Question

Explain about the Hadamard transformation

Answer 1

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)

Fast Walsh–Hadamard transform, a faster way to calculate the Walsh spectrum of (1,0,1,0,0,1,1,0).

The original function can be expressed by means of its Walsh spectrum as an arithmetical polynomial.
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 numbers, although the Hadamard matrices themselves are purely real).