# 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  
A comparison between linear arrays and planar arrays (concentric rings) shows that linear antenna arrays exhibit radiation limited to the half-space and axial radiation. This presents a significant drawback for applications requiring omnidirectional coverage. The synthesis approach using the Taguchi method, developed for linear arrays, can be extended to two-dimensional or planar arrays. For a linear array, the synthesis reduces to finding the feeding law and/or spatial distribution along an axis, while for a planar array, the synthesis involves searching for the complex weighting of the feed and/or the spatial distribution of sources within a plane. Planar antenna arrays (in this case, concentric rings) offer the advantage of 3D scanning of the main lobe, which is essential for a wide variety of applications such as wireless communications, aerial navigation, radar, and radio astronomy.  
In a concentric ring antenna array, the elements are distributed across several circular concentric rings. These rings vary in radius and the number of elements. Figure 10 represents the general configuration of this array with \(M\) concentric circular rings, where the \(m\)th ring (for \(m=1,2,\ldots,M\)) has a radius of \(r_{m}\) and contains \(N_{m}\) elements. Assuming that all elements (in all rings) are isotropic sources, the radiation pattern of this array can be expressed in terms of the array factor only [25].  
Figure 8: Circular antenna array with 16-elements at 2.45 GHz. (**a**) Uniform excitation (16 antennas). (**b**) Taguchi excitation (16 antennas).  
Figure 9: Circular antenna array with 24-elements at 2.45 GHz. (**a**) Uniform excitation (24-antennas). (**b**) Taguchi excitation (24-antennas).  
For a concentric ring array with \(M\) rings and \(N_{m}\) elements in the \(m\)th corresponding ring, the array factor is given by the following Equation (6):  
\[AF(\theta,\phi)=\sum_{m=1}^{M}\sum_{n=1}^{N_{m}}w_{mm}e^{i\mathcal{k}r_{m}}(\sin( \theta)\cos(\phi-\phi_{mn})-\sin(\theta_{0})\cos(\phi_{0}-\phi_{mn})) \tag{6}\]  
Considering the symmetry of the sources with respect to the center of the circle, we can express the array factor in the following form (7):  
\[AF(\theta,\phi)=2\sum_{m=1}^{M}\sum_{n=1}^{N_{m}}w_{mn}\cos(kr_{m}(\sin( \theta)\cos(\phi-\phi_{mn})-\sin(\theta_{0})\cos(\phi_{0}-\phi_{mn}))) \tag{7}\]  
where:  
* \(k\): wave number.
* \(\lambda\): wavelength.
* \(r_{m}=\frac{N_{m}d_{m}}{2\pi}\): radius of the \(m\)th ring.
* \(d_{m}\): spacing between the arc element of the \(m\)th ring.
* \(\phi_{mn}=2\pi\frac{n-1}{N_{m}}\): angular position of the \(n\)th element of the \(m\)th ring.
* \(w_{nm}\): excitation current of the \(n\)th element of the \(m\)th ring.
* \(\phi\) and \(\theta\): azimuth and zenith angles, respectively.
* \(\phi_{0}\): value of \(\phi\) where the main beam should be directed.
* \(\theta_{0}\): value of \(\theta\) where the main beam should be directed.  
From Equation (7), we can observe that there are three parameters controlling the array factor (AF): amplitude, phase, and positions of the elements.