# 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
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Work** & **Type** & \begin{tabular}{c} **Array** \\ **Aperture** \\ \end{tabular} & \begin{tabular}{c} **Aperture** \\ **Requirement** \\ \end{tabular} & \
\begin{tabular}{c} **Iterations** \\ **Quantity** \\ \end{tabular} &
\begin{tabular}{c} **Particle** \\ **(dB)** \\ \end{tabular} \\ \hline
[31] & AOA & \(9.5\lambda\times 4.5\lambda\) & \(d\geq 0.5\lambda\) & 1000 & \(100\) & \(-34.78\) \\
[32] & CGA & \(24\lambda\times 24\lambda\) & \(d\geq 0.5\lambda\) & 200 & \(-34.36\) \\ This work & QPSO & \(10.5\lambda\times 6.5\lambda\) & \(\lambda\geq d\geq 0.5\lambda\) & 300 & \(50\) & \(-40.26\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison diagrams of several planar sparse array arrangements. Figure 7: (a) The 3D pattern of a sparse rectangular array; (b) top view of the pattern. By substituting the coordinates of the elements in a uniform concentric circular array with polar coordinates, we have:

\[ \begin{cases}x=R\cos(\varphi)\\ y=R\sin(\varphi)\\ R^{2}=x^{2}+y^{2}\end{cases} \Leftrightarrow \begin{cases}&ku=\xi\cos(\psi)\\ &kv=\xi\sin(\psi)\\ &g^{2}=(ku)^{2}+(kv)^{2}\end{cases}  \tag{27}\]

For a concentric circular array with \(M\) rings and \(N_{m}\) elements on the \(m\)-th ring, the array factor in polar coordinates can be written as:

\[F(\xi,\psi)=\sum_{m=0}^{M}N_{m}I(R_{m},\varphi_{n})\sum_{p=-\infty}^{+\infty}J _{N_{m}p+k}(R_{m}\xi)e^{j(N_{m}p+k)\psi} \tag{28}\]

where \(J_{N_{m}p+k}(R_{m}\xi)\) denotes the zeroth-order Bessel function, defined as:

\[J_{n}(x)=\frac{1}{2\pi}{\int_{-\pi}^{\pi}e^{j(x\sin\theta-n\theta)}d\theta}= \frac{1}{\pi}{\int_{0}^{\pi}\cos(x\sin\theta-n\theta)d\theta} \tag{29}\]

Based on the Fourier relationship between the excitation of the antenna circular array elements and the pattern, a Fourier-Bessel transform exists between them. Using this time-frequency conversion, the position problem of concentric circular array elements can be transformed into an excitation energy problem involving array antennas. By considering the three optimization objectives of sparse interleaved arrays--the approximate uniformity of subarray patterns, the low sidelobe levels of individual subarray patterns, and the interleaved distribution of each antenna element--the optimal mathematical model is established as follows:

\[\begin{cases}&subarray{c}\par\par\par\par\par\par\par\par\par\par\par\par \par\par\par\par\par\par\par\par\par\par\par\par\par\par\par\par\par\par\par



A new element position optimization model is established to ensure sparse arrangement:

\[\left\{ \begin{aligned} d_{i}^{\min}=0.5\lambda\leq d_{c}\leq d_{i}^{ \max}=0.8\lambda\\ d_{\min}=0.5\lambda\leq d\leq d_{\max}=0.8\lambda\\ N_{\min}=6\leq N_{n}\leq N_{m}\leq N_{\max}=40\end{aligned}  \right.\quad\quad n\leq m \tag{31}\]

Here, \(d_{c}\) is the spacing between elements in the same ring, randomly distributed between 0.5 and 0.8 \(\lambda\); \(d\) represents the spacing between different rings; and \(N\) is the number of elements on each ring, which increases gradually with the increase in the ring spacing. In simulation tests, the particle number is set to 50, and the iteration number to 300. The number of concentric circular rings is set to 8, and the number of elements increases nonlinearly from 6 to 40. The above optimization model is incorporated into four algorithms, and the array factor patterns and sidelobe level iteration curves are obtained as shown in Figure 9.

During the sidelobe level iteration of the four algorithms, the maximum sidelobe levels of all algorithms are below \(-20\) dB, but only the grey wolf optimizer and quantum-behaved particle swarm optimization algorithms achieve a value below \(-25\) dB. As shown in Figure 9b, the proposed hybrid optimization algorithm shows superior search ability and stability compared to the grey wolf optimizer in the late iteration stage. Figure 10a shows the element position layout of the array. The array, centered at the origin, has its elements distributed sparsely across 8 rings.