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

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 AC = C[x]/p(x). A fast algorithm is obtained as a stepwise decomposition of AC. In order to reveal the connection between derivation of fast DCT algorithms and Galois theory we define A over the field of rational numbers Q instead of complex C. The decomposition of AQ requires the extension of the base field Q to splitting field E of polynomial p(x). Galois theory is used to find intermediate subfields Li in which polynomial p(x) is factored. Based on this factorization fast DCT algorithm is derived.