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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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:

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

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 2024-06-25,

fourier,inverse,transform,function

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

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