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Hadamard transform

Involutive change of basis in linear algebra / From Wikipedia, the free encyclopedia

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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 2m real numbers (or complex, or hypercomplex numbers, although the Hadamard matrices themselves are purely real).

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The product of a Boolean function and a Walsh matrix is its Walsh spectrum:[1]
(1, 0, 1, 0, 0, 1, 1, 0) × H(8) = (4, 2, 0, −2, 0, 2, 0, 2)
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Fast Walsh–Hadamard transform, a faster way to calculate the Walsh spectrum of (1, 0, 1, 0, 0, 1, 1, 0).
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The original function can be expressed by means of its Walsh spectrum as an arithmetical polynomial.

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 × 2 × ⋯ × 2 × 2.[2] It decomposes an arbitrary input vector into a superposition of Walsh functions.

The transform is named for the French mathematician Jacques Hadamard (French: [adamaʁ]), the German-American mathematician Hans Rademacher, and the American mathematician Joseph L. Walsh.