Characterizing and Enumerating Walsh-Hadamard Transform Algorithms

TL;DR

This paper introduces a new characterization and enumeration method for Walsh-Hadamard transform algorithms based on sequences of butterfly arrays and linear permutations, enhancing understanding of their structure.

Contribution

It provides a novel framework for characterizing and systematically enumerating Walsh-Hadamard transform algorithms using linear permutations and butterfly arrays.

Findings

New characterization of Walsh-Hadamard algorithms

Method for enumerating all such algorithms

Enhanced understanding of algorithm structure

Abstract

We propose a way of characterizing the algorithms computing a Walsh-Hadamard transform that consist of a sequence of arrays of butterflies ($I_{2^{n-1}}\otimes \text{DFT}_2$) interleaved by linear permutations. Linear permutations are those that map linearly the binary representation of its element indices. We also propose a method to enumerate these algorithms.