Z-Transform: Definition, ROC, and Transform Pairs

Why Do We Need Another Transform?

The Z-transform is to discrete-time signals what the Laplace transform is to continuous-time signals. It converts:

• Convolution → Multiplication (algebraic)
• Difference equations → Algebraic equations
• Time-domain analysis → Complex frequency-domain analysis

It’s the primary tool for analyzing, designing, and implementing discrete-time systems.

---

Definition

For a discrete-time signal $x[n]$, the bilateral Z-transform is:

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

where $z = re^{j\omega}$ is a complex variable.

Region of Convergence (ROC)

The sum converges only for certain $z$. The ROC is the set of $z$ where $

X(z)

< \infty$.

The Z-transform is NOT complete without its ROC. Two different signals can have the same $X(z)$ formula but different ROCs.

---

ROC Properties: The “Must-Know” Rules

Signal Type	ROC
Right-sided ($x[n]=0$ for $n<N$)	$	z	> r_{\text{max}}$ (exterior of circle)
Left-sided ($x[n]=0$ for $n>N$)	$	z	< r_{\text{min}}$ (interior of circle)
Two-sided (infinite both ways)	$r_{\text{min}} <	z	< r_{\text{max}}$ (annulus)
Finite-duration (nonzero only for $N_1 \le n \le N_2$)	Entire $z$-plane except $z=0$ or $z=\infty$

Key ROC Rules:

1. ROC cannot contain poles
2. ROC is a connected region (annulus, disc, or exterior)

3. If $x[n]$ is causal → ROC is $

   z

   > r_{\text{max}}$

4. If $x[n]$ is anti-causal → ROC is $

   z

   < r_{\text{min}}$

5. For stability: ROC must include unit circle $

   z

   =1$

---

Common Z-Transform Pairs

Signal $x[n]$	$X(z)$	ROC
$\delta[n]$	$1$	All $z$
$\delta[n-k]$	$z^{-k}$	$z \ne 0$ ($k>0$) or $z \ne \infty$ ($k<0$)
$u[n]$	$\frac{1}{1-z^{-1}} = \frac{z}{z-1}$	$	z	> 1$
$-u[-n-1]$	$\frac{1}{1-z^{-1}}$	$	z	< 1$
$a^n u[n]$	$\frac{1}{1-az^{-1}} = \frac{z}{z-a}$	$	z	>	a	$
$-a^n u[-n-1]$	$\frac{1}{1-az^{-1}}$	$	z	<	a	$
$n a^n u[n]$	$\frac{az^{-1}}{(1-az^{-1})^2}$	$	z	>	a	$
$\cos(\omega_0 n) u[n]$	$\frac{1-\cos(\omega_0)z^{-1}}{1-2\cos(\omega_0)z^{-1}+z^{-2}}$	$	z	> 1$

Notice: $a^n u[n]$ and $-a^n u[-n-1]$ have identical $X(z)$ but different ROCs. The ROC tells you which signal you have!

---