# 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| 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| 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=\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