Алгебраические ограничения при решении задачи идентификации корректирующих кодов Текст научной статьи по специальности «Компьютерные и информационные науки»

ALGEBRAIC LIMITATIONS IN SOLVING THE PROBLEM OF IDENTIFICATION OF CORRECTING CODES

DOI 10.24411/2072-8735-2018-10145

Timofey A. Chadov,

MTUCI, Moscow, Russia, Keywords: system of linear algebraic equations,

[email protected] correcting codes, discrete channel.

The peculiarities of solving the problem of signal identification with a correcting codes in a discrete channel in the time domain are considered.

Modern information transmission systems have increased requirements for ensuring the reliability of data transmission, high-capacity and structural secrecy emissions of radio-electronic means (REM). One of the most effective ways to meet these requirements is the use of correction coding in transmission system.

Correcting codes allow to greatly enhance noise immunity and provide energy and structural secrecy of the transmitted signal, which greatly complicates the process of intercepting messages and its technical analysis. However, existing approaches to technical analysis have fundamental limitations.

The effectiveness and efficiency of existing methods of technical analysis, based on the solution of linear systems are not sufficient. This leads to various modifications and changes in the methods of solutions which are inherently slightly improve the characteristics of these methods, the critical extent dependent both on the nature of the analyzed code sequences, and from arising because of the presence of noise in the communication channel errors when receiving code symbol.

Information about author:

Timofey A. Chadov, head of the research department, Moscow Technical University of Communications and Informatics, Moscow, Russia

Для цитирования:

Чадов Т.А. Алгебраические ограничения при решении задачи идентификации корректирующих кодов // T-Comm: Телекоммуникации и транспорт. 2018. Том 12. №9. С. 54-58.

For citation:

Chadov T.A. (2018). Algebraic limitations in solving the problem of identification of correcting codes. T-Comm, vol. 12, no.9, pр. 54-58.

General information about solving a system of linear algebraic equations with the technical analysis of signals

In analysis of digital sequences in transmission systems is often necessary to solve a system of linear algebraic equations (SLAE).

The set of all code words of linear codes can be obtained by

using expression k

; = 1

where ajr coefficients (for binary codes a^a^ € (0,1})

In ease of representation of code words in the form of rows of matrix n x k a generating matrix of code n x kG(nk) takes place

G(n,k) -

■yu y 12 ■ ' yi*

y2i y22 ■ • y 2?!

.y*i y*2 ' ■ ykn

wneri presenting me eo matrix is written as follows

Gr =

3n 321

012 922

9\r

32r

Lo 0 •■■ 0 1 gk2 gkr J

In this case, the right side of the matrix contains the coefficients for the parity check equations

yt+i = ^«jiyy. i = ICI)'', i=i

which in the absence of errors in reception are performed for all the code words of the code.

Since the code is uniquely defined by its generation matrix or by coefficients g^ of parity cheek equations, then to identify the code it is sufficient to determine the value of these coefficients, wherein g^g^ = 1(1 )k

yu +y2i X02i + - + yfci xfiki = &a+i,iJ yiz xin +yZ2 *g2i + --- + y*i xgki = yti+1,2;

yim x 9ll + y2m x 521 + + ykm x 9k\ ~ ykl+l,m* Similarly, we can make even (r — 1) system of linear equations for the coefficients gwhere j — l(l)/c, t = l(l)r. Note that all systems have the same coefficients of the unknown and differ only in the column of free terms.

The system of linear equations has a unique solution if the determinant of the matrix of the coefficients of the k variables in the equations arc linearly independent, non-zero, i.e,

bit m - ym'

D =

y 21 y22

y2rt

* 0.

Gauss method described in detail in the literature, its essence is sequential elimination of unknowns of the system of equations. To solve a system of equations is necessary lo write an extended matrix comprising in addition to the coefficients of the unknown column of free terms. Then, produce elementary transformations on the rows of the matrix:

- reorder the rows that correspond to change the order of

equations;

- multiply rows to any non-zero number, and addition to any

row of any of its rows, multiplied by any number.

Using these transformations each lime a new extended matrix system is obtained, equivalent to the original.

Due lo the fact that r systems of linear equations differ only in the column of free terms, they can be dealt with at the same time, which in the expanded matrix include all of the columns of the free terms of equations.

Solving procedure is divided into two stages, commonly referred lo as forward and reverse motion.

