## 4 Numerical Modeling  
This section presents the results of a mathematical model of the modified MFNN algorithm and compares its effectiveness with the use of the SL0 algorithm. It solves the problem of determining the DOA for an equidistant antenna array with linear frequency modulation, where the input data are complex signals received by the elements of this array from several point sources. The received data can be represented as follows:  
\[\sum_{j}^{J}b_{m,j}=\sum_{j}^{J}a_{j}e^{2\pi d(m-1)\gamma_{j}+\psi_{j}} \tag{4}\]  
where \(\gamma_{j}=\sin(\theta_{j})j=1,2,\ldots J\) is the number of point external sources, \(d\) is the lattice element pitch normalized to the wavelength, \(m=1,2,\ldots M\), \(M\) is the number of receiving elements of the antenna array, \(\theta_{j}\) is the angular direction to the \(j\)-th source, \(a_{j}\) is the amplitude of the \(j\)-th source, \(\psi_{j}\) is the phase of the \(j\)-th source, and \(i=\sqrt{-1}\). Then, the elements of the matrix, which comprise an overcomplete basic dictionary, can be represented as follows:  
\[A_{m,n}=\mathrm{e}^{j2\pi d(m-1)\Theta_{n}} \tag{5}\]  
where \(\Theta_{n}=\sin(\theta n)\), \(\theta_{n}=\frac{180^{\circ}\cdot n}{N}-90^{\circ}\), \(n=1,2,\ldots\), \(N\) is the dimension of the angular direction grid of the overcomplete basis dictionary, and \(m=1,2,\ldots,M\) (\(M\) is the number of receiving elements of the antenna array). To form the redundant basis A, a grid by angle N = 512 points was selected. The number of elements of the antenna array \(M=16\), and the step of the elements \(d=0.5\) wavelength. Numerical modeling was carried out using an Intel(R) Core (TM) i7-5960X CPU @ 3.00 GHz. and MATLAB ver. R2021b. Values from (7) were selected as elements of the excess basis matrix with \(N\) = 512, \(M\) = 16, and \(d\) = 0.5. The calculation results were output at the end of the cycle with a given number of iteration process steps, which varied from 250 to 10,000, with the total number of cycles being 100. The convergence rate of Algorithm 1 depends significantly on the coefficients that determine the parameters of the iteration process. The values of these coefficients were chosen as \(\mu=2.0\div 3.9\), \(\alpha=0.3\div 1.0\) to ensure the maximum convergence rate without a loss of stability. Figure 1: Neural network architecture for MFNN. In Figure 2a,b, Figures 3, 4 and 5a,b, the processing result (the vector of the coefficient modules \(|\,x|\) of the basis components upon completion of the algorithms) is shown in blue, and the result of standard processing using FFT with the Chebyshev weight distribution (a sidelobe level of \(-30\) dB) is shown in black. The direction to the point sources, taking into account their amplitude, is depicted in red. The amplitude is plotted along the vertical axis, and the angular direction in degrees is represented along the horizontal axis. Figure 2c,d, Figures 3, 4 and 5c,d show the maximum change for the current step of the solution process \(t\) of the vector elements of the basis components' coefficients x. It is normalized from step to step to its value at the previous step \(\Delta_{x}(t)=\max_{x}\Bigl{(}\frac{abs(|x(t)|-|x(t+1)|)}{|x(t)|}\Bigr{)}\) and the current error in representing the input signals, which was defined as the loss function \(P(t)=\left\|\,b-Ax(t)\,\right\|_{2}\) for norm \(L2\). The current value is plotted along the vertical axis, and the current step number is plotted along the horizontal axis. Figure 2e,f, Figures 3, 4 and 5e,f show the changes in the values of the real Re(\(x(t)\)) (red) and imaginary Im(\(x(t)\)) (blue) constituent elements of the vector \(x(t)\) of coefficients of the basis components by iteration steps. The current values of Re(\(x(t)\)) and Im(\(x(t)\)) are plotted along the vertical axis, and the current step number is shown along the horizontal axis.