A Generalised Hadamard Transform

TL;DR

This paper introduces a generalized Hadamard transform encompassing various known transforms, formalizes its jacket construction, and presents new primary matrices, expanding the mathematical framework for multi-phase signals.

Contribution

It formalizes the jacket construction for generalized Hadamard transforms and introduces new primary matrices of orders 8 and 12.

Findings

Unified framework for multiple transforms including Fourier and Walsh-Hadamard

Tensor product decomposition of jacket construction

New primary matrices of orders 8 and 12

Abstract

A Generalised Hadamard Transform for multi-phase or multilevel signals is introduced, which includes the Fourier, Generalised, Discrete Fourier, Walsh-Hadamard and Reverse Jacket Transforms. The jacket construction is formalised and shown to admit a tensor product decomposition. Primary matrices under this decomposition are identified. New examples of primary jacket matrices of orders 8 and 12 are presented.