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ELECTRONICS. RADIO ENGINEERING
APPLICATION OF TRANSFORMATIONS IN GALOIS FIELDS TO FAST ACQUISITION OF PN SEQUENCES ENSEMBLES
Vladimir Y. Mikhaylov,
Moscow Aviation Institute (National Research University), Moscow, Russia, mihvj@mai.ru
Roman B. Mazepa,
Moscow Aviation Institute (National Research University), Moscow, Russia, mrb402@mai.ru
Keywords: asynchronous-address systems, command-measuring systems; structural stealth, fast acquisition, quasi-orthogonal signal ensembles, Galois fields, fast conversions.
Object of consideration - asynchronous-address, command-measuring systems, using code division of channels. Subject of the analysis - the quasi-orthogonal signals ensembles providing reliable division of channels at influence of imitation interference and delay acquisition devices. The analysis purpose - optimization of procedures and delay acquisition devices capable to adapt to change the structure of signals constituting ensembles. The solution is based on the principles and mathematical models developed by the authors. There are the principle of fast conversions in Galois fields; algebraic representation model of considered subclass of code sequences and corresponding delay acquisition devices structure; the principle of increasing the structural secrecy of radio systems under consideration by applying of isomorphic ensembles sets. A delay acquisition problem of broadband signals based on code sequences is considered. The problem arises when using long code sequences in conditions of limited signal observation time and when high requirements to reliability and synchronization accuracy are exists. In these conditions use of various quazi-optimal methods and search algorithms, in particular, parallel and sequential-parallel search, consecutive algorithms of decision based on a sequence segment, separate processing of accepted combined sequences components is limited because of decrease in energy efficiency or because of system implementation complexity. The problem becomes significantly more complicated when it is necessary to enter into synchronism with the set (ensemble) of quasi-orthogonal signals under the action of signal-like interference.
This paper is devoted to the construction of a processing method and a search device model based on fast transformations that are close in energy efficiency to the optimal scheme and capable of promptly adapting to a change in the structure of the signal entering the ensemble. Key features of the described method of processing are implementation of decision based on entire received sequence in two stages and accumulation of symbols from multiple copies of short sequences algebraically contained within the source coding sequence. A two-stage processing and the fast conversions are the principles to provide a wide variety of implementations of the search scheme that can be used to provide a compromise between speed and complexity of search, how this is achieved in the known parallel and sliding schemes. These principles are applicable for processing a wide class of code sequences based on maximal length codes. These include, in particular, Gold codes and Gordon-Mills-Welch (GMW) codes and other, more complex encoding constructions. It was shown that the proposed model of the search device for a particular subclass of signals provides a high level of adaptation to the signal structure change. In the proposed version of the search scheme, there are no memory devices for storing the received implementation during the detection phase, which is usually required when performing fast transformations by other methods. In addition, in contrast to the conventional scheme of the correlation receiver, the capacity of accumulators (reversing counters) decreased to half. Based on the essential properties analysis of fast transformations in Galois fields, the prospects for an additional reduction in the time spent on delay search for complex two- and three-component constructions based on codes of maximum length are defined. The results will be useful to specialists engaged in the design of complex coded signals and their processing devices to ensure structural stealth and protection against interference.
Information about authors:
Vladimir Y. Mikhaylov, Professor, Dr. Sc. (Tech.), Moscow Aviation Institute (National Research University), Moscow, Russia Roman B. Mazepa, Head the Department, Ph. D., Moscow Aviation Institute (National Research University), Moscow, Russia
Для цитирования:
Михайлов В.Ю., Мазепа Р.Б. Применение преобразований в полях Галуа для быстрого поиска по задержке ансамблей квазиортогональных кодовых последовательностей // T-Comm: Телекоммуникации и транспорт. 2017. Том 11. №12. С. 64-70.
For citation:
Mikhaylov V.Yu., Mazepa R.B. (2017). Application of transformations in Galois fields to fast acquisition of PN sequences ensembles. T-Comm, vol. 11, no.12, рр. 64-70.
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Introduction
To achieve high reliability of information delivery ihe comma nil-measuring, asynchronously-address information transmission systems require to create and to use large ensembles of high quality quasi-Orthogonal code sequences. The class of information and telecommunication systems, including command-measuring, asynchronously-address systems of information transmission with code division of channels, developing intensively, are the most demanding to the choice of signals ensembles 11 ]. They must be complex (have a large signal base) and at the same time capable of fast and energy efficient processing in the absence of dedicated synchronization channels.
