  The task of finding a sparse signal decomposition in an overcomplete
dictionary is made more complicated when the signal undergoes an unknown
modulation (or convolution in the complementary Fourier domain). Such
simultaneous sparse recovery and blind demodulation problems appear in many
applications including medical imaging, super resolution, self-calibration,
etc. In this paper, we consider a more general sparse recovery and blind
demodulation problem in which each atom comprising the signal undergoes a
distinct modulation process. Under the assumption that the modulating waveforms
live in a known common subspace, we employ the lifting technique and recast
this problem as the recovery of a column-wise sparse matrix from structured
linear measurements. In this framework, we accomplish sparse recovery and blind
demodulation simultaneously by minimizing the induced atomic norm, which in
this problem corresponds to the block $\ell_1$ norm minimization. For perfect
recovery in the noiseless case, we derive near optimal sample complexity bounds
for Gaussian and random Fourier overcomplete dictionaries. We also provide
bounds on recovering the column-wise sparse matrix in the noisy case. Numerical
simulations illustrate and support our theoretical results.
