Методи машинного навчання 
© M. Kosovets, L. Tovstenko, 2020 
304 ISSN 1727-4907. Проблеми програмування. 2020. № 2–3. Спеціальний випуск 
UDC 517.9:621.325.5:621.382.049.77  https://doi.org/10.15407/pp2020.02-03.304 
THE PRACTICAL ASPECT  
OF USING THE ARTIFICIAL INTELLECTUAL TECHNOLOGY FOR 
BUILDING A MULTIDIMENSIONAL FUNCTION CFAR  
FOR SMART-HANDLED LPI RADAR 
M. Коsovets, L. Tovstenko 
The problem of the development of modern mobile smart-handled LPI radars using artificial intelligence technologies, the main difference of 
which is the construction of the CFAR function, which takes into account the influence of external and internal factors and requirements for 
the purpose, also distinguishes the developed radar among others in its class. The analysis of the publications was showed a great interest in 
modern radar systems and the lack of a unified approach to solving this problem. The purpose of the article is to reduce this gap, from 
collecting information from radar sensors and internal sensors to construct a generic multidimensional CFAR function and for organize its 
effect on the receiving and transmitting part of the radar. The application of artificial intelligence technologies in the construction of a 
modeling complex of LPI radars with CFAR function and their debugging in real time is covered.  
Key words: Smart-Handled Radar, LPI (Low Probability of Intercept) Radar, CFAR (Constant False Alarm Rate), IoT (internet of things), 
artificial Intelligence, Deep learning, Signal  processor, Signal-to-Noise Ratio (SNR). 
Розглянуто проблему розробки сучасних мобільних інтелектуальних LPI радарів використовуючи технології штучного інтелекту, 
основна відмінність яких полягає в побудові функції CFAR, що враховує вплив зовнішніх та внутрішніх факторів і вимоги по 
призначенню, також вона вирізняє розроблюваний радар серед інших у своєму класі. Проведений  аналіз публікацій показав велику 
зацікавленість сучасними радарними системами і відсутність єдиного підходу до розв’язання цієї проблеми. Мета статті – 
зменшити цю прогалину, починаючи зі збору інформації від радарних сенсорів та внутрішніх датчиків для побудови 
узагальнюючої багатовимірної функції CFAR та організації її впливу на приймальну і передавальну частину радару. Висвітлено 
застосування технологій штучного інтелекту при побудові моделюючого комплексу LPI радарів з функцією CFAR та їх 
відлагодження в реальному масштабі часу.     
Ключові слова: Переносний розумний радар, LPI (з низькою ймовірністю  перехоплення) радар, CFAR (функція постійної 
ймовірності хибних тривог),  IoT (інтернет речей),  штучний інтелект, глибоке навчання, сигнальний процесорr, коефіцієнт сигнал-
шум (коеф. с/ш). 
Рассмотрена проблема разработки современных мобильных интеллектуальных LPI радаров, используя технологии искусственного 
интеллекта, основное отличие которых заключается в построении функции CFAR, учитывающий влияние внешних и внутренних 
факторов и требования по назначению, также она отличает разрабатываемый радар среди других в своём классе. Проведённый 
анализ публикаций показал большую заинтересованность современными радарными системами и отсутствие единого подхода к 
разрешению этой проблемы. Цель статьи - уменьшить этот пробел, начиная по сбору информации от радарных сенсоров и 
внутренних датчиков для построения обобщающей многомерной функции CFAR и организации её влияния на приёмную и 
передающую часть радара. Освещены применения технологий искусственного интеллекта при построении моделирующего 
комплекса LPI радаров с функцией CFAR и их отладки в реальном масштабе времени. 
Ключевые слова: переносной умный радар, LPI (с низкой вероятностью перехвата) радар, CFAR (функция постоянной вероятности 
