# 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
This experimental design not only ensures statistical significance but also objectively reflects the differences in the optimal solution search capabilities of the various algorithms. ### Sparse Linear Array Simulation Test

Sparse arrays, known for their simple structure and easy implementation, are widely used in engineering applications. Taking linear arrays as an example, common sparse arrays usually contain an odd number of array elements, positioned symmetrically around the central unit within a fixed aperture. For arrays with fixed excitation amplitude and varying phases, the array factor is expressed as follows:

\[AF(\theta)=\sum_{n=0}^{N-1}I_{n}e^{i(nk\sin(\theta)+\phi_{n})} \tag{18}\]

where the default excitation amplitude of each array element is assumed to be \(I_{n}\); \(d\) is the spacing between adjacent antenna elements, \(\theta\) is the incident beam angle, \(\phi_{n}\) is the phase deviation of the nth antenna, \(k\) is the wavenumber, defined as \(k=2\pi/\lambda\) where \(\lambda\) is the wavelength, and \(\theta\) is the observation angle. Figure 2 is a schematic diagram of a common linear array:

This paper presents an innovative antenna array design, employing a series-fed approach [28]. The spacing between every two antenna elements varies, eliminating the need for an overall symmetrical layout while achieving low sidelobe performance. The relevant schematic is shown in Figure 3.

The entire antenna array is fed by a single source. Multiple one-to-two power dividers facilitate hierarchical feeding. One branch passes through amplitude and phase compensators to ensure consistent excitation and random phase compensation. The other branch feeds subsequent elements via a feedline running through the entire radiating cavity, enabling convenient feeding. Figure 3: Schematic diagram of the string feed array transmission system. Figure 2: Diagram illustrating a common antenna array. The optimization models for element positions and phases are defined as follows:

\[\left\{ \begin{aligned} d_{\text{min}}=d_{c}=0.5\lambda\leq d_{i} \leq d_{\text{max}}=\lambda\\ \sum\limits_{i=1}^{N}d_{i}=L\\ \text{min}(PSLL(d_{1},d_{2},\ldots,d_{n}))\end{aligned} \right. \tag{19}\]

where PSLL (peak sidelobe level) is defined as the maximum radiation intensity in the sidelobe region of the array radiation pattern. Its mathematical expression is:

\[PSLL=20\log_{10}^{(\text{max}(|AF(0)|)/|AF(\theta_{0})|)} \tag{20}\]

To compare algorithm performance, all optimization models were implemented on the MATLAB 2021(b) simulation platform under uniform conditions of equal amplitude excitation (\(I_{n}=1\)) and fixed aperture size (\(D=10.43\)\(\lambda\)). Figure 4a presents the comparison results of normalized radiation patterns, and Figure 4b shows the PSLL convergence curves of the algorithms. As shown in Figure 4, the quantum-behaved particle swarm optimization (QPSO) algorithm proposed in this paper is compared with three typical optimization algorithms: classic particle swarm optimization (PSO), the grey wolf optimizer (GWO), and the whale optimization algorithm (WOA). Figure 4a indicates that the QPSO algorithm achieves the best sidelobe suppression performance (PSLL = \(-20.46\) dB), surpassing the WOA (\(-19.6\) dB), the GWO (\(-18.86\) dB), and PSO (\(-18.15\) dB). The convergence analysis in Figure 4b shows that the WOA algorithm exhibits the fastest initial convergence speed, while the QPSO algorithm attains the global optimal solution after 200 iterations. The element spacing distribution optimized by QPSO is presented in Table 1. Its non-uniform characteristics contribute to the aforementioned sidelobe suppression effect.