Non-stationary blind super-resolution

In this paper, we propose a new framework for parameter estimation of complex exponentials from their modulations with unknown waveforms via convex programming. Our model generalizes the recently developed blind sparse spike deconvolution framework by Y. Chi [1] to the non-stationary scenario and encompasses a wide spectrum of applications. Under the assumption that the unknown waveforms live in a common random subspace, we recast the problem into an atomic norm minimization framework by a lifting trick, and this problem can be solved using computationally efficient semidefinite programming. We show that the number of measurements for exact recovery is proportional to the number of degrees of freedom in the problem, up to polylogarithmic factors. Numerical experiments support our theoretical findings.