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A METHOD FOR IDENTIFYING LDPC CODES FOR ADAPTIVE COMMUNICATIONAL SYSTEMS
DOI 10.24411/2072-8735-2018-10281
Nikita Yu. Rumenko,
Moscow technical university of communication and informatics, Moscow, Russia, mtuci@mtuci.ru
Alexander V. Kostyuck,
Moscow technical university of communication and informatics, Keywords: identification, error-correcting codes,
Moscow, Russia, tigran 201094@ mail.ru LDPC, systems of equations.
Ever-increasing requirements for high-quality and high-speed communications lead to more and more sophisticated, noise-tolerant waveforms and communication protocols. Recent advances in power control, adaptive coding and modulation now makes it possible to transmit signal under much worse channel conditions than as far as twenty years ago. In cognitive radio systems, as well as in non-cooperative communication channels, the parameters of error-control code often are not known a priori at the receiver. Hence, the development of blind code recognition systems is essential for future cognitive communications. Among other known error-correcting codes one of the most important class is iteratively decodable codes, especially low-density parity check ones. Excellent error-correcting performance of LDPC made them part of several communicational standards, including Wi-Fi, DVB-S2(x) and DVB-T2. Accordingly, there exist a lot of works considering LDPC blind identification methods, all of which being in fact methods of solving systems of linear equations [1-5].
In this work, we will study one special case, namely, we will consider a code with relatively dense parity checks used in a good, but rather low-rate channel. Note that in this case exhaustive search techniques including "birthday" algorithms cannot be used due to high computational complexity, and information set decoding algorithms will not have enough codewords. To handle this problem we adapt information-set decoding to few-codewords scenario and show, that for good enough channels even parity checks of weight larger than 10 can be relatively easy to find. Although all the results of this paper are not limited to the binary case, we will mainly focus on systems of boolean equations, due to the fact that most of LDPC in modern communications are binary.
Information about authors:
Nikita Yu. Rumenko, Moscow technical university of communication and informatics, research engineer, Moscow, Russia Alexander V. Kostyuck, Moscow technical university of communication and informatics, PhD, senior research engineer, Moscow, Russia
Для цитирования:
Руменко Н.Ю., Костюк А.В. Метод распознавания LDPC-кодов для адаптивных систем связи // T-Comm: Телекоммуникации и транспорт. 2019. Том 13. №6. С. 60-64.
For citation:
Rumenko N.Yu., Kostyuck A.V. (2019). A method for identifying LDPC codes for adaptive communicational systems. T-Comm, vol. 13, no.6, pр. 60-64. (in Russian)
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Previous results
J us i like turben codes, L D PC allow for efficient iterative decoding. Decoding LDPC is based on the different versions of "belief-propagation" algorithm, which works on a graph built from the parity-check matrix of the code. Its good performance for LDPC accounts for a special structure of parity check matrix, that has to be sparse, according to the name of the code. Note that this relaxed definition lets one consider many codes having sparse parity checks as LDPC, so all the results of ihis paper are not limited to LDPC proper. Technically, LDPC belong to the class of linear block codes.
Let us call parity check matrix some matrix H e ,runk(H)-n-k, which defines a code C(«,A) as vector space of c e F" satisfying He = 0.
Linear combinations of rows of// are called parity checks of C(n.k). Define |mj the Hamming weight Of jr. Now, LDPC can be defined as codes having (n-k) linearly independent parity checks h. satisfying ||/;J < w,i e l,(/i — k), vv « n .
Since LDPC are linear, their codewords represent linear equations with respeel to parity checks, i.e. every parity check makes a solution for any c e C( n, k). From that, finding (n — k) linearly independent solutions allows one to form parity check matrix of the code, i.e. identifying LDPC is equivalent to solving a system of linear equations, every equation being a codeword.
in practice, when solving these systems of equations, several problems can occur, mainly channel distortions and low channel rate. In this case we can use exhaustive search techniques []], which can handle large probabilities of distortion, but for rather small vt'. In case of bigger W one can use information-set decoding [1], which does variable substitution to change initial unknowns for unknowns defined by errors and therefore change the weight of the solution. However, in its existing form information-set decoding requires a large number of codewords for variable substitution, which in many cases is just impossible, for instance with low-rate channels.
From that we see that for low levels of distortion and large enough solution weight we cannot use existing methods to identity a code, so let us focus on solving this problem.
We study systems of linear Boolean equations consisting of codewords received from a memoryless binary symmetric channel with crossover probability p. Note that for a code C{n,k) any such system represents (n-k) different systems with the same letl-hand side, so without loss of generality one can consider just one of these systems and assume single parity check.
