## 4 Reduction of Mutual Coupling by a Neural Network Approach  
As mentioned before, the aim of this study was to reduce mutual coupling between each antenna element of a cross dipole antenna. The mutual coupling effect corresponds to the transmission value in antenna engineering and a resonator plane was designed in our study to reduce the transmission value (S21) in the designed cross dipole antenna. As shown in Figure 3, the designed resonator plane has a square shape and 50 mm side lengths, and it also consists of unit cells made of copper. Each unit cell has dimensions of 5 mm \(\times\) 5 mm and resonator plane consists of 6 \(\times\) 6 unit cells, as illustrated in Figure 3. According to the number of unit cells, there are 236 different artificial surface possibilities. It is hard to reduce the mutual coupling effect in FIT based microwave simulator because doing so involves the trial and error method. However, an artificial neural network model gives more accurate results with a trained data set. Thus, we used an artificial neural network model. In line of the purpose of our study, an arbitrary resonator plane was designed on a printed circuit board and integrated into the cross dipole antenna to reduce the mutual coupling effect. Each unit cell is represented by a binary bit \"1\" or \"0\" state. Therefore, the \"1\" state corresponded to a copper cell while \"0\" corresponded to an empty state. Although there were 36 bits, the designed surface was represented by 18 bits to set up a symmetric design procedure, as illustrated in Figure 4. Each symmetric column is named as X, Y, or Z for identification of the relevant cell number. A total of 48 various resonator planes were designed and tested with cross dipole antenna. The obtained return losses and transmission values are listed in Table 1.  
Figure 2: (a) Cross-dipole antenna design with dimension parameters and (b) Simulated results for S\({}_{11},\) S\({}_{21}\). These return loss and transmission values were exported to the Matlab environment and the artificial neural network toolbox was used for training processes. One of the relevant designed surface and working processes for artificial neurons is given in Figure 4a. A total of 48 various results were trained by artificial neurons and neural network tools in the Matlab environment, and mean square errors (MSE) were calculated and plotted as shown in Figure 4. As shown in Figure 4b, the artificial neural network was fully trained at the sixth iteration. Afterwards, the trained neural network was tested by eight different unknown resonator planes, as listed in Table 2. According to the obtained isolation (S21) values, the trained neural network was well matched with the results of the FIT based microwave simulator, as the error rate was less than 2%. For example, the isolation value was obtained as \(-15.5183\) dB for the resonator plane (X, Y, Z) represented by (00000, 000000, 111110) and the isolation value was \(-15.5\) dB in the FIT based simulator. As a consequence, the artificial neural network was well trained by different resonator planes and the obtained results are plotted in Figure 5 with eight unknown test planes. In addition, according to Figure 4, there was a good match in isolation values with a mean error rate of less than 2%, which is listed in Table 2.  
Figure 4: **(a)** Resonator plane and artificial neural network working principle and (**b**) neuron training results. Figure 3: Cross-dipole antenna and resonator plane in a binary system. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c c c c c c} \hline \hline \multicolumn{11}{c}{Variables} & \multicolumn{1}{c}{Output1} & \multicolumn{1}{c}{Output2} & \multicolumn{1}{c}{Output3} \\  
Furthermore, random resonator planes were created and imported into the artificial neural network to obtain an optimum design with a reduced mutual coupling effect (S21).