# Low-Frequency Square Kilometer Array Pattern Optimization via Convex Programming
Due to reciprocity arguments, this goal is equivalent to a classical synthesis problem in array antenna design [12]. In fixed-geometry arrays, where the elements' positions are defined in advance and only the elements' weights are subject to optimization, globally optimal solutions have been developed for both pencil [12; 13; 14; 15; 16; 17] and difference [18] beams, even under arbitrary sidelobe constraints. These methodologies are applicable to a wide variety of array configurations, including linear, planar, and conformal geometries, provided that the element locations remain fixed. In particular, the framework proposed in [12; 14] reformulates the synthesis task, i.e., maximizing the field (or its derivative) in a target direction while constraining the field elsewhere, as a Convex Programming (CP) [12; 14] or Linear Programming (LP) [13] problem. This formulation guarantees global optimality and is naturally extendable to directivity maximization under prescribed sidelobe constraints. Moreover, recent developments have enabled the use of phaseless data in synthesis problems [19], leveraging phase retrieval techniques [20] to achieve optimal excitation design for arbitrary radiation footprints. This work applies the methodology presented in prior studies on optimal field focusing under arbitrary Upper Bounds (UB) [12; 14] to enhance the electromagnetic performance of SKA-Low stations. The analysis addresses constrained optimization scenarios in which the degrees of freedom are amplitudes and phases of the element weight coefficients, under the assumption of a fixed-geometry array. As previously demonstrated in [12; 14], this approach enables a well-balanced trade-off between key performance metrics, such as directivity and sidelobe suppression, while ensuring globally optimal solutions tailored to the operational requirements of SKA-Low. Figure 2: Station layouts for SKA-Low: (**a**) element distribution in the XY plane for the AAVS1 and AAVS2 configurations; (**b**) element distribution in the XY plane for the AAVS3 configuration. It is worth noting that direct experimental validation is infeasible, as full-scale testing would require the deployment of all 256 SKALA 4.1 antennas over a 35 m-diameter station in a low-interference environment, as well as a real-time digital beamforming system. Furthermore, simplifying the setup by reducing the number of antennas would significantly misrepresent the mutual coupling and the other electromagnetic interactions, compromising the relevance of the results. As an accepted and practical alternative, this study relies on full-wave electromagnetic simulations using realistic embedded element patterns, providing high-fidelity emulation of the SKA-Low's physical behavior. This forms a sound conceptual validation framework for the proposed method. Additionally, this study lays the groundwork for future experimental activities, which will be explored in an upcoming international project that integrates numerical optimization with practical calibration strategies for SKA-Low. The remainder of the paper is organized as follows. Section 2 describes the proposed synthesis framework, while Section 3 details the methodology. Section 4 presents and discusses the achieved numerical results, and Section 5 concludes the paper.