jeffrey@penn:~/blog$ glow z-transform.md

The Z transform

November 1, 2026 #signal-processing #math #tutorial #control

The Z transform is the discrete-time version of the Laplace transform. It is the tool for sampled signals and digital filters.

Definition

For a discrete-time signal $x[n]$ ,

X(z) = \mathcal{Z}\{x[n]\} = \sum_{n=-\infty}^{\infty} x[n]\, z^{-n}, \qquad z \in \mathbb{C}.

Example: a geometric sequence

For $x[n] = a^{n} u[n]$ , the geometric series gives

X(z) = \sum_{n=0}^{\infty} \left(a z^{-1}\right)^{n} = \frac{1}{1 - a z^{-1}}, \qquad |z| > |a| .

There is a pole at $z = a$ , and the region of convergence is everything outside the circle of radius $|a|$ .

Relation to the Fourier and Laplace transforms

On the unit circle $z = e^{j\Omega}$ the Z transform becomes the discrete-time Fourier transform (DTFT), the discrete counterpart of the Fourier transform.

Sampling $x(t)$ every $T$ seconds and taking the Laplace transform of the samples gives the Z transform with

z = e^{sT}.

This maps the $j\omega$ axis onto the unit circle and the left half-plane onto the inside of the unit circle. So the stability rule “poles in the left half-plane” becomes “poles inside the unit circle”.

The s-plane maps to the z-plane: the jω axis becomes the unit circle

Properties

Property	Time domain	$z$ domain
Linearity	$a x_1[n] + b x_2[n]$	$a X_1(z) + b X_2(z)$
Delay	$x[n - k]$	$z^{-k} X(z)$
Difference	$x[n] - x[n-1]$	$(1 - z^{-1}) X(z)$
Convolution	$(x * h)[n]$	$X(z) H(z)$

The delay rule turns difference equations into algebra. The filter $y[n] = a\, y[n-1] + x[n]$ becomes $Y(z) = a z^{-1} Y(z) + X(z)$ , so $H(z) = \frac{1}{1 - a z^{-1}}$ : stable exactly when the pole $|a| < 1$ .

---
title: "The Z transform"
date: 2026-11-01
tags: [signal-processing, math, tutorial, control]
---

The Z transform is the discrete-time version of the [Laplace transform](/blog/laplace-transform/). It is the tool for sampled signals and digital filters.

## Definition

For a discrete-time signal $x[n]$,

$$
X(z) = \mathcal{Z}\{x[n]\} = \sum_{n=-\infty}^{\infty} x[n]\, z^{-n}, \qquad z \in \mathbb{C}.
$$

## Example: a geometric sequence

For $x[n] = a^{n} u[n]$, the geometric series gives

$$
X(z) = \sum_{n=0}^{\infty} \left(a z^{-1}\right)^{n} = \frac{1}{1 - a z^{-1}}, \qquad |z| > |a| .
$$

There is a pole at $z = a$, and the region of convergence is everything outside the circle of radius $|a|$.

## Relation to the Fourier and Laplace transforms

On the unit circle $z = e^{j\Omega}$ the Z transform becomes the discrete-time Fourier transform (DTFT), the discrete counterpart of the [Fourier transform](/blog/fourier-transform/).

Sampling $x(t)$ every $T$ seconds and taking the Laplace transform of the samples gives the Z transform with

$$
z = e^{sT}.
$$

This maps the $j\omega$ axis onto the unit circle and the left half-plane onto the inside of the unit circle. So the stability rule "poles in the left half-plane" becomes "poles inside the unit circle".

![The s-plane maps to the z-plane: the jω axis becomes the unit circle](s-to-z.svg)

## Properties

| Property | Time domain | $z$ domain |
|----------|-------------|------------|
| Linearity | $a x_1[n] + b x_2[n]$ | $a X_1(z) + b X_2(z)$ |
| Delay | $x[n - k]$ | $z^{-k} X(z)$ |
| Difference | $x[n] - x[n-1]$ | $(1 - z^{-1}) X(z)$ |
| Convolution | $(x * h)[n]$ | $X(z) H(z)$ |

The delay rule turns difference equations into algebra. The filter $y[n] = a\, y[n-1] + x[n]$ becomes $Y(z) = a z^{-1} Y(z) + X(z)$, so $H(z) = \frac{1}{1 - a z^{-1}}$: stable exactly when the pole $|a| < 1$.