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METHODS OF LEXICOGRAPHICAL OPTIMIZATION OF PROPERTIES OF AMBIGUITY FUNCTION OF COMPLEX SIGNALS WITH DESCRETE FREQUENCY MODULATION
Oleg I. Voronin,
senior researcher, Ph.D, Military Academy of Strategic Missile Forces of Peter the Great (VA Strategic Missile Forces of Peter the Great), Moscow, Russia
Valeriy I. Volokhov,
senior researcher, Ph.D, Military Academy of Strategic Missile Forces of Peter the Great (VA Strategic Missile Forces of Peter the Great), Moscow, Russia, val.volohov@yandex.ru
Galina A. Shepilova,
junior researcher, Military Academy of Strategic Missile Forces
of Peter the Great (VA Strategic Missile Forces of Peter the Great), Keyw0rds: DFM (discrete frequency
Moscow, Russia modulation), (AF) ambiguity function.
Для цитирования:
Воронин О.И., Волохов В.И., Шепилова Г.А. Методика лексикографической оптимизации свойств функции неопределенности сложных сигналов с дискретной частотной модуляцией // T-Comm: Телекоммуникации и транспорт. 2017. Том 11. №1. С. 62-66.
For citation:
Voronin O.I., Volokhov V.I., Shepilova G.A. (2017). Methods of lexicographical optimization of properties of ambiguity function of complex signals with descrete frequency modulation. T-Comm, vol. 11, no.1, pр. 62-66.
Development of complexes of communication systems is inseparably connected with appearing of new or updating the existing research and methodology instruments to determine the parameters of demolition objectives motion. One of development directions is a method of selection of optimal form of probing signal. Mathematical formula of radar signal is generally accepted as optimal if Woodward's ambiguity function (AF) of the given signal has a key-driven form.
Currently complex radiolocation signals with frequency, phase and combined modulation are commonly used to determine parameters of motion - speed and range. The given set of forms of signals is unlimited, thus the task of selection of optimal form of signal should be performed within the specific class of signals.
In the given article the method of optimization of AF properties of complex radio signals with DFM according to two sequentially used criteria is presented: minimizing the coefficient of mutual correlation of assessment errors of frequency and delay parameters; minimizing levels of AF side-lobes. Famous methods of B. Glazov and J.P. Costas are used as methods of bicriterial optimization. Methodology, based on sequential use of these methods allowed to obtain PE-2 signals, whose ambiguity function is of the desirable key-driven form. Complete sets of PE-2 signals for certain periods of numerical modulating sequences were found.
With the advent and development of communication systems the task of selection of optimal form of probing signals becomes more crucial. Among the most often used signals in practice are complex signals with frequency and phase modulation (PM). Resolution of complex PM signals delay is limited by elementary radio pulse duration (one discrete), that's why in order to reach higher resolution of delay it is reasonable to use signals with DFM [2J. Retransmitting jamming of these signals represents more difficult task, than the one of PM signals [2]. Spectrum of signals with DFM unlike signals with PM is even along F frequency band, therefore, the quality of frequency signal sorting with DFM is higher, and communication channel capacity will
Analysis of formula (2) shows that in its right part j} of terms appears, each of which maximum equals to —. Location
of maxima is characterized with difference (/ — k), as well as differences (AT - NM) and (jV. - Ni+k+l) - Terms with i = k, whose amount equals to L, determines the central part of AF when |r|<r0, |v|< —- All these terms reach maximum value
when r = 0, v — 0 and make up isolated height of the main AF peak. We note that all the rest peaks, situated in the area, limited
P
be maximum and equal to Cmax = 1,44—— [2]. In view of the by r] < r„, |v| < —, do not depend on the selection of the
above in the given article signals with DFM were selected for analysis and optimization purposes. The complex envelope of signals with DFM s(t) = S(t)eA^+Po\ t0 < t < T, [1-4] has a form of
m=0c/ -1-, >-!«-, (i >
Where L is a period of frequency modulating numerical sequence /Ay,- {Ni} - periodic sequence formed from 1,2,,,.,L
2 n
numbers; ti = irn; r0 =-i T0 ~ one symbol interval of
Act)
DFM signal; Aco = InAf - the shortest discrete of frequency; T — Lrtl - signal duration.