ложных тревог), IoT (интернет вещей), искусственный интеллект, глубокое обучение, сигнальный процессор, коэффициент сигнал-
шума (коэфф. с/ш). 
Introduction 
Today nobody will have surprised application of radar technologies in different industries of the economy: 
medicine, armament, security, agriculture, geology, IoT and others like that. Their further development is impossible 
without the use of artificial intelligence, cognitive technologies, deep learning, cloud computing, real-time 
multiprocessors, which are being rebuilt in the process of solving the problem, neural networks. All these modern 
technologies do not diminish the importance of knowledge of antenna theory, radio engineering, signal processing, but 
make them revisit, more meticulously. And a separate task, no less important - real-time setup of the complex. This is 
often a stumbling block, since debugging LPI radars is sometimes more difficult than designing the radars themselves, 
since it requires the construction of modelling complexes but often sufficient resources (time, funding) are not provided 
in the contract.  
Features of development of CFAR function for LPI radar 
The main requirement for LPI radars, depending from the using – is invisibility to the surrounding electronic 
devices, low radiation power (less than 1 W), minimizing the negative impact on human health. This makes it difficult 
to isolate the signal from the noise, which explains the unsuccessful attempts to create a ground-based scanning radar 
(GRP), a mine-finding radar, a portable intelligent through the wall radar (PSTW), near- and far-range radars. The use 
of technology in the IoT environment is rapidly increasing as demand for such a service built into existing gadgets such 
as iPhones, clocks, and so on increases, allowing for unlimited access to cloud computing, which facilitates deep 
Методи машинного навчання 
305 
learning of cognitive functions. Historically, this new class of radar has been called the Low Probability of Interception 
(LPI) radar. 
For nonlinear approach (NLFM) the CHIRP modulation as the SIDELOBE (side petals) can improve level has 
the greatest advantage and to receive losses <1 dB in comparison with windowing configuration. (NLFM) modulation 
can be considered of course as "window weightings". As a lack of nonlinear modulation in relation to linear it is 
possible to note difficulties with elimination of hindrances and big sensitivity to Doppler shift (providing Doppler 
compensation filters). 
The radar has to operate from 20 m, at fine resolution, out to 96 nmi, at reduced resolution, with the same solid 
state transmitter, relying on waveforms and processing to achieve the necessary performance.  The underlying method 
by which this is achieved is to use short pulses for close ranges, where the returned energy is adequate for detection, and 
then to integrate the power over longer pulses or sequences of pulses, preferably coherently, to allow adequate SNR for 
the detection of targets at longer ranges. 
This leads to the use of signals with a complex signal structure. Consider the key metrics that characterize radar: 
resolution by distance, azimuth, area, volume. 
The distance resolution  on range is estimated by distance ΔD between two separately observed targets 1 and 2 
located on one direction concerning the radar. Separate reception of the reflected signals from these targets it is possible 
if the reflected impulse from the first targets ends before, than there will be acceptances an impulse which is reflected 
off from the second target. 
Since the reflection from the first target lasts for τ pulse time, and the signal from the second target is delayed by 
time 2 Dt s
 
