Estimating the Frequency of a Clustered Signal

Authors Xue Chen, Eric Price

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Xue Chen
  • Northwestern University, Evanston, IL, USA
Eric Price
  • The University of Texas at Austin, USA


We thank Daniel Kane and Zhao Song for many helpful discussions. We also thank the anonymous referee for the detailed feedback and comments.

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Xue Chen and Eric Price. Estimating the Frequency of a Clustered Signal. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 36:1-36:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We consider the problem of locating a signal whose frequencies are "off grid" and clustered in a narrow band. Given noisy sample access to a function g(t) with Fourier spectrum in a narrow range [f_0 - Delta, f_0 + Delta], how accurately is it possible to identify f_0? We present generic conditions on g that allow for efficient, accurate estimates of the frequency. We then show bounds on these conditions for k-Fourier-sparse signals that imply recovery of f_0 to within Delta + O~(k^3) from samples on [-1, 1]. This improves upon the best previous bound of O(Delta + O~(k^5))^{1.5}. We also show that no algorithm can do better than Delta + O~(k^2). In the process we provide a new O~(k^3) bound on the ratio between the maximum and average value of continuous k-Fourier-sparse signals, which has independent application.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • sublinear algorithms
  • Fourier transform


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