In the case of SLE in binary field GF (2) the need for division and multiplication operations is eliminated, and all the transformations are carried out only by adding the rows of extended matrix.

Example, over a plurality of5 code words of systematic code {1011100,1101010,0111001,0100011,1001001} determine the coefficients of the parity equations.

An expanded matrix of the code words: -1011100" 0110110

Y = 0111001

0100011 Looioioi-

Let's solve linear systems by Gauss method.

I. Forw ard motion

a) add second and fifth row, containing units in the first position, to the first row:

1011100] rlOll100

0110110 0110110

Y = 0111001 t Y = 0111001 ; 0100011 0100011 looiooi-l Looioioi

6)

add third and fourth row to the second row

Ly*i y*2 - yjtnJ

In this system a solution can be found by Cramer's rule:

9ji=J. J = 1(1)*,

where Dj - the determinant obtained from D by replacement of the /-th column of free members of a column.

In practice, Cramer's formula is rarely used because of the rapid growth in complexity of the computation values of the corresponding identifiers with increasing k. More frequently linear systems is solved by a variety of modifications of the Gauss method.

1011100-1 1011100

0110110 0110110

V = 0001111 , Y = 0001111 ; 0100011 0010101 0010101-1 0010101J

b) c) swap the third and fourth lines, and then added to the third row fifth:

1011100] rlOlllOO"

0110110 0110110

Y = 0010101 , V = 0010101 . 0001111 0001111 0010101-1 L0000000J

The appearance of the zero row indicates that from the five equations in SLAE only four is linearly independent, as it should be for the code (7,4) having exactly 4 independent information symbols. In the future, all the changes will be made only with linearly independent rows of the matrix Y.

2. Reverse motion:

a) add first row containing unit in fourth position to the fourth row:

Y =

1010011' 0110110 0010101 ' 0001111

6) add second and first rows to the third row;

V =

1010011 0100011 0010101 0001111

V =

"1000110 0100011 0010101 " .0001111.

Matrix, obtained after transformation, corresponds to the systematic generator matrix code. Hence, the parity check equations are:

yi + y3 + y* + ys = o, yj + y2 + y4 + y6 = o, y2 + y3 + + y? = flIC the row of the matrix contains not one, but v code words, then, respectively, after conversion by ihe Gaussian matrix will comprise!? x k linearly independent rows, the independent information symbols are followed periodically up in positions ¿xti + 1 to iXn + ¿where i = G(l)t?— 1- This allows to determine from conversion results characteristics not only codes oC length equal to the length of the rows, but with lengths s multiple code words. This technique is often used in processing the eonvolutional codes having a relatively small value n, and accordingly, do not require a large increase in the size oC the transformation matrix associated with excessive increase in the calculation amount.

Fxample, According to the code sequence

0111111010000111001101110000

k 1

eonvolutional code with relative coding rate R = - — - fill

matrix Y by code fragments which length is 10 symbols, choosing these pieces of every n character:

-0111111010 1111101000 1110100001 1010000111 1000011100 0001110011 0111001101 1100110111 0011011100 1101110000

After Gaussian conversion we obtain

1000010001 0100010001 0010010101 0001000100 0000110100 0000000000 0000001101 0000000000 0000000011 LOOOOOOOOOO

Y =

V =

Front the result of the transformation, it is seen that the sixth symbol depends linearly on the previous five, i.e., the parity check equation for this code is:^ + y2 + + ys + yf, = 0.

Repetition of linearly independent rows with period 2 indicates that lbr a given code as well k = In = 2.

Linear relationship length Cor eonvolutional codes exceed n, as a check symbol is generated using not only current, but also several previous simbols. The length of the linear relationship for a eonvolutional code called a constraint length NA = L x n, where L - the length of the register encoder, which determines the number of data symbols involved in the formation of the check.

Thus, it is possible to determine during the technical analysis of eonvolutional codes such parameters as L ,n and k.

The revealed features of (lie process of solving a system

of linear algebraic equations

For the system of linear equations has been defined, i.e. has a unique solution, the following conditions must be met:

- the number of linear systems of equations (code words of correcting code) k is not less than the number of unknowns;

O > k)

— in the test set were present k linearly independent sequences.

In addition to this, it is necessary that in the test sample, no distortion.

If random vector Crom the received set oC vectors can be chosen linearly dependent vectors, leading to the formation of the basic matrix oC the subspace oC the space of the desired code word.