The modem conditions for the functioning of the systems of the classes in question dictate additional stringent requirements to ere ale the signals ensembles having high structural redundancy, which makes it possible to parry the effect of signal-like interference. All this stimulates the development or improving of methods, algorithms and devices for efficient processing of code sequences with large length. A lot of work is devoted to different aspects of solving problems in this direction. Note the paper [2[ in which a fast Walsh transform method was proposed to search for M-sequenccs by delay. Methods that use the adaptive threshold [3, 4], methods using composite (combined) sequences [5, 6], various variants of implementing a two-stage, parallel-sequential search are also useful [7-11]. In |12], the principle of fast transformations in Galois fields was described. Then it was used as the basis of the theoretical justification of the method to fast acquisition of a subclass of M-sequences. The maximum gain in search time is achieved by using M-sequences generated by shift registers with an even number of digits n - 2p. In [13] approach to performance estimation of considered search device is presented. The paper [14] was devoted to the problems of constructing of quasi-orthogonal sequences ensembles having high structural redundancy.
Formulation of the problem
The main elements ensuring the feasibility and efficiency of the method described in [12] arc the Demultiplexer and the control circuit in the fonn of a memory device (RAM) in Fig. 1 [12]. These elements perform synchronization of short sequences samples for the purpose of their accumulation in Accumulators. The logic (algorithm) of these dev ices operation found in [12] is subject to the rules for mapping of the sequence symbols to abstract elements of the Galois field and its subfields. These rules are such that all possible sequences derived by a given Galois field are mapped to the same elements of the Galois subfield, but in a different order. It is this property that is defined as an isomorphism of the representation of the Galois fields, and, consequently, of the M-sequences generated by it. It follows that in order to delay acquisition of M-sequences with other structure in the general case, one needs to have its own logic for synchronizing short-sequence samples with the aim of accumulating them in accumulators. Of course, it can be found, as it was done in [12]. The purpose of this paper is to substantiate the existence and find of M-sequences subset for which the logic of short sequences synchronization does not change, or is quickly modified, and therefore the general structure of fast delay acquisition can be applied to a whole ensemble of sequences.
This paper develops the obtained results in the direction of increasing the efficiency of the use of isomorphic sets of quasiorthogonal code sequences ensembles.
Structure and options for fast transformations performing
As shown in [11 ], the key to applying fast transformations to a sequence with a length of N = 2r" -1 symbols is equation
n^'^'^l^ir.....(1)
where 7"(.v) - conventional binary trace function of element x e GF(22p);
S(.v) - multiplicative form of binary trace function; Tp(y) -binary trace function of element y £ GF(2P);
2r + 1 - step of sampling (decimation parameter); i = (2'' +1)/ + j - integer argument of the original sequence;
1 = 0,2'' -2 - integer argument of the short sequence; j = 0, 2'' - integer sampling parameter; t = (2P + l)fd +1"| — integer total delay of the original sequence;
r„ - short sequence shift component; r, - sample offset component;
/3 = , (k, N) = 1; a - primitive elements of base Galois field GF( 21');
y = K" —primitive elements of Galois subfield GF{ 2''); y^f+p" - (2)
- the transform function to convert the element ¡3'~ of the original field GF(22'') into an element y. of the subfield GF(2'').
