Engineering Tables/Z Transform Properties

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

• Initial value theorem

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

• Final value theorem

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