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

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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.

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*Fourier Transform* on dCode.fr [online website], retrieved on 2024-11-05,

fourier,transform,function,frequency

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