Fractional Fourier Analysis Using the Möbius Inversion Formula

Hongxia Miao, Feng Zhang*, Ran Tao

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

26 Citations (Scopus)

Abstract

Compared with the Fourier analysis, the fractional Fourier analysis is more suitable to process linear frequency modulation type non-stationary functions. To the best of our knowledge, the theoretical framework of the fractional Fourier analysis has not well established yet, especially for the fractional Fourier series (FrFS) and the discrete fractional Fourier transform (DFrFT) algorithms. To tackle with these problems, the efficient FrFS and DFrFT algorithms based on the Möbius function are proposed. First, the existence and applicability of the FrFS are analyzed basing on the Möbius inversion formula. Second, two kinds of fast algorithms for the infinite/finite FrFS are proposed. Then, based on the amplitude scaling relationship between the FrFS and samples of the FrFT, two efficient DFrFT algorithms are obtained, which are noted as the arithmetic discrete fractional Fourier transform (ADFrFT)-I and the ADFrFT-II. Importantly, the multiplication complexity of the two proposed ADFrFT algorithms is reduced to O(M), which is less than that of the state-of-the-art DFrFT algorithms. The parallel butterfly structure of the ADFrFT-II algorithm is suitable for the very large scale integration implementation. Finally, the simulations justify the efficiency of the ADFrFT algorithms in filtering and parameter evaluation of radar and optical signals.

Original languageEnglish
Article number8695861
Pages (from-to)3181-3196
Number of pages16
JournalIEEE Transactions on Signal Processing
Volume67
Issue number12
DOIs
Publication statusPublished - 15 Jun 2019

Keywords

  • ADFrFT algorithms
  • Fractional Fourier analysis
  • Möbius inversion formula
  • VLSI implementation
  • fractional Fourier series

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