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PRELIMINARY GENETIC-LIKE CODING OF RANDOM AND DETERMINISTIC MESSAGE SOURCES STRUCTURES
This paper is a research into features of preliminary genetic-like coding of random and deterministic message source's structures. The main principle underlying the encoding method developed by the authors of this research is to maintain the compliance of the smallest distances between the messages in the original metric space of the message source and code blocks with the smallest Hamming distance. In previous studies, it was shown that preliminary genetic-like coding of random message source's structures can significantly increase noise immunity; however, deterministic structures are the "worst" from the perspective of the developed algorithm. The comparative analysis in terms of minimizing the distances between the messages in the original metric space of the message source, which have the smallest Hamming distance between their code blocks, was performed between preliminary genetic-like coding and random uniform coding as a part of this research.
The comparative analysis of deviations caused by decoding error, i.e. mean deviation or mean distance between sent and garbled received messages in the original metric space of the message source, was performed between preliminary genetic-like coding and random uniform coding as a part of this research. In this case, there were considered source's messages, which code blocks were changed due to a single-bit error.
A comparison of the results obtained during the research provides the most complete picture of the pros and cons of the preliminary genetic-like coding, but also points to the need for further study of developed method in terms of applying it on random and deterministic message source's structures. The authors believe that one of the main results of this research is that close enough performance between genetic-like coding for modulation codes and modulation codes used in DVB-S2 digital television broadcast standard was achieved.
Mikhail M. Fenchuk,
postgraduate student, Moscow Technical University of Communications and Informatics, Mosscow, Russia, mikhail.fenchuk@gmail.com
Irina S. Sineva,
Associate professor, Moscow Technical University of Communications and Informatics, Mosscow, Russia, iss@mtuci.ru
Anna V. Bott,
engineer, Research Institute of Precision Instruments, Mosscow, Russia, bottane4ka@yandex.ru
Keywords: genetic algorithm, noise immunity, genetic-like coding, uniform coding, error correction.
Для цитирования:
Фенчук М.М., Синева И.С., Ботт А.В. Предварительное кодирование генетического типа для случайных и детерминированных пространств источника сообщений // T-Comm: Телекоммуникации и транспорт. - 2016. - Том 10. - №10. - С. 60-65.
For citation:
Fenchuk М.М., Sineva I.S., Bott A.V. Preliminary genetic-like coding of random and deterministic message source's structures. T-Comm. 2016. Vol. 10. No.10, рр. 60-65.
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Genetic algorithms applications
Over the last few decades, methods based on the genetic algorithms (GA), have been widely used in practice. Because of its important features, relying on the principles of natural evolution [I |, GA is a powerful tool for solving various classes of problems, including error-correcting coding problems. Analysis of international experience in this area shows a growing number of scientific publications, including Russian papers, in which the following areas of GA application arc reflected: optimization of error-correcting decoding algorithms, generation of the error correction codes, multiclass classification's problem solving using ECOC codes, analysis and recovery of digital images and video, and many others.
With the participation of the authors of this research there was developed an algorithm of genetic-like coding of message source with a defined metric [2]. This is a research into the genetic-like coding of random and deterministic message source's structures.
Message source as a metric space
Before describing the genetie-like coding procedure il is necessary to give an idea of the message source with a defined metric. It is assumed that the messages, transmitted over a communication channel, are code combinations of N encoded points A. of multidimensional metric space with known
coordinates x(i) - (*,(/), x2(i),,.., Xm(i)), \<i<N. For a
better understanding, this article describes a case of N = 2" points, so it requires n -bit code combinations to perform a nonredundant uniform binary coding; error type: a single-bit error. However, the authors of this study have developed a various modifications of the algorithm for the message source with an arbitrary finite number of initial messages [3], as well as modifications that minimize the effect of different types of errors [4-5].
A method to define a specific metric for a given message source is beyond the scope of this discussion, so the object of the genetic-like coding algorithm's application is the original array of messages which is provided with a matrix of pairwise distances. The described technique could be easily generalized to any metric space of a message source.
