# Taguchi Method-Based Synthesis of a Circular Antenna Array for Enhanced IoT Applications
## 2 Synthesis of Radiation Patterns of a Concentric Ring Antenna Array Using the Taguchi Method
### Amplitude Synthesis
Concentric Ring Antenna Array with 30 Elements (\(N_{1}=8\), \(N_{2}=10\), and \(N_{3}=12\))  
The array factor is described by the following Equation (12):  
\[AF(\theta,\phi)=2\sum_{m=1}^{3}\sum_{n=1}^{15}w_{mn}\cos(kr_{m}(\sin(\theta) \cos(\phi-\phi_{mn})-\sin(\theta_{0})\cos(\phi_{0}-\phi_{mn}))) \tag{12}\]  
To apply the Taguchi method, determine the number of parameters (\(k=15\)), define the number of levels (\(s=3\)), set the strength (\(t=2\)), select the OA experimental design (\(27,15,3,2\)), calculate the reduced function (\(rr=0.6\)), and set the convergence value (\(0.01\)). Figure (a)a shows that we have successfully minimized the side-lobe level (SLL) down to \(-25.1911\) dB, where the gain is approximately \(14.57\) dB compared to the side-lobe level of the uniform concentric ring antenna array (SLL = \(-10.62\) dB). The optimization objective is achieved after 16 iterations, as indicated in Figure (b)b. The excitation weights for this optimized antenna array using the Taguchi method are presented in Table 7.  
1.5. Concentric Ring Antenna Array with 36 Elements (\(N_{1}=6\), \(N_{2}=8\), \(N_{3}=10\), and \(N_{4}=12\))  
The array factor is described by the following Equation (13):  
\[AF(\theta,\phi)=2\sum_{m=1}^{4}\sum_{n=1}^{18}w_{mn}\cos(kr_{m}(\sin(\theta) \cos(\phi-\phi_{mn})-\sin(\theta_{0})\cos(\phi_{0}-\phi_{mn}))) \tag{13}\]  
Figure 14(a) shows that we have successfully minimized the side-lobe level (SLL) down to \(-27.5831\) dB, where the gain is approximately \(17.2831\) dB compared to the side-lobe level of the uniform concentric ring antenna array (SLL = \(-10.3\) dB). The optimization objective is achieved after 19 iterations, as indicated in Figure 14(b). The excitation weights for this optimized antenna array using the Taguchi method are presented in Table 8.  
Figure 16 shows the reduction of the side-lobe level (SLL) for concentric ring arrays with \(12,18,24,30\), and \(36\) elements:  
* \(SLL=-11.1418\) dB for 12 elements (\(N_{1}=2\), \(N_{2}=4\), and \(N_{3}=6\)). * \(SLL=-17.6870\) dB for 18 elements (\(N_{1}=4\), \(N_{2}=6\), and \(N_{3}=8\)). * \(SLL=-21.2004\) dB for 24 elements (\(N_{1}=6\), \(N_{2}=8\), and \(N_{3}=10\)). * \(SLL=-25.1911\) dB for 30 elements (\(N_{1}=8\), \(N_{2}=10\), and \(N_{3}=12\)). * \(SLL=-27.5831\) dB for 36 elements (\(N_{1}=6\), \(N_{2}=8\), \(N_{3}=10\), and \(N_{4}=12\)). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Elements** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** \\ \hline
**Weights** & 0.2161 & 0.5458 & 0.0633 & 0.3158 & 0.7458 & 0.9851 & 0.1374 & 0.1892 & 0.9548 \\ \hline
**Elements** & **10** & **11** & **12** & **13** & **14** & **15** & & & \\ \hline
**Weights** & 0.1995 & 0.7903 & 1.0000 & 0.9712 & 0.9810 & 0.2068 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Optimized excitations of the array (\(N_{1}=6\), \(N_{2}=8\), \(N_{3}=10\), and \(N_{4}=12\)). Figure 15: Simulation results of a concentric ring array (\(N_{1}=6\), \(N_{2}=8\), \(N_{3}=10\), and \(N_{4}=12\)). Figure 17 presents the optimal amplitude values obtained using the Taguchi method with \(rr=0.6\) for 12, 18, 24, 30, and 36 elements. Table 9 summarizes the simulation results for various concentric ring array configurations, demonstrating the impact of element count on antenna performance. As the number of elements in the array increases, achieved by adding more rings, significant performance improvements are observed. These enhancements are reflected in two main factors: a reduction in the side-lobe level (SLL) and improved directivity. Specifically, as the number of rings and elements increases, the side-lobe level decreases, indicating better suppression of unwanted radiation. For example, with a 36-element array (using four rings), the side-lobe level reaches \(SLL=-27.5831\) dB, showing a substantial reduction compared to arrays with fewer elements. Additionally, the half-power beamwidth (HPBW) improves with more elements, resulting in enhanced directivity and a more focused main lobe.