# Fourier Transform

Fourier Transform
Field: Algebra
Image Created By: [[Author:| ]]

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 '"UNIQ--math-0 [...]

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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# Teaching Materials

A more mathematical explanation and proof of the formula.