As an example DFM signal with amplitude an = 1, numerical sequence period L = 1, symbol interval r„ = 0,1 and sequence modulating the frequency {A^ }= {1,2,3,4,5,6,7} are taken with graphic representation in Figure 1.
modulating sequence {A'.} [2]. However, in the general case measurement errors of frequency and delay parameters of such signals (p:v ^ 0) correlate. Maxima of terms with i k may be placed only in the units when r is multiple of r(l. and when
v is multiple of J- . Their position in certain units depends only
on selection of modulating sequence }. It is shown in [I] that at the plane (r, v) all the maxima may be placed without mutual superposition. In that case AF has side-lobes with a minimum
possible level, equal to — [I, 3]. As an example, we visualize
CAF of the signal with DFM, formed by ramp sequence {A^ {1,2,3,4,5,6,7} in Figure 2
R№)>
V^ibo
jSOQ
Fig. 1. View of DFM signal when a0 = 1, L = 7, ru = 0,01, {N,} = {1,2,3,4,5,6,7}
Bivariate correlation function of signals with DFM [2] has a form of
„ L L sin[Affi<N, ~Nlyt)+2ttv] r°—^ _ ^
2 v -i v ', L v ' 1 ■) /[anW-IW+lm-Itfn,—
w r,F) = -> > {-e 1 -
¿trtT AclKNl-Ni+t)+2xv
(2)
&&>(N, - Nl+k+l) + 2/rv where e - r-(i-A)r0, 0 <£< t0.
-OQi D 0.05
a)
0
^ :. \\ "
m -
-0.004 -0 002
0.002 0.00*
6)
Fig, 2. View of CAF surface of the signal with DFM when {,¥,} = {1,2,3,4,5,6,7} (a) and its applicative sections (6)
ELECTRONICS. RADIO ENGINEERING
In the context of the analysis of Fig. 2 the surface of AF of the examined signal has relatively high side-lobes significantly
exceeding value —. Visual analysis of Fig. 25 shows that ap-L2
plicative sections ofCAF of the given signal are not symmetric against the axis of reference r h v. It tells about the presence of correlation dependency (coefficientp~iy ^0) between estimate
errors of frequency and delay.
Correlation properties of signals with DFM mainly depend on selection of sequence structure fjV,} modulating the frequency.
Currently there are two approaches to properties optimization of CAF signals with DFM, connected with validation of proper sequence structures {jV,-}.
First approach, offered by professor B, Glazov [2] is connected with full minimization of coefficient of mutual correlation of estimate errors of frequency and delay )mjM =0). In
order to erase this correlation and to use energy of DFM signals in optimal receiver more completely it is necessary that the second mixed derivative of two-dimensional correlation function y/(v,v) equalled zero, which is possible when the following equality holds
¿r„
¡tf{t)dt = 0.
(3)
-L r.i
The use of the given criterion to DFM signals, formed by periodic numerical sequences {N{ } of period L let us synthesize
the sequences {jVJ, imparting the property of independence of
estimate errors of frequency and delay to DFM signals. The main property of these sequences is the following. In order to provide the independence of estimates of frequency and delay of signals with DFM, formed by periodic sequences {/V, }, il is necessary
and enough that the number search order in the sequence |,Yy}
satisfied the Diophantine equation
L(L + \)2
(4)
The equation (4) has one trivial solution: jV. - NI+l_r i.e. when N-(t) even time function, however, the use of such sequences only significantly limits possible DFM signal ensembles. That's why solutions (4) - unlike trivial ones, but the ones describing the structure of periodical sequences called
pseudoeven (PE) [2], as they impart the property of independency of frequency and delay measurement, typical for even functions modulating the frequency, to the DFM signal - are of main practical interest. PE sequences can be found by checking the
implementation of parity condition /.(¿ + 1)" for each
Zj i *
¡al
sequence {/V;} of total amount of L). sequences for the given L . As an example in Fig. 3 we visualize CAF signal, formed by PE sequence {/V, } = {5,1,4,7,3,6,2}.
¥
to-
-0.0 J
a)
■10-
H'
¡am o.503 0.801 o.;o2.
