Fourier transform

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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

X(f) = \int _{-\infty}^{\infty} x(t)\ e^{-i 2\pi f t}\,dt, ,

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

X_k = \sum_{n=0}^{N-1} x_n e^{-\frac{2 \pi i}{N} k n} \quad \quad k = 0, \dots, N-1

where x0, ..., xN-1 is the input sequence, and X0, ..., XN-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.

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