1) 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.

2) It performs an orthogonal, symmetric, involutional, linear operation on real numbers (or complex numbers, although the Hadamard matrices themselves are purely real).

3) The Hadamard transform Hm is a 2m × 2m matrix, the Hadamard matrix (scaled by a normalization factor), that transforms 2mreal 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.

4) Recursively, we define the 1 × 1 Hadamard transform H0 by the identity H0 = 1, and then define Hm for m > 0 by:

$$ =\begin{bmatrix} \ 1.[H(2)]&1.[H(2)] \\ \ 1.[H(2)] & -1.[H(2)] \\ \end{bmatrix} $$

5) $H(4) = \begin{bmatrix} \ 1 & 1 & 1 & 1 \\ \ 1 & -1 & 1 & -1 \\ \ 1 & 1 & -1 & -1 \\ \ 1 & -1 & -1 & 1 \\ \end{bmatrix} $