A Modified Split-Radix FFT With Fewer Arithmetic Operations

@article{Johnson2007AMS,
  title={A Modified Split-Radix FFT With Fewer Arithmetic Operations},
  author={Steven G. Johnson and Matteo Frigo},
  journal={IEEE Transactions on Signal Processing},
  year={2007},
  volume={55},
  pages={111-119}
}
Recent results by Van Buskirk have broken the record set by Yavne in 1968 for the lowest exact count of real additions and multiplications to compute a power-of-two discrete Fourier transform (DFT). Here, we present a simple recursive modification of the split-radix algorithm that computes the DFT with asymptotically about 6% fewer operations than Yavne, matching the count achieved by Van Buskirk's program-generation framework. We also discuss the application of our algorithm to real-data and… 

Figures and Tables from this paper

" A RADIX-4/8/SPLIT RADIX FFT WITH REDUCED ARITHMETIC COMPLEXITY ALGORITHM"
TLDR
Modifications to Radix-4/8 and split radix FFT’s based on DIF (decimation in frequency) version are presented and their implementation issues are discussed that further reduces the arithmetic complexity of power-of-two discrete Fourier Transform.
A new variant of Radix-4 FFT
TLDR
Two levels of saving ideas are proposed to apply the scaling operation to the Twidlle Factors(TF) similar to Tangent FFT like one proposed by Frigo for split radix so that the net computational complexity is of the order of 4Nlog2N computation, where N is the size of FFT.
VLSI Implementation of Split-Radix Fast Fourier Transform: ASurvey
TLDR
The purpose of this paper is to give review of work done on split-radix algorithm and its advantages when compared to other radix implementations.
The split-radix fast Fourier transforms with radix-4 butterfly units
  • Sian-Jheng Lin, W. Chung
  • Computer Science
    2013 Asia-Pacific Signal and Information Processing Association Annual Summit and Conference
  • 2013
TLDR
A split radix fast Fourier transform (FFT) algorithm consisting of mixed radix butterflies, whose structure is more regular than the conventional split Radix algorithm, and is fewer operations than the radix-4 algorithms.
The tangent FFT
TLDR
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.
The Tangent FFT
TLDR
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.
Generating and Searching Families of FFT Algorithms
TLDR
A Boolean Satisfiability-based proof of the lowest operation count for certain classes of DFT algorithms, and a novel way to choose new yet valid twiddle factors for the nodes in flowgraphs generated by common power-of-two fast Fourier transform algorithms, FFTs.
CORDIC based implementation of Fast Fourier Transform
  • Pooja Choudhary, A. Karmakar
  • Computer Science
    2011 2nd International Conference on Computer and Communication Technology (ICCCT-2011)
  • 2011
TLDR
It is proved that CORDIC is most suitable alternative to conventional FFT architecture and results in the elimination of multipliers, saves area, power and cost.
with reduced number of arithmetic operations
We present algorithms for the discrete cosine trans- form (DCT) and discrete sine transform (DST), of types II and III, that achieve a lower count of real multiplications and additions than
A unified expression for split-radix DFT algorithms
  • G. Bi, Gang Li, Xiumei Li
  • Computer Science
    2010 International Conference on Communications, Circuits and Systems (ICCCAS)
  • 2010
TLDR
This paper presents a unified expression that covers all previously reported split-radix-2/2m, where m is an integer larger than one, algorithms, and shows that the split- Radix- 2/4 algorithm requires a smaller computational complexity compared to other split- radix algorithms and the prime factor algorithms.
...
...

References

SHOWING 1-10 OF 56 REFERENCES
On computing the split-radix FFT
TLDR
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.
Split-radix algorithms for discrete trigonometric transforms
TLDR
New split-radix DCT-algorithms of radix-2 length are derived, which are based on real factorization of the corresponding cosine matrices into products of sparse, orthogonal matrices, which have a very low arithmetical complexity.
Implementation of "Split-radix" FFT algorithms for complex, real, and real-symmetric data
  • P. Duhamel
  • Computer Science
    IEEE Trans. Acoust. Speech Signal Process.
  • 1986
TLDR
This algorithm belongs to that class of recently proposed 2n-FFT's which present the same arithmetic complexity (the lowest among any previously published one) and can easily be applied to real and real-symmetric data with reduced arithmetic complexity by removing all redundancy in the algorithm.
A new matrix approach to real FFTs and convolutions of length 2k
TLDR
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.
A new radix-2/8 FFT algorithm for length-q×2m DFTs
In this paper, a new radix-2/8 fast Fourier transform (FFT) algorithm is proposed for computing the discrete Fourier transform of an arbitrary length N=q/spl times/2/sup m/, where q is an odd
Modified FFTs for fused multiply-ADD architectures
TLDR
This work shows how to compute a complex discrete Fourier transform (DFT) of length n = 2m with \nm-^-n + 2-\{-\)m real multiply-adds and describes efficient multidimensional FFTs.
`Split radix' FFT algorithm
A new N = 2n fast Fourier transform algorithm is presented, which has fewer multiplications and additions than radix 2n, n = 1, 2, 3 algorithms, has the same number of multiplications as the
The Design and Implementation of FFTW3
TLDR
It is shown that such an approach can yield an implementation of the discrete Fourier transform that is competitive with hand-optimized libraries, and the software structure that makes the current FFTW3 version flexible and adaptive is described.
...
...