## 1 Introduction  
Classical methods for presenting data based on a selected basis include fast Fourier transform algorithms, cosine transform, wavelet transform, etc. The concept of an overcomplete basis is the use of a basis with a significantly larger number of components compared to the number of components in the analyzed data, which leads to the so-called SR (sparse representation). Unlike classical methods, the elements of the basis may not necessarily be functions orthogonal to each other, but the method of data representation is based on their decomposition into a number of optimal basic components, which are found from an overly complete basis dictionary using optimization algorithms such as, for example, MP (matching pursuit) and MOF (method of frame), using different norms [1; 2; 3; 4; 5; 6; 7; 8; 9; 10]. The overcomplete basis or basis components are specified a priori or found using some optimization algorithms, such as correspondence search, frame methods, and basis search, depending on the current characteristics of the received data. One of the important classes of norms [11] is the so-called \(l_{p}\) norms for vector \(x\). representing dimension \(l\):  
\[l_{p}=\left(\sum_{i=1}^{l}\lvert x_{i}\rvert^{p}\right)^{1/p}\text{for }p\geq 1 \tag{1}\]  
For the occasion \(p=0\):  
\[l_{0}=\left(\sum_{i=1}^{l}\lvert x_{i}\rvert^{0}\right),\ \ \ \lvert x_{i}\rvert^{0}= \begin{cases}0,&\text{if }x_{i}=0\\ 1,&\text{otherwise}\end{cases}  \tag{2}\]  
Thus, \(l_{0}\) is the number of non-zero elements in \(x\). Minimizing the \(l_{0}\) norm leads to maximizing the number of zero values and to sparse data [12]. For the occasion \(p=1\):  
\[l_{1}=\left(\sum_{i=1}^{l}\lvert x_{i}\rvert^{1}\right) \tag{3}\]  
Norm \(l_{1}\) represents the sum of the absolute values of elements in \(x\) and is widely used in algorithms such as LASSO (least absolute shrinkage and selection operator) in statistics BP (basis pursuit) in signal processing [12]. It should be noted that methods using SR can effectively separate coherent sources because data evaluation is based on an optimization method, such as \(l_{1}\) norm minimization, rather than on sub-space orthogonality, as in MUSIC (multiple signal classification) [13] or MVDR (minimum variance distortion less response) [14] and their use [15]. In addition, if the noise variance value used as the regularization parameter does not match the environmental noise variance, the evaluation performances of the corresponding methods are significantly degraded. An essential point is that when solving highly sparse linear optimization problems, the \(l_{1}\) norm is equivalent to the \(l_{0}\) norm, including for complex data [16; 17]. The solutions given in [18; 19] also allow the representation of data in an overcomplete dictionary basis. Along with the methods mentioned earlier, the decomposition of data into weighted dictionary components can be presented in the form of the so-called \"fuel minimization\" problem or MFNN [19], which is an optimization problem in a basis with norm \(l_{1}\) linear restrictions. Extending it to the complex plane \(\mathbb{C}\), we obtain:  
\[ \begin{array}{l}\{max_{y\in\Omega}\ y^{T}x\}\to min\\ \text{given that }Ax=b,\end{array}  \tag{4}\]  
where \(\Omega=\left\{\omega\in\mathbb{C}^{N}\lvert\lvert\omega_{i}\rvert\leq 1,i-1,2, \ldots,N\right\}\subset\mathbb{C}^{N},b\in\mathbb{C}^{M}\) represents the input data,  
\(x=\left[\alpha_{\gamma_{1}}\alpha_{\gamma_{2}}\ldots\alpha_{\gamma_{N}} \right]^{T}\in\mathbb{C}^{N}\) represents the basic components,  
\(A=\left[\phi_{\gamma_{1}},\phi_{\gamma_{2}},\ldots,\phi_{\gamma_{N}}\right] \in\mathbb{C}^{M\times N}\) represents the overcomplete dictionary matrix.