Fast Walsh–Hadamard transform

In computational mathematics, the Hadamard ordered fast Walsh–Hadamard transform (FWHT_h) is an efficient algorithm to compute the Walsh–Hadamard transform (WHT). A naive implementation of the WHT of order ${\displaystyle n=2^{m}}$ would have a computational complexity of O(${\displaystyle n^{2}}$). The FWHT_h requires only ${\displaystyle n\log n}$ additions or subtractions.

The FWHT_h is a divide-and-conquer algorithm that recursively breaks down a WHT of size ${\displaystyle n}$ into two smaller WHTs of size ${\displaystyle n/2}$. This implementation follows the recursive definition of the ${\displaystyle 2^{m}\times 2^{m}}$ Hadamard matrix ${\displaystyle H_{m}}$:

${\displaystyle H_{m}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}H_{m-1}&H_{m-1}\\H_{m-1}&-H_{m-1}\end{pmatrix}}.}$

The ${\displaystyle 1/{\sqrt {2}}}$ normalization factors for each stage may be grouped together or even omitted.

The sequency-ordered, also known as Walsh-ordered, fast Walsh–Hadamard transform, FWHT_w, is obtained by computing the FWHT_h as above, and then rearranging the outputs.

A simple fast nonrecursive implementation of the Walsh–Hadamard transform follows from decomposition of the Hadamard transform matrix as ${\displaystyle H_{m}=A^{m}}$, where A is m-th root of ${\displaystyle H_{m}}$.