Специфика блочных турбокодов при анализе в частотной области Текст научной статьи по специальности «Компьютерные и информационные науки»

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SPECIFICITY OF BLOCK TURBO CODES IN THEIR FREQUENCY DOMAIN ANALYSIS

DOI 10.24411/2072-8735-2018-10146

Konstantin Yu. Ryumshin,

MTUCI, Moscow, Russia, [email protected]

Timofey A. Chadov,

MTUCI, Moscow, Russia, [email protected]

Keywords:

Turbo code, identification, discrete transformation in the Galois field, the spectrum of the code word.

Basic fundamental work devoted to the correcting coding, were published in the 60-70s of the last century. Since 80-s the main work in the field of error-correcting codes was carried out not in the direction of the development and usage of new codes, but to establish a concatenation of the existing codes and find effective signal-code constructions. Currently, however, interest in the developments in the field of design schemes and encoding and decoding algorithms, which make it possible to approach the Shannon bound, is renewed. Recently, much attention in the theory and practice of coding is given to the so-called turbo codes, which were first described in 1993 [1]. They are immediately attracted the attention of experts, because this technology makes it possible to provide high noise immunity signaling and increase data transfer speed, unattainable in the implementation of any of the other existing methods.

In communication systems with block turbo codes for synchronization between reception and transmission units sync pattern of 20 bits or more in length, is used, which are arranged between the code words of the block turbo code. Using the sync pattern reduces the data rate. New property of a code word of a block turbo codes is revealed and indications for phasing of block turbo codes, based on properties of the spectra after Discrete Fourier Transform in the Galois field, are offered.

Information about authors:

Konstantin Yu. Ryumshin, leading employee, doctor of Technical Sciences, Moscow Technical University of Communications and Informatics, Moscow, Russia

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

Для цитирования:

Рюмшин К.Ю., Чадов Т.А. Специфика блочных турбокодов при анализе в частотной области // T-Comm: Телекоммуникации и транспорт. 2018. Том 12. №9. С. 59-62.

For citation:

Ryumshin K.Yu., Chadov T.A. (2018). Specificity of block turbo codes in their frequency domain analysis. T-Comm, vol. 12, no.9, pр. 59-62.

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Analysis parameters and structure. Turbo codes, built on the basis of block turbo codes {BTC), are "the product of" extended block codes. A systematic representation of the code obtained from systematic representations of composite codes. A two-dimensional matrix model BTC is presented in Figure 1.

Consider presented in Figure 1 refer to:

- block of information bits forms a matrix I;

- each row of the matrix / is encoded by the encoder A'l forming a matrix of parity bits PI;

- common parity bits from rows of the data matrix / and row parity bit matrix P\ forms a matrix H1;

- each column of the matrix (I \J P\^J H\) is encoded by the encoder K2 forming a matrix of parity bits P2;

- common parity bits of columns of the matrix (/O PI kJi/1) and colums of matrix P2 forms a matrix H2.

Turbo word of block turbo code can be represented as a set of transmitted elementary code words of unexpanded block codes, as shown in Figure 3,

Figure 1 shows that the matrix PI is formed of the columns of matrix (IKJ P\yj H\). Each row element of P2 represents the sum of strictly defined elements of rows of matrix (! <J P1 <jH1} And, hence, each row of matrix P2 is formed of sum of strictly specific elements of the rows of the matrix (J yj P1 U HI )■ Thus, the rows of the matrix P2 also are the code words of the encoder K \. Matrix H2 is formed by the sum of all columns of the matrix (/ uPl U H\) and PI and hence also a code w ord of encoder K1.

Note that the parameter is the length of non-expanded

code word encoderK\, and determines the length of the elementary code word, and the parameter n2 the length of non-

expanded code word encoder K2, wich determines the number of elementary code words in the transformed model.

Consider the code words transformed model block turboslova.

Turbo code matrix formed after encoding operation is transmitted to the communication channel. Thus the communication channel arrives turbo word. Turbo word model of the communication channel is represented in Figure 2.

1 codeword =C,(a:) = /,(.x:)g](.v);

2 code word = C2(.y) = [/2(.v)^(a-)]a-"i+i;

3 code word = C,(x)=[/j(x)g1(x)]x2<"'tli;

Where C(x) —y-th code word of code A'l; / (,v) - information symbols ofy-th codeword; gt(jc) _ generating polynomial of code K\.

P1

H1

P2

H2

Fig, 1. Matrix model block turbo code

Bits of parity, in turn, can be represented as a kind of code word with spaced bits:

1 simbol = u^"1;

2 simbol =Uj,je2(",">';

3 simbol = u,x3("l+IH;

k simbol =uiy("l+IH; (¿ + 1) simbol

«, simbol = u, x"i("l+IH,

2 II,

Since we analyze the code words of unexpanded block codes, the bit(/j^ -f- I)excluded from consideration. An exception of

this bit from analysis of turbo word does not lead to any significant change, since the bit is a parity cheek bit. Analyzed turbo word linear model is presented in Figure 4.