Preliminary genetic-1 ike coding procedure
The main principle underlying the genetic-like encoding method is to maintain the compliance of the smallest distances between the messages in the original metric space of the message sourcc and code blocks with the smallest Hamming distance, it should be noted that the «smallest» Hamming distance is defined by the type of error in each individual ease. For example (and further), if the most probable type of error is a single-bit error then the smallest Hamming distance is assumed to be equal to I. Thus, the use of preliminary genetie-like coding allows to minimize the decoding error. The results obtained during the research support the use of genetic-like coding as a mean to improve noise immunity.
The first step of genetic-like encoding is to find an initial point of algorithm. There're different approaches to do il, but the most appropriate is the following: to calculate the sum of
2
Euclidean distances D^^d^, dtJ 0')) '
d. . = d, du- 0, (I . > 0, / * / to the rest of points for each
point A. of the original array. Initial point is the one with
minimal value of D; : = argmin {£).; !<i'<Ar} - it will be
encoded with /j-bit code combination which consists of all zeros: (000...00).
The second step of genetie-like encoding is to find the first category's points which are closest to the initial point; it is a group of n points. They will be encoded with n -bit code combination which consists of one "1" and zeros: (100. ..00),
(010...00), (001...00)..... (000. ..10) and (000...01).
Assignment of code combinations for the points of the first category could be performed explicitly or arbitrarily, unlike the next categories' points - their code combinations will be generated based on code combinations of previous categories' points.
f.
The third step of genetic-like encoding is to find
points
that are collectively less distant from the first category's points -these are the second category's points. On this step, algorithm compares every possible pair of the first category's points with all the reaming unassigned points relatively to their mutual distance. Code combinations of the second category's points will be generated based on code combinations of their "parental" pairs of the first category's points. For example, the second category's point, which is close to the pair of the first category's points with code combinations (100...00) and (000...01) will be encoded with code combination ( 100...01 ), i.e. by an operation of exclusive disjunction.
The forth step of genetic-like encoding is to find and encode " I points of the third category. This step is similar to the
V
previous, but algorithm compares every triple of the second category's points with Hamming distance of 2; the same applies to assignment of code combinations.
A similar method is used to find points of the forth and the next categories as long as on the last step of genetic-like encoding the only one unassigned point (furthest from the initial point) will be left. It'll be encoded with code combination which consists of all "1": (111...11).
A detailed mathematical description of preliminary genetic-like coding procedures and the analysis of the effectiveness presented in the article [6-7].
Genetic-likc coding of random structures
The genetic-like coding algorithm enables to greatly minimize the effect of à single-bit error, because the points with the smallest mutual distance between them in the original metric space of the message source have code combinations with Hamming distance of I. This property does not hold for random coding, which is confirmed by the results of comparative analysis of the deviations in the original metric of the message source space caused by a single-bit error.
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Technique that been used to evaluate the effectiveness of genetic-like coding is described in |6|. For each type of distribution F = {F(x,/3),j3<=B} (F(x,/3) - the distribution
of a parametric family F \ B - the set of admissible values of the parameter p) were generated the following configurations:
© - {x j= (x{ ,xJ2,...,xJk) eRk,\<j< 2,:}
During the genetic-like encoding procedures every source's message X: would be assigned with a binary code combination
g(x{) ■ The total number of code combinations with I lamming
distance r of I is equal to n. The sum of the distances S — ^ g^'^ in the original metric of the
message source space, related to the number of pairs of these n configurations, is that value, we will compare with the results of statistical modeling,
/ lo
aGA ~ »©<
n
A binary codc combinations um(x')> \<j<2n ("? -
sample number, I < m < M) would be randomly generated for each configuration. The sum of the distances
Z PiuJ^luJx1))
would be calculated and averaged over the number of code pairs with unit Hamming distance. Estimate of the average and standard deviation would be calculated for the obtained arrays
dm,\<rn<M :
M
Ave = -—-Yy,„ M ,
ivt m=I
S=J X—Y(d-Avef
VM-lf-f, "'
An indicator of the coding efficiency is the number of standard deviation that fit between the mean unit distance of genetic-like coding and the estimate of the mean unit distance of a random coding;
EF dGA-Ave
clusters with more noticeable condensation were generated in accordance with Laplace distribution L with different parameters of the location and scale;
clusters with the most noticeable localization were generated in accordance with Normal distribution N with different parameters.
