翻訳と辞書
Words near each other
・ "O" Is for Outlaw
・ "O"-Jung.Ban.Hap.
・ "Ode-to-Napoleon" hexachord
・ "Oh Yeah!" Live
・ "Our Contemporary" regional art exhibition (Leningrad, 1975)
・ "P" Is for Peril
・ "Pimpernel" Smith
・ "Polish death camp" controversy
・ "Pro knigi" ("About books")
・ "Prosopa" Greek Television Awards
・ "Pussy Cats" Starring the Walkmen
・ "Q" Is for Quarry
・ "R" Is for Ricochet
・ "R" The King (2016 film)
・ "Rags" Ragland
・ ! (album)
・ ! (disambiguation)
・ !!
・ !!!
・ !!! (album)
・ !!Destroy-Oh-Boy!!
・ !Action Pact!
・ !Arriba! La Pachanga
・ !Hero
・ !Hero (album)
・ !Kung language
・ !Oka Tokat
・ !PAUS3
・ !T.O.O.H.!
・ !Women Art Revolution


Dictionary Lists
翻訳と辞書 辞書検索 [ 開発暫定版 ]
スポンサード リンク

Walsh–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\times2\times\cdots\times2\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:
:H_m = \frac \begin H_ & H_ \\ H_ & -H_ \end = H_ \otimes H_
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 = \sum^_ = k_ 2^ + k_ 2^ + \cdots + k_1 2 + k_0
and
: n = \sum^_ = n_ 2^ + n_ 2^ + \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:
:\left( H_m \right)_ = \frac} (-1)^
This is exactly the multidimensional \scriptstyle 2 \,\times\, 2 \,\times\, \cdots \,\times\, 2 \,\times\, 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.
:\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).)
:\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 \cdot j is the bitwise dot product of the binary representations of the numbers i and j. For example, if \scriptstyle n \geq 2 , then \scriptstyle ()_ \;=\; (-1)^ \;=\; (-1)^ \;=\; (-1)^ \;=\; (-1)^1 \;=\; -1, agreeing with the above (ignoring the overall constant). Note that the first row, first column of the matrix is denoted by \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.