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 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, involutary, 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.

Definition
The Hadamard transform $$H_m$$ is a $$2^m \times 2^m$$ matrix, the Hadamard matrix (scaled by a normalization factor), that transforms $$2^m$$ real numbers $$x_n$$ into $$2^m$$ real numbers $$X_k$$. We can define the Hadamard transform in two ways: recursively, or by using the binary (base-2) representation of the indices $$n$$ and $$k$$.

Recursively, we define the $$1\times1$$ Hadamard transform $$H_0$$ by the identity $$H_0 = 1$$, and then define $$H_m$$ 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 $$1/\sqrt2$$ is a normalization that is sometimes omitted. Thus, other than this normalization factor, the Hadamard matrices are made up entirely of 1 and &minus;1.

Equivalently, we can define the Hadamard matrix by its $$(k,n)$$-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 $$k_j$$ and $$n_j$$ 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 multi-dimensional $$2\times2\times\cdots\times2\times2$$ DFT, normalized to be unitary, if we regard the inputs and outputs as multidimensional arrays indexed by the $$n_j$$ and $$k_j$$, 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 $$H_1$$ is precisely the size-2 DFT.  It can also be regarded as the Fourier transform on the two-element additive group of Z/(2).)


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


 * $$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}.$$

The rows of the Hadamard matrices are the Walsh functions.

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

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 $$|0 \rangle$$ to a superposition of all 2n orthogonal states in the $$ |0 \rangle, |1 \rangle $$basis with equal weight.
 * Hadamard gate operations:
 * $$H|1\rangle = \frac{1}{\sqrt{2}}|0\rangle-\frac{1}{\sqrt{2}}|1\rangle$$.
 * $$H|0\rangle = \frac{1}{\sqrt{2}}|0\rangle+\frac{1}{\sqrt{2}}|1\rangle$$.
 * $$H( \frac{1}{\sqrt{2}}|0\rangle-\frac{1}{\sqrt{2}}|1\rangle )= \frac{1}{2}( |0\rangle+|1\rangle) - \frac{1}{2}( |0\rangle - |1\rangle) = |1\rangle $$;
 * $$H( \frac{1}{\sqrt{2}}|0\rangle+\frac{1}{\sqrt{2}}|1\rangle )= \frac{1}{\sqrt{2}} \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle) + \frac{1}{\sqrt{2}}( \frac{1}{\sqrt{2}}|0\rangle-\frac{1}{\sqrt{2}}|1\rangle)= |0\rangle $$.

Other applications
The Hadamard transform can also be used to generate random numbers with a Gaussian distribution by the central limit theorem. Or you can combine a series of Hadamard transforms with random permutations to transform data into Gaussian noise.

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