### Abstract

A polynomial transform is the multiplication of an input vector x ∈ ℂ^{n} by a matrix P_{b α} ∈ C ^{n×n}, whose (k; l)th element is defined as p_{l}(α _{k}) for polynomials pl(x) ∈ ℂ [x] from a list b = {p _{0}(x);..., p_{n-1}(x)} and sample points α _{k}∈ℂ from a list α - {a_{0}...., α_{n-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(n log n) general-radix algorithms for the discrete Fourier transform and the discrete cosine transform of type 4.

Original language | English (US) |
---|---|

Pages (from-to) | 364-384 |

Number of pages | 21 |

Journal | SIAM Journal on Matrix Analysis and Applications |

Volume | 32 |

Issue number | 2 |

DOIs | |

State | Published - Aug 15 2011 |

### Fingerprint

### Keywords

- Algebra
- Discrete cosine transform
- Discrete Fourier transform
- Discrete sine transform
- Fast algorithm
- Fast Fourier transform
- Matrix factorization
- Module
- Polynomial transform

### ASJC Scopus subject areas

- Analysis

### Cite this

*SIAM Journal on Matrix Analysis and Applications*,

*32*(2), 364-384. https://doi.org/10.1137/100805777

**Algebraic signal processing theory : Cooley-tukey-type algorithms for polynomial transforms based on induction.** / Sandryhaila, Aliaksei; Kovacevic, Jelena; Püschel, Markus.

Research output: Contribution to journal › Article

*SIAM Journal on Matrix Analysis and Applications*, vol. 32, no. 2, pp. 364-384. https://doi.org/10.1137/100805777

}

TY - JOUR

T1 - Algebraic signal processing theory

T2 - Cooley-tukey-type algorithms for polynomial transforms based on induction

AU - Sandryhaila, Aliaksei

AU - Kovacevic, Jelena

AU - Püschel, Markus

PY - 2011/8/15

Y1 - 2011/8/15

N2 - A polynomial transform is the multiplication of an input vector x ∈ ℂn by a matrix Pb α ∈ C n×n, whose (k; l)th element is defined as pl(α k) for polynomials pl(x) ∈ ℂ [x] from a list b = {p 0(x);..., pn-1(x)} and sample points α k∈ℂ from a list α - {a0...., 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(n log n) general-radix algorithms for the discrete Fourier transform and the discrete cosine transform of type 4.

AB - A polynomial transform is the multiplication of an input vector x ∈ ℂn by a matrix Pb α ∈ C n×n, whose (k; l)th element is defined as pl(α k) for polynomials pl(x) ∈ ℂ [x] from a list b = {p 0(x);..., pn-1(x)} and sample points α k∈ℂ from a list α - {a0...., 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(n log n) general-radix algorithms for the discrete Fourier transform and the discrete cosine transform of type 4.

KW - Algebra

KW - Discrete cosine transform

KW - Discrete Fourier transform

KW - Discrete sine transform

KW - Fast algorithm

KW - Fast Fourier transform

KW - Matrix factorization

KW - Module

KW - Polynomial transform

UR - http://www.scopus.com/inward/record.url?scp=80051513363&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80051513363&partnerID=8YFLogxK

U2 - 10.1137/100805777

DO - 10.1137/100805777

M3 - Article

VL - 32

SP - 364

EP - 384

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

SN - 0895-4798

IS - 2

ER -