# Engineering Tables/Z Transform Properties

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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|
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|
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