, then the inequality of the individual receiving signals will be inequality  2
pulse
D
c
  
. 
Two targets will be displayed separately on the radar screen if the distance between the targets is: 
2
pulse
resolution
cD D    
. 
With decreasing ΔD resolution, the value of the range resolution increases. Thus, to increase the resolution of the 
range should reduce the length of the probe pulses. 
The azimuth resolution is estimated by the minimum value of the angle α0 between the directions to two 
equidistant point targets 1 and 2, inat which the reflected signals from these targets are received separately. The 
azimuth resolving power value α0 is determined by the horizontal antenna width αH at half power. Thus, in order to 
increase the azimuth resolution, one should narrow the antenna pattern in azimuth. 
The resolving area at a distance D is estimated by the size of the area limited by azimuth by the beam width αH of 
the antenna at half power, and in range — by range resolution. It should be noted that the range resolution due to the 
finite sliding angle β is 1/cosβ times larger than the generally accepted value
2
pulsec . 
The slip angle is the angle in the vertical plane between the direction of maximum radiation from the output of 
the radar antenna and the sea surface. 
Area resolution   
0.008757.3 2cos cosH pulse H pulseresolution
D c D cS       
 
For small values of β, this holds 
0.0087resolution H pulseS D c  
. 
 Estimated expression for resolving area     
150 2.657.3H pulseresolution H pulse
D cS D    
 [м2], 
were αH [град]; τpulse [mks]; D [м].  
Within the area resolution, point targets cannot be displayed separately on the radar screen 
If    D = 50⋅103 м; αH = 1°; τpulse = 1 мкс, то Sresolution = 13⋅104 м2. 
The resolving volume at a distance D is estimated by the value of a volume numerically equal to the cross-
sectional area of the antenna beam at this distance, multiplied by the range resolution 
2
pulsec .  
Get     
2 4 2
2 1.5 102(57.3)
pulseH
resolution H pulse
cV D D c       
. 
Методи машинного навчання 
306 
Calculation formula for volume resolution: 
2 2
2 150 0.045(57.3)
H
resolution pulse H pulseV D D      
 [м3],    
were D [м]; αH [град]; θ [град]; τpulse [mks]. 
We get:  
2 4 2
2 1.5 102(57.3)
pulseH
resolution H pulse
cV D D c       
. 
Clouds, rain, fog and snow interfere with the radar, which manifests itself in the form of false radar reflections 
(weather clutter, volume clutter). Thus, difficult meteorological conditions lead to a weakening of the reflected signals 
from the targets and the appearance of additional re-reflected interference. If the sea surface is mirrored, then the probe 
signal reflected from the sea surface propagates only towards the target. 
With an excited sea surface, the geometry of the reflecting surface of the sea is constantly changing, part of the 
power of the probing signal is reflected from the sea waves, arrives not only towards the target, but also returns to the 
side of the locator, creating additional interference (sea clutter). The reflected signals in this case are diffuse in nature; 
the probability distribution of instantaneous noise values has a normal distribution, the same as that of thermal noise.  
LPI Radar emites electromagnetic waves with impulse modulation low power of up to 1 watt. Depending on 
application , modulation can be phase, frequency, pseudorandom, noise-like, and the like. For determine goals with 
good resolution requires complex mathematical processing of the received signal, controlling process of formation of 
probing impulses, interaction with the receiving part and control of the processing of received information. Similar non 
LPI radars have a power of about 1 kW. 
The task of collecting information from the sensors of the electromagnetic environment, the internal sensors, 
isolating of signal from noise at the input of the radar receiver, signal processing, using of artificial intelligence 
technology, deep learning, classification of signals, support of cognitive functions  of radar are provided by a 
multiprocessor  with architecture which in during of processing is rebuilt with forming a multidimensional function 
dependent on the application of the radar. 
This feature is called: Constant False Alarm Rate (CFAR). The developer's task is to minimize false alarms at 
different signal, interference, and noise levels. In this article, we will focus on the formation of the CFAR function and 
will cover, beyond the scope of consideration, the formation of interference elements, the effect of internal noise and the 
methods of forming controls on the components of the radar LPI and the process of organizing calculations. The rapid 
growth in the number of scientific publications in domestic and foreign publications confirms the importance of the 
CFAR function, as well as the lack of a clear concept for its construction and the presence of a small number of 
practical successful applications related to the most understandable and widely using military field: marine and 
surveillance radars. 
The main task of creating LPI radars with CFAR is to build a CFAR processor that processes an adaptive 
algorithm for noise detection and interference problems. This must be the original architecture. We begin the process of 
developing radar by formalizing radar requirements, building a complex to model a multidimensional CFAR function 
with sufficient resources, and generating an impact function. This dispels the myth of the ease of constructing LPI 
radars as a transmitter and superheterodyne receiver, as well as exaggerated affinity with basic telecommunication 
wireless stations. 
In most real systems, the level of interference changes over time. To maintain the constant probability of false 
alarms, we change their threshold. A series of discrete values, each of which is the average signal level over a 
corresponding time interval, enters the CFAR detection circuit. The detection scheme must decide whether the signal in 
the study interval indicates the presence of a target or whether it is caused by external and internal noises.  In most 
simple CFAR detection schemes, the threshold level is calculated as the average noise value over several time intervals 
before and after the test. We consider that the signal level in the test interval is conditioned by the presence of a target if 
it exceeds the signal level at adjacent intervals. This approach is called cell-averaging CFAR (CA-CFAR). 
