=Paper= {{Paper |id=Vol-1987/paper37 |storemode=property |title=About Use of Methods of Convex Programming for Synthesis of Conformal Arrays with Matched Dual-polarized Patterns |pdfUrl=https://ceur-ws.org/Vol-1987/paper37.pdf |volume=Vol-1987 |authors=Viktor S. Izhutkin,Anton A. Sharapov,Viktor S. Izhutkin,Alexander Zonov,David Kalaj,Arsen Zlatičanin,Maksat Kalimoldayev,Maksat Akhmetzhanov,Assel Abdildayeva,Fariza Galiyeva,Igor Kaporin,Alexander Kel'manov,Alexander Kel'manov,Vladimir Khandeev,Alexander Kel'manov,Anna Motkova,Alexander Kel'manov,Artem Pyatkin,Michael Khachay,Katherine Neznakhina,Michael Khachay,Vasiliy Pankratov,Daniel Khachay,Dmitry V. Khlopin,Vladimir E. Krivonozhko,Andrey V. Lychev,Eugene A. Kalashnikov,Konstantin N. Kudryavtsev,Irina S. Stabulit,Viktor I. Ukhobotov,Konstantin N. Kudryavtsev,Vladislav I. Zhukovskiy,Alexander Lazarev,German Tarasov,Dmitry Arkhipov,Alexander Lazarev,Nail Khusnullin,Elena Musatova,Denis Yadrentsev,Konstantin Ponomarev,Sergey Lupin,Aye Min Thike,Hein Tun,Valeriy M. Marakulin,Olga N. Masina,Olga V. Druzhinina,Alexey A. Petrov,Vladimir D. Mazurov,Alexander I. Smirnov,Nevena Mijajlović,Milojica Jaćimović,Igor E. Mikhailov,Ivan A. Suvorov,Leonid I. Minchenko,Daniil E. Berezhnov,Nataliia K. Obrosova,Alexander A. Shananin,Nicholas N. Olenev,Andrei V. Orlov,Nicolás Majorel Padilla,Pedro Araujo,Adrián Will,Sebastián Rodríguez,Alexander V. Pesterev,Lev F. Petrov,Nikolay Pogodaev,Boris T. Polyak,Andrey A. Tremba,Yuri S. Popkov,Larisa Rybak,Elena Gaponenko,Viktoria Kuzmina,Olga N. Samsonyuk,Natalya Sedova,Vadim I. Shmyrev,Ruslan Yu. Simanchev,Inna V. Urazova,Alexander K. Skiba,Stepan P. Sorokin,Maxim V. Staritsyn,Alexander S. Strekalovsky,Anna Tatarczak,Nikolay P. Tikhomirov,Dmitry V. Aron,Alexey A. Tret'yakov,Sergey Trofimov,Aleksey Ivanov,Yury Fettser,Tatiana S. Zarodnyuk,Alexander Yu. Gornov,Anton S. Anikin,Evgeniya A. Finkelstein,Elena S. Zasukhina,Sergey V. Zasukhin,Vitaly Zhadan,Anna V. Zykina,Olga N. Kaneva }} ==About Use of Methods of Convex Programming for Synthesis of Conformal Arrays with Matched Dual-polarized Patterns== https://ceur-ws.org/Vol-1987/paper37.pdf

About Use of Methods of Convex Programming
for Synthesis of Conformal Arrays
with Matched Dual-polarized Patterns

Viktor S. Izhutkin Anton A. Sharapov
Moscow Power Engineering Institute Moscow Power Engineering Institute
Krasnokazarmennaya Ulica 14, Krasnokazarmennaya Ulica 14,
111250 Moscow, Russia. 111250 Moscow, Russia.
izhutkin@yandex.ru deadkingser@gmail.com

Abstract
The solution of a task of the analysis and collecting polarizing infor-
mation can improve considerably possibilities of radars in various ap-
pendices, such as: detection, assessment and tracking of radar targets.
This task for a cage antenna lattice with the standard dual-polarized
patterns is formulated in terms of convex optimization. The possibili-
ties of the solution of an objective by means of a special Matlab CVX
toolbox and various classical algorithms of convex optimization are con-
sidered. Also comparison of the results of the solution of an optimizing
task received in the diﬀerent ways is presented.