0.305 n era 0 901
V
0.106
< V \ \ X 0.106 \ \ \\ \ 0.205
0.305
0.901 0.603 0.205 0.801 Û--503
0.106
-1-
-0.001
0.106"
-1—
0.001
6)
Fig. 3. View of the surface of PE CAF signal when {Ar(} st {1,7,3,5,6,4,2} (a) and its applicative sections (6)
The analysis of Fig. 35 shows, that high sections of PE CAF
signal when |y/(r, v)j" > 0,5 have forms of ellipses, whose axes
coincide with the axes of reference r and v. This ts a geometric
sign of condition (/c£)min =0 fulfillment, determining a linear
dependency of small errors of frequency and delay measurement in optimal receivers. Fig. 3a shows that single lateral peaks of
AF of the given signal exceed the minimum possible level —.
L1
It indicates the necessity of further optimization of the structures of modulating sequences {iV,}-
Second approach to optimization of properties of CAF signals with DFM, offered by J. Costas [3]. is connected to minimization of CAF side-lobes levels (|y/(r,v)|"". —> min ■
Maxima of terms of two-dimensional correlation function y/{r,v) of Costas signals with i ^ k are placed on the surface (r,v) without mutual superposition, and AF has side-lobes with
equal minimum possible height (j\y/(T,vf6, _)_,)„„ = ~
evenly distributed on the surface On'. Sequences {//,}, by which the given signals are formed - are called [3] Costas arrays. Belonging of the sequence } to Costas arrays is determined on the base of construction and analysis of difference matrix A=|||tt||=«|jVi-Arjt| i,k = \X [3. 4]. Necessary and sufficient condition of existence of optimal sequence {/V.} according
to Costas is that each line of numerical triangle T should not contain equal difference. In Fig. 4 CAF of the signal formed by the given Costas sequence 1^,} = {1,3.6,2,7,4,5} is visualized.
i-
0.5-
-0.05 o o o;
a)
6)
Fig. 4. View of CAF surface of the Costas signal when {A^ I = {1,3,6,2,7,4,5} (a) and its applicative sections (6)
ELECTRONICS. RADIO ENGINEERING
seen in Fig. 43. It tells about the linear dependence of errors of frequency and delay measurement in optimal receivers
As a result of analysis of the studied methods the question arose: if such sequences {A'y}, which were pseudoeven and and
related to Costas arrays at the same time - exist. If the given sequences exist, their search is of peculiar practical interest. In accordance with the given question the bicriterial task of lexicographical optimization of CAF signals with DFM is formulated.
The solution of this task is the selection of signal S" (0 DFM signal form at the given period L with optimal sequence {;V,} by two sequentially applied criteria: {p~TV)min = 0
— equation of coefficient of mutual correlation of frequency and delay estimate errors and minimal level of AF side-lobes
1 a 1
((w(t,v)\ ) ) = —- to zero.
V * 'mm /mm
Methods of optimization of properties of CAF signals with DFM reduce to the search of such sequences {A';} , that AK of
signals, fonned on their base, would satisfy both specified criteria. It consist of three sequently completed steps.
The first step is to select the period of sequence L and to form the sequences |Ar1- }. determining the order of enumerating
the frequencies in DFM signal (of all ¿!).
The second step is to search for PE sequences by enumerative method: fulfillment of pseudoeven condition l LiL +1)2
y^ i A'. =—---— is checked for each further «-type sem 4
quence {a^ } («<£!). The result of the given step is a multitude of the ascertained PE sequences jjA^ }*, {A', }*{.'V, J" j
- optinial for the first criterion of the given task, M — power of the multitude equal to the total amount of certain PE sequences.
The third step is to search for Costas sequences among the PE sequences, found in the previous step, by simple enumerative method. Each further /«-type sequence j/V} (m < M) is
checked for its belonging to J. Costas arrays on the base of construction and analysis of properties of the numerical triangle T. If each of its lines has no repeating numbers, the further sequence {a^ } is the Costas array. The result of the given step
and solution of the given bicriterial task is, in general, the pareto-
optimal multitude ^ ' 'i ' ^ Z —is a power of
the multitude, equal to the total amount of certain sequences, which was determined to call LPE-2 sequences'. The main practical result of use of the developed methods are ascertained PE-2 sequences for L=7, L=8, 1=9 periods. Generalized quantitative results arc presented in Table 1.
ЭЛЕКТРОНИКА. РАДИОТЕХНИКА
According to Fig. 6a we may conclude that the view of CAF surface tend to the desired key-driven form.
Visual analysis of Fig. 66 lets us talk of absence of correlation of estimate errors of frequency and delay, as high values of CAF has an even symmetry against the axes of reference T w v. Fig. 6a illustrates the minimal level of CAF side-lobes of the
similar height-L.
