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Formal Sciences
Mathematics
1822
Advanced

Fourier Transform

f^(ξ)=∫−∞∞f(x) e−2πixξ dx\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\, e^{-2\pi i x \xi}\, dx

Decomposes any signal into its constituent frequencies—revealing the hidden spectrum within data.

By Joseph Fourier

Formal Sciences
Fourier Transform
1822 · Joseph Fourier
Expert Reviewed
89%

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Story PortalWho discovered this?Discover how Joseph Fourier and others shaped this equation.Visual PortalWhat does it look like?See the equation come alive in the Visual Studio.Machine PortalWhere is it used?Explore machines powered by this equation — MRI scanners and smartphones.Math PortalWhat does it derive from?Trace the mathematical lineage from integrals and sine waves.Future PortalWhere is it going?Quantum Fourier transform and neural signal processing
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Why it matters: Foundation of modern telecommunications, imaging, audio, and data compression.

Discoverers: Joseph Fourier (1822)

What does it mean?

Decomposes any signal into its constituent frequencies—revealing the hidden spectrum within data.

Why should I care?

Foundation of modern telecommunications, imaging, audio, and data compression.

Equation Compass

North — Prerequisites

West — History

East — Applications

South — Derivations

Variables & Units

SymbolNameUnitMeaning
f(x)f(x)Signal—Time-domain function
ξξFrequencyHzFrequency variable
f^(ξ)f̂(ξ)Spectrum—Frequency-domain representation

Worked Example

A pure tone sin(2πft) transforms to peaks at ±f in frequency domain.

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Continue your trail

MRI ScannerWi-Fi RouterSmartphoneTelecommunications

Pictures & video

Portrait of Joseph Fourier
Joseph Fourier, whose analysis of heat flow gave us the Fourier transform.Wikimedia Commons · Public domain

But what is the Fourier Transform? A visual introduction

3Blue1Brown

Sources & further reading

Fourier transform — WikipediaSearch WikipediaSearch Wolfram MathWorldOpen in Wolfram|AlphaWatch explainers on YouTube

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Equation Universe

Fourier Transform

f^(ξ)=∫−∞∞f(x) e−2πixξ dx\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\, e^{-2\pi i x \xi}\, dx

Real-world impact

JPEG, MP3 & 5G

Frequency decomposition compresses media and shapes wireless signals.

Photo: Unsplash — data center

Decomposes any signal into its constituent frequencies—revealing the hidden spectrum within data.

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