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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 DFT matrix ： ウィキペディア英語版 DFT matrix A DFT matrix is an expression of a discrete Fourier transform (DFT) as a transformation matrix, which can be applied to a signal through matrix multiplication. ==Definition== An ''N''-point DFT is expressed as the multiplication $X\; =\; W\; x$, where $x$ is the original input signal, $W$ is the ''N''-by-''N'' square DFT matrix, and $X$ is the DFT of the signal. The transformation matrix $W$ can be defined as $W\; =\; \backslash left(\backslash frac\}\backslash right)\_$, or equivalently: :$$ W = \frac 1&1&1&1&\cdots &1 \\ 1&\omega&\omega^2&\omega^3&\cdots&\omega^ \\ 1&\omega^2&\omega^4&\omega^6&\cdots&\omega^\\ 1&\omega^3&\omega^6&\omega^9&\cdots&\omega^\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 1&\omega^&\omega^&\omega^&\cdots&\omega^\\ \end, where $\backslash omega\; =\; e^\}$ is a primitive $N$th root of unity in which $i=\backslash sqrt$. This is the Vandermonde matrix for the roots of unity, up to the normalization factor. Note that the normalization factor in front of the sum ($1/\backslash sqrt$) and the sign of the exponent in ω are merely conventions, and differ in some treatments. All of the following discussion applies regardless of the convention, with at most minor adjustments. The only important thing is that the forward and inverse transforms have opposite-sign exponents, and that the product of their normalization factors be 1/''N''. However, the $1/\backslash sqrt$ choice here makes the resulting DFT matrix unitary, which is convenient in many circumstances. Fast Fourier Transform algorithms utilize the symmetries of the matrix to reduce the time of multiplying a vector by this matrix, from the usual $O(N^2)$. Similar techniques can be applied for multiplications by matrices such as Hadamard matrix and the Walsh matrix. 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「DFT matrix」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.