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