It immediately follows from relation (!) that the result of this representation can be considered and used as a set of short sequences of integer argument i with length of N = 2'' -1 symbols and the same structure (the sequences are generated by the same polynomial) having a common unknown shift in the form of a component r„ and various shifts, depending from component rt, short sequence number (integer sampling parameter) /, and the structure of the original sequence:
In this case, it is convenient to represent the initial sequence and the transformation method in the form of a matrix of dimensionality (2P-1)x(2'' +1) (Fig. 1),
x 0 1 1'-2
G #.+/60) «(t + r. + zM u(2'-2 + t„ + /(t,))
I w(r„ + /(l + r,» "(i+r^/a + r,)) u(2'-2+r„+/(1 + r|))
V * 1 - r. 0 0 D
2* "(r„ + /<2' + rj) H(l+rj+/(2'+r,)) «(2'-2+r„+/<2' + r,))
Fig, 1. Matrix representation of the original sequence
in which the columns represent blocks of 2'' +1 consecutive symbols of the original sequence, and all the lines, except one -short sequences. This single sequence consists of all symbols 0 and is a mapping of the samples taken at times multiples 2'' +1 to the zero element of the subfield. This follows from relation (2) for z sO mod{2/' +1). The first two parameters of the argument of the sequences - r„ and T, - elements of the desired delay, the third parameter-J sample parameter, and the last - /(v) - composite algebraic function, depending not only from structure of the initial Galois field, but also from option of its specific representation as the order of the elements listing. It is obvious that for the fixed Galois field the values of function f{y) can be found by means of transformation function (2) and the method of their calculation becomes the main, but not the only problem. Exception is the argument y = 0 mod(2'' +1), at which value /(>') it is not defined, and u(i + T0 + /00) = 0 at any i. For the given Galois field /(v) values can be found rather easily algorithmically - it is much more difficult to define the general solution way and to offer options of effective hardware-software implementation. The first task along this path is to find the way (algorithm) of mutual synchronization of the short sequences set, i.e. their reduction to such look that in each column of the matrix there were the same named symbols. It is logical to consider that synchronization has to be reached at r, = 0 .
The easiest way to do this is to shift the short sequences (rows of a matrix) by the sizes compensating values/(/). However, this requires that you first lake the whole sequence, writing it into memory. В таком виде модель представлена па рис.2. Another way is to distribute the received digitized symbols to the accumulative "pockets" corresponding to the numbers of the short sequence symbols, for each current sample with the number j. Advantage of such synchronization is that after acceptance of one period of the initial sequence all "poekets" will be filled with the same named short sequence symbols.
This way was offered in [9]. In 112] the model version of the device realizing the specified way was presented. Conditionally this model version consists of two sequentially connected "decoders", first of which searches for the original sequence for the component r,, and the second - for r„. The implementation of both decoders generally depends on structure of the initial sequence and option of fast transformations performance. In this form, the model is shown in Fig. 2.
In this figures the following blocks according to their numbers are presented.
1. (Control scheme) - the scheme providing synchronous accumulation of the short sequence symbols at r, = 0 . in more detail the scheme is considered in Fig. 3,
2. (Demultiplexer) - scheme that performs the distribution of the received symbols over the accumulated "pockets" - channels. The scheme works under control of the scheme 1 and its logic does not depend on structure of the initial sequence and the transformation method.
Decoder I
,v(í + r)¡
2"-2
2'-]
Dccodcr 2
"3 2
С и
■3.5 £ M
с 3 .2 §
Lh
■I §
tí о я
Control schcmc
search on i]
search
i" != (2^ + !)r„ +r.
Fig, 2. Search device model in the form of two consecutive decoders
3. (Accumulators block), which can be used as the reversible counters of bit capacity p, accumulating the same named short sequence symbols for r, = 0. The operation logic of this block also does not depend on structure of the initial sequence and the method of transformation. Let's note that the result of accumulation does not depend also on existence of synchronization on the component r0 of full delay that allows to divide search into two stages,
4. (The decision-making block) - the unit providing management of search for delay. Its logic, obviously, does not depend on structure of the original sequence and transformation method. This device is critical and problematic in the sense that its performance affects the efficiency of the entire device as a whole. In particular, it can be performed as close as possible to the optimal parallel search scheme [15], making a decision only after completing the entire search for r, by storing the locally maximum level of accumulated energy. Alternative option is implementation of the so-called two-stage search method [7-10] which feature is preservation of the most perspective conditions of consecutive search for further final verification. The idea of option of such search based oil fast transformations in Galois fields is described in [9,10].
5. (The fast correlation processing unit) - the block providing acquisition of the full delay component r0. The logic of this block operation generally depends on structure of the original sequence and the chosen transformation method. It can be performed according to any known scheme including by matched filtering method. In this ease the device becomes searcliless. Since the length of short sequences significantly (approximately in Viv) smaller than the length of the original sequence, the matched filtering gains interest. Anyway, as the result of accumulation in the block 3 is stored, decoding of the short sequence can be performed much faster, than real-time processing.
In Fig. 3 discloses the structure of the control circuit, an essential element of the search device that provides synchronous accumulation of short sequence symbols.
