Derivation of fast DCT algorithms using algebraic technique based on Galois theory

TL;DR

This paper introduces an algebraic method based on Galois theory within algebraic signal processing to derive fast discrete cosine transform algorithms, connecting polynomial algebra decomposition with field extensions.

Contribution

It presents a novel approach using Galois theory to systematically derive fast DCT algorithms through polynomial algebra over rational numbers.

Findings

Galois theory facilitates polynomial factorization in DCT derivation.

The method reveals algebraic structures underlying fast DCT algorithms.

New algorithms are derived from intermediate field extensions.

Abstract

The paper presents an algebraic technique for derivation of fast discrete cosine transform (DCT) algorithms. The technique is based on the algebraic signal processing theory (ASP). In ASP a DCT associates with a polynomial algebra C[x]/p(x). A fast algorithm is obtained as a stepwise decomposition of C[x]/p(x). In order to reveal the connection between derivation of fast DCT algorithms and Galois theory we define polynomial algebra over the field of rational numbers Q instead of complex C. The decomposition of Q[x]/p(x) requires the extension of the base field Q to splitting field E of polynomial p(x). Galois theory is used to find intermediate subfields L_i in which polynomial p(x) is factored. Based on this factorization fast DCT algorithm is derived.