In solving technical analysis following the detector correcting codes objectives must take into account certain aspects of algebraic below.

Proposition 1.Suppose that there are k arbitrary binary sequences. Then the probability that the fcequations of 7i binary sequences are independent, is given by

ft = 1(1)00, (1)

¡=ix /

Evidence. The validity of this statement stems Crom obtaining the approximate values oCthe product (I) and the convergence oC

the series In (l — ^ on the basis of the DAIembert.

Transform the infinite product oCthe form

p = PJ(1_JL) = = e^M^-M. (2)

n=l

A necessary and sufficient condition Cor convergence is infinite product convergence numerical series

I

n = l

In

1 *

(3)

Numerical series (3) is a constant sign, all oC its terms are negative values. This number can be positive or negative sign to endure the summation sign.

The convergence of the series (3) on the basis of the D'Alembert determined Crom the expression 1

(4)

The values in the numerator and denominator of the limit (4) arc infinitely small, whereby, there is an equivalence. Here then

the series (4) takes the form ]n(l -for) I — I —>0.

v / ia—»u n n <n—►so

Matrix 2x2

Matrix 3x3

lim -

1

2"+I _

= lim

rt-í

I I

-X~ 1

2" 2 _ 1

= -<1.

2

1 n-i 1

~~T V

Thus, the series converges on the basis of the d'Alembert. Due to the fact that this series converges, and infinite product converges. Its value can be calculated approximately using the private product of the form P„

HYtiH^"-

i

Fig. I. Probability of linearly independence of rows

Thus, the probability of finding a single arbitrary SLR solutions will not exceed the value

P = lim PJ

2'

s= 0,28879.

Illustration number of all binary combinations of matrices, and the number of non-singular and singular matrices are presented in Figure 2 for matrices 2 x 2,3 x 3,4 x 4,

Thus it can be argued that the probability of solving the linear and convergent, respectively, and the number of matrices to other dimensions will also satisfy the expression (1) P„.

16

10

512

68

w

Matrix 4x4

Number of non-singular matrices

100% 100% 100%

6 062,S0% 0=7,19% 0.9,24%

^^p016^ ^0,76%

In this way, P\^~P„

Draw the absolute error of assessment by using the Cauchy criteria. If, on the absolute error. The assertion is proved.

| P- P„ \< e(e = 10"3) j P - P„ \< a ■ 10 ■ £.

Proposition 2.Suppose that P„ when n - k( 1) is a sequence of numbers determined by the expression (1). Then the sequence P„ is decreasing bounded sequence 0< P„< 0,5.

Evidence. The validity of this proposition follows from the fact that each member of the sequence P„ can be defined as P ,=PxA .. While P„ > 0 for any n<=N where

/1 + 1 H U + l *

where o<A,<1, therefore

< Pm for any n e N.

Therefore this sequence P„ is decreasing bounded sequence 0<Pn< 0,5. The proposition is proved.

Dependence probabilities of linear independence of rows from the matrix SLAE dimension is shown in Figure I.

Fig, 2. Quantitative assessment nonsingularity: first column - the total number of matrices; second column - number of non-singular matrices; third column - number of matrices of singular

Evaluation methods of solving the system

of linear algebraic equations

Widespread various modifications Gauss method as algebraic method of technical analysis of linear codes due to the fact that its implementation requires an acceptable quality for most operations codes used in practice. This algorithm requires approximately operations. Thus memory requirements proportional (0,7k3 + I ,5k2 - 0,5t?)/72.

For solving linear systems by Cramer's rule is required to make transactions [(« -l)x»! + »].

The number of arithmetic operations when solving linear systems of equations Cramer and Gauss method shown in Table I.

Table I

The volume of arithmetic operations

n Cramer's formula Gauss method

3 51 31

4 364 67

5 2885 123

10 3.6h 108 Shi 02

50 6.7h 1067 4.5 Ii 104

2048 7.211105900 5,6lil09

Table 1 shows that even for frequently used Reed-Solomon code with a codeword length 2040 binary symbols necessary volume calculations with the help of modern computer can be made in a reasonable time.

