Publication Date:
2017-04-06
Description:
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.
Print ISSN:
1069-5869
Electronic ISSN:
1531-5851
Topics:
Mathematics
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