Deterministic sparse fourier approximation via approximating arithmetic progressions

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We present a deterministic algorithm for finding the significant Fourier frequencies of a given signal f ∈ C N and their approximate Fourier coefficients in running time and sample complexity polynomial in N, L1(f̂)/||f̂2, and 1/τ , where the significant frequencies are those occupying at least a τ-fraction of the energy of the signal, and L1(f̂) denotes the L1-norm of the Fourier transform of f. Furthermore, the algorithm is robust to additive random noise. This strictly extends the class of compressible/Fourier sparse signals efficiently handled by previous deterministic algorithms for signals in CN. As a central tool, we prove there is a deterministic algorithm that takes as input N , ε and an arithmetic progression P in ZN, runs in time polynomial in ln N and 1/ε, and returns a set AP that ε-approximates P in ZN in the sense that equation presented}. In other words, we show there is an explicit construction of sets AP of size polynomial in ln N and 1/ε that ε-approximate given arithmetic progressions P in ZN. This extends results on small-bias sets, which are sets approximating the entire domain, to sets approximating a given arithmetic progression; this result may be of independent interest.

Original languageEnglish
Article number6657776
Pages (from-to)1733-1741
Number of pages9
JournalIEEE Transactions on Information Theory
Issue number3
StatePublished - Mar 2014
Externally publishedYes


  • Algorithm design and analysis
  • Combinatorial mathematics
  • Computational efficiency
  • Discrete Fourier transforms
  • Fast Fourier transforms
  • Noise
  • Signal processing algorithms
  • Signal reconstruction
  • Signal sampling

ASJC Scopus subject areas

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences


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