin{cases}1,&n=0\\0,&n\neq 0\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfbd93ec15d6d9070adb434706fe249b5f944bdc)
![{\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 \,e^{-\alpha n}u[n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea4acaa4f561a314f2cb97656a91950895450a99)
![{\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)
