A theory of super-resolution from short-time Fourier transform measurements

Authors

Céline Aubel, David Stotz, and Helmut Bölcskei

Reference

Journal of Fourier Analysis and Applications, Vol. 24, No. 1, pp. 45-107, Feb. 2018.

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, D, between spikes is not too small. Specifically, for a measurement cutoff frequency of fc, Donoho [2] showed that exact recovery is possible if the spikes (on R) lie on a lattice and D > 1/fc, but does not specify a corresponding recovery method. Candès and Fernandez-Granda [3,4] provide a convex programming method for the recovery of periodic spike trains (i.e., spike trains on the torus T), which succeeds provably if D > 2/fc and fc >= 128 or if D > 1.26/fc and fc >= 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 [3] for pure Fourier measurements. For a STFT Gaussian window function of width 1/(4fc) this method succeeds provably if D > 1/fc, without restrictions on fc. 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 R and T. The case of spike trains on R comes with significant technical challenges. For recovery of spike trains on 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

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