Gröbner bases and wavelet design

Jérôme Lebrun; Ivan Selesnick

doi:10.1016/j.jsc.2002.06.002

Gröbner bases and wavelet design

Jérôme Lebrun, Ivan Selesnick

Electrical and Computer Engineering

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper, we detail the use of symbolic methods in order to solve some advanced design problems arising in signal processing. Our interest lies especially in the construction of wavelet filters for which the usual spectral factorization approach (used for example, to construct the well-known Daubechies filters) is not applicable. In these problems, we show how the design equations can be written as multivariate polynomial systems of equations and accordingly how Gröbner algorithms offer an effective way to obtain solutions in some of these cases.

Original language	English (US)
Pages (from-to)	227-259
Number of pages	33
Journal	Journal of Symbolic Computation
Volume	37
Issue number	2
DOIs	https://doi.org/10.1016/j.jsc.2002.06.002
State	Published - Feb 2004

Keywords

Conjugate quadrature filters (CQFS)
Discrete wavelet transform (DWT)

ASJC Scopus subject areas

Algebra and Number Theory
Computational Mathematics

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10.1016/j.jsc.2002.06.002

Cite this

@article{f1b7b1ee5647465db619aa162a8beccc,

title = "Gr{\"o}bner bases and wavelet design",

abstract = "In this paper, we detail the use of symbolic methods in order to solve some advanced design problems arising in signal processing. Our interest lies especially in the construction of wavelet filters for which the usual spectral factorization approach (used for example, to construct the well-known Daubechies filters) is not applicable. In these problems, we show how the design equations can be written as multivariate polynomial systems of equations and accordingly how Gr{\"o}bner algorithms offer an effective way to obtain solutions in some of these cases.",

keywords = "Conjugate quadrature filters (CQFS), Discrete wavelet transform (DWT)",

author = "J{\'e}r{\^o}me Lebrun and Ivan Selesnick",

note = "Funding Information: J.L. warmly thanks Martin Vetterli for having made Gr{\"o}bner Bases unavoidable from the start of his Ph.D. thesis. A big credit has to be given also to Jean-Charles Faug{\`e}re and Fabrice Rouillier for making their wonderful software available and a very special thank you goes to Gil Strang and Thierry Blu for their useful remarks and fruitful help. This work was partly supported by the Swiss National Science Foundation under grant 2100-043136.95/1. I.S. would like to thank Richard Baraniuk and Rob Nowak for bringing Gr{\"o}bner bases to his attention in 1995. He also gratefully acknowledges support by the NSF under CAREER grant CCR-987452. ",

year = "2004",

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doi = "10.1016/j.jsc.2002.06.002",

language = "English (US)",

volume = "37",

pages = "227--259",

journal = "Journal of Symbolic Computation",

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AU - Lebrun, Jérôme

AU - Selesnick, Ivan

N1 - Funding Information: J.L. warmly thanks Martin Vetterli for having made Gröbner Bases unavoidable from the start of his Ph.D. thesis. A big credit has to be given also to Jean-Charles Faugère and Fabrice Rouillier for making their wonderful software available and a very special thank you goes to Gil Strang and Thierry Blu for their useful remarks and fruitful help. This work was partly supported by the Swiss National Science Foundation under grant 2100-043136.95/1. I.S. would like to thank Richard Baraniuk and Rob Nowak for bringing Gröbner bases to his attention in 1995. He also gratefully acknowledges support by the NSF under CAREER grant CCR-987452.

PY - 2004/2

Y1 - 2004/2

N2 - In this paper, we detail the use of symbolic methods in order to solve some advanced design problems arising in signal processing. Our interest lies especially in the construction of wavelet filters for which the usual spectral factorization approach (used for example, to construct the well-known Daubechies filters) is not applicable. In these problems, we show how the design equations can be written as multivariate polynomial systems of equations and accordingly how Gröbner algorithms offer an effective way to obtain solutions in some of these cases.

AB - In this paper, we detail the use of symbolic methods in order to solve some advanced design problems arising in signal processing. Our interest lies especially in the construction of wavelet filters for which the usual spectral factorization approach (used for example, to construct the well-known Daubechies filters) is not applicable. In these problems, we show how the design equations can be written as multivariate polynomial systems of equations and accordingly how Gröbner algorithms offer an effective way to obtain solutions in some of these cases.

KW - Conjugate quadrature filters (CQFS)

KW - Discrete wavelet transform (DWT)

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JO - Journal of Symbolic Computation

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ER -

Gröbner bases and wavelet design

Abstract

Keywords

ASJC Scopus subject areas

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