The Algebraic Approach to the Discrete Cosine and Sine Transforms and Their Fast Algorithms

Abstract

Abstract. It is known that the discrete Fourier transform (DFT) used in digital signal processing can be characterized in the framework of representation theory of algebras, namely as the decomposition matrix for the regular module C[Zn] = C[x]/(x n − 1). This characterization provides deep insight on the DFT and can be used to derive and understand the structure of its fast algorithms. In this paper we present an algebraic characterization of the important class of discrete cosine and sine transforms as decomposition matrices of certain regular modules associated to four series of Chebyshev polynomials. Then we derive most of their known algorithms by pure algebraic means. We identify the mathematical principle behind each algorithm and give insight into its structure. Our results show that the connection between algebra and digital signal processing is stronger than previously understood.

Alternatives to the discrete fourier transform

“…Note that this class of transforms does not contain the discrete cosine and sine transforms, which can be captured in the algebraic framework by using Chebyshev polynomials [6,4]. In our example, the zeros of…”

Fourier transform for the spatial quincunx lattice

“…1, with symmetric or antisymmetric b.c. 's (that depend on the DCT or DST chosen and are not shown) [1,2]. Intuitively, "undirected" implies that the associated "Fourier transforms" ( In this paper we derive a Fourier transform for spatial signals residing on the quincunx lattice shown in Fig.…”

The discrete triangle transform

“…This theory provides the foundation for understanding the DFT, the DCTs and DSTs, their properties, their associated signal models, and their fast algorithms [1,2]. The particular case DCT, type III, will be explained in Section 3.…”

“…There are various ways of deriving fast algorithms for a polynomial transform [1]. In each case, the algorithms are derived by decomposing the underlying algebra in steps, rather than manipulating the transform's matrix entries.…”

“…Recent research has shown that the 16 types of discrete cosine and sine transforms (DCTs and DSTs), can, like the discrete Fourier transform (DFT), be characterized in the framework of polynomial algebras in one variable [1,2]. The polynomial algebra for a DCT or DST explains the interaction between the underlying signal model, its boundary conditions (b.c.…”