Optimized Fast Algorithms for the Quaternionic Fourier Transform

  • Michael Felsberg
  • Gerald Sommer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1689)


In this article, we deal with fast algorithms for the quaternionic Fourier transform (QFT). Our aim is to give a guideline for choosing algorithms in practical cases. Hence, we are not only interested in the theoretic complexity but in the real execution time of the implementation of an algorithm. This includes floating point multiplications, additions, index computations and the memory accesses. We mainly consider two cases: the QFT of a real signal and the QFT of a quaternionic signal. For both cases it follows that the row-column method yields very fast algorithms. Additionally, these algorithms are easy to implement since one can fall back on standard algorithms for the fast Fourier transform and the fast Hartley transform. The latter is the optimal choice for real signals since there is no redundancy in the transform. We take advantage of the fact that each complete transform can be converted into another complete transform. In the case of the complex Fourier transform, the Hartley transform, and the QFT, the conversions are of low complexity. Hence, the QFT of a real signal is optimally calculated using the Hartley transform.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Michael Felsberg
    • 1
  • Gerald Sommer
    • 1
  1. 1.Christian-Albrechts-University of KielInstitute of Computer Science and Applied Mathematics Cognitive SystemsKielGermany

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