# Meander Structure Analysis Techniques Using Artificial Neural Networks
## 4 Discussion  
Meander structures can be used in many different microwave devices to delay the signal or to increase the signal path while keeping or reducing the occupied area. The main parameters characterizing meander structures are \(S_{21}\) (transmission coefficient), \(S_{11}\) (reflection coefficient), and \(t_{\text{d}}\) (delay time). Usually, meander structures are analyzed using commercial software based on numerical methods. On the other hand, numerical methods are time-consuming, and calculations using numerical methods can last hours or even days. Therefore, three different intelligent methods were used to replace traditional numerical methods with methods based on neural networks. The obtained prediction results were compared with numerical results from Sonnet(r) and also the baseline machine learning LSBoost technique. The FFNN is already widely used in the synthesis and analysis of microwave devices. However, each new dataset for every different MS is unique and requires an individual FFNN network structure to achieve the best characteristic prediction results. Quite different dynamics for \(t_{\rm d}\), \(S_{21}\), and \(S_{11}\) resulted in the requirement of predicting each parameter with a separately trained FFNN network. The average MAE with its SD did not exceeded \(3.3\pm 1.5\) ps for \(t_{\rm d}\) in the 0-10 GHz frequency range. If only the passband were considered in the 0-3 GHz frequency range, then the size of the MAE would decrease even more to \(3\pm 1.5\) ps. The MAEs of the individual characteristics without averaging did not exceed 20 ps. The predictions for \(S_{21}\) and \(S_{11}\) were slightly worse but insignificant from the perspective of the design of microwave devices. The MAE of \(S_{21}\) did not exceed \(0.64\pm 0.42\) dB in the overall frequency range. The average MAE of \(S_{11}\) did not exceed \(2.47\pm 1.55\) dB. The maximum peak errors of MS1 for \(S_{21}\) and \(S_{11}\) were equal to 5.36 dB and 14.55 dB, respectively. The obviously worse prediction results were obtained due to the nature of the characteristics of \(S_{11}\). The reflections from the output port were clearly visible in the bandwidth, and as a result, the fluctuation in \(S_{11}\) was quite fast. The worst prediction results for \(S_{21}\) and \(S_{11}\) were comparable to \(t_{\rm d}\) and could already be expected in the FFNN training stage when the calculated MSE values were equal to 0.086 and 8.618, respectively, while the MSE of \(t_{\rm d}\) was equal to \(2.23\times 10^{-5}\). It could be seen that the FFNN predicted the desired MS characteristics accurately enough to design and analyze an MS (Figure 6). For example, the error margin of 3.3 ps while working in the range of ns is highly desirable. On the other hand, the predictions with the FFNN were worse in the section where the characteristics changed quickly, such as the 7-8 GHz or 8.5-9.5 GHz range of frequencies for \(S_{21}\). For these reasons, the time delay neural network was also investigated. The TDNN is structurally similar to the FFNN, except that the TDNN has delay buffers in the input. These delay buffers allow the TDNN to consider information from previous time steps. The base of the TDNN consists of the previously discussed FFNN. The same FFNN structures were used in the TDNN for predicting \(t_{\rm d}\), \(S_{21}\), and \(S_{11}\). Regarding the TDNN, the MAEs of the predictions for \(t_{\rm d}\), \(S_{21}\), and \(S_{11}\) for MS2 are presented, taking into account the different delay buffer size \(N_{\rm d}\) of the TDNN (Figure 7). The results show (Figure 7a) that by increasing the delay buffer from 1 to 8, the MAE of \(t_{\rm d}\) could be reduced from \(2.64\pm 2.35\) ps to \(0.9\pm 0.64\) ps. An obviously lower MAE in fast-changing regions can be seen from \(S_{21}\) (Figure 7b). The MAE of \(S_{21}\) decreased from \(0.59\pm 0.68\) dB to \(0.11\pm 0.12\) dB.