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

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, ∆, between spikes is not too small. Specifically, for a measurement cutoff frequency of f c , Donoho [2] showed that exact recovery is possible if the spikes (on R) lie on a lattice and ∆ > 1/f c , 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 ∆ > 2/f c and f c 128 or if ∆ > 1.26/f c and f c 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 σ = 1/(4f c ) this method succeeds provably if ∆ > 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 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.

Point Source Super-resolution Via Non-convex $$L_1$$ Based Methods

“…Note that both aforementioned works [6,18] are formulated in terms of off-grid spikes. Another article about off-grid spikes was [1] by Aubel et al, who also arrived at MSF > 1 if windowed Fourier (or short-time Fourier transform) measurements are available. Furthermore, there are two works that do not require MS. De Castro and Gamboa [10] showed that K spikes can be resolved from 2K +1 Fourier samples; and with additional positive assumption of point sources, Donoho et al [14] showed that 2K noiseless measurements are sufficient to yield exact recovery of K positive spikes.…”

Generalized Line Spectral Estimation via Convex Optimization

“…A similar statement continues to hold when only a random subset of the uniformly spaced samples are available [Tan+13]. Related convex programs have been studied for denoising [BTR13], signal recovery from short-time Fourier measurements [ASB15], deducing time-frequency components of a signal with applications in super-resolution radar [HMS16], and line spectral estimation from multiple measurements [LC16].…”

Radar Emitter Recognition Based on the Energy Cumulant of Short Time Fourier Transform and Reinforced Deep Belief Network

“…It converts the time-domain signals into two dimensional TFD, which presents a frequency variation trend with time. Recently, many studies about STFT were conducted, getting good results [ 32 , 33 , 34 , 35 ]. In [ 32 ], Aubel developed a theory of super-resolution from STFT measurements and presented a new recovery method for pure Fourier measurements.…”