Abstract: Due to the constraints of the bounded lemma, direction-of-arrival (DOA) estimation methods based on atomic norm minimization (ANM) usually necessitates exhaustive search over the entire angular domain, resulting in substantial computational redundancy. To address this challenge, this paper proposes a dual-variable processing framework involving bandpass filtering and downsampling. This approach enables the conversion of dual trigonometric polynomial inequality constraints into semidefinite constraints through the bounded lemma, thereby transforming the dual problem into a semidefinite programming (SDP) formulation. Consequently, an ANM-based DOA estimation method with angular sector constraints is developed. To enhance computational efficiency, this paper further propose a fast algorithm that leverages primal-dual interior-point methods to accelerate computation. Simulation results demonstrate that, compared to the original ANM algorithm, the proposed constrained area ANM algorithm significantly reduces computational time. Additionally, the proposed fast algorithm further increases computation speed while maintaining the same estimation accuracy as the original algorithm.