The two-modular Fourier transform of binary functions

Yi Hong, Emanuele Viterbo, Jean-Claude Belfiore

Research output: Contribution to journalArticleResearchpeer-review


In this paper, we provide a solution to the open problem of computing the Fourier transform of a binary function defined over n-bit vectors taking m-bit vector values. In particular, we introduce the two-modular Fourier transform (TMFT) of a binary function f : G → ℜ, where G = (F2n, +) is the group of n bit vectors with bitwise modulo two addition +, and ℜ is a finite commutative ring of characteristic 2. Using the specific group structure of G and a sequence of nested subgroups of G, we define the fast TMFT and its inverse. Since the image ℜ of the binary functions is a ring, we can define the convolution between two functions f : G → ℜ. We then provide the TMFT properties, including the convolution theorem, which can be used to efficiently compute convolutions. Finally, we derive the complexity of the fast TMFT and the inverse fast TMFT.
Original languageEnglish
Article number7439833
Pages (from-to)2813 - 2826
Number of pages14
JournalIEEE Transactions on Information Theory
Issue number5
Publication statusPublished - May 2016


  • Two-modular Fourier transform
  • Binary functions
  • Binary groups
  • Group ring

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