Examples of CFAR implementation 
There are many options for forming a CFAR function, depending on the purpose. Basically, it comes down to 
determining the threshold level. For example, some algorithms average the signal at predetermined intervals before and 
after the test, and then take more or less of the values obtained. These approaches are called maximum-of-CFAR, GO-
CFAR and least-of-CFAR, LO-CFAR. 
More sophisticated CFAR algorithms can adaptively adjust the threshold level based on radar statistics. This is 
especially true for shore-based radars, where noise due to reflection from the sea surface is poorly approximated by the 
additive white Gaussian noise. This makes it difficult to detect objects such as submarines, yachts, swimmers, etc. 
Article [1] examines critical interferences in the form of an unwanted echo called clutter. These include echoes 
of the earth, weather (especially rain, snow), the sea, and deliberate disturbances. Therefore, the developer of shore-
Методи машинного навчання 
307 
based radars faces the problem of how to eliminate external interference. This article discusses troubleshooting methods 
such as Doppler processing and Log-FTC, and then introduces an algorithm for protection against accidental changes 
that are suitable in a transition state and a steady state. Also, this technique does not depend on the type of interference 
and center frequency. 
Article [2] presents the original structure of a CFAR detector and the estimation of moving marine targets under 
natural interference conditions using bistatic radar scattering. Studies confirm that the amplitude of the marine target 
signal is distributed by Rayleigh, and the amplitude of the reflected signal from the sea is distributed by Weibul. In this 
situation, an adaptive approach to the formation of a statistical rule was used to suboptimal detect the unknown length 
of the target at sea. The purpose of this approach is to build a structure of decision-making rules to determine unknown 
parameters in situations where part of the statistical parameters of two hypotheses is unknown: power or length of 
target. 
A comparison of the performance of the CFAR API and the Hauff detector with incoherent multilayer 
integration is investigated in [3]. For comparison, the Rolling approach was used to calculate losses. This article unifies 
the results of an extensive loss study of several types of detectors. This study examines the results of a comparative 
analysis of a Hauff detector with incoherent integration in a multilayer situation. Losses are defined as a statistical 
estimate using the probabilistic characteristics of detectors of both types. The efficiency of the Hauf detectors is 
calculated for different values of the false alarm probability with different observations in the reference window, the 
average interference ratio, the noise, and the probability of occurrence of a multi-path medium-length situation in the 
cell range. Our results show that Hauf transformation is effective if the multipath situation is reduced. 
Article [4] investigates the automatic averaging (ACCA) of a CFAR detector based on ordered variable data 
(ODV) for heterogeneous background environments. The ACCA-ODV detector dynamically selects, by successive test 
checks, the appropriate set of ranked cells to estimate the unknown background level. The proposed detector does not 
require any prior information about the background environment and uses the variability index statistics as a form 
parameter to reject or accept the ordered ordered cells. For implementation, a two-tier architecture is proposed, in which 
both ODV-based sequential statistics match the hypotheses that are being processed simultaneously. The performance 
of the proposed detector is evaluated and compared with those such as OS-CFAR and modified index-CFAR (VI-
CFAR) detectors in different background environments. The results show that the ACCA-ODV detector acts as a CA-
CFAR on a homogeneous background and operates reliably in heterogeneous environments. 
Article [5] examines the received signals in a radar system, which are always accompanied by noise and 
interference, such as echoes from the earth, sea, rain, birds, insects, wind gusts and atmospheric turbulence. These 
disturbances can cause a serious deterioration in the performance of radar systems, leading them to conclude that these 
echoes are targets (false alarms). To overcome this problem and make the right decision, the receiver in the radar 
system must achieve a constant false alarm rate (CFAR) and a maximum probability of target detection. Modern radars 
typically make the detection decisions automatically, using an adaptive threshold based on the CFAR architecture, 
where the threshold is determined dynamically based on local background noise / clutter rather than the constant value 
of a powerful signal. 
Formation of CFAR function and its influence in modern LPI councils is impossible without the use of digital 
filtering. We offer its construction on a multiprocessor, which is being rebuilt in the process of operation. Using 
multiprocessor with neurocomputer elements allows you to achieve multiple types of signal processing in real time. The 
processing is easy to do in the baseband, not the bandwidth. Then the low pass filter performs the task without taking 
into account the external band. Increasing the signal-to-noise ratio improves the detection of targets in the middle of the 
noise, if this filter is consistent with the expected radar signal, even when it has distorted the signal. Doppler processing 
excludes signals from fixed and slow targets. The proposed filter, developed as an adaptive filter, is used for spectrum 
formation and interference elimination, based on the least-squares algorithm, which monitors interferences and 
eliminates the "blinding" of moving targets that occur during digital Doppler smoothing, which are more likely to be 
caused. 
After the electrodynamic formalization of the electromagnetic field, we construct an operator equation of the 
first kind, where the nucleus is a Green function. We display the instantaneous characteristics of the observed object, 
the set of values of which determines the image. This is the information that underlies the training and recognition of 
radar information. 
Today, there are many options for building CFAR algorithms based on linear and nonlinear operations. Analysis 
of publications confirmed the complexity of building LPI radars. The impasse of the situation is seen in the non-