The parameters of the location and structure of clusters are the shift factor a and the scale factor h for each of the coordinates. Shift factors of the first cluster are all equal to 0 and the first scale factor is equal to 1 because of the results' invarianee relative to shift family and sealing over the standard deviation in all the cases. The number and proportion of clusters, as well as their location, were also variable parameters so that there were investigated message sources with both explicit (strongly remote) clusters, and merging regions. In addition, the dimension of the features' original space k was a variable parameter; all calculations arc performed using parallel computing in multi-threaded mode [8]. For reliability, the generation of the points following a certain type of distribution with fixed set of parameters was performed for 10 times. The genetic-like encoding of each configuration was permed once and the random encoding was performed for 35 limes.
In order to systematize obtained results we introduce the notation of random source models as RSM(k, q, pc. dc), where: k - the dimension of the message source's space; q - the number of clusters; pc — the proportion of clusters for q > 1, if all clusters have the same proportion, this parameter will be equal to «es» (equal share); dc - the distribution of points in the cluster. If al l clusters have the same type of distribution (U. C. L, N), it will be specified as a parameter (either in general form as dsc -distributions of the same class), otherwise it will be specified as vd(various distributions). The following results were obtained:
1) RSM(2,1, dc)
Cluster configuration was varied by changing the scale factor
b e [lj 100] • The coding efficiency regression:
is described by the
EF = Jc, -> -Jq,
■ co
with a coefficient of determination of more than 70%. Table I shows asymptotic efficiency EFas=Jc\ - Stable asymptotic
behavior has not been detected for the Cauchy distribution, but the efficiency was higher than the value of 100 in all cases.
The higher the value of EF, the better genetic-like encoding compared with random arrangement of the code combination and the unit of scale is the estimation S of standard deviation.
Generated arrays were composed of both single and multiple clusters of different types. Clusters can be more or less well-defined localization, so they were generated in accordance with the various distributions:
clusters without accumulation points were generated in accordance with a Uniform distribution V with different parameters;
clusters with weak condensation were generated in accordance with Cauchy distribution C with different parameters ofthe location and scale;
Table Ï
Asymptotic efficiency for RSM(2,1, dc) model
dc EFM
U IIS
N 130
I 137
Thus, single-cluster models in R~ show an average efficiency of 127, i.e. genetic-like encoded message sources belter than randomly encoded on an average of 127a.
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2) RSM{3,1, dc)
One cluster is centered: ¿>,=1, b = b2 - ¿>3 e [l, 100] ■
The following regression describes the coding efficiency for the Laplace distribution:
-Vc.
•CO
with asymptotic efficiency EF -107- For other distrihuiions
dependence of the efficiency of the variable parameter increases monotonically and is well described by some linear regressions EF s- c{ + c\b with the coefficients of detemiination of 77.2%
(10, 78.7% (AO and 92.2% (Q and different values of coefficients c, and c2 ■ EF > 100 in all cases.
3) RSM(2, q, es, dsc)
Clusters have the same distribution with the same scale factor coefficient b — 1. their arrangement differs only by a shift factor £/e[l, 100]■ Considered values of q = 1,2,..„7 allow to obtain
representative results for each cluster.
EF > 100 in all cases, table 2 shows asymptotic efficiency obtained in the experiment.
Table 2
Asymptotic efficiency for RSM (2, q, es, dsc) model in the lattice centroid's location
dc EFm
C 204
L 201
N 177
U 155
These results show high efficiency for the genetic-like coding of ASM (2, t\. es. dsc) models.
4) RSM(2.2, es, dsc)
There are two identical clusters that differ only by the shift factors and all the scale factor arc equal to 1. it has been found that the closer the clusters are, the more effective gcnctic-likc coding is. The following repression reflects the asymptotic behavior:
ef=4
C, + 1
a-,
■ to
The effectiveness depends on the location of the clusters more than ¡1 depends on the distribution of cluster points. Table 3 shows asymptotic efficiency.
