## 1 Introduction
The input data \(b\) can be represented as follow:  
\[b=\sum_{i=1}^{N}\alpha_{\gamma_{i}}\phi_{\gamma_{i}}+r^{(N+1)}, \tag{2}\]  
where \(r^{(N+1)}\in\mathbb{C}^{M}\)--is the residual vector, or error vector, the result of approximating \(b\) by the basis functions \(\phi_{\gamma_{1}},\phi_{\gamma_{2}},\ldots,\phi_{\gamma_{N}}\) from the dictionary \(D\), and \(\alpha_{\gamma_{i}}\) is the \(i\)-th weighting factor of the corresponding basis component. The solution to the optimization problem can be obtained by minimizing the norm of the loss function:  
\[P=\left\|r^{(N+1)}\right\|_{p}=\left\|b-Ax\right\|_{p}, \tag{3}\]  
where \(\left\|\ \right\|_{p}\)--p-the norm, in general, is p = 2 (norm \(L2\)) or p = 1 (norm \(L1\)). For the occasion \(p=0\): \(L0=\left(\sum_{i=1}^{l}\left|x_{i}\right|^{0}\right)\), \(\left|x_{i}\right|^{0}=\left\{ \begin{array}{l}0,\ if\ x_{i}=0\\ 1,\ otherwise\end{array} \right.\). Thus, \(L0\) is the number of non-zero elements in \(x\). For the occasion \(p=1\): \(L1=\left(\sum_{i=1}^{l}\left|x_{i}\right|^{1}\right)\). Norm \(L1\) represents the sum of the absolute values of elements in \(x\). The main difficulty in using the algorithms [7; 8; 9] is that they require large computational costs and the regularization parameter, which is associated with the cost function, strongly depends on the noise variance. If the noise variance used for the corresponding parameter does not meet the real noise variance, the efficiency of DOA estimation of the corresponding methods is reduced significantly. In [10; 11; 12; 13; 14; 15; 16; 17], the SL0 algorithm is presented, which implements a very fast method for searching for a sparse solution to an indefinite system of linear equations, including for data belonging to a complex space. The SL0 algorithm implements an estimated solution based on approximating the discontinuous function of the \(L0\) norm with an arbitrary continuous function. In this case, the solution can be obtained from the global maxima of the corresponding function and the gradient method with a step-by-step projection of the result onto the space of admissible vectors. This method requires fewer computationsthan solving the problem of minimizing the \(L1\) norm and does not require the exact noise variance. However, this algorithm is sensitive to the choice of the initial values of the coefficients and does not guarantee convergence to the optimal solution. The variant proposed in [14] makes it possible to significantly reduce the problem dimension without degrading the performance. In [15], a new model for representing a sparse signal using the lower left diagonals of the covariance matrix is proposed, and it demonstrates high performance in DOA estimation. An improved SL0 algorithm which uses correlation matrices and several signal snapshots instead of one is given in [16]. For performing sparse reconstruction using a single snapshot, traditional methods are ineffective due to the many neighboring targets and low signal-to-noise ratio. In [18], the problem of ill-conditioning when applying SL0 to Doppler range-angle estimates for a MIMO radar was studied and an efficient approach for MIMO radars based on using SL0 for this type of estimates was presented. A method that improves the efficiency of Doppler range imaging for a noisy radar was also presented in [19].