systematic consideration of the design of modern radar systems. The key problem is the generalization of the CFAR 
function, the construction of a modeling complex to work out the options for constructing the function, the allocation of 
computing resource to implement the function in real time. The CFAR processor architecture is offered as a specialized 
module synthesized for multiple FPGA Xilinx Virtex VII configurations. 
The optimum choice of architecture is ensured by its maximum approximation to the class of solved problems. 
To implement the CFAR function, the architecture of a fault-tolerant real-time m-cluster multiprocessor for 
multidimensional signal processing [6] with a high-speed radio channel is developed. High performance and reliability 
are achieved through the use of high-speed radio to exchange between individual clusters, channel modulation and 
fault-tolerant technologies. Such an architecture is optimally oriented to solve multidimensional field processing 
problems in radar. The spatial orientation of the information sources most fully reflects the natural parallelization of the 
tasks being solved in the multiprocessor whose architecture corresponds to this class. A set of functional modules 
Методи машинного навчання 
308 
designed for radar field processing has been developed, which is characterized by a large amount of information. The 
central unit generates CFAR functions, organizes deep learning, provides diagnostics and initial start-up. There are 
three stages of the operation of the CFAR calculator, aimed at managing the process of obtaining information and the 
formation of probing signals through the CFAR interface: the task of recognizing the radar situation, developing a 
decision strategy, implementing the decision. 
The dynamics of the change of the radio frequency field of the radar can be analyzed in the abstract space of 
states – the phase space in which you can enter coordinates describing the state of the system, such as the 
electromagnetic field strength of points at different distances from the antenna at each time. We plot the trajectory in the 
phase space, indicating the direction of field change along the phase trajectory over time. Note that the introduction of 
such a phase space makes it easier to analyze the field if you move from a normal coordinate space to a phase space. 
For example, if the propagation of oscillations in the space-time diagram of variables x, t is represented by a curve line, 
then on the phase plane P, x (where P is the Power, x is the Coordinate), such motion is represented by a point and a 
line respectively (first order curve). In phase space it is also easier to analyze the stability of the solution of the wave 
propagation problem and to investigate the problem of stability and instability of the system. The basis for the 
classification of the wave propagation and their models is the condition of the reproduction of solutions under given 
initial conditions. Analysis of the radar system shows that over time, the phase trajectories of certain areas of space 
concentrate around some points - the system seems to be drawn to these points in the course of its development. The 
points attracting the trajectory are called attractors. An example of an attractor would be the propagation of energy from 
the antenna to the target and in reverse order. Power and distribution are constantly changing, but the energy flow is 
generally stable in space and does not go beyond certain limits. 
To better understand the radar's internal links and, most importantly, the links between the information gathering 
systems, namely, the sources of noise, both internal and external, the effects of random and directional interference, 
antenna information and others, we will examine the "reticular" aspect of the radar LPI. Depending on the course of the 
radar situation, the information content of the radar and its computing environment changes. The radar system has 
reticular property due to the fact that its processors of different architecture and computing power have many contacts 
with external sources of information, the ability to scale the number of computers and links). In the course of training, 
different computers have different activity (priority when processing interrupts), blocking connections after operator 
intervention, or receiving a directional interference. The role of the arbiter in the reticular system of the radar takes on a 
central module that interacts directly and indirectly with almost all radar structures and systems, affecting its various 
functions. One of the most important is the radar activation function associated with the process of changing the radar 
environment. 
The assumption is made that functional metastable structures of the neural network phase space are model 
representations of the radar system. It is noted that the process of relaxation of the system initiates the reflection of 
metastable structures of model representations of images in the cognitive space in the form of functional modes. 
Cognitive memory space is defined as the phase space of functional image modes. The mechanism of influence of 
functional modes on the process of forming model representations of images is investigated and of creation of model 
representation of an image with consideration of chaotic change of parameters of system with time is studied.  
Forming a generalized multidimensional CFAR function 
The effectiveness of the radar recognition system as a whole depends directly on the efficiency of the technical 
means and its mathematical support - software-implemented algorithms for constructing descriptions of classes of 
objects and phenomena in sign language. The obtained radar information is divided into groups of similar but not 
identical phenomena. We create a radar image of the object against the background of distorted information in real 
conditions. The characteristic property of the images is manifested in the fact that after getting acquainted with the finite 
number of manifestations of the object, we learn about any number of its representatives. Also, when teaching a radar 
neurocomputer (rebuilding multiprocessor) at different observation sites, the same objects are classified equally and 
independently of each other. The objective nature of the basic property of the images allows us to model the recognition 
process. 
