Calculus
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Fourier Transform

Decomposes a function or signal into the frequencies that make it up — converting between the time (or space) domain and the frequency domain.

Formula

f^(ξ)=∫−∞∞f(x) e−2πixξ dx\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\, e^{-2\pi i x \xi}\, dx

Variables

f(x)Original signal
f̂(ξ)Frequency-domain representation
ξFrequency
iImaginary unit

Example

A musical chord in time → sharp peaks at each note’s frequency

Did You Know?

The Fourier transform underpins JPEG images, MP3 audio, MRI scans and Wi-Fi — almost all modern signal processing.

fouriertransformsignalfrequencyanalysis

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