Table 3
Asymptotic efficiency for RSM (2, 2, es, dsc)
dc EFas
U 125
C 120
L 115
N 87
In this case, the effectiveness is also very high. 5) RSM(2.2.st:s2,dse)
The parameter s — sj(s^ ) is equal to the share of the
first cluster, e(0,1) - The dependence of the efficiency from
the shares of clusters is insignificant. The regression equation shown in tabic 4, the coefficient of determination varies from 71.9% to 93.2%. C| >0, e, >0 in all cases.
Table 4
The regression equation and the effectiveness for RSM (2,2, sb Si, dsc)
dc Regression equation mill EF
U EF -ic, — 181
C EF = 50
L EF = (c, -c, Ins) 85
N EF = yjc] + c2 92
As can be seen from table 4, even in the worst case, generated in accordance with distribution of Cauchy with large emissions, efficiency more than 50a.
6) RSM(2, 2,es,dc)
There isn't a significant dependence of the results on the distributions for RSM (2.2, es, dc) models. Independent
varying of coordinates' distributions allows to reduce the results into a single array that most adequately describes by the logistics distribution.
exp -
(x-M)
a
"M-^jï
with parameters // = 116.263 -the shift factor and cr = 21 .K076 — the scale factor. This array also fits the Normal distribution with parameters //=117,842 and a = 22.0865. In this ease, the three sigma rule gives a lower bound of the efficiency of 51.6 (actual observations give a value greater than 70), which shows the high efficiency of the genetic-like coding.
7) RSM(3,2, es,dc)
There isn't statistically significant differences between RSM(3,2, es.de) and RSM( 2, 2, es.de) models.
Summarizing for a single-cluster models:
coding efficiency decreases with increasing of shift factor of one coordinate. On average, the efficiency tends to 127;
coding efficiency insignificantly decreases with increasing of message source's space. For the studied parameter's values, it is more than 94.
For a multi-cluster models:
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coding efficiency is not significantly changed hy increasing the number of clusters with a random shift factor, hi this case, coding efficiency is always greater than 80.
coding efficiency increases with increasing number of clusters with deterministic shift factor. Coding efficiency asymptotically tends to 182;
coding efficiency decreases with increasing of distance between the clusters. On average, the efficiency tends to 112;
coding efficiency decreases, but still significantly better than random coding, with increasing the percentage of cluster; for tested parameters' values the efficiency is greater then 84
coding efficiency decreases with increasing scale factor of one cluster's coordinate by the same way as for a single-cluster models. Coding efficiency asymptotically tends to 122.
the dependence of the coding efficiency from the distribution is not essential. Genetic-like coding is equally copes well with all distributions. Estimation of the mean efficiency is 116 with a standard deviation of 21.8;
encoding efficiency does not change with an increase in the dimension of message source's space. Estimation of the mean efficiency is 119.2 with a standard deviation of 22.83.
Thus, it can be argued that the genetic-like coding is many times more effective than random coding for any field of message source.
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This can be seen on the example of the genetic-like encoding of constellation points, fig. î (16APSK and 32APSK constellations, coding rate 9/10). For comparison, fig. 2 shows constellation points with code combinations used in standard of DVB-S2. On both, fig. 1 and fig. 2, constellation points, which have Hamming distance of I between their code combinations, are connected with lines. Due to the presence of matching elements in the matrix of pairwise distances between the constellation points, it is difficult for the algorithm to perform distance-based categories assigning properly. Nevertheless, the results arc quite close to the results for codc combinations used in standard of DVB-S2 in terms of minimizing q, table 5.
Table 5
Comparative analysis of Qmin and deviations caused by a single-hit error
I6APSK Gcnctic-like coding I6APSK Standard of DVB-S2 32 A PS K Genetic-like coding 32APSK Standard of DVB-S2
On 'i 0,957053502094 0,853905820597 0.746387005736 0.678843950636
Deviation 0,959127735482 0,852389499389 0.74721 ¡¡793029 0.677649964553
Genetic-like coding of deterministic structures
As mentioned above, one of the characteristic features of the genetic-like coding algorithm is the ongoing analysis of the distances between the points. This fact is of great importance when the message source's structure has deterministic nature. For example, there is a set of matching elements in the matrix of pairwise distances between the points when the algorithm is applied to the regular structure of the message source, which makes the process of distance-based categories assigning indeterminate. This, in turn, affects the effectiveness of coding in terms of minimizing the distances between the messages in the original metric space of the message source, which have the smallest Hamming distance between their code combinations (this distance will hereinafter be referred to as Q ). Thus, it is
possible that noise immunity will be substantially reduced.
