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Journal of Fourier Analysis and Applications

, Volume 24, Issue 1, pp 45–107 | Cite as

A Theory of Super-Resolution from Short-Time Fourier Transform Measurements

  • Céline Aubel
  • David Stotz
  • Helmut Bölcskei
Article
  • 541 Downloads

Abstract

While spike trains are obviously not band-limited, the theory of super-resolution tells us that perfect recovery of unknown spike locations and weights from low-pass Fourier transform measurements is possible provided that the minimum spacing, \(\Delta \), between spikes is not too small. Specifically, for a measurement cutoff frequency of \(f_c\), Donoho (SIAM J Math Anal 23(5):1303–1331, 1992) showed that exact recovery is possible if the spikes (on \(\mathbb {R}\)) lie on a lattice and \(\Delta > 1/f_c\), but does not specify a corresponding recovery method. Candès and Fernandez-Granda (Commun Pure Appl Math 67(6):906–956, 2014; Inform Inference 5(3):251–303, 2016) provide a convex programming method for the recovery of periodic spike trains (i.e., spike trains on the torus \(\mathbb {T}\)), which succeeds provably if \(\Delta > 2/f_c\) and \(f_c \ge 128\) or if \(\Delta > 1.26/f_c\) and \(f_c \ge 10^3\), and does not need the spikes within the fundamental period to lie on a lattice. In this paper, we develop a theory of super-resolution from short-time Fourier transform (STFT) measurements. Specifically, we present a recovery method similar in spirit to the one in Candès and Fernandez-Granda  (2014) for pure Fourier measurements. For a STFT Gaussian window function of width \(\sigma = 1/(4f_c)\) this method succeeds provably if \(\Delta > 1/f_c\), without restrictions on \(f_c\). Our theory is based on a measure-theoretic formulation of the recovery problem, which leads to considerable generality in the sense of the results being grid-free and applying to spike trains on both \(\mathbb {R}\) and \(\mathbb {T}\). The case of spike trains on \(\mathbb {R}\) comes with significant technical challenges. For recovery of spike trains on \(\mathbb {T}\) we prove that the correct solution can be approximated—in weak-* topology—by solving a sequence of finite-dimensional convex programming problems.

Keywords

Super-resolution Sparsity Inverse problems in measure spaces Short-time Fourier transform 

Mathematics Subject Classification

28A33 46E27 46N10 42B10 32A10 46F05 

Notes

Acknowledgements

The authors are indebted to H. G. Feichtinger for valuable comments, in particular, for pointing out an error in an earlier version of the manuscript, J.-P. Kahane for answering questions on results in [24], M. Lerjen for his technical support with the numerical results, and C. Chenot for inspiring discussions. We also acknowledge the detailed and insightful comments of the anonymous reviewers.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Dept. IT & EEETH ZurichZurichSwitzerland
  2. 2.Kantonsschule am BurggrabenSt. GallenSwitzerland

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