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|<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