1 Introduction
At present, the directions of using mathematical optimization methods in radar problems are actively developing.
Obtaining and analyzing polarization information in backscattered scattered waves can greatly improve the radar
capabilities in various applications such as: detection, evaluation and tracking of various objects. To capture
polarization information, the radar must measure two components of the orthogonal polarization of the target,
which is the so-called polarization diversity. Taking into account the diversity of polarizations, the synthesis of
a beam with the desired level of polarization and low side lobes is a new topic in studies for polarimetric radars
that has attracted much attention in recent years. In the solution of this problem, various approaches have been
applied in various works such as the iterative method and the method of least squares.
The idea of the above methods is to synthesize a beam with a high degree of polarization purity and low
lateral lobes, but in some situations, for example high-frequency polarimetric radar measurements, alignment of
beams between the double polarization directivity patterns is required. The authors of [Wanqiu Hu et al., 2015]
attempted to transform the task to the problem of convex programming for the synthesis of conformal arrays in
antenna arrays of arbitrary conﬁguration with matched dual-polarization beams. This approach proved to be
more eﬀective for solving the problem of the synthesis of conformal arrays, in comparison with those considered
in earlier works. In connection with these, the task arose, in more detail to open the topic of developing and
projecting software for the synthesis of conformal arrays for cylindrical type antennas.

Copyright ⃝
c by the paper’s authors. Copying permitted for private and academic purposes.
In: Yu. G. Evtushenko, M. Yu. Khachay, O. V. Khamisov, Yu. A. Kochetov, V.U. Malkova, M.A. Posypkin (eds.): Proceedings of
the OPTIMA-2017 Conference, Petrovac, Montenegro, 02-Oct-2017, published at http://ceur-ws.org

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2 Formulation
2.1 General Case
The formulation of the optimization problem is described in [Wanqiu Hu et al., 2015]. The object of research in
the problem of synthesis of conformal arrays is the functions of co- and crosspolarization:
[ ]
∑
M ϕ
Emϕ (θ, ϕ)
ϕ
E (θ, ϕ) = ωm exp(jkar · Rm )
ϕ
ϕ
m=1
Emθ (θ, ϕ)

∑
M [ θ
]
Emϕ (θ, ϕ)
E θ (θ, ϕ) = θ
ωm exp(jkar · Rm ) θ
Emθ (θ, ϕ)
m=1

Here ϕ and θ indicate the direction of the angle in the spherical coordinate system with respect to which the
polarization is measured. M is the total number of elements of the antenna array. ω ϕ and ω θ are complex vectors
whose elements denote the excitation of the corresponding elements of the antenna array. Both these quantities
ϕ ϕ θ θ
are the desired characteristics in the original problem. Emϕ , Emθ , Emϕ and Emθ is components the polarization
functions of the element with the number m. For diﬀerent types of antenna arrays, these functions will be slightly
diﬀerent. Similarly, the polarization functions of a single element depend on the type of antenna.

2.2 Cylindrical Antenna Array
The authors of the article set and solved the task of implementing software for the synthesis of conformal arrays
in the case of a cylindrical antenna array. By changing the parameters of the antenna array, we can obtain
solutions for all possible conﬁgurations. In the case of a cylindrical antenna array [Voskresensky, 2012], the
formulas take the following form:

∑
M ∑
N
ϕ
E ϕ (θ, ϕ) = ϕ
ωm,n exp(j · 2π(a · sin(θ) · cos(ϕ − ϕm ) − cos(θ) · zn )) · Em (θ, ϕ)
m=1 n=1

∑
M ∑
N
E θ (θ, ϕ) = θ
ωm,n exp(j · 2π(a · sin(θ) · cos(ϕ − ϕm ) − cos(θ) · zn )) · Em
θ (θ, ϕ)

m=1 n=1
ϕ θ
ω and ω we represent in the form of a matrix since the conﬁguration of a rectangular grid is natural for the
arrangement of elements on a cylindrical antenna array. a is the diameter of the base of the antenna cylinder.

