User:Vanished user fijw983kjaslkekfhj45/Z-Transform Tables

Bilateral Z-transform

 * $$X(z) = \mathcal{Z}\{x[n]\} = \sum_{n=-\infty}^{\infty} x[n] z^{-n} $$

where n is an integer and z is, in general, a complex number.

Unilateral Z-transform
Alternatively, in cases where x[n] is defined only for n ≥ 0, the single-sided or unilateral Z-transform is defined as


 * $$X(z) = \mathcal{Z}\{x[n]\} = \sum_{n=0}^{\infty} x[n] z^{-n}. \ $$

Inverse Z-transform

 * $$ x[n] = \mathcal{Z}^{-1} \{X(z) \}= \frac{1}{2 \pi j} \oint_{C} X(z) z^{n-1} dz \ $$

where $$ C \ $$ is a counterclockwise closed path encircling the origin and entirely in the region of convergence (ROC). The contour or path, $$ C \ $$, must encircle all of the poles of $$ X(z) \ $$.

Properties

 * 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 the unit circle

Table of common Z-transform pairs
Here:
 * $$u[n] = \begin{cases} 1, & n \ge 0 \\ 0, & n < 0 \end{cases}$$
 * $$\delta[n] = \begin{cases} 1, & n = 0 \\ 0, & n \ne 0 \end{cases}$$