# discrete Fourier transform

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## discrete Fourier transform

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(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 -1ƒ(r) exp (-inr / 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).

## FFT

(Fast Fourier Transform) 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.

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 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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References in periodicals archive ?
The discrete Fourier transform is a one-to-one correspondence between the spaces [L.sub.2](h[Z.sup.m]) and [L.sub.2](h[T.sup.m]) with norms
The discrete Fourier transform is used for processing the received signal, and the sparse representation of the original signal will be obtained as been shown in Figure 3.
Therefore, based on the translation property of the two-dimensional discrete Fourier transform, and the equivalent form of the fractional order partial differential in the frequency domain can be obtained:
The discrete Fourier transform described above is used for the image registration.
In contrast to that, its counterpart the discrete Fourier transform produces a 2[pi]-periodic spectrum, which is defined for a positive number of samples 0 [less than or equal to] n < N and 0 [less than or equal to] m frequencies.
If amplitude spectra (computed by means of the discrete Fourier transform) are transformed to polar coordinates, only a half of the domain on the angular axis is sufficient.
Equation (25) implies that [m.sub.i] is the inverse discrete Fourier transform of [[PSI].sub.AN](n).
This method uses spectrum analysis of sessions by Discrete Fourier Transform (DFT).
Like the Discrete Fourier Transform (DFT), a DCT operates on a function at a finite number of discrete data points.
Allen, "Short term spectral analysis, synthesis, and modification by discrete Fourier transform," IEEE Transactions on Acoustics, Speech, and Signal Processing, vol.
The DCT is based on the Discrete Fourier Transform which replaces the complex analysis with real numbers by a symmetric signal extension.

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