# Low-Frequency Square Kilometer Array Pattern Optimization via Convex Programming
## 4 Results and Discussion
(**c**) Superposition of \(\left|E(\theta)\right|^{2}\) for a fixed \(\varphi\)-cut at \(197^{\circ}\), comparing the patterns achieved with unitary (red) and optimized (blue) element weights. Figure 7: Optimized weight coefficients for each array element: (**a**) Amplitude; (**b**) Phase. Blue crosses indicate the positions of the array elements, while the blue circle outlines the overall array boundary. To provide a more comprehensive performance metric, the Integrated Sidelobe Ratio (ISR), computed over the entire radiation sphere, was employed. This well-established metric is defined as follows:  
\[\text{ISR}(\text{dB})=10 log10\frac{\int_{sidelobes}\lvert E(\theta,\varphi) \rvert^{2}sin\theta d\theta d\varphi}{\int_{main\,\,lobe}\lvert E(\theta,\varphi) \rvert^{2}sin\theta d\theta d\varphi} \tag{7}\]  
ISR quantifies the ratio of total radiated power in the sidelobe region to that within the main lobe. As such, it provides a more holistic assessment of the pattern quality, particularly under the assumption of isotropic radio frequency noise, where the overall sidelobe suppression is more critical than peak values alone. Using this metric, the ISR improved from \(-11.45\) dB before the optimization to \(-13.57\) dB after optimization. The \(2.12\) dB improvement in ISR confirms that the proposed synthesis technique effectively reduces the total sidelobes' energy across all directions, providing an improved signal-to-noise ratio performance in noise-affected radio astronomy environments. Table 1 summarizes the key performance metrics for the two excitation strategies. The only notable trade-off observed was a slight reduction in directivity, from \(32.2\) dB in the uniform case to \(30.1\) dB with optimized weights. This reduction is consistent with expectations, as the optimization redistributes energy to suppress sidelobes, slightly broadening the energy footprint of the main lobe. Overall, the obtained results confirm the effectiveness of the proposed synthesis framework in improving the radiation characteristics of SKA-Low stations. The significant reduction in sidelobe levels, achieved without compromising beamwidth or introducing beam distortion, is particularly advantageous for radio astronomy applications. The improved sidelobe suppression achieved here enhances the station's dynamic range, mitigates susceptibility to radio frequency interference, and contributes to more accurate sky imaging and calibration. Furthermore, the minimal loss in directivity suggests that the trade-off between main lobe gain and sidelobe suppression is well-balanced. This demonstrates the robustness of the optimization method, especially considering the constraints imposed by the physical layout and electromagnetic behavior of real-world antenna systems. In conclusion, the presented synthesis methodology provides a scalable and effective solution for large-scale array calibration, and its application to next-generation radio telescopes such as the SKA can support the stringent performance requirements demanded by deep-space observation missions. Furthermore, to complement the scalability claim, we have considered the computational features of the proposed optimization strategy. The convex formulation ensures consistent convergence even with large-scale arrays, such as the 256-element SKA-Low station examined in this work. Nonetheless, as the array size increases, both memory requirements and solver runtime naturally grow with the problem dimensionality. These effects remain manageable in the current framework (which can be integrated by dedicated numerical strategies and parallelized solvers) for future deployments involving even \begin{table}
\begin{tabular}{c c c c c c} \hline
**WEIGHTS** & **HPBW** & **NULL-TO-NULL BW** & **PSL** & **DIRECTIVITY** & **ISR** \\ \hline Constant & \(\pm 0.62^{\circ}\) & \(\pm 1.5^{\circ}\) & \(-14.6\) dB & \(32.2\) dB & \(-11.45\) dB \\ Optimized & \(\pm 0.62^{\circ}\) & \(\pm 1.5^{\circ}\) & \(-18.2\) dB & \(30.1\) dB & \(-13.57\) dB \\ \hline \end{tabular}
\end{table}