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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Bibliographic Details
Published in:SIAM journal on matrix analysis and applications Vol. 32; no. 2; pp. 364 - 384
Main Authors: Sandryhaila, Aliaksei, Kovačević, Jelena, Püschel, Markus
Format: Journal Article
Language:English
Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.04.2011
Subjects:
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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Summary: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]
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ISSN:0895-4798
1095-7162
DOI:10.1137/100805777