©
search on r,
i
beal number
3
J
Zr + ]
/—s
i' © 1 —►
2r +1
J =J+n
/ =0mod(2'+l}
L j * 0 mod(2i> + i)
n
©
/+/</) mod(2*-t) ► l
x
a
~3 E
Fig. 3. Structure of control scheme
The block 1, calculates sample parameter./ - the shift of the current symbol in the bloek with number i calculated by the block 1The block 13 calculates transformation function value /(/} taking into account the current value r, of the full delay component. This value together with the block number i is used by the block 14 to determine the channel number of the Accumulators block 4 for accumulation of the current short sequence symbol. In a particular case. Block I3 selects the zero channel number. As you can see, the structure of the control circuit fully corresponds to the representation of the original sequence in Fig. I. In [12] complete mathematical justification of the possibility of synchronization and method to create the transformation f( j) providing accumulation of the short sequence symbols was given. In the same place the proof of the correctness of transformation in case of lack of synchronization is provided, i.e. at r, ^ 0. In other words, it was proved the existence of such a synchronization method when, in the absence of noise and interfering signals, a accumulation effect in each channel of block 3, matches w ith the autocorrelation function of the short sequence. The channel numbered 0 is similar to the other channels, but in it, if there is synchronization, an unsigned symbol always accumulates. So, in this sense transformation /(./) is optimum. In Table I the example of the control scheme logic for distribution of received symbols over accumulative "pockets" - channels (see Fig. 1) for original sequence w itli length of N ~ 2 — 1 (parameterp - 5) is given.
Table 1
Example of transformation function /( /) for original sequcncc with length of N = 2'° -1
Sample tin mbery /(/) values Offset inside the block with length of 2P +1 Channel number
1 1 1 0
2 32 32 0
3 58 25 !
4 77 11 2
5 79 13 2
6 85 19 2
7 102 3 3
8 109 10 3
Sample number/ fU) values Offset inside the block with length of 2^ + 1 Channel number
9 177 12 5
10 195 30 5
11 205 7 6
12 266 2 8
13 328 31 9
14 339 9 10
15 380 17 11
16 418 22 12
17 419 23 12
18 422 26 12
19 434 5 13
20 468 6 14
21 477 15 14
22 482 20 14
23 549 21 16
24 589 28 17
25 618 24 18
26 654 27 19
27 674 14 20
28 796 4 24
29 833 8 25
30 907 16 27
31 920 29 27
32 942 18 28
Justification of existence of the M-sequences subset invariant
In relation to the fast transformations in Galois fields
Following the purpose of this paper, the main conclusion of the previous section is the list of the search device model blocks (Fig. 2) which operation logic generally depends on structure of the original sequence: these are blocks 1 and 5. The purpose of further analysis is the mathematical justification of the possibility of reducing the dependence of blocks 1 and 5 on the structure of the original code sequence and identifying the necessary conditions for this.
The representation of the search device model in the form of two consecutive decoders obtained above allows to offer (still hypothetically) two separate research directions related to operation logic of the specified blocks. It is obvious that completely it is impossible to achieve the desirable goal as it would lead to destruction of initial premises about use of detailed properties of Gaiois fields.
Let's consider at first the possibilities of unifying the logic for constructing block 1 (Fig. 2) - the control scheme.
Justification of existence of the code sequences subset, invariant in relation to the Decoder 1 operations
As the basis of justification we use the method to create the sets of transformations f(j), obtained in |12J. For the purposes of this section, any set can be used, including the one used in the construction of Table ! and given below. Let's consider element p' at j - I, i.e. p as the basic element to create the transformation, It is the most natural (though not only) crcation option as corresponds to the first nondegenerate sample of the short sequence under the condition r, = 0 . Let's designate the set of all
values f(j) as J' = {f0, j',,.,, /'* }, in which elements arc defined from algebraic relations: P'" = P (j'o = 0; pJ' =l + p;
Pi[ = y + P',
where y^p2** e GF(2'').
