## 2 Materials and Methods
### Synthetic Data Generation  
The main aspect that determined the use of synthetic data for our study of NN capabilities was the lack of objects available for measurement: it is unrealistic to gather many objects of different thickness (varying from 200 to 500 mm), different relative dielectric permittivity \(\varepsilon_{r}\) (varying from 1 to 8), and different absorption \(\tan\delta\) (varying from 0 to 0.1) in order to evaluate NN performance using real measurement signals. At the start of the study, the team did not have any reliable tool or method for EM wave propagation synthetic data generation; thus, at the start, the UWB IR EM wave propagation model had to be developed and tested. Meanwhile, the EM wave propagation simulationpars software had to be tested in order to determine whether its simulated signals of impulse radar scenes corresponded to the signals of the real impulse radar. For this purpose, the open source EM field solver OpenEMS was chosen [26]. For the simulation of EM wave propagation, it uses the finite difference time domain (FDTD) method, which is mandatory in the case of impulse radars. For the needs of synthetic data generation, the simulation models of the EDI radar signal and antennas had to be developed. To estimate dielectric material properties with the addition of AI, the regression problem must be solved. In machine learning, the regression problem is characterized as prediction calculus. Unlike the classification problem, the regression problem occurs when numerical value prediction of a parameter is required. In our case, if we take raw data, the synthesized UWB signals with an input length of 1024 data points from a single channel are determined as input features of the NN model. Then, the linear regression implements these data as inputs for the NN, and therefore, is defined in terms of a linear function [27]. In our research, multiple inputs (which feature further in the text) are simply UWB signal data (echoed signal data points gathered on the receiving antenna/s in the simulation) of raw or any other representative form. Single output refers to the single output layer, excluding any activation, and with the numerical value as a prediction of the parameter of \(e_{r}\) or \(tan\delta\). By gathering all features, for example, into a vector \(\mathbf{x}\in R^{k}\), and all weights into a vector \(\mathbf{w}\in R^{k}\), assumptions can be expressed as  
\[\widetilde{y}=\mathbf{w}^{\top}\mathbf{x}+b \tag{1}\]  
In Expression (1), the \(\mathbf{x}\) corresponds to the features of a single UWB signal data point. To refer to the features of the entire data set of \(n\) examples, the design matrix is used as \(\mathbf{X}\in R^{n\times k}\). The \(\mathbf{X}\) contains one row for every signal and one column for every feature (data point) of the signal. For the collection of features \(\mathbf{X}\), the predictions \(\mathbf{\tilde{y}}\in R^{n}\) are expressed via the matrix-vector product:  
\[\mathbf{\tilde{y}}=\mathbf{X}\mathbf{w}+b, \tag{2}\]  
where broadcasting is applied during the summation [28]. To sum up, the goal of the linear regression problem is to find the weight vector \(\mathbf{w}\) and bias \(b\) so that the features of the new and unseen UWB signal data will be predicted with the lowest error possible. The weights in the presented linear regression model are the coefficients of the independent variables that determine the slope of the line for each feature. These coefficients learn, during the training process, to make predictions and minimize the difference between the predicted and the true output. _Convolution Neural Network Model_  
During the research, several problems were defined for successful prediction of material dielectric properties with the help of the NN:  
* Choice of the NN architecture;
* Selection of raw signal preprocessing methods;
* Methodology of the model development and its hyperparameter tuning;
* Selection of loss function;
* Impact of noise.