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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 Rader's FFT algorithm ： ウィキペディア英語版 Rader's FFT algorithm Rader's algorithm (1968) is a fast Fourier transform (FFT) algorithm that computes the discrete Fourier transform (DFT) of prime sizes by re-expressing the DFT as a cyclic convolution (the other algorithm for FFTs of prime sizes, Bluestein's algorithm, also works by rewriting the DFT as a convolution). Since Rader's algorithm only depends upon the periodicity of the DFT kernel, it is directly applicable to any other transform (of prime order) with a similar property, such as a number-theoretic transform or the discrete Hartley transform. The algorithm can be modified to gain a factor of two savings for the case of DFTs of real data, using a slightly modified re-indexing/permutation to obtain two half-size cyclic convolutions of real data (Chu & Burrus, 1982); an alternative adaptation for DFTs of real data, using the discrete Hartley transform, was described by Johnson & Frigo (2007). Winograd extended Rader's algorithm to include prime-power DFT sizes $p^m$ (Winograd 1976; Winograd 1978), and today Rader's algorithm is sometimes described as a special case of Winograd's FFT algorithm, also called the ''multiplicative Fourier transform algorithm'' (Tolimieri et al., 1997), which applies to an even larger class of sizes. However, for composite sizes such as prime powers, the Cooley–Tukey FFT algorithm is much simpler and more practical to implement, so Rader's algorithm is typically only used for large-prime base cases of Cooley–Tukey's recursive decomposition of the DFT (Frigo and Johnson, 2005). ==Algorithm== Recall that the DFT is defined by the formula :$X\_k\; =\; \backslash sum\_^\; x\_n\; e^\; nk\; \}$ \qquad k = 0,\dots,N-1. If ''N'' is a prime number, then the set of non-zero indices ''n'' = 1,...,''N''–1 forms a group under multiplication modulo ''N''. One consequence of the number theory of such groups is that there exists a generator of the group (sometimes called a primitive root), an integer ''g'' such that ''n'' = ''g''''q'' (mod ''N'') for any non-zero index ''n'' and for a unique ''q'' in 0,...,''N''–2 (forming a bijection from ''q'' to non-zero ''n''). Similarly ''k'' = ''g''–''p'' (mod ''N'') for any non-zero index ''k'' and for a unique ''p'' in 0,...,''N''–2, where the negative exponent denotes the multiplicative inverse of ''g''''p'' modulo ''N''. That means that we can rewrite the DFT using these new indices ''p'' and ''q'' as: :$X\_0\; =\; \backslash sum\_^\; x\_n,$ :$X\_^\; x\_\; e^\; g^\; \}$ \qquad p = 0,\dots,N-2. (Recall that ''x''''n'' and ''X''''k'' are implicitly periodic in ''N'', and also that ''e''2πi=1. Thus, all indices and exponents are taken modulo ''N'' as required by the group arithmetic.) The final summation, above, is precisely a cyclic convolution of the two sequences ''a''''q'' and ''b''''q'' of length ''N''–1 (''q'' = 0,...,''N''–2) defined by: :$a\_q\; =\; x\_$ :$b\_q\; =\; e^\; g^\; \}.$ 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「Rader's FFT algorithm」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.