Having in mind isomorphism property of Galois fields and its role in creation of isomorphic ensembles [14], we will put at first k -1. Let's designate the set of all values /( /), got for the basic
clement /i = a*=cr as J*. Now choose k * 1;(k,N) = 1 such that k'e Ji . We further take the element ft = ak as the reference to obtain the values set Jj of transformation functions fk(j) ■ To obtain the desired result it is necessary that sets J, , J] contain the same elements, i.e. J, =Jj. In order to estimate the possibility to meet this condition at least for one value k & 1;(&,N) = 1, it must be kept in mind that property of isomorphism of the Galois field leads to other order of its elements listing according to the chosen value k, and, therefore, and to other symbols order of the corresponding M-sequence. On the other hand, the logic of the block 1 (Fig. 2) - the control scheme - performs sampling of the sequence having another structure in the invariable look. At the same time the algebraic relations defining structure of sets J", J j. look as follows;
So, it is proved that the block 1 {Fig. 2) - the control scheme -for the code sequences, different on structure, must have different logic. However, having the set J" , il is possible easily obtain the
a'" = a\
ah = 1 + a; a h +
ak>" =ak-akj' =l + cr*; a*" =y* + a*;
correct set J, :
a'" = a;
a" =1 + gt; ah =y + a\
ß'" =ß'\ ßf' =! + /?';
ß"'-=y' + ß'\
As sets by definition completely match, for any s conditions j] = j* (mod N) have to be satisfied. Considering pairs of such comparisons, finally we will come to the ratio j] = j* k (mod ,V), what is impossible in view of restriction (k,N) = 1.
I lere I, k correspond to the condition Ik = l(mod N). This conclusion in program (algorithmic) implementation allows to quickly rebuild the block 1 operation logic (Fig. 2). having adjusted the Decodcr 1 for work with other code sequence.
Let's now consider the possibility of the block 5 (Fig. 2) construction unification - the block of fast correlation processing of the short sequence.
Justification of existence of the code sequences subset, invariant in relation to the Decoder 2 operations
Unlike the previous section, the existence of the required code sequences in the proof does not need, since the transformation of long sequences into short sequences is not unique. Set of numbers k\ (k,N) = \ having desired property, was used in [16-18] for analytical estimation of cross-correlation functions of the M-sequences. The basis to find the desired set is the relation (1}. 1 laving substituted in il expression for the element p = ak, wc will get
where y = a1,12'. Suppose that the logic of the Decoder 2 is constructed according to this relation and it is required to provide its use for decoding of the sequence specified by value k - 1, i.e. the sequence
For this purpose will be required performance of following condition
k(2p +1) = (2P + 1) modp" -1), or
k = 1 (mod (2P -1)). (3)
It remains now to find the subset of numbers k:(k,N) = ], having at the same time property (3) and ihe ability to create quasi-orthogonal ensemble together with other code sequences of the same subset. Problems to create the quasi-orthogonal ensembles including having structural redundancy are in detail considered in [14, 19]. Using (3), the example of the sequence with length JV = 2'°—1 symbols from the previous section, as well as results of searching for the best ensembles by the technique given in [14, 19] for the allowable level of cross-correlation au„ = 3.0, we
will receive the following greatest possible set ofthc code sequences having invarianey in relation to Decoder 2 logic in Fig. 2: k e {1,35, 95,101,157} .
Certainly, in according with the results [14, 18], it is possible to construct rather large number of isomorphic sets of such ensembles. Note that the power of the best ensemble with the characteristic of allowable level of cross-correlation alltl = 3.0 for
N = 2"' -1 it is equal to 10. Thus the requirement to adapt the Decoder 2 to change of code sequence structure halves the ensemble power.
So, the only way to unify structure of the search device mode! in Fig. 2 in relation to processing of all sequences of ensemble is the way to unify the structure of the Decoder 2.
Conclusion
Application of fast transformations in Galois fields of the type considered allows to significantly simplify and accelerate the delay acquisition of the whole ensemble of the M-sequences. Properties of the code sequences ensembles having such abilities are found. At the same time complexity of the search device, which is usually determined by length of the code sequence N, also significantly decreases. Options of key components of search devices model implementation, and also examples of specific algorithms parameters and the code sequences ensembles are given. The specified advantages allow to recommend the method for application in the asynchronous-address and command-measuring radio systems using the complex coded signals with large length.
The described fast transformations are potentially applicable to the compound code sequences w ith large and superlarge length where as components the M-sequences, in particular, two-component {Gold codes) and to three-component codes [20-22] are used.
