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

The Fourier transform

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

The Fourier transform writes a continuous-time signal $x(t)$ as a sum of pure frequencies $e^{j\omega t}$ and tells you how much of each frequency it contains. It is the first of three related transforms; the Laplace transform and the Z transform extend the same idea.

Definition

X(\omega) = \mathcal{F}\{x(t)\} = \int_{-\infty}^{\infty} x(t)\, e^{-j\omega t}\, dt

and the inverse transform rebuilds the signal from its frequencies:

x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} X(\omega)\, e^{j\omega t}\, d\omega .

$X(\omega)$ is complex: $|X(\omega)|$ is how strong frequency $\omega$ is, and $\angle X(\omega)$ is its phase.

Example: a rectangular pulse

Let $x(t) = 1$ for $|t| < T/2$ and $0$ otherwise. Then

X(\omega) = \int_{-T/2}^{T/2} e^{-j\omega t}\, dt = \frac{2 \sin(\omega T / 2)}{\omega} = T \,\operatorname{sinc}\!\left(\frac{\omega T}{2\pi}\right).

A pulse that is short in time is wide in frequency, and the other way around. This trade-off is the uncertainty principle of signal processing.

Useful properties

Property	Time domain	Frequency domain
Linearity	$a x_1(t) + b x_2(t)$	$a X_1(\omega) + b X_2(\omega)$
Time shift	$x(t - t_0)$	$e^{-j\omega t_0} X(\omega)$
Modulation	$e^{j\omega_0 t} x(t)$	$X(\omega - \omega_0)$
Derivative	$\dot{x}(t)$	$j\omega X(\omega)$
Convolution	$(x * h)(t)$	$X(\omega) H(\omega)$

The last row is why the transform matters for linear time-invariant (LTI) systems: filtering, which is a convolution in time, becomes a simple multiplication in frequency.

When it doesn’t exist

The integral only converges when $x(t)$ dies out fast enough, for example when $\int_{-\infty}^{\infty} |x(t)|\, dt < \infty$ . A growing signal like $e^{t}u(t)$ has no Fourier transform. The Laplace transform fixes this.

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

The Fourier transform writes a continuous-time signal $x(t)$ as a sum of pure frequencies $e^{j\omega t}$ and tells you how much of each frequency it contains. It is the first of three related transforms; the [Laplace transform](/blog/laplace-transform/) and the [Z transform](/blog/z-transform/) extend the same idea.

## Definition

$$
X(\omega) = \mathcal{F}\{x(t)\} = \int_{-\infty}^{\infty} x(t)\, e^{-j\omega t}\, dt
$$

and the inverse transform rebuilds the signal from its frequencies:

$$
x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} X(\omega)\, e^{j\omega t}\, d\omega .
$$

$X(\omega)$ is complex: $|X(\omega)|$ is how strong frequency $\omega$ is, and $\angle X(\omega)$ is its phase.

## Example: a rectangular pulse

Let $x(t) = 1$ for $|t| < T/2$ and $0$ otherwise. Then

$$
X(\omega) = \int_{-T/2}^{T/2} e^{-j\omega t}\, dt = \frac{2 \sin(\omega T / 2)}{\omega} = T \,\operatorname{sinc}\!\left(\frac{\omega T}{2\pi}\right).
$$

A pulse that is short in time is wide in frequency, and the other way around. This trade-off is the uncertainty principle of signal processing.

## Useful properties

| Property | Time domain | Frequency domain |
|----------|-------------|------------------|
| Linearity | $a x_1(t) + b x_2(t)$ | $a X_1(\omega) + b X_2(\omega)$ |
| Time shift | $x(t - t_0)$ | $e^{-j\omega t_0} X(\omega)$ |
| Modulation | $e^{j\omega_0 t} x(t)$ | $X(\omega - \omega_0)$ |
| Derivative | $\dot{x}(t)$ | $j\omega X(\omega)$ |
| Convolution | $(x * h)(t)$ | $X(\omega) H(\omega)$ |

The last row is why the transform matters for linear time-invariant (LTI) systems: filtering, which is a convolution in time, becomes a simple multiplication in frequency.

## When it doesn't exist

The integral only converges when $x(t)$ dies out fast enough, for example when $\int_{-\infty}^{\infty} |x(t)|\, dt < \infty$. A growing signal like $e^{t}u(t)$ has no Fourier transform. The [Laplace transform](/blog/laplace-transform/) fixes this.