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arXiv · 1211.1340

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

Maxim Vashkevich·Alexander Petrovsky
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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.

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Maxim Vashkevich, Alexander Petrovsky. 2012-11-06. Derivation of fast DCT algorithms using algebraic technique based on Galois theory. https://arxiv.org/abs/1211.1340

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