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, \(\Delta \), 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.