C,(x) u, C{x) u2

C(n2*1)(X)

(«3 + 1) codeword = C. +1,(x)-[i, t)(x)^*)]*"2

Fig. 4. The analyzed linear model of turbo word

h ph hh 12 ph h12 P2i H2

Fig, 2. Turbo word model in the communication channel

C,(x) u, C2(x) <* Cn2+l(x) Un2+1

J

V-

"V"

~Ph\\h±\\ '> \ph\ [Ä7

Fig. The transformed model block turboslova communications transmitted in the channel

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Cyclic code generating polynomial K 2 can be represented as follows:

g2 (x) = g0x° + g^x"^ + • (1)

Therelore, we write the polynomial expression for the code word of the second code in the process of character encoding/.. (x)

Contained in submatrix H1:

ich,ck (x)g2(x) = u{x°+u2x*+--- + u>i2x'h--1. (2)

The resulting polynomial is divisible by the polynomial o, (x), Which is the generating polynomial of the second code.

These symbols are analyzed in a linear model (Figure 4) as follows:

w/'1 +«,je2("'tlH+-" + zi,„jr'Mfl|+IH. (3)

Performing transformations we obtain: if,** = j4)

iw* +U,x< + Uyx-f* = ^ (V (|«t(-v))",+l. Consequently, on the basis of formulas I -4, distributed characters i/ i/ represent a cyclic code, which can be represented as iclKck(x)g2(x).

Then turbo word can be represented as a combination of several different code words:

C„,K (x) = /, (x)gl (x)+[i2(x)gl (x)]xn,+l +••■

+ (VOB(-v)g2(-v))"l+l

If gt (x) = g^ (x) ~ g{x) turbo word can be represented as follows:

Cl>IK(x) = g(x)[ii(x) + i2{x)xn+l + ■■■]■ Turbo word representation as a combination of several code words of different codes valid for the turbo code consisting of a large number of encoders.

Thus, one can conclude that if the generating polynomials of elementary encoders gt (x) ■ ■••;£"„(*) are relatively

prime, the common word can not be represented as a cyclic code. If the generator polynomials of elementary coders gt (x) ■

g2(x) —.gn(x) have a common divisor, the general turbo word

divided by the GCD of polynomials. Ifg( (x) = g-, (.v) =

turboslovo total is divided by the polynomial tj(x) •

Property. To turbo word could be seen as a code word of a cyclic code, it is necessary and sufficient that the generating polynomials elementary coders are not relatively prime, that is, have a common divisor, or were equal to each other.

If necessary and sufficient conditions are met, the turbo word can be seen as an extended cyclic code. Based on this property it follows that turbo word represents a row vector of a cyclic multiplicative (Unite) group of the order L = (/?"' — I) + I where

L — turbo word length. Turbo word has a characteristic cyclotomic class of modulo ¿, formed by zero spectral components (roots) of discrete Fourier Galois transform (DFGT).

Possible applications. Cyclical property of turbo word formed the basis of the phasing by successive shifts, based on the

difference between synchronous and asynchronous receiver state where a synchronous meant a state in which only the received code word (excluding the effect of errors in the communication line). Usually the hallmark used in phasing algorithms is the zero syndrome. Note that it is the complexity of calculating the syndrome in most cases is the determining factor in assessing the computational and time-consuming, since this operation is performed on each phasing step.

For the block turbo code satisfying the necessary and sufficient conditions, equality syndrome to zero is equivalent to the presence of the respective spectra! components at DFGT of turbo word. Therefore, the phasing of block turbo codes can be produced by highlighting the moments of occurrence of zero spectral components forming a certain class cyclotomic DFGT when the test signal by successive shifts, and itself receiving cyclotomic class zero spectra! components is the feature of achieving synchronous state.

Phasing by successive shifts not on generally turbo word, but on code words elementary encoders, on one hand accelerates the process of calculating the syndrome, but does not generally applicable because firstly, does not allow to lind turbo word borders without using the syne pattern, secondly , this method does not lake into account features of the internal structure of turbo words, i.e. it makes it impossible to define the boundaries of the elementary coders, when polynomials are elementary coders. Therefore, the phasing of block turbo codes by allocating zero moments of occurrence of the spectral components, forming a certain cyclotomic class, must be performed on turbo word.

