Fast Fourier Transform and Convolution Algorithms

Nussbaumer, Henri J.

doi:10.1007/978-3-662-00551-4

Cited by 396 publications

(120 citation statements)

References 0 publications

Supporting

Mentioning

116

Contrasting

Unclassified

Order By: Relevance

“…[52,36] give an overview of FFT algorithms. [5] uses the Kronecker product formalism to describe many different FFT algorithms, including parallel and vector algorithms.…”

Section: Complexity and Analysismentioning

confidence: 99%

How to Write Fast Numerical Code: A Small Introduction

Chellappa¹,

Franchetti²,

Püschel³

2008

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Abstract. The complexity of modern computing platforms has made it extremely difficult to write numerical code that achieves the best possible performance. Straightforward implementations based on algorithms that minimize the operations count often fall short in performance by at least one order of magnitude. This tutorial introduces the reader to a set of general techniques to improve the performance of numerical code, focusing on optimizations for the computer's memory hierarchy. Further, program generators are discussed as a way to reduce the implementation and optimization effort. Two running examples are used to demonstrate these techniques: matrix-matrix multiplication and the discrete Fourier transform.

show abstract

“…[52,36] give an overview of FFT algorithms. [5] uses the Kronecker product formalism to describe many different FFT algorithms, including parallel and vector algorithms.…”

Section: Complexity and Analysismentioning

confidence: 99%

How to Write Fast Numerical Code: A Small Introduction

Chellappa¹,

Franchetti²,

Püschel³

2008

Lecture Notes in Computer Science

View full text Add to dashboard Cite

show abstract

“…Cooley and Tukey [2] were the first to propose an FFT algorithm in 1965 (known as 2-radix FFT) with time complexity O(n log n). A range of other FFT algorithms have been discovered since then: besides the Cooley-Tukey, different r-radix, mixed-radix, and multidimensional versions [14].…”

Section: Introductionmentioning

confidence: 99%

Parallel FFT with Eden Skeletons

Berthold

Dieterle

Lobachev

et al. 2009

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Abstract. The notion of Fast Fourier Transformation (FFT) describes a range of efficient algorithms to compute the discrete Fourier transformation, frequency distribution in a signal. FFT plays a major role both for pure mathematical applications and for real-life scenarios such as digital signal processing. The paper investigates and compares skeletonbased parallel Haskell implementations of different FFT-algorithms on workstation clusters with distributed memory. Our experiments show that the original divide-and-conquer versions suffer from an inherent input distribution and result collection problem, because huge amounts of data have to be communicated. Advanced approaches like distributable homomorphism FFT or multidimensional FFT provide more flexibility to overcome these problems. Assuming a distributed access to input data and re-organising computation in such a way that the results can be returned in a distributed way leads to versions with an acceptable parallel runtime behaviour.

show abstract

“…Much of this work was carried out during the late 1970s and early 1980s by S. Winograd [19][20][21][22] and H. J. Nussbaumer [12,13] and placed in a mathematical framework by several of their collaborators including L. Auslander, E. Feig [3] and at a later time by C. S. Burrus [5,6], I. Gertner [9], and M. Rofheart [15]. Additional results and references can be found in texts [4,8,11,17,18].…”

Section: Introductionmentioning

confidence: 99%

“…The convolution theorem which diagonalizes cyclic convolution mod AE relative to the AE-point [12,13], both for multidimensional convolution and Fourier transform (polynomial transform). One goal in these works is to rely as much as possible on shifts and cyclic shifts to carry out these computations.…”

Section: Introductionmentioning

confidence: 99%

Lesser Known FFT Algorithms

Tolimieri¹,

An²

2001

Twentieth Century Harmonic Analysis — A Celebration

View full text Add to dashboard Cite

ABSTRACT. The introduction of the Cooley-Tukey Fast Fourier transform (C-T FFT) algorithm in 1968 was a critical step in advancing the widespread use of digital computers in scientific and technological applications. Initial efforts focused on realizing the potential of the immense reduction in arithmetic complexity afforded by the FFT for computing the finite Fourier transform and convolution. On existing serial butterfly architectures, this limited implementations of the FFT to transform sizes a power of two.The original C-T FFT algorithms and its many extensions rested mainly on additive structures of the data indexing set and divide-and-conquer strategies for reducing complexity. The relative cost of multiplications as compared with additions on these early machines motivated the study of new algorithms for reducing multiplications usually at the cost of increasing additions. These new algorithms were based on multiplicative structures of the data indexing set. They resulted in greater flexibility as compared to earlier efforts by allowing for fast small prime size computations which could be embedded in a wide collection of large transform sizes.During the 1980s, the increasing importance of RISC and parallel computers removed some of the initial motivation for these multiplicative FFTs. Many RISC architectures featured a hardwired multiply and accumulate which permitted multiplications to be nested in additions. The goal was to arrange the computations so that most multiplications were followed by an addition. Parallel architectures placed the major algorithmic burden on controlling the data flow. The exotic data flow of the multiplicative FFTs were often incompatible or at least required an immense coding effort for efficient parallel computation. Only the need for flexible transform sizes justified continued efforts.The recent importance of FPGAs and reconfigurable hardware renews the need to reduce multiplication counts. In this talk, we will review some of the work on multiplicative approaches introduced mainly at IBM in the 1970s and 1980s and place these results within the realm of harmonic analysis.

show abstract

Fast Fourier Transform and Convolution Algorithms

Cited by 396 publications

References 0 publications

How to Write Fast Numerical Code: A Small Introduction

How to Write Fast Numerical Code: A Small Introduction

Parallel FFT with Eden Skeletons

Lesser Known FFT Algorithms

Contact Info

Product

Resources

About