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

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Pschel, Markus Moura, Jos M. F.
Copyright Year	2003
Abstract	It is known that the discrete Fourier transform (DFT) used in digital signal processing can be characterized in the framework of the representation theory of algebras, namely, as the decomposition matrix for the regular module ${\mathbb{C}}[Z_n] = {\mathbb{C}}[x]/(x^n - 1)$. This characterization provides deep insight into 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 with 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.
Starting Page	1280
Ending Page	1316
Page Count	37
File Format	PDF
ISSN	00975397
DOI	10.1137/S009753970139272X
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	5
Volume Number	32
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2003-01-01
Access Restriction	Subscribed
Subject Keyword	Fourier series in special orthogonal functions Data analysis Applications Chebyshev polynomial Orthogonal functions and polynomials, general theory Factorization of matrices discrete Fourier transform (DFT) Discrete and fast Fourier transforms discrete cosine transform (DCT) Connections with groups and algebras, and related topics group representation discrete sine transform (DST) symmetry FFT polynomial transform fast algorithm algebra representation discrete trigonometric transform (DTT)
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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