Stable signal recovery from the roots of the short-time Fourier transform

This paper presents a method to recover a bandlimited signal, up to an overall multiplicative constant, from the roots of its short-time Fourier transform. We assume that only finitely many sample values are non-zero. To generate the number of roots needed for recovery, we use a type of aliasing, a time-frequency quasi-periodization of the transform. We investigate the stability of the recovery algorithm under perturbations of the signal, in particular under low-pass filtering, and verify the stability results with numerical experiments. In these experiments we implement a deconvolution strategy for sparse bandlimited signals, whose non-zero sample values are interspersed with vanishing ones. The recovery from roots of such signals is insensitive to the effect of random echoes. In addition, we study the effect of aliasing by the time-frequency quasi-periodization on such sparse signals. If the signal is convolved with white noise, then the number of roots generated with the quasi-periodized short-time Fourier transform can be adjusted to be proportional to the number of non-vanishing samples to give recoverability with overwhelming probability.