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ПРИМЕНЕНИЕ ПРЕОБРАЗОВАНИЙ В ПОЛЯХ ГАЛУА ДЛЯ БЫСТРОГО ПОИСКА ПО ЗАДЕРЖКЕ АНСАМБЛЕЙ КВАЗИОРТОГОНАЛЬНЫХ КОДОВЫХ ПОСЛЕДОВАТЕЛЬНОСТЕЙ
Михайлов Владимир Юрьевич, Московский авиационный институт (Национальный исследовательский университет), Москва, Россия, mihvj@mai.ru Мазепа Роман Богданович, Московский авиационный институт (Национальный исследовательский университет), Москва, Россия, mrb402@mai.ru
Aннотация
Объект рассмотрения - асинхронно-адресные, командно-измерительные системы, использующие кодовое разделение абонентов. Предмет анализа - квазиортогональные ансамбли сигналов, обеспечивающие разделение абонентов при воздействии имитационных и сигналоподобных помех, устройства их обработки. Цель анализа - оптимизация процедур и устройств поиска по задержке кодовых последовательностей, способных адаптироваться к изменению структуры сигналов, составляющих ансамбли. Решение базируется на принципах и математических моделях, разработанных авторами: принципе быстрых преобразований в полях Галуа, алгебраической модели представления и структуре устройств быстрого поиска по задержке рассматриваемого подкласса кодовых последовательностей, способе повышения структурной скрытности рассматриваемых систем путем применения множеств изоморфных ансамблей. Рассмотрена задача поиска по задержке широкополосных сигналов на основе кодовых последовательностей. Проблема возникает при использовании кодовых последовательностей большой длины в условиях ограниченного времени наблюдения сигнала и высоких требованиях к надежности и точности синхронизации. В этих условиях применение разнообразных квазиоптимальных методов и алгоритмов поиска, в частности, параллельного и последовательно-параллельного поиска, последовательных алгоритмов принятия решений по сегменту последовательности, раздельной обработки компонентов принимаемых комбинированных последовательностей ограничено из-за снижения энергетической эффективности, или усложнения реализации систем. Проблема существенно усложняется при необходимости вхождения в синхронизм с множеством (ансамблем) квазиортогональных сигналов в условиях воздействия сигналоподобных помех. Статья посвящена построению метода обработки и модели устройства поиска на основе быстрых преобразований, близкого по энергоэффективности к оптимальной схеме и способного оперативно адаптироваться к изменению структуры сигнала, входящего в ансамбль. Ключевыми особенностями описанного метода обработки являются реализация принципа приема "в целом" в два этапа и накопление символов множества копий короткой последовательности, алгебраически составляющих исходную кодовую последовательность. Принципы двухэтапной обработки и быстрых преобразований обеспечивают большое разнообразие реализаций схем поиска, что может быть использовано для обеспечения компромисса между скоростью и сложностью поиска, как это достигается в известных последовательно-параллельных схемах. Эти принципы применимы для обработки широкого класса кодовых последовательностей, основанных на кодах максимальной длины. Он включает коды Голда и коды Гордона-Миллса-Уэлча (GMW коды) и другие, более сложные конструкции. Показано, что предложенная модель устройства поиска для определенного подкласса сигналов обеспечивает высокий уровень адаптации к изменению структуры сигнала. В предлагаемом варианте схемы поиска отсутствуют устройства памяти для хранения принятой реализации на этапе обнаружения, что обычно требуется при выполнении быстрых преобразований другими методами. В отличии от обычной схемы корреляционного приемника, разрядность сумматоров-накопителей вдвое меньше. На основе анализа существенных свойств быстрых преобразований в полях Галуа определены перспективы дополнительного снижения временных затрат на поиск по задержке сложных двух- и трехкомпонентных конструкций, базирующихся на кодах максимальной длины. Результаты работы будут полезны специалистам, занимающимся проектированием сложных кодированных сигналов и устройств их обработки для обеспечения безопасности и защиты от помех.
Ключевые слова: асинхронно-адресные системы, командно-измерительные системы, структурная избыточность, быстрый поиск по задержке, квазиортогональные ансамбли сигналов, поля Галуа, быстрые преобразования.
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Информация об авторах:
Михайлов Владимир Юрьевич, Московский авиационный институт (национальный исследовательский университет), профессор, д.т.н., доцент, Москва, Россия Мазепа Роман Богданович, Московский авиационный институт (национальный исследовательский университет), заведующий кафедрой "Радиосистемы и комплексы управления, передачи информации и информационная безопасность", к.т.н., профессор, Москва, Россия