As a result of deep learning, the recognition system acquires the ability to respond with the same reactions to all 
objects of the same image and different - to all objects of different images. It is very important that the learning process 
be completed only by displaying a finite number of objects, without any “clues”. 
The objects of training are radar objects, interference, radar specifications. It is also important that only the 
objects themselves and their affiliation with the image are indicated in the learning process. The training is followed by 
the process of recognizing new objects, which characterizes the actions of the already learned system. Automation of 
these procedures is a problem of pattern recognition training. 
The most comprehensible to us is the spatial-temporal interpretation of tasks. Hence the problem of learning 
pattern recognition has received its geometric (spatial) and temporal (dynamic) interpretation. Any radar image 
resulting from the observation of an object in the course of training or testing can be represented as a vector, and hence 
as a point of some space. 
If it is claimed that radar images can be uniquely assigned to one of two (or more) images, then it is claimed that 
in some space there are two (or more) areas that have no common points, and the images shown are points from these 
areas. Each such area can be assigned a name, that is, give a name to the corresponding image. 
Методи машинного навчання 
309 
During deep training, points randomly selected from these areas are presented and information is given to which 
area these points belong. No additional information about these areas, ie the location of their borders, is reported during 
the training. 
The purpose of the training is either to construct a surface that separates not only the points shown in the 
learning process, but all other points belonging to these areas, or the construction of surfaces that limit these areas of 
points so that each of them is only one point insult. That is, deep learning is about constructing such features from 
image vectors that would be, for example, positive at all points of one and negative at points of another image. 
Due to the fact that the areas do not have common points, there are always a whole lot of such separating 
functions, and as a result of training one of them must be built. If the displayed images do not belong to two, but to a 
larger number of images, then the task is to build on the points shown in the course of training, separating all the areas 
corresponding to these images from each other. This problem can be solved, for example, by constructing a function 
that takes over the points of each area the same value and the points over the different areas different. 
A deep learning mechanism is used to build the CFAR function. First of all, we solve the problems of object 
classification. Let there be some set of investigated objects { }   , each of which is determined by the vector of its 
values in the phase space of some m given signs: 
1 1 1 11 1 2{ , , , }mx x x x   ; 
2 2 2 22 1 2{ , , , }mx x x x   ; 
………………………; 
1 2{ , , , }t t t tt mx x x x   . 
We divide objects into homogeneous classes (their number can be specified or may be unknown), which we call 
clusters, and methods for finding them - by cluster analysis. There is no training sample in this case, that is, it is 
unknown to which classes the objects in the set belong { }   . We define patterns of structure of objects in clusters. 
In particular, a "compactness hypothesis" can be used as a similar pattern, ie the requirement that objects belonging to 
the corresponding classes be located in a given characteristic space 
1 2{ , , , }mx x x x "compactly". This means that 
the "distance" between objects assigned to this class is no more than specified. 
Other requirements may be formulated in the automatic classification, namely, the generated classes in the sign 
space must be spaced apart at certain distances. Essentially, with the help of a multidimensional function defined on an 
arbitrary grid, we construct some sign or a generalized image of a class, by which one can uniquely determine the 
belonging to one of two points of a point in the phase space of measurements of the values of some predefined features. 
To separate the sets of points of two different visions, we use hyperplanes or their sets, but only if the sets of 
points of the visions are so far apart in the phase space of the signs that a separating hyperplane can be constructed 
between them. When this is not present, then we move on to a new signs space. As a rule, we are going to add new 
signs to existing ones and to increase the dimension space of the signs space. In this case, to obtain a guarantee of 
separation reliability, it is necessary to significantly add the number of training sample points. 
The main task of training is not to keep all the available information about the object, but to minimize the 
irrelevant, highlight the most informative and leave only the essential, that is, what can only be called information about 
the object, not just a random set of data. Sets of image points can also be reliably separated in spaces with fewer 
features when the surfaces of the sets are of rather complex configuration and do not intersect. For this case, we write an 
algorithm for constructing a separating hypersurface in the form of a multidimensional function defined on an arbitrary 
grid. But first, let's write an algorithm for determining a multidimensional function on an arbitrary grid, which we will 
use to construct the function itself. To do this, we: 
1) We set the points of the grid of ordinates in the multidimensional coordinate space - the number of these 
points and the values of the coordinate vector of each point; 
2) Determine the values of the parameters for all components: Green's function for solving the wave equation 
from grid points plus elements of the polynomial kernel. 
Let some arbitrary grid of ordinates P  be given in the form:  
(1) (2) ( )( , ,..., )NP P P P  
The system of randomly arranged at nR different unique points is given in the form: 
( ) ( ) ( )1( ,..., ) , 1,k k k nnP P P R k N   
Define as follows on the grid P a multidimensional function ( ; ; )P P  , which at points of the grid takes some 
Методи машинного навчання 
310 
already set values: , 1,ir i N : 
( )( ; ; ) , 1,i iP P r i N   ,      (1) 
and has a simple look - as a sum of some functions: 
( )
,1 1
( ; ; ) ( )N im ni mP P G P P Pi

