## 4 Numerical Modeling
### The Case of Two Close Sources  
Next, Figure 5 (similar to Figures 2 and 4) 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 \(a_{1}=a_{2}=1\), angles of arrival of \(\theta_{1}=-22.50^{\circ}\), \(\theta_{2}=-19.69^{\circ}\) (angle difference \(2.8^{\circ}\)), and the corresponding phase values for sources are \(\varphi_{1}=0^{\circ}\) and \(\varphi_{2}=-0.3\pi\). On the left, vertically, are the values for Algorithm 2 and on the right, vertically, are the values for Algorithm 2.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Algorithm** & \(\mu\), \(\alpha\) & \multicolumn{3}{c}{**Coefficient Quantities**} \\ \hline  & & **>0.1** & **>0.01** & **>0.001** \\ \hline
1 & - & 2 & 0 & 0 \\
2 & \(\mu=2\), \(\alpha=0.3\) & 4 & 9 & 13 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Number of excess basis coefficients with a level exceeding 0.1, 0.01, and 0.001 for two separated sources.  
1. For example, Figure 5b shows the plot obtained for FFT (black) and Algorithm 1 and resulting basis components coefficients \(\left|x\right|\) (blue).  
In Figure 5a,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 \(\left|x\right|\), 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 received in this case, the radiation pattern formed using FFT does not allow resolving sources at an angular distance between them equal to \(5.38^{\circ}\).  
It is evident from the results 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 5a,h. In this case, the loss function characterizing the error in representing the input signals P(t) becomes less than \(10^{-14}\), a 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 50 cycles of 15,000 steps.  
At the same time, for Algorithm 1, with the selected values of \(\mu=1.5\), \(\alpha=0.3\), the rate of change of the coefficients \(\Delta_{x}(t)\) decreases to \(10^{-9}\), and the loss function \(P(t)\) becomes less than \(10^{-15}\), as can be seen from Figure 5b,g, and the results themselves practically stop changing after 30 cycles of 15,000 steps. Thus, for the case under consideration, the convergence time of Algorithm 2 is approximately three times less than for Algorithm 1. Algorithm 1, similar to the case in Section 4.3, did not allow an exact solution to be found to the problem, which led to the appearance of a larger number of non-zero coefficients with their own values of the amplitudes of angular directions and phases that do not coincide with the values for two sources. In this case, the number of non-zero coefficients significantly exceeds 2, and the angular shift of the two main coefficients was \(0.50^{\circ}\) and \(-0.25^{\circ}\), respectively.  
Table 4 shows the number of excess basis coefficients, the value of which is normalized to the maximum value \(\max\limits_{i}\left|x_{i}\right|\) exceeding 0.1, 0.01 and 0.001.  
It follows from the obtained results that Algorithm 1 also does not allow the finding of a global minimum in accordance with the L0 norm for the paired close sources. This suggests a solution with a rather significant error; this is in contrast to Algorithm 2, which finds an exact solution with two non-zero coefficients, the real and imaginary values of which coincide with the corresponding values of the sources, while the speeds of the algorithms are close.