## 4 Numerical Modeling
### Two Separated External Sources  
Next, Figure 4 (similar to Figure 2) show the results of a numerical simulation of the operation of Algorithms 1 and 2 if there are two point sources with an amplitude of \(a_{1}=a_{2}=\)1, with angles of arrival of \(\theta_{1}=-22.50^{\circ}\), \(\theta_{2}=-16.88^{\circ}\), (angle difference \(5.6^{\circ}\)) and the corresponding phase values for sources are \(\varphi_{1}=0^{\circ}\) and \(\varphi_{2}=-0.3\pi\). On the  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Algorithm** & \(\mu\), \(\alpha\) & \multicolumn{3}{c}{**Coefficient Quantities**} \\  & & **\textgreater{}0.1** & **\textgreater{}0.01** & **\textgreater{}0.001** \\ \hline
1 & - & 1 & 0 & 0 \\
2 & \(\mu=2\), \(\alpha=1\) & 1 & 0 & 0 \\
2 & \(\mu=3.9\), \(\alpha=0.99\) & 1 & 260 & 512 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Number of excess basis coefficients with a level exceeding 0.1, 0.01, and 0.001 for single source.  
left, vertically, are the values for Algorithm 2 and on the right, vertically, the values for Algorithm 1 are depicted. For example, Figure 4b shows the plot obtained for an FFT (black) and Algorithm 1 and resulting basis components coefficients \(|x|\) (blue).  
In Figure 4a,b the angular directions in degrees are plotted along the horizontal axis (in the grid of angular directions of an overcomplete basic dictionary); along the vertical axis are the module values of the resulting basic components \(|x|\), obtained as a result of processing using Algorithms 1 and 2. The modules of the radiation pattern values obtained using FFT (black) are shown in the same angular grid. The position of the point sources with the amplitudes \(a_{1}\) and \(a_{2}\) is represented by red dots.  
From the result obtained in this case, the radiation pattern formed using FFT does not allow resolving sources at an angular distance equal to 5.38\({}^{\circ}\).  
The received results show that the use of Algorithm 2 leads to a solution with only two main non-zero basis coefficients, which correspond to the amplitude, direction, and phase of the sources, as shown in Figure 4a,h. In this case, the loss function characterizing the error in representing the input signals \(P(t)\) becomes less than \(10^{-14}\), and the rate of change of the coefficients \(\Delta_{x}(t)\) drops to \(10^{-16}\), that is, actually to zero, and the results stop changing after 10 cycles of 6000 steps.  
At the same time, for Algorithm 1, with the selected values \(\mu=2\), \(\alpha=0.3\), the rate of change of the coefficients \(\Delta_{x}(t)\) decreases to \(10^{-11}\), and the loss function \(P(t)\) becomes less than \(10^{-14}\), as can be seen from Figure 4b,g, and the results themselves practically stop changing after 30 cycles of 6000 steps. Thus, for the case under consideration, the convergence time of Algorithm 2 is approximately three times less than for Algorithm 1. But, what is more important is that Algorithm 1 did not allow an exact solution to be found to the problem, which led to the appearance of a larger number of coefficients different from zero with their values of the amplitudes of angular directions and phases that do not coincide with the values for the two sources. In this case, the number of non-zero coefficients significantly exceeds 2.  
Table 3 shows the number of excess basis coefficients, the value of which is normalized to the maximum value \(\underset{i}{\max}|x_{i}|\) exceeding 0.1, 0.01 and 0.001.  
From the obtained results, it follows that for paired sources, Algorithm 1 does not allow the finding of a global minimum in accordance with the L0 norm and gives a solution with a fairly significant error; this is unlike Algorithm 2, which gives two non-zero coefficients, which is the exact solution whose real and imaginary values coincide with the corresponding values of the sources.