$\mathcal{Z}$ -Transforms

Definition

$\large x[n]$ be a discrete-time signal. Then,

Z {x [n]} = \sum_{k = 0}^{\infty} x [k] z^{- k}

$\large \mathcal{Z}\{x[n]\}= X(z)$ $\mathcal{Z}$ -Transform $\large x[n]$ , assuming the sum converges.

Properties

Linearity

$\Large \mathcal{Z}\{c_1x_1[n] + c_2x_2[n]\} = c_1X_1(z) + c_2X_2(z)$

Time Advance

$\Large \mathcal{Z}\{x[n+k]\} = z^{k}X(z)$

Time Delay

$\Large \mathcal{Z}\{x[n-k]\} = z^{-k}X(z)$

$n$

$\Large \mathcal{Z}\{nx[n]\} = -z\frac{dX(z)}{dz}$

$a^n$

$\Large \mathcal{Z}\{a^n x[n]\} = X(\frac{z}{\alpha})$

Folding

$\Large \mathcal{Z}\{x[-n]\} = X(\frac{1}{z})$

Convolution

$\Large \mathcal{Z}\{x*y\} = X(z)Y(z)$

Conjugation

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

Real & Imaginary Components

$\Large \mathcal{Z}\{Re\{x[n]\}\} = \frac{1}{2}[X(z) + X^*(z^*)]$
$\Large \mathcal{Z}\{Im\{x[n]\}\} = \frac{1}{2}[X(z) - X^*(z^*)]$

Initial Value Theorem

$\Large x[0] = \lim\limits_{z \to \infty} X(z)$

Final Value Theorem

$\Large \lim\limits_{n \to \infty} x[n] = \lim\limits_{z \to 1} (z-1)X(z)$

$\large x[n]$ $\large n \to \infty$ .

$\mathcal{Z}$ -Transform Table

$\large x[n]$	$\mathcal{Z}$ $X(z)$	Region of Convergence
$\Large \delta [n]$	$\Large 1$	$\large \text{All } z$
$\Large \delta_k [n]$	$\Large z^{-k}$	$\large z \ne 0, \: z \ne \infty$
$\Large u[n]$	$\huge \frac{z}{z-1}$	$\large \abs{z} \gt 1$
$\Large \alpha^n u[n]$	$\huge \frac{z}{z-\alpha}$	$\large \abs{z} \gt \alpha$
$\Large - \alpha^n u[- n - 1]$	$\huge \frac{z}{z-\alpha}$	$\large \abs{z} \lt \alpha$
$\Large n \alpha^n u[n]$	$\huge \frac{\alpha z}{(z-\alpha)^2}$	$\large \abs{z} \gt \alpha$
$\Large - n \alpha^n u[- n - 1]$	$\huge \frac{\alpha z}{(z-\alpha)^2}$	$\large \abs{z} \lt \alpha$
$\Large u[n] sin(\omega_0 n)$	$\huge \frac{z\:sin(\omega_0)}{z^2-2z\:cos(\omega_0) + 1}$	$\large \abs{z} \gt 1$
$\Large u[n] cos(\omega_0 n)$	$\huge \frac{z^2-z\:cos(\omega_0)}{z^2-2z\:cos(\omega_0) + 1}$	$\large \abs{z} \gt 1$
$\Large a^n u[n] sin(\omega_0 n)$	$\huge \frac{\alpha z\:sin(\omega_0)}{z^2-2\alpha z\:cos(\omega_0) + \alpha^2}$	$\large \abs{z} \gt \alpha$
$\Large a^n u[n] cos(\omega_0 n)$	$\huge \frac{z^2-\alpha z\:cos(\omega_0)}{z^2-2\alpha z\:cos(\omega_0) + \alpha^2}$	$\large \abs{z} \gt \alpha$

Power Series

$\large \rho$ $\large |\rho| < 1$ . Then,

\sum_{k = 0}^{\infty} ρ^{k} = \frac{1}{1 - ρ}

By taking the derivative, we can also derive:

\sum_{k = 0}^{\infty} k ρ^{k} = \frac{ρ}{(1 - ρ)^{2}}

Z\mathcal{Z}-Transforms