It is known that if {/}<-» jj/7, | j are a pair of DFGT, the pairs of DFGT are also

where f . - Mh symbol of a codeword, f' -y-th spectral component, a - a primitive element, and the double brackets indicate that the indices are calculated modulo length of turbo word L. From the expression (5) it can be shown that the non-zero spectral components changc during cyclic shift of converted turbo word whereas zero components and hence cyclotomic class not changed. Since the change does not affect the spectrum forms the basis of the phasing block turbo feature, it is possible to significantly speed up the computation by use of the process of determining the number of zero spectral components instead of expression ii-1

(6)

i=0

or a fast algorithms recursion formula obtained by taking into account relations ( 1 ),

F J (5) - F J (,-]) + (/v+„_, - 1 )«(M))i-

j = 0(1)X -1 • (7)

Where F¡(s) -_/-th spectral component to s-th step of phasing, f ! - first character convert the signal fragment in m s-

phasiug step, s - phasing step number.

Zero spectral components function in the calculation of DFGT by formulas (6) and (7) will be identical, and when the synchronous conditions will occur emergence of the cyclotomic class, with its appearance period is equal to the turbo word length L.

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If necessary, the analysis of non-zero components of the true shape of the spectrum can be restored on the basis of (5). However, in solving the problem of phasing of block turbo codes waiver of recovery can reduce the required amount of computation in log ¿/2 times and thus reduce the total time phasing.

Furthermore, the method of the phasing block turbo codes eliminates the need for syne pattern between turbo words thereby increase the data transfer rate.

References

I. Berrou C., Glavieux A. (1993). Near Shannon limit error correction coding and decoding. Turbo codes. Proc. Int.conf. Comm., pp. 1064-1070.

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

3. Ryumshin K.Yu. (2011).Analysis of digital signals with corrective coding in a discrete channel. Proceedings of the MSA n.a Mozhaisky. SPb.: MSA n.a A.F. Mozhaisky. Vol. 633. 220 p.

4. Ryumshin K.Yu. (2013). Models and methods for identification of signals with error correcting coding in a discrete channel, methods for constructing and decoding modern code constructions. SPb.: MSA n.a A.F. Mozhaisky. 173 p.

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

СПЕЦИФИКА БЛОЧНЫХ ТУРБОКОДОВ ПРИ АНАЛИЗЕ В ЧАСТОТНОЙ ОБЛАСТИ

Рюмшин Константин Юрьевич, МТУСИ, Москва, Россия, [email protected] Чадов Тимофей Александрович, МТУСИ, Москва, Россия, [email protected]

Аннотация

Одним из основных направлений развития телекоммуникационных систем в последние годы вновь стало повышение надежности передачи информации за счет совершенствования канального кодирования. Роль корректирующих кодов значительно возросла в связи с повсеместным переходом к цифровым способам передачи информации. Основные фундаментальные работы, посвящён-ные корректирующему кодированию, были опубликованы в 60-70-е годы прошлого века. С 80-х годов в области развития корректирующих кодов в основном велись работы не в направлении разработки и использования новых кодов, а по созданию кон-катенаций существующих кодов и нахождению эффективных сигнально-кодовых конструкций. Однако в настоящее время вновь возник интерес к разработкам в области создания конструктивных схем и алгоритмов кодирования и декодирования, дающих возможность приблизиться к границе Шеннона. В последнее время большое внимание в теории и практике кодирования уделяется так называемым турбокодам, которые впервые были описаны в 1993 году [1]. Они сразу же привлекли внимание специалистов, поскольку применение этой технологии дает возможность обеспечить высокую помехоустойчивость передачи сигналов и увеличить скорость передачи данных, недостижимую при реализации любого другого из существующих методов. В системах связи, использующих блочные турбокоды, для синхронизации приёмных и передающих узлов используются синхрокомбинации длиной 20 бит и более, которые располагаются между кодовыми словами блочного турбокода. Использование синхрокомбинации снижает скорость передачи данных. Выявлено новое свойство кодового слова блочного турбокода и предложены признаки для фазирования блочных турбокодов, базирующиеся на свойствах спектров, получаемых при дискретном преобразовании Фурье в поле Галуа.

Ключевые слова: турбокод, идентификация, дискретное преобразование в поле Галуа, спектр кодового слова. Литература

1. Berrou C., Glavieux A. Near Shannon limit error-correcting codingand decoding / Turbo codes, in Proc. Int. Conf. Comm. 1993, рр. 1064-1070.

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

3. Рюмшин К.Ю. Анализ цифровых сигналов с корректирующим кодированием в дискретном канале / Труды Военно-космической академии имени А.Ф. Можайского. СПб.: ВКА имени А.Ф. Можайского. 2011. Вып. 633. 220 с.

4. Рюмшин К.Ю. Модели и методы идентификации сигналов с корректирующим кодированием в дискретном канале, методы построения и декодирования современных кодовых конструкций. Монография. СПб.: ВКА имени А.Ф. Можайского, 2013. 173 с.

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 авторах:

Рюмшин Константин Юрьевич, ведущий сотрудник, д.т.н., Московский технический университет связи и информатики, Москва, Россия

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

T-Comm ^м 12. #9-2018