  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.
