# Enhancing Beamforming Efficiency Utilizing Taguchi Optimization and Neural Network Acceleration
## 3 Taguchi Method
### Amplitude Synthesis of Antenna Array Radiation Pattern under Constraints
#### 3.1.1 Ten-Element Antenna Array  
The array factor is described by the following equation:  
\[AF(\theta)=2\sum_{n=1}^{5}a(n)e^{j\phi(n)}cos[\beta d(n)sin(\theta)]. \tag{10}\]  
To minimize the sidelobes, the fitness function is chosen based on the optimization objective:  
\[fitness=min(max\{20 log|AF(\theta)|\}) \tag{11}\]  
subject to constraints \(\theta\in[0^{\circ},76^{\circ}],[104^{\circ},180^{\circ}]\). To simplify the task and enhance understanding of this application, we elaborate on the Taguchi optimization procedure in the following subsections. First Step: Initialization Problem  
The first step involves selecting an appropriate experimental design and a well-structured fitness function. There are five parameters requiring optimization. Therefore, the chosen orthogonal experimental design should have five columns (\(k=5\)) to represent these parameters. To capture nonlinear effects, three levels (\(s=3\)) for each parameter are deemed sufficient. Typically, an orthogonal experimental design with a strength of 2 (\(t=2\)) is effective for most problems. In summary, an orthogonal experimental design with five columns, three levels, and a strength of 2 is required. Second Step: Designating Input Parameters  
After accessing the online database OA, an orthogonal array (OA) (Z7, 5, 3, 2) is available, as indicated in Table 2. The fitness function is chosen based on the optimization objective. In this optimization example, the function is selected to achieve a low sidelobe level. The input parameters need to be selected before conducting the experiments. When the orthogonal experimental design is chosen, numerical values corresponding to the three levels of each input parameter must be determined in the first iteration. The value for level 2 is selected at the midpoint of the optimization range. The values for levels 1 and 3 are calculated using the following equations:  
\[DN_{1}=\frac{A  
Third Step: Conducting Experiments and Building a Response Table  
After determining the input parameters, the fitness function for each experiment can be calculated. For instance, the fitness value for experiment 1 (i.e., the first row of Table 3) is computed using Equation (14), resulting in 12.97. Then, the fitness value in the Taguchi method is converted to a signal-to-noise ratio (S/N), denoted as (\(\eta\)), using Formula (14).  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Experiment** & \(\mathbf{a_{1}}\) & \(\mathbf{a_{2}}\) & \(\mathbf{a_{3}}\  
process is repeated in the subsequent iteration while reducing the optimization interval using Equation (25).  
\[\frac{DN_{i+1}}{DN_{1}}<0.01 \tag{23}\]  
\[a_{n}\mid_{i+1}^{2}=a_{n}\mid_{i}^{opt} \tag{24}\]  
\[DN_{i+1}=RR(i)\times DN_{1}=rr^{i}\times DN_{1} \tag{25}\]  
In this optimization example, the reduced function (_rr_) is set to 0.9. This first type of synthesis (amplitude synthesis under constraint) aims to minimize the secondary lobes as much as possible using our optimization technique. In Figure 3a, the obtained results demonstrate successful minimization of the secondary lobe level (SLL), also knows as the sidelobe level, down to \(-25.2722\) dB, where the gain is approximately 12.3 dB higher compared to the secondary lobe level of the uniform linear antenna array (SLL = \(-12.9651\) dB). The optimization objective is achieved after 80 iterations, as indicated in Figure 3b. \begin{table}
\begin{tabular}{c c} \hline \hline
**Optimized Values of \(a_{i}\)** & **Maximum SLL** \\ \hline
1.0000 & \(-13.1526\) \\
0.8413 & \(-22.8982\) \\
0.9322 & \\
0.7675 & \\
0.6049 & \\
0.5715 & \\
0.4746 & \\
0.4877 & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Optimized values of \(a_{i}\) and maximum SLL for networks. Figure 3: Radiation-pattern of ten-element linear antenna array and fitness function convergence.