# discrete Fourier transform

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## discrete Fourier transform

[di¦skrēt für·yā ′tranz‚fȯrm] (mathematics)

A generalization of the Fourier transform to finite sets of data; for a function ƒ defined at

*N*data values, 0, 1, 2, …,*N*- 1, the discrete Fourier transform is a function, ƒ, also defined on the set (0, 1, 2, …,*N*- 1, the discrete Fourier transform is a function, ƒ, also defined on the set (0, 1, 2, …,*N*- 1), whose value at*n*is the sum over the variable*r*, from 0 through*N*-1, of the quantity*N*^{-1}ƒ(*r*) exp (-*i*2π*nr*/*N*).McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

## discrete Fourier transform

(mathematics)(DFT) A Fourier transform, specialized to the
case where the abscissas are integers.

The DFT is central to many kinds of signal processing, including the analysis and compression of video and sound information.

A common implementation of the DFT is the Fast Fourier Transform (FFT).

See also discrete cosine transform.

The DFT is central to many kinds of signal processing, including the analysis and compression of video and sound information.

A common implementation of the DFT is the Fast Fourier Transform (FFT).

See also discrete cosine transform.

This article is provided by FOLDOC - Free Online Dictionary of Computing (

**foldoc.org**)## FFT

(**F**ast

**F**ourier

**T**ransform) A computer algorithm used in digital signal processing (DSP) to modify, filter and decode digital audio, video and images. FFTs commonly change the time domain into the frequency domain.

**Myriad Recognition Uses**

FFTs are widely used in voice recognition and myriad other pattern recognition applications. For example, noise-cancelling headphones use FFT to turn unwanted sounds into simple waves so that inverse signals can be generated to cancel them. FFTs are used to sharpen edges and create effects in static images and are widely used to turn a number series into sine waves and graphs.

**Fast DFT Processing**

The FFT quickly performs a discrete Fourier transform (DFT), which is the practical application of Fourier transforms. Developed by Jean Baptiste Joseph Fourier in the early 19th century, the Fourier equations were invented to transform one complex function into another. One of Fourier's primary goals was to predict the rate of heat transfer based on temperature, mass and proximity. In practice, the terms FFT, DFT and Fourier transform are used synonymously. See DSP.

An FFT Transform |
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A Fourier transform was used to chart the power levels at different frequencies from the half second of digital samples (top). |

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