It would be also convenient to consider all the distortions concentrated in the right-hand side of the system. With this simplification we do not change the solution and only increase the probability of distortion [7j.
Suppose m codewords have been received. Let's form a system of linear Boolean equations
4%-f, U)
where A is a random Boolean rectangular matrix of size in x n, tank(A) = til, i' - (^i,.. .,ir„) is the unknown Boolean distortion vector with independent elements taking its values with probabilities
Pr(£ = 1) = 1 - Pr(£ = 0) = p,p e[0;l/2),/ e[l;w] -
Systems like (!) are cal led systems with corrupted right-hand side and vector JCn is called the true solution. It is assumed that tbfJ < vi\|i|| < t, where / depends on m andp, H' is fixed.
Now let us pass on to our method. Form a homogeneous system Xe = 0, (2)
where X — [/ | , e1 - [i x<, | and I is identity matrix.
Clearly, with this structure finding e allows one to find-Y„ and
solve system (I}. However, since system (2) is even more undetermined than (I). we still don't have not enough equations to make full variable substitution. Instead, we substitute as many variables as possible, updating variables with random permutations. For accelerating the search, we additionally use "meet-in-the-middle" approach with low memory requirements on the right-hand side of the system.
Let us denote F"" = ¡re F:",||i-| = w} the set of all binary
vectors of length n and weight w. We propose an algorithm based, on [6], with numeric parameters (w, /7, w,qj): Algorithm 1
Assign qt = |q/4\, q2 = / 4j,
/ =
log;
q/2
log,
<h)
I) form system A'according to (2) Procedure 1
- choose random permutation P of length (m + n)
- assign X <— XP
- bring A'to the form X = [/1 A] Procedure 2
- choose submatrix L formed by / first rows of matrix A
- form lists Q{t) = | a « Lv,y € K""'} and
Procedure 3
for all s 6 [0;2'' -1]
- form a list
0131 = i(e,v) | cmn®b,v = y®z,(a,y) e Q{V\(b,z) e £>(3'} of pairs (e, v), s.t. c - s on the first I. bits for all
- calculate vector x e V{q) ,x = v- © Vj
- if | ,'lvj| > vv + / go to Procedure 1
- else assign e = [Ax | x\-P~l and return ,v(l - ,.. .cj„t(I1 ].
Let us estimate for this algorithm the probability of success. Theorem /.
Proposed algorithm finds right solution for system (1) with probability not less than (1 - S), where
F(-x,y) = jfJtt-+fl - *) In ^ and
y (i-.v)
V{yM) :={veF: :||if<w + f}.
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Proof
Note that ones in vector £ are Bernoulli variables with parameter/?, and ones in any "wrong1' sum of columns of A' are Bernoulli variables with parameter 1/2.
Let lis calculate the probability of success from its complement A: {not finding the right solution!.
The A event represents a union of three compatible events
1) A, - all the / permutations were bad;
2) JL - weight of e is more than (w+f}j;
3) A3 - at least one of |l<"'n"+"| -1) wrong sums x of
olumns Jfhas weight not more than (w+f) , Since probability of compatible events is not greater than the sum of their probabilities, we have P(A)< P(A,)+ P(A2) + P(A,)
and
P( Aj)< (|Khv+"| - 1 )/>{|*|< M' + /} •
The probability of a good permutation is mainly defined by the threshold I. thus, after / iterations, we have
P(A)<
Denoting
f 1 _ ( HI ~ I 1 V ' « + m ^
1 [w+t-qj ^w+tj
' V
m
\w + t-q
fn\
n + m w+t
and choosing
Theorem 2.
Choosing I —
n
q/2)
and I] > 6 computational complexity
of the proposed algorithm is not greater than
W <expuvt'+/)ln
n + m
m—l—w+t + q
+q In
q(m-l-w-t + q)
+ q /2 In n + 2 In m > {W+t~q\n-q) ) J
Proof
Every iteration consists of three stages:
1) elimination
2) lists calculation
3) weight checking.
Thereby, following [6], computational complexity of one:
iteration equals
„, (n + mY(n~m) (i) ~ Ö +/
n
[q/2
+2(w + f)
' n *
{q/2)
/2'.
Since we better choose q> 6 because of the elimination step, minimum complexity for one iteration will be of order (?(«)» from that, the complexity is defined by the two last terms, hence
choosing / =
t „ \
wv
we will have
W[U<m-
n
and for overall complexity
/ = \m / one gets
According to the C hem off bound, probability of A2 satisfies
P{A2)<e'mf'[p"m).
