Blind Two-Dimensional Super Resolution in Multiple Input Single Output Linear Systems

Abstract: In this paper, we consider a multiple-input single-output (MISO) linear time-varying system whose output is a superposition of scaled and time-frequency shifted versions of inputs. The goal of this paper is to determine system characteristics and input signals from the single output signal. More precisely, we want to recover the continuous time-frequency shift pairs, the corresponding (complex-valued) amplitudes and the input signals from only one output vector. This problem arises in a variety of applications such as radar imaging, microscopy, channel estimation and localization problems. While this problem is naturally ill-posed, by constraining the unknown input waveforms to lie in separate known low-dimensional subspaces, it becomes tractable. More explicitly, we propose a semidefinite program which exactly recovers time-frequency shift pairs and input signals. We prove uniqueness and optimality of the solution to this program. Moreover, we provide a grid-based approach which can significantly reduce computational complexity in exchange for adding a small gridding error. Numerical results confirm the ability of our proposed method to exactly recover the unknowns.