Properties of the Z-Transform

Properties of the Z-Transform#

	Property	Discrete Time Domain	$Z$ Transform
1	Linearity	$a f_{1} [n] + b f_{2} [n] + \dots$	$a F_{1} (z) + b F_{2} (z) + \dots$
2	Shift of $x [n] u_{0} [n]$	$f [n - m] u_{0} [n - m]$	$z^{- m} F (z)$
3	Left shift	$f [n - m]$	$z^{- m} F (z) + \sum_{n = 0}^{m - 1} f [n - m] z^{- n}$
4	Right shift	$f [n + m]$	$z^{m} F (z) + \sum_{n = - m}^{- 1} f [n + m] z^{- n}$
5	Multiplication by $a^{n}$	$a^{n} f [n]$	$F (\frac{z}{a})$
6	Multiplication by $e^{- n s T_{s}}$	$e^{- n s T_{s}} f [n]$	$F (e^{s T_{s}} z)$
7	Multiplication by $n$	$n f [n]$	$- z \frac{d}{d z} F (z)$
8	Multiplication by $n^{2}$	$n^{2} f [n]$	$- z \frac{d}{d z} F (z) + z^{2} \frac{d^{2}}{d z^{2}} F (z)$
9	Summation in time	$\sum_{m = 0}^{n} f [m]$	$\frac{z}{z - 1} F (z)$
10	Time convolution	$f_{1} [n] * f_{2} [n]$	$F_{1} (z) F_{2} (z)$
11	Frequency convolution	$f_{1} [n] f_{2} [n]$	$\frac{1}{j 2 π} \oint x F_{1} (v) F_{2} (\frac{z}{v}) v^{- 1} d v$
12	Initial value theorem	$f [0] = lim_{z \to \infty} F (z)$
13	Final value theorem	$lim_{n \to \infty} f [n] = lim_{z \to 1} (z - 1) F (z)$

See also: Wikibooks: Engineering_Tables/Z_Transform_Properties and Z-Transform—WolframMathworld for more complete references.