Generating and Searching Families of FFT Algorithms

  title={Generating and Searching Families of FFT Algorithms},
  author={Steve Haynal and Heidi Haynal},
A fundamental question of longstanding theoretical interest is to prove the lowest exact count of real additions and multiplications required to compute a power-of-two discrete Fourier transform (DFT). For 35 years the split-radix algorithm held the record by requiring just 4n log n - 6n + 8 arithmetic operations on real numbers for a size-n DFT, and was widely believed to be the best possible. Recent work by Van Buskirk et al. demonstrated improvements to the split-radix operation count by… 

Figures and Tables from this paper

Brute-force search of fast convolution algorithms
  • S. Haynal, Heidi Haynal
  • Computer Science
    2013 IEEE International Conference on Acoustics, Speech and Signal Processing
  • 2013
This paper finds that the combination of FFT and IFFT algorithms in fast convolution permits greater freedom when selecting valid twiddle factor assignments, and exploits this freedom and uses SAT solvers to find new fast Convolution algorithms with the lowest operation counts known.
Improved QFT algorithm for power-of-two FFT
This paper shows that it is possible to improve the computational cost, the memory requirements and the accuracy of Quick Fourier Transform (QFT) algorithm for power-of-two FFT (Fast Fourier
Parallel Computing for the Radix-2 Fast Fourier Transform
  • G. Xie, Yangbo Li
  • Computer Science
    2014 13th International Symposium on Distributed Computing and Applications to Business, Engineering and Science
  • 2014
The one dimensional and two dimensional continuous and discrete Fourier transform is formulates, their parallel algorithms are considered, and the speed up of parallel computing in both shared memory and distributed memory modes is reported.
On the real complexity of a complex DFT
It is shown that the DFT of a complex vector of length N is performed with complexity of 3.76875N log2N real operations of addition, subtraction, and scalar multiplication.
Enhanced Measurements in Fourier Analysis of MEMS Dynamics


The Tangent FFT
The tangent FFT is presented, a straightforward in-place cache-friendly DFT algorithm having exactly the same operation counts as Van Buskirk's algorithm, and it is pinpoints how the tangentFFT saves time compared to the split-radix FFT.
Algorithms meeting the lower bounds on the multiplicative complexity of length-2n DFTs and their connection with practical algorithms
  • P. Duhamel
  • Computer Science
    IEEE Trans. Acoust. Speech Signal Process.
  • 1990
This work shows that an algorithm that computes a length-2/sup n/ discrete Fourier transform using 2/sup 2+1/-2n/Sup 2/+4n-8 nontrivialcomplex multiplications actually provides the attainable lower bound on the number of complex multiplications and provides useful information on the possibility of further improvements of the SRFFT.
A Modified Split-Radix FFT With Fewer Arithmetic Operations
A simple recursive modification of the split-radix algorithm is presented that computes the DFT with asymptotically about 6% fewer operations than Yavne, matching the count achieved by Van Buskirk's program-generation framework.
An economical method for calculating the discrete Fourier transform
  • R. Yavne
  • Computer Science
    AFIPS '68 (Fall, part I)
  • 1968
This article describes another algorithm for computing the Discrete Fourier Transform where the required number of additions and subtractions is the same as in the Cooley-Tukey Algorithm; but therequired number of multiplications is only one half of that in the cooley- Tukey Al algorithm.
A new matrix approach to real FFTs and convolutions of length 2k
A new matrix, scaled odd tail, SOT, is introduced and a compromise is reached between Fourier transform and polynomial transform methods for computing the action of cyclic convolutions.
An algorithm for the machine calculation of complex Fourier series
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.
On computing the split-radix FFT
This paper presents an efficient Fortran program that computes the Duhamel-Hollmann split-radix FFT, which seems to require the least total arithmetic of any power-of-two DFT algorithm.
The FFT: an algorithm the whole family can use
It is concluded that the FFT is both parent and child of the digital revolution, a computational technique at the nexus of the worlds of business and entertainment, national security and public communication.
Polynomial evaluation via the division algorithm the fast Fourier transform revisited
This method is useful if the coefficient sequence of d(x) can be chosen to be sparse, thus simplifying the construction of r(x), the fast Fourier transform algorithm.
Automatic generation of customized discrete Fourier transform IPs
A parameterized soft core generator for the discrete Fourier transform (DFT) that could yield DFT cores over a range of different performance/cost tradeoff points that are not available from the library.