Z-transform - Properties

Properties

Properties of the z-transform
Time domain Z-domain Proof ROC
Notation ROC:
Linearity \begin{array} {lcl} X(z) = & \\ \sum_{n=-\infty}^{\infty} (a_1x_1(n)+a_2x_2(n))z^{-n}\ & \\ = a_1\sum_{n=-\infty}^{\infty} (x_1(n))z^{-n} + & \\ a_2\sum_{n=-\infty}^{\infty}(x_2(n))z^{-n} & \\ = a_1X_1(z) + a_2X_2(z)\end{array} At least the intersection of ROC1 and ROC2
Time expansion

: integer

 \begin{array} {lcl} X_k(z)=\sum_{n=-\infty}^{\infty} x_k(n)z^{-n} = & \\ = \sum_{r=-\infty}^{\infty}x(r)z^{-rk} = & \\ =\sum_{r=-\infty}^{\infty}x(r)(z^{k})^{-r} = & \\ = X(z^{k}) \end{array} R^{1/k}
Time shifting \begin{array} {lcl} Z\{x\} = & \\ \sum_{n=0}^{\infty} xz^{-n}\& \\ \text{ ,let }j = n - k & \\ = \sum_{j=-k}^{\infty} xz^{-(j+k)}& \\ = \sum_{j=-k}^{\infty} xz^{-j}z^{-k}& \\ = z^{-k}\sum_{j=-k}^{\infty}xz^{-j}& \\ = z^{-k}\sum_{j=0}^{\infty}xz^{-j} & \\ \text{, since }x=0 \text{ if }\beta<0 & \\ = z^{-k}X(z)& \\ \end{array} ROC, except if and if
Scaling in

the z-domain

 \begin{array} {lcl} Z \{a^n x\} = & \\ \sum_{n=-\infty}^{\infty} a^{n}x(n)z^{-n}& \\ = \sum_{n=-\infty}^{\infty} x(n)(a^{-1}z)^{-n} & \\ = X(a^{-1}z) & \\ \end{array}
Time reversal \begin{array} {lcl} \mathcal{Z}\{x(-n)\} = & \\ \sum_{n=-\infty}^{\infty} x(-n)z^{-n}\ & \\ = \sum_{m=-\infty}^{\infty} x(m)z^{m}\ & \\ = \sum_{m=-\infty}^{\infty} x(m){(z^{-1})}^{-m}\ & \\ = X(z^{-1}) & \\ \end{array}
Complex conjugation \begin{array} {lcl}Z\{x^*(n)\} = & \\ \sum_{n=-\infty}^{\infty} x^*(n)z^{-n}\ & \\ = \sum_{n=-\infty}^{\infty} ^*\ & \\ = ^* & \\ = X^*(z^*)& \\ \end{array} ROC
Real part ROC
Imaginary part ROC
Differentiation \begin{array} {lcl}Z\{nx(n)\} = & \\ \sum_{n=-\infty}^{\infty} nx(n)z^{-n}\ & \\ = z \sum_{n=-\infty}^{\infty} nx(n)z^{-n-1}\ & \\ = -z \sum_{n=-\infty}^{\infty} x(n)(-nz^{-n-1})\ & \\ = -z \sum_{n=-\infty}^{\infty} x(n)\frac{d}{dz}(z^{-n})\ & \\ = -z \frac{dX(z)}{dz}& \\ \end{array} ROC
Convolution \begin{array} {lcl}\mathcal{Z}\{x_1(n)*x_2(n)\} = & \\ \mathcal{Z} \{\sum_{l=-\infty}^{\infty} x_1(l)x_2(n-l)\}\ & \\ = \sum_{n=-\infty}^{\infty} z^{-n}\ & \\ =\sum_{l=-\infty}^{\infty} x_1(l) \sum_{n=-\infty}^{\infty} x_2(n-l)z^{-n} ]\ & \\ = \ & \\ =X_1(z)X_2(z)& \\ \end{array} At least the intersection of ROC1 and ROC2
Cross-correlation At least the intersection of ROC of and
First difference At least the intersection of ROC of X1(z) and
Accumulation \begin{array} {lcl}\sum_{n=-\infty}^{\infty}\sum_{k=-\infty}^{n} x\cdot z^{-n}\\ =\sum_{n=-\infty}^{\infty}(x+x+\\ x\cdots x)z^{-n}\\ =X(1+z^{-1}+z^{-2}+z^{-3}\cdots )\\ =X\sum_{j=0}^{\infty}z^{-j} \\ =X \frac{1}{1-z^{-1}}\end{array}
Multiplication -
Parseval's relation
  • Initial value theorem
, If causal
  • Final value theorem
, Only if poles of are inside the unit circle

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