Fourier transform

lastmod 2026-01-11created 2014-10-1223 edits

The Fourier transform decomposes a function of time (e.g. a signal) into its frequencies. This is similar to expressing a musical chord by the amplitudes (loudness) of the notes that make it up. The Fourier transform of a function of time is a complex-valued function of frequency, whose absolute value represents how much of that frequency is present in the original function.

The Fourier transform is a linear transform that takes one function as an argument and transforms it into another function. The transformed function is generally called the frequency-domain representation of the original function.

Fast Fourier transform

• Wikipedia (en) explanation of the fast Fourier transform
• Fourier transform formulas
• Fast Fourier Transform

The fast Fourier transform (FFT) is an efficient algorithm that quickly computes the discrete Fourier transform (DFT) and its inverse. FFT is used in many fields, from digital signal processing to algorithms for finding the roots of partial differential equations.

Discrete Fourier transform

• Wikipedia (en) explanation of the DFT

The discrete Fourier transform (DFT) is the Fourier transform of a discrete input signal, used in fields such as digital signal analysis. The discrete Fourier transform can be computed quickly using the fast Fourier transform.

Documentation

Libraries

• Benchmarked FFT Implementations

List of libraries:

• FFTW
• cuFFT
• Kiss FFT
• OpenCV:FourierTransform

See also

• Fourier series
• DWT
• Spectrogram

Favorite site

• Wikipedia (en) explanation of the Fourier transform
• Collection of posts on the Fourier transform
• ICT terminology glossary - Fourier Transform ¹
• The beginning of the Fourier series (Fourier series) ²
• A younger brother better than the elder: the Fourier transform (Fourier transform) ³
• [Recommended] Fourier transform (fourier transform) (basic explanation, with Python code) ⁴
• [Recommended] An interactive introduction to the Fourier transform
• The unreasonable effectiveness of the Fourier transform | GeekNews
  • A talk showing how powerfully the Fourier transform works in real-world technical applications

Project

• JTransforms: multithreaded FFT library written in pure Java

Sample code

• Simple DFT in C
• cips2edsrc DFT.C
• How to implement the discrete Fourier transform
• nolandda.org: dft.c
• [Recommended] DFT in c code ⁵
  • FFT C code
• FFT Source

References