# Low-Frequency Square Kilometer Array Pattern Optimization via Convex Programming
## 4 Results and Discussion  
This section presents a comprehensive analysis of the numerical results obtained to evaluate the performance of the proposed synthesis strategy. The objective was to assess the enhancement in radiation characteristics achieved by transitioning from a conventional uniform excitation scheme to an optimized excitation profile derived through convex optimization. The simulation campaign was conducted at a central frequency of 350 MHz using the EEPs generated from full-wave electromagnetic modeling, as described in Section 3. Two excitation configurations were analyzed: (i) uniform excitation, in which all elements are driven with equal amplitude and phase; and (ii) optimized excitation, where complex weights were determined by solving the constrained optimization problem defined in Section 2, imposing \(UB=-25\) dB for \(\theta\in\left[2^{\circ},90^{\circ}\right]\). To ensure that the optimization addressed the entire radiation space, the synthesis procedure was executed over a dense spherical grid consisting of 720 azimuthal samples and 181 elevation points. Although the main beam was steered towards the zenith (\(\theta=0^{\circ}\)), the optimization did not focus on a single angular slice. Instead, the sidelobe suppression constraints were applied globally across all azimuthal cuts and elevation angles, resulting in a pattern that is uniformly optimized over the full 3D angular domain. The formulation aimed to maximize the real part of the field in the zenith direction while simultaneously imposing two critical constraints: the imaginary part at the zenith was constrained to zero to ensure phase coherence, and the field amplitude elsewhere was required to remain below a predefined UB mask. This approach guarantees that sidelobe suppression is spatially consistent and not limited to specific directions or planes. Moreover, the constraining mask was carefully designed to preserve the main features of the main lobe, such as peak amplitude, Half Power Beamwidth (HPBW), and null-to-null beamwidth, thereby isolating the effects of sidelobe reduction from any distortion of the primary beam. Figure 6 illustrates the radiated power patterns obtained for both excitation configurations. Under uniform excitation, Figure 6a, the pattern exhibits recurring and prominent sidelobes, particularly around 32\({}^{\circ}\) elevation. These features are a consequence of the quasi-periodic nature of the pseudo-random element layout, which, despite its aperiodicity, imposes spectral content that manifests as directional sidelobe artifacts. Figure 6: _Cont._  
To address this, the convex optimization approach was implemented using MATLAB's _fmincon_ solver with the Active-Set algorithm. The resulting complex weight coefficients, including both amplitude and phase, are shown in Figure 7, alongside the geometric distribution of the antenna elements. Following optimization, the synthesized radiation pattern, Figure 6b, demonstrates a substantial improvement in sidelobe performance. The Peak Sidelobe Level (PSL) was reduced from \(-14.6\) dB in the uniform case to \(-18.2\) dB, representing an improvement of approximately \(3.6\) dB. Notably, this enhancement was achieved without degradation of the main beam. Both the HPBW and the null-to-null beamwidth remained consistent at approximately \(\pm 0.62^{\circ}\) and \(\pm 1.5^{\circ}\), respectively, for both configurations. Furthermore, Figure 6c presents a comparison between the radiation pattern at \(\varphi=197^{\circ}\) for unitary and optimized element weights. This cut, representing the worst-case scenario in terms of PSL, confirms that the sidelobe reduction does not adversely affect the main lobe of the antenna array's radiation pattern. Figure 6: Superposition of \(\left|E(\theta)\right|^{2}\) along all \(\varphi\)-cuts: patterns respectively achieved by using unitary (**a**) and optimized (**b**) element weights. The red line indicates the best sidelobe performance achieved using the synthesis method described in Section 2.