# Enhancing Beamforming Efficiency Utilizing Taguchi Optimization and Neural Network Acceleration
## 3 Taguchi Method
### Phase Synthesis of the Antenna Array Radiation Pattern  
Phase synthesis allows for the realization of directed lobes with a moderately controllable level of sidelobes. Using this technique, it is possible to control the received level in the direction of useful and interfering radiation. The linear antenna array has \(N\) equally spaced elements along the \(z\)-axis. The spacing between elements is half the wavelength (\(\lambda/2\)), and the excitations of the array elements are symmetric and uniform with respect to the \(x\)-axis. Hence, the excitation phases of the antenna array are optimized within the interval \([-\pi,\pi]\). The array factor of this linear array is defined by  
\[AF(\theta)=2\sum_{n=1}^{N}a(n)e^{i\phi(n)}cos[\beta d(n)sin\theta], \tag{30}\]  
where:  
* \(d(n)\): distance between sources \(=\frac{\lambda}{2}\)
* \(a(n)\): amplitude \(=1\)
* \(\phi(n)\): phase \(\in[-\pi,\pi]\)
* \(\beta=\frac{2\pi}{2}\)  
Ten-Element Antenna Array  
The Taguchi method used in this type of synthesis aims to minimize the sidelobes as much as possible. The array factor is described by the (31):  
\[AF(\theta)=2\sum_{n=1}^{5}cos[\beta d(n)sin(\theta)+\phi(n)] \tag{31}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Elements** & **Weights** & & \\ \hline
1 & 1.0000 & 7 & 0.5292 \\
2 & 0.9717 & 8 & 0.4203 \\
3 & 0.9171 & 9 & 0.3182 \\
4 & 0.8399 & 10 & 0.2275 \\
5 & 0.7454 & 11 & 0.1512 \\
6 & 0.6397 & 12 & 0.1262 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Taguchi-optimized weights for the 24 elements. Figure 6: Radiation- pattern of the 24-element linear antenna array and fitness function convergence. The array factor in decibels is  
\[AF_{dB}(\theta)=20.log\bigg{|}\frac{AF(\theta)}{AF(\theta_{0})}\bigg{|}, \tag{32}\]  
with \(\theta_{0}\) being the main beam direction = \(0^{\circ}\). The objective (fitness) function is chosen based on the optimization objective:  
\[fitness=k_{1}\times f_{SL}(\theta)+k_{2}\times f_{NS}(\theta). \tag{33}\]  
\[f_{SL}:\text{Region of SLL minimization}; f_{SL}(\theta)=\max\{AF_{dB}(\theta)\}\] \[\theta_{NS}:\text{Region of interference}; f_{NS}(\theta)=\sum_{k}AF_{dB}(\theta_{null}^{k})\]  
To use the Taguchi method, the following steps are required:  
* Step 1: Determine the number of parameters (\(k=10\)). * Step 2: Determine the number of levels (\(s=3\)). * Step 3: Determine the strength (\(t=2\)). * Step 4: Determine the OA experimental design (\(27,10,3,2\)). * Step 5: Determine the reduced function (\(rr=0.9\)). * Step 6: Determine the convergence value \(=0.001\). We succeeded in minimizing the secondary lobes by approximately 3 dB compared to those of the uniform array. To achieve beamforming in a desired direction, phase synthesis aims to determine the relative phases applied to each individual antenna element in the array. Let us start with the case of a uniform amplitude array where the sources are fed with the same amplitude; then, the array factor is as follows. After optimization with the Taguchi algorithm, we obtain directive radiation patterns with moderately controllable levels of secondary lobes sweeping across the entire useful area [\(-70^{\circ}\), \(70^{\circ}\)] (Figure 7). Table 1 in Appendix A.1 contains all the synthesized weights at different pointing angles. The figures below show a reduction of the secondary lobes to \(-20\) dB (Figure 7b), \(-29\) dB (Figure 7c), and \(-38\) dB (Figure 7d) (please refer to Appendix A.2 for further details). If is is desirable to switch between various beam pointing directions, either active feeding systems (variable amplifiers for amplitudes and phase shifters) or passive circuits can be used. In the latter case, for each desired lobe direction, a different distribution circuit would theoretically be required. In practice, circuits are used that apply the desired phases to the antennas depending on the chosen input. These are known as passive beamformers.