  Spatial frequency estimation from a mixture of noisy sinusoids finds
applications in various fields. While subspace-based methods offer
cost-effective super-resolution parameter estimation, they demand precise array
calibration, posing challenges for large antennas. In contrast, sparsity-based
approaches outperform subspace methods, especially in scenarios with limited
snapshots or correlated sources. This study focuses on direction-of-arrival
(DOA) estimation using a partly calibrated rectangular array with fully
calibrated subarrays. A gridless sparse formulation leveraging shift
invariances in the array is developed, yielding two competitive algorithms
under the alternating direction method of multipliers (ADMM) and successive
convex approximation frameworks, respectively. Numerical simulations show the
superior error performance of our proposed method, particularly in highly
correlated scenarios, compared to the conventional subspace-based methods. It
is demonstrated that the proposed formulation can also be adopted in the fully
calibrated case to improve the robustness of the subspace-based methods to the
source correlation. Furthermore, we provide a generalization of the proposed
method to a more challenging case where a part of the sensors is unobservable
due to failures.
