# 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} \hline \hline
**Elements** & **Unit Spacing (\(\lambda\))** & **Elements** & **Unit Spacing (\(\lambda\))** \\ \hline
1-2 & 1 & 9-10 & 0.5000 \\
2-3 & 0.7597 & 10-11 & 0.5000 \\
3-4 & 0.7599 & 11-12 & 0.5670 \\
4-5 & 0.5136 & 12-13 & 0.5683 \\
5-6 & 0.5000 & 13-14 & 0.7792 \\
6-7 & 0.5000 & 14-15 & 0.7821 \\
7-8 & 0.5065 & 15-16 & 0.9157 \\
8-9 & 0.5000 & 16-17 & 0.7761 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Spacing table of each unit in a sparse array. Figure 4: (**a**) Sparsely distributed linear array pattern; (**b**) minimum sidelobe\(-\)level iteration number diagram. Table 2 provides a comparison of the proposed algorithm with classical particle swarm optimization (PSO) and the Levy flight particle swarm optimization (LFPSO) algorithms from other works in the literature. As shown in Table 2, when the number of array elements is fixed, the quantum-behaved particle swarm optimization (QPSO) algorithm has certain advantages over the LFPSO algorithm in terms of element spacing requirements and the number of iterations. Specifically, the peak sidelobe level (PSLL) of QPSO is \(-20.46\) dB, which is significantly lower than that of LFPSO (\(-19.61\) dB). Moreover, compared with the PSO algorithm in Reference [25], QPSO achieves a lower sidelobe level than the best result (\(-19.87\) dB) reported in that study, even though the number of iterations for QPSO is \(100\) more than that used in Reference [25]. ### Sparse Matrix Simulation Test

This study constructs a rectangular sparse array model of \(M\times N\) with a physical aperture size of \(L\times H\). To avoid element-coupling effects, the minimum element spacing is set to \(d_{\min}=0.5\lambda\), where \(\lambda\) is the operating wavelength. The main beam is directed at \((\theta_{0},\varphi_{0})\). Under equal-amplitude excitation, the array pattern function is:

\[F(\theta,\varphi)=\sum_{m=1}^{M}\sum_{n=1}^{N}e^{jk[x_{mn}(\sin\theta\cos \varphi-\sin\theta_{0}\cos\varphi_{0})+y_{mn}(\sin\theta\sin\varphi-\sin \theta_{0}\sin\varphi_{0})]} \tag{21}\]

Using the first quadrant coordinate system, \((x_{mn},y_{mn})\) represents the two-dimensional coordinates of the \((m,n)\)-th element in the non-uniform grid, where \(x_{mn}\) and \(y_{mn}\) denote the horizontal and vertical distances of the \(i\)-th element from the origin, respectively, and they satisfy \(x_{mn}\leq L,Y_{mn}\leq H\). To better search for the positions of the elements, a position optimization matrix is established:

\[\mathbf{X}= \begin{pmatrix}x_{1,1}&\ldots&x_{1,n}\\ \vdots&&\vdots\\ x_{m,1}&\ldots&x_{m,n}\end{pmatrix} ,\quad\mathbf{Y}= \begin{pmatrix}y_{1,1}& \ldots&y_{1,n}\\ \vdots&&\vdots\\ y_{m,1}&\ldots&y_{m,n}\end{pmatrix}  \tag{22}\]

For the sum of the dimensions of the matrix, we find:

\[\mathbf{X}\times\mathbf{Y}\geq(M\times N),\quad\frac{L}{x_{\max}}\ \leq \mathbf{X}\leq\frac{L}{x_{\min}},\quad\frac{H}{y_{\max}}\leq\mathbf{Y}\leq \frac{H}{y_{\min}} \tag{23}\]

The fitness values of the two matrices are used as a reference for measuring the sidelobe level. To maintain the aperture size, elements are fixed at the four corners of the array:

\[ \begin{cases}N_{1}(x_{1}=0,y_{1}=0);N_{2}(x_{2}=L,y_{2}=H)\\ N_{3}(x_{3}=0,y_{3}=L);N_{4}(x_{4}=H,y_{4}=0)\end{cases}  \tag{24}\]

The relevant schematic is shown in Figure 5. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Work** & **Type** & **Array** & **Array** & **Iterations** & **Spacing** & **PSLL** \\  & & **Elements** & **Type** & & **Requirement** & **(dB)** \\ \hline
[29] & LFPSO & & Symmetry & 500 & \(d\geq 0.5\lambda\) & \(-19.61\) \\
[30] & PSO & 17 & Symmetry & 100 & \(d\geq 0.5\lambda\) & \(-19.87\) \\ This work & QPSO & & Asymmetry & 300 & \(\lambda\geq d\geq 0.5\lambda\) & \(-20.46\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison table of several algorithms.