D.V. Panov – Ph.D.(Eng.), Associate Professor, Professor of Department, pdvlist@list.ru
A.V. Ashikhmin – Post-graduate Student, ashi_hmin@mail.ru​​​​​​​
R.A. Yuldazbaev – Post-graduate Student,  yuldazbaev@mail.ru

Low characteristics of existing radars for resolution and recognition of objects of observation (due to the available time, frequency and energy resources) reduce their information capabilities and do not allow consumers to provide reliable information about objects of observation located in a given area of space.
One of the ways to improve the ability of radars to recognize and resolve objects of observation is to use estimates of their radar portraits (RLP). At the same time, the above restrictions prevent the radars from obtaining portraits of the required quality.
In [2], a projection scheme for solving the inverse scattering problem is proposed, which is more General in relation to classical problems of primary signal processing. That is, obtaining a stable assessment of the object's portrait on the considered parameter field from its echo signal means solving the problems of detecting, measuring the coordinates (motion parameters) of the object, and resolving individual sources of secondary radiation in its composition.
Projection multigrid method for solving the inverse scattering problem – it is a quasi-optimal method for estimating discretely distributed RLP of concentrated observation objects (that is, RLP characteristic of groups of observation objects formed by a limited number of individual objects). It allows you to repeatedly overcome the Rayleigh limit on the radar resolution and come close to the Helstrom limit. The main feature of the method is that based on some a priori information about the RLP configuration of concentrated observation objects, there are many acceptable solutions y Ax n (x and y belong to complex Hilbert spaces 2x и 2y , A – a completely continuous (compact) linear operator, a completely continuous (compact) linear operator with a 2x -dense domain of definition;  n – noise monitoring) discussed up to a finite-dimensional space Km , a subspace of 2x . Elements of the Hilbert space are not used as a mathematical model for the portrait of an aerial object x 2x , and their projections x%  Px for some trial space Km – m-dimensional subspace 2x .
The mathematical model of the portrait of a group object is represented as x Px ГГГ Г x ГE%   , 1 ,  , где E&i – complex coefficients of decomposition of the portrait of an air object in a solid basis Г or reflection coefficients corresponding to the selected test functions.
Noise-resistant observation estimation of reflection coefficients Eˆ , corresponding to the projection assessment of the object's portrait x ГEˆ  ˆ , have the form E Q Zˆ  1 .
In the process of implementing a PMM to estimate the number of individual objects in a group, the dimension of the sample space increases sequentially, starting with the one-dimensional one. Modules of the element vector Eˆ they are compared with the threshold values calculated for the m-th hypothesis about the number of individual objects according to the expression Eпорm k Qm1 , where k – coefficient that determines the level of false alarms; Qm – the gram matrix of a system of echo signals of trial functions that approximate a group object consisting of m separate ones.
In the course of research, it was determined that when solving the problem of estimating discretely distributed RLP of concentrated observation objects, the optimal value of the coefficient k is k =2,2.
A modification of the projection multigrid method for solving the inverse scattering problem was proposed in order to optimize the estimation of the RLP of concentrated observation objects. The value of the square of the error rate averaged over the observation noise and normalized to the echo signal energy is used as an indicator of the effectiveness of evaluating the RLP of a concentrated observation object Ψнорм2  .
Studies have shown that the minimum averaged error values Ψнорм2 for estimating continuously distributed HRRP models of observation objects correspond to the value k≈2,35. Thus, the PMM with a refined value of the coefficient that determines the level of false alarms is proposed to be considered as a modified PMM for solving the inverse problem in a radar channel with scattering in order to optimize the evaluation of observation objects with continuously distributed HRRP.
The evaluation of the effectiveness of the PMM allows us to conclude that the average increase in the accuracy of estimating continuously distributed HRRP of concentrated observation objects due to the use of the proposed modified PMM is approximately 15% for SNR values less than 20 db. At higher SNR values, the accuracy of estimating continuously distributed RLP objects of observation using both the existing PMM and the proposed modified PMM actually coincides with the potentially achievable for primary echo processing.
One of the features of radar recognition is the length of the object being observed (for example, radial or transverse). In conditions of limited SNR values, the estimated length of the RLP (taking into account the lack of reliable information about the angle of the observation object) is only indirectly related to its true geometric dimensions. A study was conducted on the effectiveness of using the PMM modification to recognize different classes of observation objects with different radial lengths. In the course of the study, models of observation objects with relative lengths were used δ, 0,5δ и 0,1δ and continuously distributed HRRP. On the basis of «radial length», objects of observation were classified into: large -, medium - and small-sized.
Comparison of the proposed and existing methods showed that the use of the proposed modified projection multigrid method for solving the inverse problem in a radar channel with scattering increases the probability of correct recognition by 20−30% when the number of cycles of averaging the recognition feature is less than 10 and by 5−10% when the number of cycles of averaging the recognition feature is more than 10 compared to the situation when using the ratio of the maxima of the average values of the squares of responses consistent with limited wideband and narrowband probing processing signals as a recognition feature; the formation of a recognition feature associated with the length of concentrated observation objects will require a coordinated processing response with a fivefold increase in the spectrum width (coherent accumulation interval) at a SNR of 15−20 dB and a 10−20 – fold increase in the spectrum width (coherent accumulation interval) at a SNR of more than 20 dB to achieve the same recognition efficiency as when using the proposed modified projection multigrid method for solving the inverse problem in a radar channel with scattering.
Thus, we can conclude that the modified projection multigrid method for solving the inverse problem in a radar channel with scattering provides a quasi-optimal estimation of continuously distributed radar portraits of concentrated observation objects. Based on the results of mathematical simulation, a comparative assessment of the effectiveness of using this modified projection method in the interests of recognizing concentrated observation objects by their radial length showed its advantage in comparison with other known approaches to solving the problem under consideration.

Panov D.V., Ashikhmin A.V., Yuldazbaev R.A. Modified projection multigrid method using for inverse problem solving in a radar channel with scattering in the interest of lumped objects recognition by their extension. Electromagnetic waves and electronic systems. 2020. V. 251. № 5. P. 28−40. DOI: 10.18127/j15604128-202005-04. (in Russian) <