Blind Super-resolution in Two-dimensional Parameter Space

In this paper, we provide a new mathematical framework for identifying the parameters of a linear system from its response to multiple unknown waveforms. We assume that the system response is given by an unknown number of scaled versions of time-delayed and frequency-shifted unknown waveforms. Then, we develop a blind two-dimensional super-resolution framework that is based on the convex atomic norm frame-work to recover the continuous time-frequency shifts as well as the unknown waveforms. We prove that under a minimum separation condition between the time-frequency shifts and with a certain lower bound on the total number of the observed samples, all the unknowns in the system can be recovered precisely and with high probability. Simulation results that confirm the theoretical findings in the paper are provided.