1}
Thus, the methodology of optimization of properties of AF signals with DFM, letting us synthesize PE-2 signals with the best correlation properties, is studied in the given article. It appears that the given signals are appropriate to place into software of high-precision weapon complexes.
References
1. Varakin L. (1970), Theory of complex signals. Moscow: Sov.radio. 376 p.
2. Glazov B. (1972). Numerical periodic sequences forming noise-type signals with frequency modulation. Radiotekhnika, vol. 27, no.3.
3. Costas J.P. (1984). Properties of signals with almost perfect ambiguity function in 'range-doppler frequency' coordinates. TUER. No.S.
4. Varakin L. ( 1985). System of communication with noise-type signals. Moscow: Radio i svyaz. 384 p.
5. Pestryakov V., Afanas'ev V., Gurvits V. (1973/ Shumopodobnye signaly v sistemakh peredahi informatsii. Moscow: Sov.radio. 424 p.
6. Tuzov G.I., Sivov V.A., Prytkov V.l. etc. (1985). Pomekhozash-chishchennost' radiosistem so slozhnymi signal ami. Moscow: Radio i svyaz'. 264 p.
МЕТОДИКА ЛЕКСИКОГРАФИЧЕСКОЙ ОПТИМИЗАЦИИ СВОЙСТВ ФУНКЦИИ НЕОПРЕДЕЛЕННОСТИ СЛОЖНЫХ
СИГНАЛОВ С ДИСКРЕТНОЙ ЧАСТОТНОЙ МОДУЛЯЦИЕЙ
Воронин Олег Игоревич, ВА РВСН им. Петра Великого, старший научный сотрудник, к.т.н., Москва, Россия Волохов Валерий Иванович, ВА РВСН им. Петра Великого, старший научный сотрудник, к.в.н., Москва, Россия Шепилова Галина Александровна, ВА РВСН им. Петра Великого, младший научный сотрудник, Москва, Россия
Аннотация. Развитие комплексов систем связи неразрывно связано с появлением нового или совершенствованием существующего научно-методического аппарата для процессов определения параметров движения объектов поражения. Одним из направлений развития являются методики выбора оптимальной формы зондирующего сигнала. Математическую форму радиолокационного сигнала принято считать оптимальной, если функция неопределенности Ф. Вудворда (ФН) данного сигнала имеет кнопочный вид. В настоящее время в практике определения параметров движения объекта - скорости и дальности используются сложные радиолокационные сигналы с частотной, фазовой и комбинированной модуляцией. Данное множество форм сигналов является бесконечным, поэтому задачу выбора оптимальной формы сигналов целесообразно проводить внутри конкретного класса сигналов. Изложена методика оптимизации свойств ФН сложных радиолокационных сигналов с ДЧМ по двум последовательно применяемым критериям: минимизации коэффициента взаимной корреляции погрешностей оценок параметров частоты и запаздывания; минимизации уровней боковых лепестков ФН. В качестве методов двухкритериальной оптимизации используются известные методы Б. Глазова и Дж. П. Костаса. Методика, основанная на последовательном применении этих методов позволила получить ПЧ-2 сигналы, функция неопределенности которых имеет желаемый кнопочный вид. Для отдельных периодов числовых модулирующих последовательностей были найдены полные наборы ПЧ-2 сигналов.
Ключевые слова: ДЧМ (дискретная частотная модуляция), ФН (функция неопределенности). Литература
1. Варакин Л.Е. Теория сложных сигналов. М.: Сов. радио, 1970. 376 с.
2. Глазов Б.И. Числовые периодические последовательности для формирования шумоподобных сигналов с частотной модуляцией // Радиотехника. 1972. Т. 27. №3.
3. Костас Дж.П. Свойства сигналов с почти идеальной функцией неопределенности в координатах "дальность-доплеровская частота" // ТИИЭР. 1984. № 8.
4. Варакин Л.Е. Системы связи с шумоподобными сигналами. М.: Радио и связь, 1985. 384 с.
5. Пестряков В., Афанасьев В., Гурвиц В. Шумоподобные сигналы в системах передачи информации. М.: Сов.Радио, 1973. 424 с.
6. Тузов Г.И., Сивов В.А., Прытков В.И. и др. Помехозащищенность радиосистем со сложными сигналами / Под ред. Г.И.Тузова.. М.: Радио и связь, 1985. 264 с.
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