  We consider the problem of direction-of-arrival (DOA) estimation in unknown
partially correlated noise environments where the noise covariance matrix is
sparse. A sparse noise covariance matrix is a common model for a sparse array
of sensors consisted of several widely separated subarrays. Since interelement
spacing among sensors in a subarray is small, the noise in the subarray is in
general spatially correlated, while, due to large distances between subarrays,
the noise between them is uncorrelated. Consequently, the noise covariance
matrix of such an array has a block diagonal structure which is indeed sparse.
Moreover, in an ordinary nonsparse array, because of small distance between
adjacent sensors, there is noise coupling between neighboring sensors, whereas
one can assume that nonadjacent sensors have spatially uncorrelated noise which
makes again the array noise covariance matrix sparse. Utilizing some recently
available tools in low-rank/sparse matrix decomposition, matrix completion, and
sparse representation, we propose a novel method which can resolve possibly
correlated or even coherent sources in the aforementioned partly correlated
noise. In particular, when the sources are uncorrelated, our approach involves
solving a second-order cone programming (SOCP), and if they are correlated or
coherent, one needs to solve a computationally harder convex program. We
demonstrate the effectiveness of the proposed algorithm by numerical
simulations and comparison to the Cramer-Rao bound (CRB).
