On Wavelet Decomposition over Finite Fields

TL;DR

This paper develops wavelet theory over finite fields, deriving orthogonal and non-orthogonal wavelets, and demonstrates their applications in multiresolution analysis and spread-spectrum sequence design.

Contribution

It introduces the foundations of wavelets over Galois fields, including derivation of FF-Haar, FF-Daubechies, and B-spline wavelets, expanding finite-field signal processing capabilities.

Findings

Derived orthogonal FF-wavelets like FF-Haar and FF-Daubechies.

Presented multiresolution analysis examples over finite fields.

Applied FF-wavelets to design spread-spectrum sequences.

Abstract

This paper introduces some foundations of wavelets over Galois fields. Standard orthogonal finite-field wavelets (FF-Wavelets) including FF-Haar and FF-Daubechies are derived. Non-orthogonal FF-wavelets such as B-spline over GF(p) are also considered. A few examples of multiresolution analysis over Finite fields are presented showing how to perform Laplacian pyramid filtering of finite block lengths sequences. An application of FF-wavelets to design spread-spectrum sequences is presented.