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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 (general) ： ウィキペディア英語版 Discrete Fourier transform (general) This article is about the discrete Fourier transform (DFT) over any ring, commonly called a number-theoretic transform (NTT) in the case of finite fields. For specific information on the discrete Fourier transform over the complex numbers, see discrete Fourier transform. ==Definition== Let $R$ be any ring, let $n\backslash geq\; 1$ be an integer, and let $\backslash alpha\; \backslash in\; R$ be a principal ''n''th root of unity, defined by:〔Martin Fürer, "(Faster integer multiplication )", STOC 2007 Proceedings, pp. 57–66. Section 2: The Discrete Fourier Transform.〕 *$\backslash alpha^n\; =\; 1$ *$\backslash sum\_^\; \backslash alpha^\; =\; 0$ for The discrete Fourier transform maps an n-tuple $(v\_0,\backslash ldots,v\_)$ of elements of $R$ to another n-tuple $(f\_0,\backslash ldots,f\_)$ of elements of $R$ according to the following formula: :$f\_k\; =\; \backslash sum\_^\; v\_j\backslash alpha^.\backslash qquad\; (2)$ By convention, the tuple $(v\_0,\backslash ldots,v\_)$ is said to be in the ''time domain'' and the index $j$ is called ''time''. The tuple $(f\_0,\backslash ldots,f\_)$ is said to be in the ''frequency domain'' and the index $k$ is called ''frequency''. The tuple $(f\_0,\backslash ldots,f\_)$ is also called the ''spectrum'' of $(v\_0,\backslash ldots,v\_)$. This terminology derives from the applications of Fourier transforms in signal processing. If $R$ is an integral domain (which includes fields), it is sufficient to choose $\backslash alpha$ as a primitive ''n''th root of unity, which replaces the condition (1) by:〔 :$\backslash alpha^\; \backslash ne\; 1$ for Proof: take $\backslash beta\; =\; \backslash alpha^k$ with . Since $\backslash alpha^n=1$, $\backslash beta^n=(\backslash alpha^n)^k=1$, giving: :$\backslash beta^n-1\; =\; (\backslash beta-1)\backslash left(\backslash sum\_^\; \backslash beta^j\backslash right)\; =\; 0$ where the sum matches (1). Since $\backslash alpha$ is a primitive root of unity, $\backslash beta\; -\; 1\; \backslash ne\; 0$. Since $R$ is an integral domain, the sum must be zero. ∎ Another simple condition applies in the case where ''n'' is a power of two: (1) may be replaced by $\backslash alpha^\; =\; -1$.〔 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 (general)」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.