   
      
,         (2) 
where *1 21 2 1... , ( ,..., )nn nP P P P P P P      , kkP ; k - coordinate in k  – degree, 
1m  

– the sum of all possible 
combinations: when 0  – is the free ratio. 
When 1   – this is a free coefficient + components in linear form
0
1
n
k k
k
P

  
, at 2   – sum at 1  + 
components at quadratic form 
1 1
n n
kl k l
k l
P P
 
 
 etc, (total  ( 1)!
!( 1)!
n m
n m
 

 components);  
, ( )m nG x P  2
2
ln , 2
, 2 1
m n
m n
x P x P n k
x P n k


      
, 
21]2
1
)([

 n
i ii
PxPx
, 
*1 2( , ,..., )nP P P P , 
*1 2( , ,..., )nx x x x . 
The continuity of Green's 
, ( )m nG x P
functions requires that the condition / 2m n  be satisfied. By 
summarizing the variation theory of constructing multidimensional functions on a chaotic grid to simple algebraic 
conditions, we can consider finding the coefficients of a multidimensional function as a solution of some system of 
linear equations. To do this, we simply introduce a continuous linear indexing for the coefficients   of formula (2). 
Then we find the values of the coefficients  
, 1,i i N  and , 1,l l    , 
where 
( 1)!
!( 1)!
m
n
n mC n m
    
, 
 
by solving two algebraic conditions: and, where ( 1)!
!( 1)!
m
n
n mC n m
    
 to solve two algebraic conditions: 
1)  coincidence of values of multidimensional - function at given points P  (1); 
2) and the orthogonality of the polynomial kernel functions at each other at the same points. In matrix form it 
has the following generalized form: 
C r  , 
where the matrix C has dimension 
 
(N ) (N )   , 
 
or 
1 2 1 2( , ,..., , , ,..., )N        , 
 
1 2( , ,..., ,0,0,...,0)Nr r r r   – vectors of dimension N   each,  – is the sign of transposition,   
 
( ) 1 2( , ,..., )i nP x x x  , 1,i N . 
 
For the system to have a solution, a condition is required N   .Consider constructing a separating hypersurface 
Методи машинного навчання 
311 
in the form of a multidimensional function defined on an arbitrary grid. Suppose two sets of m - measurable points 
X and Y with the number of points M and L respectively: 
 
1 2{ , ,..., }MX x x x , 1 2{ , ,..., }NY y y y , 
where 
1 2{ , , , }i i i imx x x x ,  1,i M ,  1 2{ , , , }i i i imy y y y , 1,i L .  
 
We find a multidimensional function on an arbitrary grid that can sufficiently reliably divide sets of points 
X and Y , that is, divide into two intersecting regions by two sets in some multidimensional space of phase features, 
which are initially represented as two arbitrary sets of multidimensional points. The automatic construction of such a 
rule is the algorithm of machine learning. This is achieved by constructing a surface that separates not only the points 
shown in the learning process, but also all other points belonging to these areas. 
To construct a multidimensional function on an arbitrary grid, we define a grid P  of Nn – dimensional points 
and the values of the function in them , 1,ir i N . Due to the fact that we have not determined the direction from which 
to build the separating hypersurface, we have the opportunity to do so in the direction of each of the m coordinates. To 
do this, we use the value of one of the m coordinates as the value of a multidimensional function, and we use all other 
(m-1) coordinates as the values of points to determine the grid of ordinates with the dimension n = (m-1). With this 
approach, there are only m options for constructing a multidimensional function on an arbitrary grid. In addition, it can 
be built both from the side of the set X , checking for Y and vice versa. You can also try moving around the points of 
one set, searching for the closest points of the other and finding some averages and already, on them to look for a 
hypersurface. In total, we have at least 2m + 1 options for constructing a multidimensional function of which we choose 
the most acceptable. 
Conclusions 
In this article, we have considered the construction of a multidimensional CFAR function for LPI radars and 
have made a comparative analysis of the most typical variants of the function construction based on materials of 
domestic and foreign publications in scientific publications. [1-5] They also noted a small number of practical 
implementation of theoretical developments. This is due to the complexity of forming a multidimensional CFAR 
function, the requirements for creating a computing environment, which is formed on the basis of a real-time 
multiprocessor, which is reconstructed in the process of processing radar information with wireless high-speed 
backbone, conveyor computing, neural networks, parallels. In addition, information for CFAR function formation is 
collected at all levels of the radar LPI: antennas, transceiver unit, primary and secondary processing units and, 
accordingly, primary processing influence generation, signaling probe generation, and radar receiver. The laboratory is 
testing the CFAR processor on a crystal (SoC). This is a promising direction for the development of a key element for 
LPI radars and the ability to modularly formulate a line of destination radars. 
Formalizing a recognition system to investigate and analyze the real processes of collecting, processing, and 
visualizing radar images whose mathematical models we are not yet able to build is very important when building 
radars and learning in the application process. To date, a formal apparatus designed to construct mathematical models of 
sufficiently complex phenomena and processes adequate to simulated phenomena or processes has not yet been 
developed, which opens up great prospects for the best solution of the problem as it is sufficient to accurately quantify 
the development of processes solely on the basis of accumulated information about processes. 
Another feature of smart radars using artificial intelligence technologies is the ease of use, just like using 
smartphones. This will simplify the work of professional users of LPI radars (marine radar operators, subsurface 
sensing, etc.) and will allow for the introduction into everyday life, such as the IoT segment. 
 
