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
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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.
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Inverse Fourier Transform on dCode.fr [online website], retrieved on 2024-07-27,
