Z Transformation Properties

	Time domain	Z-domain	ROC
Notation	${\displaystyle x[n]={\mathcal {Z}}^{-1}\{X(z)\}}$	${\displaystyle X(z)={\mathcal {Z}}\{x[n]\}}$	ROC: ${\displaystyle r_{2}<|z|<r_{1}\ }$
Linearity	${\displaystyle a_{1}x_{1}[n]+a_{2}x_{2}[n]\ }$	${\displaystyle a_{1}X_{1}(z)+a_{2}X_{2}(z)\ }$	At least the intersection of ROC1 and ROC2
Time shifting	${\displaystyle x[n-k]\ }$	${\displaystyle z^{-k}X(z)\ }$	ROC, except ${\displaystyle z=0\ }$ if ${\displaystyle k>0\,}$ and ${\displaystyle z=\infty }$ if ${\displaystyle k<0\ }$
Scaling in the z-domain	${\displaystyle a^{n}x[n]\ }$	${\displaystyle X(a^{-1}z)\ }$	${\displaystyle |a|r_{2}<|z|<|a|r_{1}\ }$
Time reversal	${\displaystyle x[-n]\ }$	${\displaystyle X(z^{-1})\ }$	${\displaystyle {\frac {1}{r_{2}}}<|z|<{\frac {1}{r_{1}}}\ }$
Conjugation	${\displaystyle x^{*}[n]\ }$	${\displaystyle X^{*}(z^{*})\ }$	ROC
Real part	${\displaystyle \operatorname {Re} \{x[n]\}\ }$	${\displaystyle {\frac {1}{2}}\left[X(z)+X^{*}(z^{*})\right]}$	ROC
Imaginary part	${\displaystyle \operatorname {Im} \{x[n]\}\ }$	${\displaystyle {\frac {1}{2j}}\left[X(z)-X^{*}(z^{*})\right]}$	ROC
Differentiation	${\displaystyle nx[n]\ }$	${\displaystyle -z{\frac {\mathrm {d} X(z)}{\mathrm {d} z}}}$	ROC
Convolution	${\displaystyle x_{1}[n]*x_{2}[n]\ }$	${\displaystyle X_{1}(z)X_{2}(z)\ }$	At least the intersection of ROC1 and ROC2
Correlation	${\displaystyle r_{x_{1},x_{2}}(l)=x_{1}[l]*x_{2}[-l]\ }$	${\displaystyle R_{x_{1},x_{2}}(z)=X_{1}(z)X_{2}(z^{-1})\ }$	At least the intersection of ROC of X1(z) and X2(${\displaystyle z^{-1}}$)
Multiplication	${\displaystyle x_{1}[n]x_{2}[n]\ }$	${\displaystyle {\frac {1}{j2\pi }}\oint _{C}X_{1}(v)X_{2}({\frac {z}{v}})v^{-1}\mathrm {d} v\ }$	At least ${\displaystyle r_{1l}r_{2l}<|z|<r_{1u}r_{2u}\ }$
Parseval's relation	${\displaystyle \sum ^{\infty }x_{1}[n]x_{2}^{*}[n]\ }$	${\displaystyle {\frac {1}{j2\pi }}\oint _{C}X_{1}(v)X_{2}^{*}({\frac {1}{v^{*}}})v^{-1}\mathrm {d} v\ }$

• Initial value theorem

${\displaystyle x[0]=\lim _{z\rightarrow \infty }X(z)\ }$, If ${\displaystyle x[n]\,}$ causal

• Final value theorem

${\displaystyle x[\infty ]=\lim _{z\rightarrow 1}(z-1)X(z)\ }$, Only if poles of ${\displaystyle (z-1)X(z)\ }$ are inside unit circle

Z Transformation of Functions

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	${\displaystyle a^{n}\sin(\omega _{0}n)u[n]\,}$	${\displaystyle {\frac {az^{-1}\sin(\omega _{0})}{1-2az^{-1}\cos(\omega _{0})+a^{2}z^{-2}}}}$	${\displaystyle |z|>|a|\,}$