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

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, , between spikes is not too small. Specifically, for a measurement cutoff frequency of , Donoho (SIAM J Math Anal 23(5):1303-1331, 1992) showed that exact recovery is possible if the spikes (on ) lie on a lattice and , but does not specify a corresponding recovery method. CandSs 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 ), which succeeds provably if and or if and , 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 CandSs and Fernandez-Granda (2014) for pure Fourier measurements. For a STFT Gaussian window function of width this method succeeds provably if , without restrictions on . 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 and . The case of spike trains on comes with significant technical challenges. For recovery of spike trains on we prove that the correct solution can be approximated-in weak-* topology-by solving a sequence of finite-dimensional convex programming problems.