Wideband DOA Estimation Using a Compact Formulation of ℓ_2,1 Norm Minimization with Multiple Dictionaries

Wideband direction-of-arrival (DOA) estimation is often formulated as a sparse signal recovery problem with multiple dictionaries, where the commonly adopted ${\ell }_{2,1}$-norm minimization framework exploits the joint sparsity shared across different frequency bins. However, the resulting optimization problem involves a large number of variables and becomes computationally expensive as the problem scale increases. In this paper, a compact reformulation of the multi-dictionary ${\ell }_{2,1}$-norm minimization problem is derived, which significantly reduces the number of optimization variables by introducing an equivalent diagonal representation. Under the special case of uniform linear arrays and harmonic sources, the proposed formulation is further extended to a gridless form, and its equivalence to wideband atomic norm minimization is discussed. For the grid-based compact formulation, an efficient block coordinate descent algorithm is developed, where each update admits a closed-form expression. For the gridless formulation, a first-order solver based on the alternating direction method of multipliers is employed to handle large-scale problems. Numerical simulations demonstrate that the proposed methods achieve substantial reductions in computational complexity, thereby enabling efficient wideband DOA estimation in large-scale scenarios.