# A Hybrid Optimization Algorithm for the Synthesis of Sparse Array Pattern Diagrams
## 2 Quantum Hybrid Particle Swarm Optimization Algorithm Combined with Quantum Behavior
### Quantum-Behaved Particle Swarm Optimization Algorithm
The calculation method for \(PSLL\) is as follows:

\[PSLL=\min\biggl{(}\max_{\phi\in S1}\biggl{(}\frac{F(\theta_{0},\varphi;\theta_{0}, \varphi_{0})}{\max F(\theta_{0},\varphi;\theta_{0},\varphi_{0})}\biggr{)}+\max_{ \phi\in S2}\biggl{(}\frac{F(\theta_{0},\varphi;\theta_{0},\varphi_{0})}{\max F( \theta_{0},\varphi;\theta_{0},\varphi_{0})}\biggr{)}\biggr{)} \tag{25}\]

In the above formula, the first term represents the sidelobe region of the array pattern at a specific azimuth angle, and the second term represents the sidelobe region in the pitch direction at a specific elevation angle. The overall peak sidelobe level is the sum of the fitness functions of the two. The array synthesis model is as follows:

\[\left\{ \begin{matrix}\max(PSLL)\Leftrightarrow sum\ \ fitness(\mathbf{X},\mathbf{Y})\\ (N_{1},N_{2},N_{3},N_{4})\\ x_{\min}=0.5\lambda\leq x_{i}\leq x_{\max}\ =\lambda,\quad y_{\min}=0.5 \lambda\leq y_{j}\leq y_{\max}=\lambda\\ x_{min}\leq L,Y_{mm}\leq H\end{matrix} \right. \tag{26}\]

The simulation experiments were conducted using the control variable method, and the key parameters of each algorithm were optimized through parameter scanning. The simulation results are as follows. In Figure 6a, among the normalized array factor patterns of the four algorithms, the sidelobe levels of classical PSO and the GWO are relatively high, exceeding \(-20\) dB, while those of the QPSO and WOA are below \(-20\) dB. Further analysis shows that in the GWO, except for the first three sidelobes, the others are below \(-40\) dB. Notably, the hybrid optimization algorithm proposed in this paper achieves excellent sidelobe suppression, with all sidelobes below \(-40\) dB. Figure 6b shows that the proposed hybrid optimization algorithm outperforms the other three in terms of

Figure 5: Schematic of a sparse rectangular array. Figure 6: (**a**) Pattern of a sparse rectangular array; (**b**) convergence of the maximum sidelobe level. global search performance and resistance to local convergence. It successfully escapes local optima in the late iteration stage and converges to the global optimum after 300 iterations, indicating its superior robustness and stability. Figure 7a presents the 3D pattern of a sparse matrix. The radiation field is distributed across all four quadrants, with maximum radiation \((\theta_{0}=0^{\circ},\varphi_{0}=0^{\circ})\) at the origin. At a specific angle, the sidelobe radiation intensity where \(\varphi=90^{\circ}\) is more significant than in other directions, when \(\varphi=0^{\circ}\). This radiation characteristic is crucial for antenna array pattern optimization as it directly impacts the antenna's radiation performance and sidelobe levels in different directions. To more intuitively illustrate the comparative performance of QPSO in the radiation pattern of sparse arrays, the results are summarized in Table 3, along with several other sparse planar rectangular array optimization algorithms. Among these three algorithms, the PSLL values are all taken as the minimum values from the elevation and azimuth planes. Compared with the arithmetic optimization algorithm (AOA), the QPSO achieves a significantly lower sidelobe level of \(-40.26\) dB under stricter aperture requirements and a smaller number of iterations, which is a much lower level than the \(-34.78\) dB obtained by the AOA. When the number of iterations is the same, QPSO demonstrates a lower PSLL than the chaotic global algorithm (CGA) under more stringent array aperture and element spacing conditions, with a PSLL of \(-40.26\) dB compared to \(-34.36\) dB with the CGA. ### Sparse Multi-Ring Concentric Circular Array

This study proposes a sparse concentric circular array. Due to the identical center position in concentric circular arrays, it can be regarded as the combination of multiple uniform concentric circular arrays, and a classic array is shown in Figure 8.