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

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.