# Engineering Tables/Z Transform Properties

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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[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