  Estimating the directions of arrival (DOAs) of multiple sources from a single
snapshot obtained by a coherent antenna array is a well-known problem, which
can be addressed by sparse signal reconstruction methods, where the DOAs are
estimated from the peaks of the recovered high-dimensional signal. In this
paper, we consider a more challenging DOA estimation task where the array is
composed of non-coherent sub-arrays (i.e., sub-arrays that observe different
unknown phase shifts due to using low-cost unsynchronized local oscillators).
We formulate this problem as the reconstruction of a joint sparse and low-rank
matrix and solve its convex relaxation. While the DOAs can be estimated from
the solution of the convex problem, we further show how an improvement is
obtained if instead one estimates from this solution the phase shifts, creates
"phase-corrected" observations and applies another final (plain, coherent)
sparsity-based DOA estimation. Numerical experiments show that the proposed
approach outperforms strategies that are based on non-coherent processing of
the sub-arrays as well as other sparsity-based methods.
