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, Δ, between spikes is not too small. Specifically, for a cutoff frequency of f_c, the work of Donoho (1992) shows that exact recovery is possible if Δ > l/f_c, but does not specify a corresponding recovery method. On the other hand, Candès and Fernandez-Granda (2013) provide a recovery method based on convex optimization, which provably succeeds as long as Δ > 2/f_c. In practical applications one often has access to windowed Fourier transform measurements, i.e., short-time Fourier transform (STFT) measurements, only. In this paper, we develop a theory of super-resolution from STFT measurements, and we propose a method that provably succeeds in recovering spike trains from STFT measurements provided that Δ > l/f_c.