## 1 Introduction  
In signal analysis, the direction of arrival (DOA) describes the direction from which a propagating wave usually arrives, reaching the point where the sensor set is located. This set of sensors forms the so-called sensor matrix. The DOA estimation is a widely used method for estimating the location of objects, for example, in a road scene formed by an automobile radar using the phase difference between signals received by different antenna elements included in an equidistant linear array. For a sensor matrix in the form of a linear array of antenna elements, the angular resolution of the signals for classical processing methods is limited by the ratio of the array size to the wavelength. This means that it is not possible to improve the resolution without increasing the size of the array. In some cases, this may not be feasible. At present, there is a sufficient number of research devoted to theoretical studies of direction-of-arrival estimation methods, their implementation options, as well as an assessment of the efficiency of using the methods to solve applied problems. Common examples of the DOA estimation method are the Capon method and the beamforming algorithm using the fast Fourier transform (FFT) [1, 2, 3, 4]. The orthogonality of the eigenvectors of the noise part and the control vectors are used in the MUSIC (multiple signal classification algorithm) [5] and the more efficient ESPRIT (estimation of the signal parameters via the rotational invariance technique) [6] for determining the DOA of targets. The mentioned DOA estimation algorithms [1, 2, 3, 4, 5, 6] are based on an overdetermined system. Therefore, they have low resistance to errors or failure of individual receiving channels in the ULA, which is a factor that significantly worsens the DOA estimation results. In addition, these algorithms are not suitable for cases of highly correlated signals, which are signals reflected from objects. In contrast, DOA estimation methods based on CS (compressed sensing) methods use the spatial sparsity of targets and are more reliable. To estimate the DOA of received signals based on the CS (compressed sensing) method, an algorithm was proposed [7], which uses a vector of measurements of one or more snapshots of the received signal. Finding a solution with a minimum \(L0\) norm is a problem whose complexity rapidly increases with increasing problem dimensionality and in the presence of noise in the input data. A fairly effective approach to solving this problem is the BP (basis pursuit) method [8; 9], which allows finding a solution with a minimum \(L1\) norm (i.e., a solution \(Ax=b\), for which the value \(\sum_{j}|\mathbf{\chi}_{j}|\) is minimal), but it is still not fast enough, and it does not allow the use of complex data. The above methods provide the decomposition of data into weight components of the dictionary, which is an optimization problem with a basis of the \(L1\) norm with linear constraints. Extending it to the complex space \(\mathbb{C}\), we obtain the following:  
\[\left\{\begin{array}{l}max_{y\in\Omega}\ y^{T}x\right\}\to min,\ \ (\text{or}\ min\ \left\|x\right\|_{1}) \tag{1}\]  
subject to \(Ax=b\),  
where \(\Omega=\left\{\omega\in\mathbb{C}^{N}||\omega_{i}|\leq 1,i=1,2,\ldots,N\right\} \subset\mathbb{C}^{N},b\in\mathbb{C}^{M}\)--input data, \(x=\left[a_{\gamma_{1}},\omega_{\gamma_{2}},\ldots,a_{\gamma_{N}}\right]^{T} \in\mathbb{C}^{N}\)--basis components, and \(A=\left[\phi_{\gamma_{1}},\phi_{\gamma_{2}},\ldots,\phi_{\gamma_{N}}\right] \in\mathbb{C}^{M\times N}\)--the matrix of the redundant dictionary. \(D=\left(\phi_{\gamma}\right)\) - the formed dictionary, where \(\phi_{\gamma}\in\mathbb{C}^{M}\) means the component of the basis with index \(\gamma\in\Gamma\)\(\Gamma\)\(\Gamma\)\(\subset\)\(\mathrm{L}^{2}(\mathbb{R})\).