## Abstract

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, L_{1}(f̂)/||f̂_{2}, and 1/τ , where the significant frequencies are those occupying at least a τ-fraction of the energy of the signal, and L_{1}(f̂) denotes the L_{1}-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 C^{N}. As a central tool, we prove there is a deterministic algorithm that takes as input N , ε and an arithmetic progression P in Z_{N}, runs in time polynomial in ln N and 1/ε, and returns a set A_{P} that ε-approximates P in Z_{N} in the sense that equation presented}. In other words, we show there is an explicit construction of sets A_{P} of size polynomial in ln N and 1/ε that ε-approximate given arithmetic progressions P in Z_{N}. 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 language | English |
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Article number | 6657776 |

Pages (from-to) | 1733-1741 |

Number of pages | 9 |

Journal | IEEE Transactions on Information Theory |

Volume | 60 |

Issue number | 3 |

DOIs | |

State | Published - Mar 2014 |

Externally published | Yes |

## Keywords

- 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