Hadamard transform

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Not to be confused with Walsh matrix.

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 is named for the French mathematician Jacques Solomon Hadamard, the German-American mathematician Hans Adolph Rademacher, and the American mathematician Joseph Leonard Walsh. It performs an orthogonal, symmetric, involutional, linear operation on 2m 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.

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[edit] Definition

The Hadamard transform Hm is a 2m × 2m matrix, the Hadamard matrix (scaled by a normalization factor), that transforms 2m real numbers xn into 2m real numbers Xk. 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 H0 by the identity H0 = 1, and then define Hm for m > 0 by:

H_m = \frac{1}{\sqrt2} \begin{pmatrix} H_{m-1} & H_{m-1} \\ H_{m-1} & -H_{m-1} \end{pmatrix},

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 (kn)-th entry by writing

k=k_{m-1} 2^{m-1} + k_{m-2} 2^{m-2} + \cdots + k_1 2 + k_0\,

and

n=n_{m-1} 2^{m-1} + n_{m-2} 2^{m-2} + \cdots + n_1 2 + n_0,

where the kj and nj are the binary digits (0 or 1) of n and k, respectively. In this case, we have:

\left( H_m \right)_{k,n} = \frac{1}{2^{m/2}} (-1)^{\sum_j k_j n_j}.

This is exactly the multidimensional 2\times2\times\cdots\times2\times2DFT, normalized to be unitary, if the inputs and outputs are regarded as multidimensional arrays indexed by the nj and kj, respectively.

Some examples of the Hadamard matrices follow.

\ H_0 = +1

H_1 = \frac{1}{\sqrt2} \begin{pmatrix}\begin{array}{rr} 1 & 1 \\ 1 & -1 \end{array}\end{pmatrix}

(This H1 is precisely the size-2 DFT. It can also be regarded as the Fourier transform on the two-element additive group of Z/(2).)

H_2 = \frac{1}{2} \begin{pmatrix}\begin{array}{rrrr} 1 & 1 & 1 & 1 \\ 1 & -1 & 1 & -1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & -1 & 1\end{array}\end{pmatrix}

H_3 = \frac{1}{2^{3/2}} \begin{pmatrix}\begin{array}{rrrrrrrr} 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{array}\end{pmatrix}.

(H_n)_{i,j} = \frac{1}{2^{n/2}} (-1)^{i \cdot j}

where  i \cdot j is the bitwise dot product of the binary representations of the numbers i and j. For example,  H_{32} = (-1)^{3 \cdot 2} = (-1)^{(1,1) \cdot (1,0)} = (-1)^{1+0} = (-1)^1 = -1 , agreeing with the above (ignoring the overall constant). Note that the first row, first column of the matrix is denoted by H00

The rows of the Hadamard matrices are the Walsh functions.

[edit] Computational complexity

The Hadamard transform can be computed in m log m operations, using the fast Hadamard transform algorithm.

[edit] Quantum computing applications

In quantum information processing the Hadamard transformation, more often called Hadamard gate in this context (cf. quantum gate), is a one-qubit rotation, mapping the qubit-basis states |0 \rangle and |1 \rangle to two superposition states with equal weight of the computational basis states |0 \rangle and |1 \rangle . Usually the phases are chosen so that we have

\frac{|0\rangle+|1\rangle}{\sqrt{2}}\langle0|+\frac{|0\rangle-|1\rangle}{\sqrt{2}}\langle1|

in Dirac notation. This corresponds to the transformation matrix

H_1=\frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}

in the |0 \rangle , |1 \rangle basis.

Many quantum algorithms use the Hadamard transform as an initial step, since it maps n qubits initialized with |0 \rangleto a superposition of all 2n orthogonal states in the  |0 \rangle , |1 \rangle basis with equal weight.

Hadamard gate operations:

H|1\rangle = \frac{1}{\sqrt{2}}|0\rangle-\frac{1}{\sqrt{2}}|1\rangle.

H|0\rangle = \frac{1}{\sqrt{2}}|0\rangle+\frac{1}{\sqrt{2}}|1\rangle.

H( \frac{1}{\sqrt{2}}|0\rangle-\frac{1}{\sqrt{2}}|1\rangle )= \frac{1}{2}( |0\rangle+|1\rangle) - \frac{1}{2}( |0\rangle - |1\rangle) = |1\rangle;

H( \frac{1}{\sqrt{2}}|0\rangle+\frac{1}{\sqrt{2}}|1\rangle )= \frac{1}{\sqrt{2}} \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle) + \frac{1}{\sqrt{2}}( \frac{1}{\sqrt{2}}|0\rangle-\frac{1}{\sqrt{2}}|1\rangle)= |0\rangle.

[edit] Other applications

The Hadamard transform is also used in many signal processing and data compression algorithms, such as HD Photo and MPEG-4 AVC. In video compression applications, it is usually used in the form of the sum of absolute transformed differences.

[edit] See also

[edit] External links

Retrieved from "http://en.wikipedia.org/wiki/Hadamard_transform"

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