A significant drawback of the algebraic method is its sensitivity to errors in the communication channel. Since all combinations with multiplicity of errors smaller than dmin is not a code words, the presence of incorrect symbols in the code words being analyzed will cause the appearance of additional linearly independent matrix rows. This will lead to the fact that even a single wrong symbol makes it impossible to determine the true structure of the code. If the matrix / contains n~ code symbols, then they should all be received correctly. The probability of error-free reception P(, = (1 - p)n2 decreases rapidly with increasing n even at relatively small p. So when Ph > 0,9 for the BCH code (31,26) admissible probability of error in the communication channel is not more than p < 10"4, and for code RS (255,239) in a field GF(2*) with a code word length n = 2048 binary symbols p < 2,5 x 10 s.

Conclusion

The effectiveness and efficiency of existing methods of technical analysis, based on the solution of linear systems are not sufficient. This leads to various modifications and changes in the methods of solutions which are inherently slightly improve the characteristics of these methods, the critical extent dependent both on the nature of the analyzed code sequences, and from arising because of the presence of noise in the communication channel errors when receiving code symbol.

References

I, Berlekemp F..R, (1980). Error correction coding technique. Proc. of the IEEE. Vol.68. No.5. Moscow: PIEEE, pp. 24-58.

2. Prokis [3. (2000). Digital communication. Moscow: Radio and communication. 800 p.

3. Ryumshin K.Yu. (2008). Possibilities of applicability of a system of linear equations in the technical analysis of numerical sequcnccs. Works of the 4th Central Research institute of the Ministry of Defense of the Russian. Federation. Yubileiny: 4 Central Rcscarch Institute of the Ministry of Defense of the Russian Federation, No. 93, pp. 182-189.

4. Ryumshin K.Yu,, Zaitscv I.E., Kryachko A,F. (2011). Technical analysis of signals in monitoring of electromagnetic environment. Proceedings of the 9th International Symposium on EMC. SPb.: ETU «LETI». Special edition of IEE Electronics Leiters, pp. 226-231.

5. Chadov T.A., Tikhonyuk A.I., Erochin S.D. (2017), Approach to the problem of identification of noise-resistant codes in telecommunication systems. Systems of signal synchronization, generating and processing. No, 1, pp. 21 -26.

АЛГЕБРАИЧЕСКИЕ ОГРАНИЧЕНИЯ ПРИ РЕШЕНИИ ЗАДАЧИ ИДЕНТИФИКАЦИИ

КОРРЕКТИРУЮЩИХ КОДОВ

Чадов Тимофей Александрович, Московский технический университет связи и информатики, Москва, Россия, [email protected]

Аннотация

К современным системам передачи информации предъявляются повышенные требования по обеспечению достоверности передачи информации, высокой пропускной способности и структурной скрытности излучений радиоэлектронных средств (РЭС). Одним из наиболее эффективных способов удовлетворения этим требованиям является использование в ССиПД корректирующего кодирования. Корректирующие коды позволяют в значительной степени повысить помехоустойчивость, а также обеспечить энергетическую и структурную скрытность передаваемого сигнала, что существенно усложняет процесс перехвата сообщений и его технического анализа. Однако существующие подходы к техническому анализу имеют принципиальные ограничения. Рассматриваются особенности решения задачи идентификации сигналов с корректирующим в дискретном канале во временной области.

Ключевые слова: астема линейных алгебраических уравнений, корректирующие коды, дискретный канал.

Литература

1. Берлекемп Э.Р. Техника кодирования с исправлением ошибок // ТИИЭР. Т. 68. №5. 1980. С. 24-58.

2. Прокис Д. Цифровая связь. М.: Радио и связь, 2000. 800 с.

3. Рюмшин К.Ю. Возможности применимости системы линейных уравнений при техническом анализе цифровых последовательностей / Труды 4 ЦНИИ МО РФ. Юбилейный: 4 ЦНИИ МО РФ. 2008. № 93. С. 182-189.

4. Рюмшин К.Ю., Зайцев И.Е., Крячко А.Ф. Технический анализ сигналов при мониторинге электромагнитной обстановки / Труды по материалам IX Международного симпозиума по ЭМС. СПб.: ЛЭТИ(ЭТУ). Спецвыпуск IEE Electronics Letters. 2011. С. 226-231.

5. Чадов Т.А., Тихонюк А.И., Ерохин С.Д. Подход к решению проблемы идентификации помехоустойчивых кодов в телекоммуникационных системах связи // Системы синхронизации, формирования и обработки сигналов. № 1. С. 21-26.

Информация об авторе:

Чадов Тимофей Александрович, начальник научно-исследовательской части, Московский технический университет связи и информатики, Москва, Россия