ϕ,θ
Em (θ, ϕ) = E0ϕ,θ (θ, ϕ − ϕm )
 [ ]

 ϕ cos(ϕ)sin(θ)
E0 (θ, ϕ) = , |ϕ| < π/2
sin(ϕ)cos(θ)
[ ]

 0
 E0ϕ (θ, ϕ) = , |ϕ| ≥ π/2
0
 [ ]

 cos(ϕ)sin(θ)
E0 (θ, ϕ) =
θ , |ϕ| < π/2
[0]

 0
 E0θ (θ, ϕ) = , |ϕ| ≥ π/2
0
It is seen that the polarization of a single element vanishes at an angle of |ϕ| ≥ π/2, this is due to the fact that
a quenching winding is located in this antenna region.
ϕm we deﬁne in such a way that to center the main beam on ϕ = 0:

(m − M2+1 )
ϕm =
2a
n
zn =
2

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When solving the problem in our case, we introduce some additional conditions:

M E = Eϕϕ (θd , ϕd ) − Eθθ (θd , ϕd )

Thus, since there is no need to consider more than one point in the region of the fundamental beam, we take
L = 1. (θd , ϕd ) is the direction angle of the main beam. We also skip the step of calculating the parameter ς θ ,
because the polarization component Eϕθ ≡ 0. The parameter τ is deﬁned as τ = max(τ ϕ , τ θ ), the parameter ς
as ς ϕ . In the rest, the implementation of the method remains pre-empted. Areas of limiting the side lobes and
the level of cross-polarization are assumed to be equivalent:
[ ] [ ]
ΩP = ΩS = ((θ, ϕ) : θ = θd , ϕ ∈ −π; −∆w ∪ ∆w ; π )

Where ∆w user-deﬁned parameter.

3 Realization
When solving the problem, we had to work with complex-number vectors. The most convenient means for this
was the CVX toolbox system [CVX Users Guide, 2012]. CVX is a modeling system based on Matlab for convex
optimization. CVX turns Matlab into a modeling language, allowing you to deﬁne constraints and objective
functions using the standard syntax of Matlab expressions. In its mode, by default it supports a special approach
to convex optimization, called disciplined convex programming. In this approach, convex sets are constructed
from a small set of rules of convex analysis starting from the base library of convex functions. Constraints and
objective functions expressed through these rules are automatically converted into a canonical form and resolved.
Disciplined convex programming is a methodology for constructing convex optimization problems proposed by
Michael Grant, Stephen Boyd, and Yinyu Ye. It is meant to support the formulation and construction of
optimization problems that the user intends from the outset to be convex.
Disciplined convex programming imposes a set of conventions or rules, which we call the DCP ruleset. Problems
which adhere to the ruleset can be rapidly and automatically veriﬁed as convex and converted to solvable form.
Problems that violate the ruleset are rejected even when the problem is convex. That is not to say that such
problems cannot be solved using DCP; they just need to be rewritten in a way that conforms to the DCP ruleset.
A detailed description of the DCP ruleset is given in The DCP ruleset. It is extremely important for anyone who
intends to actively use CVX to understand it. The ruleset is simple to learn, and is drawn from basic principles of
convex analysis. In return for accepting the restrictions imposed by the ruleset, we obtain considerable beneﬁts,
such as automatic conversion of problems to solvable form, and full support for nondiﬀerentiable functions. In
practice, we have found that disciplined convex programs closely resemble their natural mathematical forms.
CVX solves the problems of convex programming using iterative methods. This system is ﬂexible and has the
ability to work with precision. To solve the task in the Matlab environment, it is necessary to implement the
following functional modules:
• Co-polarization and cross-polarization functions;

• Functions for determining and selecting optimal parameters τ and ς;

• Basic computational function;
• Module for output and registration of graphs of the results.

