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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 Hadamard transform ： ウィキペディア英語版 Hadamard transform The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal, symmetric, involutive, linear operation on $2^m$ real numbers (or complex numbers, although the Hadamard matrices themselves are purely real). The Hadamard transform can be regarded as being built out of size-2 discrete Fourier transforms (DFTs), and is in fact equivalent to a multidimensional DFT of size $2\backslash times2\backslash times\backslash cdots\backslash times2\backslash times2$. It decomposes an arbitrary input vector into a superposition of Walsh functions. The transform is named for the French mathematician Jacques Hadamard, the German-American mathematician Hans Rademacher, and the American mathematician Joseph L. Walsh. ==Definition== The Hadamard transform ''H''''m'' is a 2''m'' × 2''m'' matrix, the Hadamard matrix (scaled by a normalization factor), that transforms 2''m'' real numbers ''x''''n'' into 2''m'' real numbers ''X''''k''. The Hadamard transform can be defined in two ways: recursively, or by using the binary (base-2) representation of the indices ''n'' and ''k''. Recursively, we define the 1 × 1 Hadamard transform ''H''0 by the identity ''H''0 = 1, and then define ''H''''m'' for ''m'' > 0 by: : where the 1/√2 is a normalization that is sometimes omitted. Thus, other than this normalization factor, the Hadamard matrices are made up entirely of 1 and −1. Equivalently, we can define the Hadamard matrix by its (''k'', ''n'')-th entry by writing : $k\; =\; \backslash sum^\_\; =\; k\_\; 2^\; +\; k\_\; 2^\; +\; \backslash cdots\; +\; k\_1\; 2\; +\; k\_0$ and : $n\; =\; \backslash sum^\_\; =\; n\_\; 2^\; +\; n\_\; 2^\; +\; \backslash cdots\; +\; n\_1\; 2\; +\; n\_0$ where the ''k''''j'' and ''n''''j'' are the binary digits (0 or 1) of ''k'' and ''n'', respectively. Note that for the element in the top left corner, we define: $k\; =\; n\; =\; 0$. In this case, we have: :$\backslash left(\; H\_m\; \backslash right)\_\; =\; \backslash frac\}\; (-1)^$ This is exactly the multidimensional $\backslash scriptstyle\; 2\; \backslash ,\backslash times\backslash ,\; 2\; \backslash ,\backslash times\backslash ,\; \backslash cdots\; \backslash ,\backslash times\backslash ,\; 2\; \backslash ,\backslash times\backslash ,\; 2$ DFT, normalized to be unitary, if the inputs and outputs are regarded as multidimensional arrays indexed by the ''n''''j'' and ''k''''j'', respectively. Some examples of the Hadamard matrices follow. :$\backslash begin$ H_0 = &+1\\ H_1 = \frac &\begin\begin 1 & 1\\ 1 & -1 \end\end \end (This ''H''1 is precisely the size-2 DFT. It can also be regarded as the Fourier transform on the two-element ''additive'' group of Z/(2).) :$\backslash begin$ H_2 = \frac &\begin\begin 1 & 1 & 1 & 1\\ 1 & -1 & 1 & -1\\ 1 & 1 & -1 & -1\\ 1 & -1 & -1 & 1 \end\end\\ H_3 = \frac}} &\begin\begin 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1\\ 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1\\ 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1\\ 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1\\ 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1\\ 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1\\ 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \end\end\\ (H_n)_ = \frac}} &(-1)^ \end where $i\; \backslash cdot\; j$ is the bitwise dot product of the binary representations of the numbers i and j. For example, if $\backslash scriptstyle\; n\; \backslash geq\; 2$, then $\backslash scriptstyle\; ()\_\; \backslash ;=\backslash ;\; (-1)^\; \backslash ;=\backslash ;\; (-1)^\; \backslash ;=\backslash ;\; (-1)^\; \backslash ;=\backslash ;\; (-1)^1\; \backslash ;=\backslash ;\; -1$, agreeing with the above (ignoring the overall constant). Note that the first row, first column of the matrix is denoted by $\backslash scriptstyle\; ()\_$. The rows of the Hadamard matrices are the Walsh functions. 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「Hadamard 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.