Journal ArticleDOI
Fast computation of discrete Fourier transforms using polynomial transforms
Henri J. Nussbaumer,P. Quandalle +1 more
TLDR
In this article, two polynomial transforms have been proposed for computing discrete Fourier transform (DFT) by polynomials, which are particularly well adapted to multidimensional DFT's as well as to some one-dimensional DFTs.Abstract:
Polynomial transforms, defined in rings of polynomials, have been introduced recently and have been shown to give efficient algorithms for the computation of two-dimensional convolutions. In this paper we present two methods for computing discrete Fourier transforms (DFT) by polynomial transforms. We show that these techniques are particularly well adapted to multidimensional DFT's as well as to some one-dimensional DFT's and yield algorithms that are, in many instances, more efficient than the fast Fourier transform (FFT) or the Winograd Fourier Transform (WFTA). We also describe new split nesting and split prime factor techniques for computing large DFT's from a small set of short DFT's with a minimum number of operations.read more
Citations
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Journal ArticleDOI
Fast fourier transforms: a tutorial review and a state of the art
Pierre Duhamel,Martin Vetterli +1 more
TL;DR: Note: V. Madisetti, D. B. Williams, Eds.
Journal ArticleDOI
Image algebra techniques for parallel image processing
Gerhard X. Ritter,Paul D. Gader +1 more
TL;DR: This paper shows how the image algebra suggests a general-purpose cellular pyramid array processor for real time image processing tasks and demonstrates how algebraic techniques can be used to develop systematic methods for deriving parallel algorithms.
Journal ArticleDOI
Algebraic Signal Processing Theory: Foundation and 1-D Time
Markus Püschel,Jose M. F. Moura +1 more
TL;DR: The paper illustrates the general ASP theory with the standard time shift, presenting a unique signal model for infinite time and several signal models for finite time and the latter models illustrate the role played by boundary conditions and recover the discrete Fourier transform and its variants as associated Fourier transforms.
Book
Fast Algorithms for Signal Processing
TL;DR: A collection of cyclic convolution algorithms and a collection of Winograd small FFT algorithms for solving Toeplitz systems are presented.
Proceedings ArticleDOI
Polynomial transform computation of the 2-D DCT
TL;DR: A 2-D DCT (discrete cosine transform) algorithm based on a direct polynomial approach is presented and it is shown that, although being mathematically involved, it possesses a clean, butterfly-based structure.
References
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Journal ArticleDOI
An algorithm for the machine calculation of complex Fourier series
J.W. Cooley,John W. Tukey +1 more
TL;DR: Good generalized these methods and gave elegant algorithms for which one class of applications is the calculation of Fourier series, applicable to certain problems in which one must multiply an N-vector by an N X N matrix which can be factored into m sparse matrices.
Journal ArticleDOI
Discrete Fourier transforms when the number of data samples is prime
TL;DR: The discrete Fourier transform of a sequence of N points, where N is a prime number, is shown to be essentially a circular correlation, which permits the discrete Fouriers transform to be computed by means of a fast Fouriertransform algorithm, with the associated increase in speed, even though N is prime.
Book
Introduction to Number Theory
TL;DR: A specific feature of this text on number theory is the rather extensive treatment of Diophantine equations of second or higher degree as discussed by the authors, and a large number of non-routine problems are given.
Journal ArticleDOI
A prime factor FFT algorithm using high-speed convolution
TL;DR: Two recently developed ideas, the conversion of a discrete Fourier transform to convolution and the implementation of short convolutions with a minimum of multiplications, are combined to give efficient algorithms for long transforms.
Journal ArticleDOI
New algorithms for digital convolution
R. Agarwal,J. Cooley +1 more
TL;DR: It is shown how the Chinese Remainder Theorem can be used to convert a one-dimensional cyclic convolution to a multi-dimensional convolution which is cyclic in all dimensions and can be more efficient, for some data sequence lengths, than the fast Fourier transform (FFT) algorithm.