  The problem of estimating a sparse signal from low dimensional noisy
observations arises in many applications, including super resolution, signal
deconvolution, and radar imaging. In this paper, we consider a sparse signal
model with non-stationary modulations, in which each dictionary atom
contributing to the observations undergoes an unknown, distinct modulation. By
applying the lifting technique, under the assumption that the modulating
signals live in a common subspace, we recast this sparse recovery and
non-stationary blind demodulation problem as the recovery of a column-wise
sparse matrix from structured linear observations, and propose to solve it via
block $\ell_{1}$-norm regularized quadratic minimization. Due to observation
noise, the sparse signal and modulation process cannot be recovered exactly.
Instead, we aim to recover the sparse support of the ground truth signal and
bound the recovery errors of the signal's non-zero components and the
modulation process. In particular, we derive sufficient conditions on the
sample complexity and regularization parameter for exact support recovery and
bound the recovery error on the support. Numerical simulations verify and
support our theoretical findings, and we demonstrate the effectiveness of our
model in the application of single molecule imaging.
