Tool to calculate the Fourier transform of an integrable function on R, the Fourier transform is denoted by ^f or F.

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*!

The Fourier transformation of a function $ f $ is denoted $ \hat{f} $ (or sometimes $ F $), its result (the transform) describes the frequency spectrum of $ f $.

Several definitions of the Fourier transform coexist, they are identical except for a multiplicative coefficient (which can simplify the calculations)

For any function $ f $ integrable on $ \mathbb{R} $, the 3 most common Fourier transforms of $ f $ are:

— $ (1) $ most used definition in physics / mechanics / electronics, with time $ t $ and frequency $ \omega $ in rad/sec:

$$ \hat{f}(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} f(t) \, \exp(i \omega t) \, \mathrm{d} t \tag{1} $$

The advantage of the factor $ \frac{1}{\sqrt{2\pi}} $ is that it can be reused for the inverse Fourier transform.

— $ (2) $ basic mathematical definition, without coefficient:

$$ \hat{f}(\omega) = \int_{-\infty}^{+\infty} f(x) \, \exp(-i \omega x) \, \mathrm{d} x \tag{2} $$

— $ (3) $ alternative definition in physics:

$$ \hat{f}(\omega) = \int_{-\infty}^{+\infty} f(t) \, \exp(-i 2 \pi \omega t) \, \mathrm{d} t \tag{3} $$

The calculation of the Fourier transform is an integral calculation (see definitions above).

On dCode, indicate the function, its variable, and the transformed variable (often $ \omega $ or $ w $ or even $ \xi $).

__Example:__ $ f(x) = \delta(t) $ and $ \hat{f}(\omega) = \frac{1}{\sqrt{2\pi}} $ with the $ \delta $ Dirac function.

dCode retains ownership of the "Fourier Transform" source code. Except explicit open source licence (indicated Creative Commons / free), the "Fourier Transform" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "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.) and all data download, script, or API access for "Fourier Transform" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!

Reminder : dCode is free to use.

The copy-paste of the page "Fourier Transform" or any of its results, is allowed (even for commercial purposes) as long as you credit dCode!

Exporting results as a .csv or .txt file is free by clicking on the *export* icon

Cite as source (bibliography):

*Fourier Transform* on dCode.fr [online website], retrieved on 2024-06-25,

fourier,transform,function,frequency

https://www.dcode.fr/fourier-transform

© 2024 dCode — El 'kit de herramientas' definitivo para resolver todos los juegos/acertijos/geocaching/CTF.

Feedback