Radix-R FFT and IFFT Factorizations for Parallel Implementation

  • Pere Marti-Puig
  • Ramon Reig Bolaño
  • Vicenç Parisi Baradad
Part of the Advances in Soft Computing book series (AINSC, volume 50)


Two radix-R regular interconnection pattern families of factorizations for both the FFT and the IFFT -also known as parallel or Pease factorizations- are reformulated and presented. Number R is any power of 2 and N, the size of the transform, any power of R. The first radix-2 parallel FFT algorithm -one of the two known radix-2 topologies- was proposed by Pease. Other authors extended the Pease parallel algorithm to different radix and other particular solutions were also reported. The presented families of factorizations for both the FFT and the IFFT are derived from the well-known Cooley-Tukey factorizations, first, for the radix-2 case, and then, for the general radix-R case. Here we present the complete set of parallel algorithms, that is, algorithms with equal interconnection pattern stage-to-stage topology. In this paper the parallel factorizations are derived by using a unified notation based on the use of the Kronecker product and the even-odd permutation matrix to form the rest of permutation matrices. The radix-R generalization is done in a very simple way. It is shown that, both FFT and IFFT share interconnection pattern solutions. This view tries to contribute to the knowledge of fast parallel algorithms for the case of FFT and IFFT but it can be easily applied to other discrete transforms.


Fast Fourier Transform Parallel algorithms Fast algorithms 


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  1. 1.
    Pease, M.C.: An adaptation of the fast Fourier transform for parallel processing. J. Assoc. Comput. 15, 252–264 (1968)zbMATHGoogle Scholar
  2. 2.
    Sloate, H.: Matrix Representations for Sorting and the Fast Fourier Transform. IEEE Trans.on Circuits and Systems 21, 109–116 (1974)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Schott, J.R.: Matrix Analysis for Statistics. John Wiley & Sons, New York (1996)Google Scholar
  4. 4.
    Magnus, J.R., Neudecker, H.: Matrix Differential Calculus with Applications in Statistics and Econometrics. John Wiley & Sons, New York (1999)zbMATHGoogle Scholar
  5. 5.
    Glassman, J.A.: A generalization of the fast Fourier transform. IEEE Trans. Comp. 19, 105–116 (1970)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Drubin, M.: Kronecker product factorization of the FFT matrix. IEEE Trans. 20, 590–593 (1971)CrossRefGoogle Scholar
  7. 7.
    Granata, J., Conner, M., Tolimieri, R.: Recursive Fast Algorithms and the Role of the Tensor Product. IEEE Trans. Signal Proc. 12, 2921–2930 (1992)CrossRefGoogle Scholar
  8. 8.
    Egner, S., Püschel, M.: Automatic Generation of Fast Discrete Signal Transforms. IEEE Trans. on Signal Proc. 49, 1992–2002 (2001)CrossRefGoogle Scholar
  9. 9.
    Marti-Puig, P.: A Family of Fast Walsh Hadamard Algorithmsw with Identical Sparse Matrix Factorization. IEEE Signal Proc. Let. 13, 672–675 (2006)CrossRefGoogle Scholar
  10. 10.
    Johnson, S.G., Frigo, M.: A Modified Slitp-Radix FFT with Fewer Arithmetic Operations. IEEE Trans. Signal Processing 55, 111–119 (2007)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Pere Marti-Puig
    • 1
  • Ramon Reig Bolaño
    • 1
  • Vicenç Parisi Baradad
    • 2
  1. 1.Department of Digital Information and TechnologiesUniversity of Vic (UVIC)Vic, BarcelonaSpain
  2. 2.Department of Electronic EngineeringPolitechnical Univerity of Catalonia (UPC)Vilanova i la Geltrú, BarcelonaSpain

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