The procedures for calculating the constraint parameters are represented by four functions that accept the con-
ﬁguration data for the antenna array and return the optimal parameters found. In the main calculation module,
the parameters are chosen as a maximum among the values of the performance results of both computational
procedures relating to each of the parameters.
The main computational module is a fully automated procedure that extracts input data from a ﬁle and writes
the result of the work to another output ﬁle.

255
Figure 1: As a ﬁrst example, let us consider the results of the program for a 7x7 antenna array with a base
diameter of 4.0 and the direction of the main ray (θd , ϕd ) = (π 3.0), ∆w = π/9.

256
Figure 2: On the graph the squares of the modules are shown, the function Eϕϕ red line, Eθθ blue line for θ = π/3.
On the abscissa axis, the angle ϕ in radians. In the following example, the antenna array conﬁguration remains
predeﬁned, but ∆w = π/18.
With decreasing deltas, we see a narrowing of the main beam and an increase in the level of the side lobes.

257
Figure 3: The running time of the algorithm in both situations is the same and is about 4 minutes. The next
graph shows the result of the program with a large number of elements. Antenna grid consists of 625 (25x25)
elements a base diameter of 5.0 and the direction of the main ray (θd , ϕd ) = (π 3.0), ∆w = π/9.

258
4 Results
The application of methods of mathematical optimization to problems of synthesizing the polarization ﬂux
is becoming increasingly popular, in view of its eﬀectiveness and the quality of the results obtained. New
approaches to the formulation of classical radar problems in the format of problems of convex programming,
show their advantage in comparison with other methods.
In this paper, a software package for the synthesis of conformal arrays with two-polarized circuits in a cylin-
drical array is designed and implemented. The results of testing the program with a large number of elements
showed an acceptable speed of operation.
We demonstrated the operation of the system with various conﬁgurations of antenna arrays. On the graphs
of the results obtained, we can see that the radiation of the main polarization ﬂux reaches a maximum at the
angles given in the input data. The side lobes of polarization, entering the area of minimization, do not exceed
the established limits. It is also possible to observe good consistency of beam diagrams in the region of the main
ray, which was required in the initial formulation of the problem.

5 Conclusion
The software project was created based on the requirements of simplicity of modiﬁcation of the source code.
It should be noted that the software has been developed by the order of the All-Russian Scientiﬁc Research
Institute of Radio Engineering, where it is now actively used to solve the corresponding problems. A user with a
minimal knowledge of the use of the MATLAB system is able to write their own functions to work with diﬀerent
versions of antenna arrays. Also, the environment provides the ability to form dynamically linked libraries (dll)
from individual functional modules, for their further use in high-level languages.
Authors are grateful to M.V. Indenbom for deﬁnition of problem and useful consultations.

References
[Wanqiu Hu et al., 2015] Wanqiu Hu, Xuesong Wang, Yongzhen Li, and Shunping Xiao, (2015). Synthesis of
Conformal Arrays With Matched Dual-Polarized Patterns. IEEE Antennas and Wireless Propagation
Letters, 1341-1344. DOI: 10.1109/LAWP.2015.2508438
[Voskresensky, 2012] Ustroistva SVCh i antenny. Proektirovanie fazirovannykh antennykh reshetok [Microwave
devices and antennas. Designing of phased array.Charter 2.7], (2012), ed. by D.I. Voskresensky. Moscow,
Radiotekhnika Publ.,. 744 p.

[CVX Users Guide, 2012] CVX Users Guide, (2012)[Online]. Available: http://web.cvxr.com/cvx/doc/
[Grant et al., 2006] Grant, Michael, Stephen Boyd, and Yinyu Ye. “Disciplined convex programming.” Global
optimization. Springer, 2006. 155-210

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