Iterative Reweighted ℓ₁ Synthesis of Sparse Antenna Arrays with Continuous Element Positions

Sparse antenna arrays are attractive for compact microwave and millimeter-wave front ends because they can achieve prescribed radiation performance with fewer radiating elements, thereby reducing the number of feeding channels, hardware cost, weight, and power consumption. However, the joint optimization of element positions and complex excitations remains challenging, since the element positions enter the array factor nonlinearly and the element-count objective is inherently combinatorial. This paper presents an iterative reweighted ℓ₁ synthesis framework for sparse antenna arrays with continuous element positions. At each iteration, position perturbations are introduced and the array factor is linearized using a first-order Taylor expansion within a trust region. The resulting non-convex sparse synthesis problem is then approximated by convex programing through an iteratively reweighted ℓ₁ relaxation, allowing the excitation amplitudes, phases, and element positions to be updated simultaneously. Additional aperture, minimum-spacing, and minimum-directivity requirements are formulated as convex constraints and incorporated when required, enabling joint control of sparsity, sidelobe level, physical layout, and radiation performance. The proposed method is validated through four representative examples, including a shaped-beam linear array, a tri-pattern reconfigurable linear array, a planar pencil-beam array, and a directivity-constrained planar array. Compared with fixed-grid reweighted ℓ₁ methods under the same specifications, the proposed approach produces sparser layouts while avoiding the grid-resolution limitation. In the directivity-constrained benchmark, it also achieves competitive element reduction while enforcing a prescribed minimum element spacing. These results indicate that the proposed framework provides a flexible and practical synthesis tool for compact and integrated sparse antenna-array design.