## 1 Introduction
In [18], approaches to solving systems of linear equations based on neural networks are presented. In [19], a solution to the fuel minimization problem is presented in the form of a neural network MFNN, an example of which, from [19] for \(M=8\) and \(N=3\), is shown in Figure 1, which is described by a system of differential equations of the form:  
\[ \begin{cases}\frac{dx}{dt}=-A^{\mathsf{T}}[Ax-y-b]-P_{\mathsf{\Omega}}\big{(} x+A^{\mathsf{T}}y\big{)},\\ \frac{dy}{dt}=-A\big{[}x+A^{\mathsf{T}}y-P_{\mathsf{\Omega}}\big{(}x+A^{ \mathsf{T}}y\big{)}\big{]}+b,\end{cases}  \tag{7}\]  
\[z=x+A^{\mathsf{T}}\,y,\,\mathbf{z}\,\in\,\mathbb{C}^{N}\]  
Then, the elements of the overcomplete basis dictionary matrix can be represented as:  
\[A_{m,i}=b_{k}\mathrm{e}^{2\pi d(m-1)\sin{(\theta_{i})}}, \tag{10}\]  
where \(\theta_{i}=\frac{180^{c}\cdot i}{N}-90^{\circ}\)), \(i=1\), \(2\), \(\ldots\), \(N\) 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.) There are two possible implementation options for the minimum fuel neural network MFNN, based on the structure presented in Figure 1.  
The first option for implementing the neural system under consideration is the option with the presentation of complex input data \(b=\mathrm{Re}\left(b\right)+j\,\mathrm{Im}(b)\), specified as a double-length real data vector \(b=[\mathrm{Re}(b_{1}),\,\ldots,\,\mathrm{Re}(b_{M}),\,\mathrm{Im}(b_{1}),\, \ldots,\,\mathrm{Im}(b_{M})]\); then, the vector of the resulting basis components (solutions) \(X=\mathrm{Re}(X)+j\,\mathrm{Im}(X)\) similarly can be represented in the form:  
\[X=[\mathrm{Re}(x_{1})\,\,\mathrm{Re}(x_{2})\ldots\mathrm{Re}(x_{N})\,\, \mathrm{Im}(x_{1})\,\,\mathrm{Im}(x_{2})\ldots\mathrm{Im}(x_{N})]^{T} \tag{11}\]  
The algorithm for forming a dictionary in this case will be as follows:  
1. A grid of angular directions to the source can be formed with a uniform step both in wave numbers \(\varphi_{i}=\frac{\pi}{N}(i-1)-\frac{\pi}{2}\) and in the angle \(\sin(\varphi_{i})\), where \(i=1\), \(2\), \(\ldots\), \(N\) is the total number of angular directions. In the first case, it turns out to be denser in the vicinity of the zero-angle value. 2. For an \(m\)-channel receiving system, two parts of the dictionary are formed in the following ways: \[A_{1}=\sin(\pi(m-1)\sin(\varphi_{i})),A_{2}=\cos(\pi(m-1)\sin(\varphi_{i}))\] (12)
3. The formed parts are combined into a common dictionary, according to the rule: \[A= \begin{pmatrix}A1&-A2\\ A2&A1\end{pmatrix} ,\] (13)  
In this case, all operations on data in the neural network will be carried out with real values. The second option for implementing a neural network is the option of replacing real variables with complex ones and performing all operations with complex values. The form of the system of Equation (7) in both the first and second cases will remain unchanged. The choice of one of the options will be determined only by the convenience of the specific hardware and software implementations. At the same time, since the absolute values of the real and complex variables are defined differently, the question arises of choosing the optimal form of the activation function \(P_{\Omega}\) for the case of using complex data.