Algebraic Signal Processing Theory: Cooley–Tukey-Type Algorithms for Polynomial Transforms Based on Induction

A polynomial transform is the multiplication of an input vector x∈Cn by a matrix Pb,α∈Cn×n, whose (k,[cursive l])th element is defined as p[cursive l](αk) for polynomials p[cursive l](x)∈C[x] from a list b={p0(x),...,pn-1(x)} and sample points αk∈C from a list α={α0,...,αn-1}. Such transforms find a...

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Vydané v:	SIAM journal on matrix analysis and applications Ročník 32; číslo 2; s. 364 - 384
Hlavní autori:	Sandryhaila, Aliaksei, Kovačević, Jelena, Püschel, Markus
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Philadelphia, PA Society for Industrial and Applied Mathematics 01.04.2011
Predmet:	Algebra Algorithms Computer engineering Data compression Decomposition Exact sciences and technology Fourier transforms General logic Linear and multilinear algebra, matrix theory Logic and foundations Mathematical logic, foundations, set theory Mathematics Numerical analysis Numerical analysis. Scientific computation Numerical approximation Numerical linear algebra Polynomials Sciences and techniques of general use Signal processing Polynomial Type theory Signal compression Multiplication Fourier transformation Double sample 65T50 Fast Fourier transformation Signal theory 65T99 algebra Cosine transform 42C10 discrete cosine transform Input 15A23 Primary List Sine transform Fast algorithm 33C90 Vector discrete sine transform matrix factorization Secondary Induction Data processing 13C05 Factorization fast Fourier transform discrete Fourier transform Algorithm theory Interpolation Area Numerical analysis 15B99 Linear algebra Signal processing Representation theory Module 33C80 Polynomial transform 42C05
ISSN:	0895-4798, 1095-7162
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Shrnutí:	A polynomial transform is the multiplication of an input vector x∈Cn by a matrix Pb,α∈Cn×n, whose (k,[cursive l])th element is defined as p[cursive l](αk) for polynomials p[cursive l](x)∈C[x] from a list b={p0(x),...,pn-1(x)} and sample points αk∈C from a list α={α0,...,αn-1}. Such transforms find applications in the areas of signal processing, data compression, and function interpolation. An important example includes the discrete Fourier transform. In this paper we introduce a novel technique to derive fast algorithms for polynomial transforms. The technique uses the relationship between polynomial transforms and the representation theory of polynomial algebras. Specifically, we derive algorithms by decomposing the regular modules of these algebras as a stepwise induction. As an application, we derive novel O(nlogn) general-radix algorithms for the discrete Fourier transform and the discrete cosine transform of type 4. [PUBLICATION ABSTRACT]
Bibliografia:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ISSN:	0895-4798 1095-7162
DOI:	10.1137/100805777