16APSK Genetic afg. 32APSK Genetic a!g.
1.0 0.5 0.0 -0.5 -1.0
-1.0 -0.5 0.0 0.5 1.0
-1.0 -0.5 0.0 0.5 1.0
Figure I, Gcnctic-like coding 16APSK DVB-S2 32AP S K DV8-S2
1.0 0 5 0.0 -0.5 -1.0
\
-1.0 -0.5 0.0 05 10 -1.0 -0.5 0.0 0-5
Figure 2. DVB-S2
Table 5 also shows the comparative analysis of deviations caused by a single bit-error, i.e. mean deviation or mean distance between sent and decoded with error constel lation points in the original metric. These values are close to the values of Qt.....,
which further confirms the effectiveness of coding methods.
Conclusion
The analysis showed an extremely high efficiency of preliminary genetic-like coding to improve the noise immunity of transmission in the presence of impulse noise. This efficiency is typically measured in the hundreds of sigma and can he obtained for a known source by analyzing its structure. Worst case is the encoding of regular structures due to the specifics of the proposed algorithm. But for these structures the results are very close to existing encodings.
The authors believe that one of the main results of this research is that close enough performance between genetic-1 ike coding for modulation codes and modulation codes used in DVB-S2 digital television broadcast standard was achieved. It should be noted that genetic-like coding algorithm has a high resistance to interference and there's a need for its further study to that enhance genetic-like coding versatility and effectiveness.
References
1. Holland J.H. Adaptation in Natural and Artificial Systems. Ann Arbor: The University of Michigan Press, 1975.
2. Sinevct I.S. Improving the quality of the transmission codes, based on the topology of the message source. In the book: Information Society Technologies: Proceedings of the Moscow branch scientific and technical conference. Moscow: Insvyazizdat, 2007, Pp. 169-170. (in Russian)
3. Fenchuk M.M.. Sineva I.S. Genetic algorithm optimization for space of arbitrary dimension. T-Comm. 2015. Vol 9. No.7, pp. 74-79. (in Russian).
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4. Fençftuk M.M.. Sineva /.S. Noise immunity analysis of genetic code using cyclic redundancy check method. T-Comm. 2014. Vol 8. No. 11, pp. 108-112. (in Russian).
5. Fenchuk M Ai, Batalov A.E., Sineva I.S. Increase noise immunity of CRC codes using preliminary genetic coding of a m et ri zed messages source / Fundamental problems of radio-electronic instrument; Proceedings of the International Scientific and Technical Conference INTERMAT1C-2013; Encrgoatomizdat. 2013 4), pp. 65-70. (in Russian).
6. Adzhemov A.S., Gorbunov N.V., Sineva IS. Evaluating the effectiveness of the genetic encoding algorithm for different
МАТЕМАТИКА
distributions of sources and a variety of metrization / Scientific Conference of faculty, scientific and technical staff, materials; Moscow, 2002: materials. Moscow: MTUCI, 2002. Pp. 106-107. (in Russian).
7. Batalov A.E., Sineva I.S. Optimization of genetic algorithms of message source coding. T-Comm. 2014. Vol 8. No.12, pp. 6-9. (in Russian).
8. Yakovlev D.A.. Sineva I.S. Parallel computing in genetic search algorithms /Fundamental problems of radio-electronic instrument; Energoatomizdat, 2014, (5), pp. 214-219. (in Russian).
