# Fourier transform

### From lingwiki

A **Fourier Transform** is a kind of transformation of one function to another. Specifically, it transforms a function from a "time domain" to its representation in a "frequency domain" in which certain operations are easier to perform.

## [edit] Integral Fourier Transforms

An **Integral Fourier Transform** takes a continuous time signal function, decomposes it into harmonics of various frequencies, and outputs a continuous spectrum of the magnitudes and phases of these frequencies.

It is defined by

,

where *f* is a real number representing frequency, and *t* represents time in seconds.

## [edit] Discrete Fourier Transforms

A **Discrete Fourier Transform** takes a discrete input of *n* complex numbers, corresponding to equally spaced points on some continuous function, and outputs *n* complex numbers, each describing a sine function of a given frequency. It is defined by

where *x*_{0}, ..., *x*_{N-1} is the input sequence, and *X*_{0}, ..., *X*_{N-1} is the output sequence.

## [edit] Use of Fourier Transforms in Speech Recognition

Discrete Fourier transforms are used in digital signal processing to analyze frequencies contained in a sample of a signal. In speech recognition, these samples are of speech sounds. The output of the transformation is similar to the output of a spectrograph; this makes it possible to identify phonetic features and determine sequences of phonemes in the original signal.