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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-time Fourier transform ： ウィキペディア英語版 Discrete-time Fourier transform In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to the uniformly-spaced samples of a continuous function. The term ''discrete-time'' refers to the fact that the transform operates on discrete data (samples) whose interval often has units of time. From only the samples, it produces a function of frequency that is a periodic summation of the continuous Fourier transform of the original continuous function. Under certain theoretical conditions, described by the sampling theorem, the original continuous function can be recovered perfectly from the DTFT and thus from the original discrete samples. The DTFT itself is a continuous function of frequency, but discrete samples of it can be readily calculated via the discrete Fourier transform (DFT) (see Sampling the DTFT), which is by far the most common method of modern Fourier analysis. Both transforms are invertible. The inverse DTFT is the original sampled data sequence. The inverse DFT is a periodic summation of the original sequence. The fast Fourier transform (FFT) is an algorithm for computing one cycle of the DFT, and its inverse produces one cycle of the inverse DFT. == Definition == The discrete-time Fourier transform of a discrete set of real or complex numbers: ''x''(), for all integers ''n'', is a Fourier series, which produces a periodic function of a frequency variable. When the frequency variable, ω, has normalized units of ''radians/sample'', the periodicity is 2π, and the Fourier series is: The utility of this frequency domain function is rooted in the Poisson summation formula. Let X(f) be the Fourier transform of any function, x(t), whose samples at some interval, T (''seconds''), are equal (or proportional to) the x() sequence, i.e. $T\backslash cdot\; x(nT)=x()$. Then the periodic function represented by the Fourier series is a periodic summation of X(f). In terms of frequency $\backslash scriptstyle\; f$ in hertz (''cycles/sec''): \sum_^ \underbrace_\ e^\; \stackrel \; \sum_^ X\left(f - k/T\right). |}} The integer ''k'' has units of ''cycles/sample'', and 1/''T'' is the sample-rate, ''fs'' (''samples/sec''). So ''X''1/''T''(''f'') comprises exact copies of ''X''(''f'') that are shifted by multiples of ''fs'' hertz and combined by addition.  For sufficiently large ''fs'' the ''k''=''0'' term can be observed in the region (''fs''/2 ) with little or no distortion (aliasing) from the other terms. In Fig.1, the extremities of the distribution in the upper left corner are masked by aliasing in the periodic summation (lower left). We also note that  $e^$  is the Fourier transform of  $\backslash scriptstyle\; \backslash delta(t-nT).$  Therefore, an alternative definition of DTFT is:〔In fact is often justified as follows: :$\backslash begin$ \mathcal\left \ T\cdot x(nT) \cdot \delta(t-nT)\right \} &=\mathcal\left \ \delta(t-nT)\right \}\\ &= X(f) * \mathcal\left \ \delta(t-nT)\right \} \\ &= X(f) * \sum_^ \delta \left(f - \frac\right) \\ &= \sum_^ X\left(f - \frac\right). \end〕 The modulated Dirac comb function is a mathematical abstraction sometimes referred to as ''impulse sampling''. 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-time 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.