ПРЕДВАРИТЕЛЬНОЕ КОДИРОВАНИЕ ГЕНЕТИЧЕСКОГО ТИПА ДЛЯ СЛУЧАЙНЫХ И ДЕТЕРМИНИРОВАННЫХ ПРОСТРАНСТВ ИСТОЧНИКА СООБЩЕНИЙ
Фенчук Михаил Михайлович, аспирант, МТУСИ, Москва, Россия, mikhail.fenchuk@gmail.com
Синева Ирина Сергеевна, доцент, МТУСИ, Москва, Россия, iss@mtuci.ru Ботт Анна Витальевна, инженер, Научно-исследовательский институт точных приборов, Москва, Россия,
bottane4ka@yandex.ru
Аннотация
Проводится исследование алгоритма кодирования генетического типа для случайных и детерминированных пространств источника сообщений. В основе разработанного авторами метода кодирования лежит следующий принцип: сохраняется соответствие наименьших расстояний между сообщениями в метрике исходного пространства источника и блоков кода с наименьшим расстоянием Хэмминга. Таким образом, использование алгоритма генетического кодирования позволяет минимизировать ошибку декодирования. Полученные результаты исследования подтверждают возможность использования генетического подхода как средства повышения помехоустойчивости. В предыдущих работах авторов было показано, что предварительное генетическое кодирование нерегулярных пространств источника может существенным образом повысить помехоустойчивость, однако, регулярные структуры в этом смысле являются "наихудшими" с точки зрения работы алгоритма. Настоящее исследование продолжает их анализ. Первый этап исследования заключается в сравнительном анализе оптимальности кодирования с точки зрения минимизации расстояния по Евклиду между сообщениями, которым соответствуют блоки кода с минимальным расстоянием Хэмминга. Данный анализ проводится между алгоритмом генетического типа и случайным кодированием. Аналогично первому, на втором этапе исследования проводится сравнительный анализ отклонений, вызванных ошибкой декодирования, в метрике исходного пространства, т.е. рассматривается среднее отклонение или расстояние "перехода" между отправленным сообщением и сообщением, принятым с ошибкой. Данный анализ проводится относительно пар сообщений, между соответствующими блоками кода которых был осуществлен "переход" на единичное расстояние Хэмминга. Сопоставление результатов, полученных в ходе двухэтапного анализа, даёт наиболее полное представление о преимуществах и недостатках предлагаемого авторами метода кодирования, а также указывает на необходимость дальнейшего изучения работы генетических алгоритмов над случайными и детерминированными структурами. Авторы считают, что одним из основных результатов данного исследования является получение достаточно близких показателей между генетическим модуляционным кодом и модуляционным кодом, использующимся в стандарте цифрового телевидения DVB-S2 по рассматриваемым критериям оптимальности.
Ключевые слова: генетический алгоритм, помехоустойчивость, кодирование генетического типа, равномерное кодирование, исправление ошибок.
Литерура
1. Holland J.H. Adaptation in Natural and Artificial Systems. Ann Arbor: The University of Michigan Press, 1975.
2. Синева И.С. Улучшение качества передачи кодами, опирающимися на топологию источника сообщений / В кн.: Технологии информационного общества: Тезисы докладов московской отраслевой научно-технической конференции. - М.: Инсвязьиздат, 2007. - С. 169-170.
3. Фенчук М.М., Синева И.С. Оптимизация алгоритма генетического кодирования для пространств произвольных размерностей // T-Comm: Телекоммуникации и транспорт. - 2015. - Том 9. - №7. - С. 74-79.
4. Фенчук М.М., Синева И.С. Анализ помехоустойчивости генетического кодирования с применением циклического избыточного кода // T-Comm: Телекоммуникации и транспорт. - 2014. - №11. - С. 108-1 12.
5. Фенчук М.М, Баталов А.Э., Синева И.С. Повышение помехоустойчивости кодов CRC при помощи предварительного
генетического кодирования метризованного источника сообщений // Фундаментальные проблемы радиоэлектронного приборостроения. - М.: Энергоатомиздат, 2013, часть 4. - С. 65-70.
6. Аджемов А.С., Горбунов Н.В., Синева И.С. Оценка эффективности генетического алгоритма кодирования сообщений при различных распределениях источников и их разнообразных метризациях // В кн.: Научная конференция профессорско-преподавательского, научного и инженерно-технического состава. - М.: МТУСИ, 2002. - С. 106-107.
7. Баталов А.Э., Синева И.С. Оптимизация алгоритма генетического кодирования источника сообщений // Т-Сотт: Телекоммуникации и транспорт. - 2014. - №12. - С. 6-9.
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