# 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  
Amplitude synthesis has a dual advantage: first, it creates symmetric directive lobes with low levels of side lobes, positively influencing signal quality; second, it creates nulls in predetermined areas to avoid undesirable signals (interference). In this type of synthesis, a concentric ring antenna array is used, as shown in Figure 10, which consists of \(M\) rings and \(N\) equally spaced elements. The spacing of the elements can be \(0.5\lambda,0.7\lambda,\lambda,\ldots\). The excitations of the array elements are symmetric with respect to the center of the circular geometry. Therefore, the excitation weights of the antenna array will be optimized within a range of [0, 1] (see Equation (6)). Figure 10: Circular antenna array in concentric rings with isotropic elements. If the number of antennas is even, the circular geometry is symmetric about the center of the circle, meaning there exists an integer \(\frac{N}{2}<i\leq N\) such that \(\phi_{i}=\phi_{n}+\pi\); for all \(1\leq n\leq\frac{N}{2}\). Taking into account the symmetry of the sources with respect to the center of the circle, we can express the array factor in the form of Equation (7) [22]. In the following examples 1, 2, 3, 4, and 5, Taguchi optimization is applied for 12, 18, 24, 30, and 36 elements; (\(N_{1}=2\), \(N_{2}=4\), and \(N_{3}=6\)), (\(N_{1}=4\), \(N_{2}=6\), and \(N_{3}=8\)), (\(N_{1}=6\), \(N_{2}=8\), and \(N_{3}=10\)), (\(N_{1}=8\), \(N_{2}=10\), and \(N_{3}=12\)), and (\(N_{1}=6\), \(N_{2}=8\), \(N_{3}=10\), and \(N_{4}=12\)). 1.1 Concentric Ring Antenna Array with 12 Elements (\(N_{1}=2\), \(N_{2}=4\), and \(N_{3}=6\))  
The array factor is described by the following Equation (8):  
\[AF(\theta,\phi)=2\sum_{m=1}^{3}\sum_{n=1}^{6}w_{mn}\cos(kr_{m}(\sin(\theta) \cos(\phi-\phi_{mn})-\sin(\theta_{0})\cos(\phi_{0}-\phi_{mn}))) \tag{8}\]  
To minimize the levels of side lobes, the fitness function is chosen based on the optimization objective:  
\[fitness=(\max\{20\log|AF(\theta)|\}) \tag{9}\]  
Subject to the constraint \(\theta\in[0^{\circ},45^{\circ}]\). In order to apply the Taguchi method, the following steps should be followed: First, determine the number of parameters \((k=6)\), the number of levels \((s=3)\), and the strength \((t=2)\). Then, establish the orthogonal experiment design OA \((27,6,3,2)\), followed by the reduced function \((rr=0.6)\). Lastly, set the convergence value to 0.01. After determining the input parameters, the objective function (fitness) for each experiment can be calculated. The performance evaluation results of the CCA method are presented in Table 2. Table 3 shows the response table for the first iteration, while Table 4 displays the optimized excitations of the network (\(N_{1}=2\), \(N_{2}=4\), and \(N_{3}=6\)) obtained using the Taguchi method. 9.14 dB compared to the side-lobe level of the uniform concentric ring antenna array (\(\text{SLL}=-1.9\) dB). The optimization objective is achieved after 59 iterations, as indicated in Figure (b)b. The excitation weights of this optimized antenna array using the Taguchi method are mentioned in Table 4.  
1.2. Concentric Ring Antenna Array with 18 Elements (\(N_{1}=4\), \(N_{2}=6\), and \(N_{3}=8\))  
The array factor is described by the following Equation (10):  
\[AF(\theta,\phi)=2\sum_{m=1}^{3}\sum_{n=1}^{9}w_{mn}\cos(kr_{m}(\sin(\theta) \cos(\phi-\phi_{mn})-\sin(\theta_{0})\cos(\phi_{0}-\phi_{mn}))) \tag{10}\]  
To apply the Taguchi method effectively, the following steps should be followed: Begin by determining the number of parameters, denoted as \((k=9)\), which represents the factors that will be varied in the experiment. Next, define the number of levels, \((s=3)\), which refers to the different values or settings each parameter can take. The next step is to determine the strength of the orthogonal array, set to \((t=2)\), which indicates the level of interaction considered between the factors.