References 
1. Saidi H., Askari G., Akhlaghi J.  An antichaff technique for tracking radar. European Radar Conference, 2005. EURAD 2005. Paris, France. 
2. Mirmohammad H.S."Using lights in a volume-oriented rendering" in Photonics Applications in Astronomy, Communications, Industry, and 
High-Energy Physics Experiments 2011, edited by Ryszard S. Romaniuk, Proceedings of SPIE Vol. 8008 
3. Kabakchiev Chr., Doukovska L., Garvanov I. API CFAR Processors with Hough transform in multipath situation. Institute of Information 
Technologies, Bulgarian Academy of Sciences “Acad. G. Bonchev”  2009. Str., bl. 2, 1113 Sofia, Bulgaria 
4. Farrouki A., Barka M. Automatic censoring CFAR detector based on ordered data variability for nonhomogeneous environments. IEE Proc.-
Radar Sonar Navig.  Vol. 152, N 1, February 2005 
5. Rosyadi I.  An FPGA Implementation of Automatic Censoring Algorithms for Radar Target Detection. CITEE 2012. Electrical Engineering 
Department, Jenderal Soedirman University. Yogyakarta, 12 July 2012. 
6. Kosovets M., Tovstenko L. Architecture design of a high-performance real-time multiprocessor with an asynchronous exchange bus for radar 
tasks. / Radar. Navigation. Communication. X International Scientific and Technical Conference. Voronezh 2004. – Vol. 1. P. 627–639.  
 
Методи машинного навчання 
312 
Література 
1. Сайді Х., Аскарі Г., Аклафі Ж. Метод боротьбиз завадами для радіолокатора стеження. Європейська радіолокаційна конференція. 
2005. EURAD 2005. Париж. Франція. 
2. Мірмохамад Х.С. Використання світла в об'ємно-орієнтованій моделі в додатках Фотоніки в астрономії, комунікаціях, промисловості і 
високоенергетичних фізичних експериментах. 2011. Під ред. Річарда С. Романюка. Proceedings of SPIE Vol. 8008. 
3. Кабакчієв Х., Дуковська Л., Гарванов І. API CFAR процесор з перетворенням Хауфа в багатопроменевій ситуації. Інститут 
інформаційних технологій, Болгарська академія наук «Акад. Г. Бончева ” 2009. вул. 2, 1113, Софія, Болгарія. 
4. Факурі A., Барка M. Автоматичне  зондування CFAR детектором на основі мінливих упорядкованих  даних для неоднорідних 
середовищ. IEE Proc.-Radar, Sonar Navig., Том 152, № 1, лютий 2005. 
5. Розіді І. Використання ПЛІС для  автоматичного виділення алгоритмів для виявлення радіолокаційних цілей. CITEE 2012. 
Електричний інженерний департамент. Державний департамент Коедіпман. Джакарта, 12 липня 2012. 
6. Косовець М., Товстенко Л. Розробка архітектури високопродуктивного відмовостійкого мультипроцесора реального часу з 
асинхронною шиною обміну для задач радіолокації.  Матеріали X Міжнародної науково-технічної конференції. “Радіолокація. 
Навігація. Зв'язок”. Воронеж 2004. Том 1. С. 627–639. 
 
 
 
Received 21.02.2020 
 
 
 
About the authors: 
 
Kosovets Mykola Andriyovych, 
Researcher 
Departments of modeling of complex systems 
Faculty of Computer Science and Cybernetics,  
Taras Shevchenko National University. 
Publications in Ukrainian journals – 40. 
Publications in foreign journals – 10. 
Hirsch  index – 2. 
https://orcid.org/0000-0001- 8443-7805, 
 
Tovstenko Lilia Mykolayivna, 
leading software engineer of the department № 115 
V.M. Glushkov Institute of Cybernetics of the NAS of Ukraine. 
Publications in Ukrainian journals –15. 
Publications in foreign journals – 5. 
Hirsch  index – 2. 
https://orcid.org/0000-0002-3348-6065. 
 
 
Authors’ place of work: 
 
Taras Shevchenko National University of Kyiv,  
03022, Kyiv-187, 4d Glushkov Avenue. 
Tel.: +38(066) 255-41-43.   
E-mail: quantor.nik@gmail.com,    
 
V.M. Glushkov Institute of Cybernetics of the NAS of Ukraine,  
03187, Glushkov Avenue, 40.  
Tel.: +38(067) 777-40-10.