Fourier Transform

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Fourier Transform
Image:FourierTransform.png
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Fourier Transform

A Fourier Transform changes a function's domain from time to frequency


Basic Description

Given a function in the time domain, f(x), a Fourier Transform produces a function F(u) where u is a frequency and F(u)returns a complex number whose real component gives the amplitude of the sine wave at frequency u and whose imaginary component gives the phase.

A More Mathematical Explanation

The equation for the Fourier Transform where f(t) is a discrete function with domain [0,N) is


F(u)=\sum_{t=0}^{N-1}f(t)e^{\frac{-2 \pi i u t}{N}}=\sum_{t=0}^{N-1}f(t)\left(\cos\left(\frac{-2 \pi u t}{N}\right)+i \sin\left(\frac{-2 \pi u t}{N}\right)\right)

Demonstration

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