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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Discrete Fourier transform ： ウィキペディア英語版 Discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite list of equally spaced samples of a function into the list of coefficients of a finite combination of complex sinusoids, ordered by their frequencies, that has those same sample values. It can be said to convert the sampled function from its original domain (often time or position along a line) to the frequency domain. The input samples are complex numbers (in practice, usually real numbers), and the output coefficients are complex as well. The frequencies of the output sinusoids are integer multiples of a fundamental frequency, whose corresponding period is the length of the sampling interval. The combination of sinusoids obtained through the DFT is therefore periodic with that same period. The DFT differs from the discrete-time Fourier transform (DTFT) in that its input and output sequences are both finite; it is therefore said to be the Fourier analysis of finite-domain (or periodic) discrete-time functions. The DFT is the most important discrete transform, used to perform Fourier analysis in many practical applications. In digital signal processing, the function is any quantity or signal that varies over time, such as the pressure of a sound wave, a radio signal, or daily temperature readings, sampled over a finite time interval (often defined by a window function). In image processing, the samples can be the values of pixels along a row or column of a raster image. The DFT is also used to efficiently solve partial differential equations, and to perform other operations such as convolutions or multiplying large integers. Since it deals with a finite amount of data, it can be implemented in computers by numerical algorithms or even dedicated hardware. These implementations usually employ efficient fast Fourier transform (FFT) algorithms;〔Cooley et al., 1969〕 so much so that the terms "FFT" and "DFT" are often used interchangeably. Prior to its current usage, the "FFT" initialism may have also been used for the ambiguous term "finite Fourier transform". ==Definition== The sequence of ''N'' complex numbers $x\_0,\; x\_1,\; \backslash ldots,\; x\_$ is transformed into an ''N''-periodic sequence of complex numbers: \ \sum_^ x_n \cdot e^, \quad k\in\mathbb\, (integers)   〔In this context, it is common to define $\backslash omega$ to be the Nth primitive root of unity, $\backslash omega\; =\; e^$, to obtain the following form: :$X\_k\; =\; \backslash sum\_^\; x\_n\; \backslash cdot\; \backslash omega^$〕|}} Because of periodicity, the customary domain of k actually computed is (). That is always the case when the DFT is implemented via the Fast Fourier transform algorithm. But other common domains are  ()  (''N'' even)  and  ()  (''N'' odd), as when the left and right halves of an FFT output sequence are swapped. The transform is sometimes denoted by the symbol $\backslash mathcal$, as in $\backslash mathbf\; =\; \backslash mathcal\; \backslash left\; \backslash$ or $\backslash mathcal\; \backslash left\; (\; \backslash mathbf\; \backslash right\; )$ or $\backslash mathcal\; \backslash mathbf$.〔As a linear transformation on a finite-dimensional vector space, the DFT expression can also be written in terms of a DFT matrix; when scaled appropriately it becomes a unitary matrix and the ''X''''k'' can thus be viewed as coefficients of ''x'' in an orthonormal basis.〕 can be interpreted or derived in various ways, for example: *It completely describes the discrete-time Fourier transform (DTFT) of an N-periodic sequence, which comprises only discrete frequency components. (Using the DTFT with periodic data) *It can also provide uniformly spaced samples of the continuous DTFT of a finite length sequence. (Sampling the DTFT) *It is the cross correlation of the ''input'' sequence, ''xn'', and a complex sinusoid at frequency ''k''/''N''.  Thus it acts like a matched filter for that frequency. *It is the discrete analogy of the formula for the coefficients of a Fourier series: :which is also N-periodic. In the domain  $\backslash scriptstyle\; n\backslash \; \backslash in\backslash \; (N-1),$  this is the inverse transform of . In this interpretation, each $X\_k$ is a complex number that encodes both amplitude and phase of a sinusoidal component  $(e^)$  of function $x\_n$. The sinusoid's frequency is ''k'' cycles per ''N'' samples.  Its amplitude and phase are: ::$|X\_k|/N\; =\; \backslash sqrt(X\_k)^2\}/N$ ::$\backslash arg(X\_k)\; =\; \backslash operatorname\backslash big(\; \backslash operatorname(X\_k),\; \backslash operatorname(X\_k)\; \backslash big)=-i\backslash operatorname\backslash left(\backslash frac\backslash right),$ :where atan2 is the two-argument form of the arctan function. The normalization factor multiplying the DFT and IDFT (here 1 and 1/''N'') and the signs of the exponents are merely conventions, and differ in some treatments. The only requirements of these conventions are that the DFT and IDFT have opposite-sign exponents and that the product of their normalization factors be 1/''N''.  A normalization of $\backslash scriptstyle\; \backslash sqrt$ for both the DFT and IDFT, for instance, makes the transforms unitary. In the following discussion the terms "sequence" and "vector" will be considered interchangeable. Using Euler's Formula, it can be derived further to the forms commonly used in Engineering and Computer Science. Fourier Transform: Inverse Fourier Transform: ::N = number of time samples we have ::n = current sample we're considering (0...N-1) ::x = value of the signal at time n ::k = current frequency we're considering (0 Hertz up to N-1 Hertz) ::X = amount of frequency k in the signal (Amplitude and Phase, a complex number) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Discrete Fourier transform」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.