arXiv:2406.16041v2 Announce Type: replace Abstract: 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.