## 5 Numerical Modeling
### The Case of Two Sources with Different Initial Signal Phases  
Next, Figures 4 and 5 (similar to Figures 2 and 3) show the results of a numerical simulation of the operation of a neural network with various activation functions (18)-(20) under the influence of two point sources with an amplitude of \(\dot{b}_{1}=\dot{b}_{2}=\)1, angles of arrival of \(\theta_{1}=-22.5^{\circ}\) and \(\theta_{2}=-18.98^{\circ}\), and two phase options for each of the two sources: \(\varphi_{1}=0^{\circ}\) and \(\varphi_{2}=0^{\circ}\), and \(\varphi_{1}=72^{\circ}\) and \(\varphi_{2}=-72^{\circ}\). In Figures 4 and 5, the activation functions used from (18) to (20) are indicated vertically on the left, and at the top, horizontally, are the phase values used in modeling sources \((\varphi_{1},\ \varphi_{2})=(0^{\circ},\ 0^{\circ})\) and \((\varphi_{1},\ \varphi_{2})=(72^{\circ},\ -72^{\circ})\). For example, Figure 4c shows the plot obtained for the activation function (19) and the phases of sources \(\varphi_{1}=0^{\circ}\) and \(\varphi_{2}=0^{\circ}\). In Figure 4a-f, the angular directions in degrees are laid along the horizontal axis (in the grid of angular directions of an excessively complete basic dictionary); along the vertical axis are the values of the modules of the resulting basic components \(\mid\!X\!\mid\), obtained as a result of processing using a neural network with the selected activation function. They correspond to the angular directions of the grid, the angles of the basic functions (blue), and the modules of the values of the radiation pattern \(\mid\!F\!\mid\), formed in the same angular grid using the standard FFT method (black). The position of point sources, taking into account amplitudes \(\dot{b}_{1}\) and \(\dot{b}_{2}\), is represented in red. From the results obtained in all cases, the radiation pattern formed using FFT does not allow resolving sources at an angular distance between them equal to \(3.52^{\circ}\). With a phase value of \(\varphi=(0^{\circ},\ 0^{\circ})\), all activation functions lead to an optimal solution with two non-zero basis coefficients, which correspond with the amplitude and direction of the sources (Figure 4a,c,e). However, like the case of one source, when a discrepancy occurs between the phases of the sources and the basis functions, the use of activation functions (18) and (20) leads to errors in determining the directions and amplitudes of the sources, as shown in Figure 4b,f. In this case, the number of non-zero values of the basic coefficients increases, and among them, there are no corresponding ones in the direction of the source. Table 2 shows the number of basic coefficients with a value normalized to the maximum value \(\max\limits_{i}\lvert x_{i}\rvert\) exceeding \(10^{-3}\). In Figure 5a-f, the horizontal axis shows the time normalized to the duration of one internal calculation cycle t, and the vertical axis (scale on the left) shows the relative change in the coefficients of the basis functions for one normalized time step, \(\Delta X(t)=\max\limits_{i}\lvert x_{i}(t-1)-x_{i}(t)\rvert/\max\limits_{i} \lvert x_{i}(t-1)\rvert,\ t=1,2,\dots,T\) (blue), and the current error in representing the input data by basis components, which was defined as the change in time of the loss function (6), as shown in the scale on the right (red). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Activation Function** & **Source Phase** & **Coefficients Quantity** \\ \hline (18) & \((0^{\circ},0^{\circ})\) & 2 \\ (18) & \((72^{\circ},-72^{\circ})\) & 5 \\ (19) & \((0^{\circ},0^{\circ})\) & 2 \\ (19) & \((72^{\circ},-72^{\circ})\) & 2 \\ (20) & \((0^{\circ},0^{\circ})\) & 4 \\ (20) & \((72^{\circ},-72^{\circ})\) & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Number of basis coefficients with a level exceeding \(10^{-3}\), two sources. From the results obtained, it follows that the time to establish the values of the coefficients of the basis functions when using activation functions (18) and (19) weakly depends on the value of the source phase.