# Low-Frequency Square Kilometer Array Pattern Optimization via Convex Programming
## 2 The Proposed Approach  
The component of interest in the receiving pattern of a generic array antenna composed of \(N\)-elements can be expressed as follows:  
\[\mathbf{E}(\theta,\varphi)=\sum_{n=1}^{N}I_{n}\mathbf{E}\mathbf{E}\mathbf{P}_{ n}(\theta,\varphi) \tag{1}\]  
where \(\theta\) and \(\varphi\) denote the elevation and azimuth angle with respect to the array's boresight, while \(I_{n}\) represents the complex weight applied to the n-th element. The term \(\mathbf{E}\mathbf{P}_{n}(\theta,\varphi)\) is the embedded element pattern of the n-th antenna element, computed with respect to a common origin shared by the entire array. As a result, the phase contribution typically associated with the array factor is inherently included in the EEP. This pattern represents the electric field radiated when the n-th element is excited with a unit weight, while all other elements are terminated with matched loads (typically 50 \(\Omega\)). Note that this formulation inherently accounts for mutual coupling effects [21]. The objective is to maximize the difference between the field amplitude in the target direction and that in the sidelobes' region by optimally controlling the array element weights. This leads to a substantial reduction in interference from undesired directions, thereby enhancing the overall electromagnetic performance of SKA-Low stations. The proposed approach is grounded on a mask-constrained power synthesis framework, which is particularly well-suited for forming a narrow pencil-beam while limiting radiation in non-target directions. This characteristic is critical for radio astronomy applications such as SKA-Low, where minimizing both terrestrial and celestial interference is a fundamental requirement. The synthesis is formulated as a convex optimization problem, drawing from established techniques for optimal field focusing under arbitrary upper bounds [12; 14]. The adopted framework is based on a number of key modeling assumptions, which reflect both the design of the SKA-Low station, as well as the constraints imposed by the available data as detailed in the following. First, the array geometry follows a Vogel spiral distribution [Figure 3b], which provides quasi-random spatial sampling while preserving rotational symmetry and is well known as an effective strategy for reducing grating lobes in large-aperture systems. Second, the antenna element model is derived from the copolar component of the SKALA 4.1 radiation pattern. This choice is consistent with the available full-wave electromagnetic data. Third, the array is modeled as a sparse one, in line with the physical design of the station. In fact, sparsity can mitigate mutual coupling while inducing some limitations on the degrees of freedom available for excitation control. These assumptions influence both the structure of the synthesis procedure and the interpretation of the results. In particular, while the framework prioritizes sidelobe level (SLL) suppression across the full angular domain (which is a key requirement for high-dynamic-range sky imaging), the corresponding trade-off is a moderate reduction in directivity. This outcome is consistent with expectations as, in sparse arrays, redistributing energy away from the main beam to suppress sidelobes typically leads to slight broadening of the beam and reduced gain. As shown in the results, the achieved sidelobe suppression and the associated directivity loss reflect a balanced and physically meaningful optimization outcome, fully aligned with the assumptions embedded in the problem formulation. To describe the optimization process in detail, let \(\theta_{T}\) denote the target direction in which the receiving pattern intensity has to be maximized.