Tool to calculate the inverse Fourier transform of a function having undergone a Fourier transform, denoted by ^f or F.
Inverse Fourier Transform - dCode
Tag(s) : Functions
dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!
Inverse Fourier Transform
Inverse Fourier Transform Calculator
Answers to Questions (FAQ)
What is the Fourier Inverse Transform? (Definition)
The inverse Fourier transform (IFT) is the reciprocal operation of a Fourier transform.
Several variants of the Fourier transform exist and differ only by a multiplicative coefficient.
For any transformed function $ \hat{f} $, the 3 usual definitions of inverse Fourier transforms are:
— $ (1) $ widespread definition for physics / mechanics / electronics calculations, with $ t $ the time and $ \omega $ in radians per second:
$$ f(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} \hat{f}(\omega) \, \exp(i \omega t) \, \mathrm{d} \omega \tag{1} $$
— $ (2) $ mathematical definition:
$$ f(x) = \frac{1}{2\pi} \int_{-\infty}^{+\infty} \hat{f}(\omega) \, \exp(i \omega x) \, \mathrm{d} \omega \tag{2} $$
— $ (3) $ alternative definition in physics:
$$ f(x) = \int_{-\infty}^{+\infty} \hat{f}(\omega) \, \exp(2 i \pi \omega t) \, \mathrm{d} \omega \tag{3} $$
How to calculate the inverse Fourier transform?
The calculation of the Fourier inverse transform is an integral calculation (see definitions above).
On dCode, indicate the function, its transformed variable (often $ \omega $ or $ w $ or even $ \xi $) and it's initial variable (often $ x $ or $ t $).
Example: $ \hat{f}(\omega) = \frac{1}{\sqrt{2\pi}} $ and $ f(t) = \delta(t) $ with the $ \delta $ Dirac function.
Source code
dCode retains ownership of the "Inverse Fourier Transform" source code. Any algorithm for the "Inverse Fourier Transform" algorithm, applet or snippet or script (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or any "Inverse Fourier Transform" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) or any database download or API access for "Inverse Fourier Transform" or any other element are not public (except explicit open source licence like Creative Commons). Same with the download for offline use on PC, mobile, tablet, iPhone or Android app.
Reminder: dCode is an educational and teaching resource, accessible online for free and for everyone.
Cite dCode
The content of the page "Inverse Fourier Transform" and its results may be freely copied and reused, including for commercial purposes, provided that dCode.fr is cited as the source.
Exporting the results is free and can be done simply by clicking on the export icons ⤓ (.csv or .txt format) or ⧉ (copy and paste).
To cite dCode.fr on another website, use the link:
In a scientific article or book, the recommended bibliographic citation is: Inverse Fourier Transform on dCode.fr [online website], retrieved on 2025-04-16,
