On generalized Walsh bases

TL;DR

This paper explores generalized Walsh bases derived from Cuntz algebra representations, establishing their orthonormality, irreducibility, and an uncertainty principle, with applications to fast transforms and signal processing.

Contribution

It introduces a new class of generalized Walsh bases from Cuntz algebra representations and analyzes their properties and applications in signal processing.

Findings

The ONB property is tied to irreducibility of Cuntz algebra representations.

A new fast generalized transform is developed based on these bases.

The generalized transform outperforms classic methods in compression and sparse recovery.

Abstract

This paper continues the study of orthonormal bases (ONB) of $L^2[0,1]$ introduced in \cite{DPS14} by means of Cuntz algebra $\mathcal{O}_N$ representations on $L^2[0,1]$. For $N=2$, one obtains the classic Walsh system. We show that the ONB property holds precisely because the $\mathcal{O}_N$ representations are irreducible. We prove an uncertainty principle related to these bases. As an application to discrete signal processing we find a fast generalized transform and compare this generalized transform with the classic one with respect to compression and sparse signal recovery.