Likewise, for A* we have
Final lyt
Substituting j one gets the statement of the
theorem □
Let us now estimate computational complexity of the algorithm according to the number of iterations defined in the proof of Theorem 1.
Following iterative structure of the algorithm, its computational complexity IV has to be a product of the computational complexity of one iteration fV(l, and the number of iterations.
W <m2-
' n + mx
w+t
q/2) fnY m-t ^ ,q){w + t-q
Let us estimate
inequality chain holds: n + 111
w + t
(n+m^ H f m-l 1
{ W + t j yW + r-qj
Following
Y m-t ^
sw+t-q
( w + t-q)\(n-q)\{n + m)\(m-l -w-t+ q)\q\
(w + /) !/j!(/) + m -w-t) l(m -l)\
n + m
m-l-w + t+ q ^
q(m-l-w-t + q)
(w + t-q)(n-q} Taking logarithm, we get an estimate on the computational, complexity of the algorithm as
r ' n + m \
+
W <exp-j (ii' + i)ln + iyln
m-l-w+t + q q( m - /'- w ~l+q)
(iW+t-q)(n-q)
+ q / 2 In n + 2 In m
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Performance evaluation
We implemented our Algorithm in C++. Since it is usually advised to use "meet-in-the-middle11 approach for identifying LDPC in our scenario, we compared proposed algorithm (Algorithm 2) with [Ij (Algorithm 1}.
All the systems were modeled using pseudo-random number generators from <random> file in Visual Studio 13.
Table 1
SLE n = 256, m = 128, - 8, q = 2_
p Mean time, see
Algorithm 1 Algorithm 2
0.001 144,284 0,062
0.005 164,131 0.120
0.010 191,219 0,355
0.020 305.235 1,625
Table 2 SLF n = 512, m = 256, w = 10. Q = 2
P Mean lime, sec
Algorithm 1 Algorithm 2
0.001 >7200 12,100
0.005 >7200 49,021
0.010 >7200 109,960
0.020 >7200 576.460
Experiments were carried out on the personal computer with Intel Core i7 (2,8 GHz) and DDR3 RAM of size 8 Gb. Mean computational time for the two competing algorithms is presented in the tables. Note that sampling reliability of solution lor all experiments bad not gone below 0,99.
Conclusion
The problem ofblind parameters identification is essential for future cognitive radio systems. Here we study the problem of identifying LDPC codes using a sample of a small number of noise-corrupted codewords. We have presented a method for identification those codes and an efficient algorithm, which outperforms previously known ones.
References
1. Cluzeau M., Finiasz M. (2009). Recovering length and Synchronisation from a noisy intercepted hitstream.
2. Chabot C. (2007). Recognition of a code in a noisy environment, IEEE Conference. IS IT'<>7, pp. 2210-2215.
3. Cluzeau M. (2006). Block eudt reconstruction using iterative decoding techniques. In IEEE Conference. IS1T06, pp. 2269-2273, Seattle, USA. IEEE Press.
4. Sicoi Ci. and Houcke S. (2005). Blind detection of interleaver parameters. In ICCASPOS.
5. Valembois A. (2001). Detection and recognition of a binary linear code Discrete Applied Mathematics, 111 (I -2); 199-218, July 2001.
6. May A.. Meurer A. and lhomae E. (2011) Decoding Random Linear Codes in O(2(l0i4,,i In Asiacrypi 2011. Springer, 2011.
7. Balakin G.V. (1997). Introduction to the theory of random systems of equations. (In Russian). Transactions in applied discrete mathematics. Vol. 1, pp. 1-18.
CJMMUNICATIONS
МЕТОД РАСПОЗНАВАНИЯ LDPC-КОДОВ ДЛЯ АДАПТИВНЫХ СИСТЕМ СВЯЗИ
Руменко Никита Юрьевич, Московский технический университет связи и информатики, Москва, Россия, mtuci@mtuci.ru Костюк Александр Владимирович, Московский технический университет связи и информатики, Москва, Россия
Аннотация
Непрерывное повышение требований к качеству связи и скорости передачи информации стимулирует постоянное совершенствование используемых сигнально-кодовых конструкций, повышение их помехозащищенности. Произошедшее в последние годы внедрение систем регулировки мощности ретранслятора, технологий адаптивной модуляции и кодирования позволяет сегодня осуществлять радиосвязь в значительно более сложных условиях передачи информации, чем это было еще около двадцати лет назад. Одним из наиболее важных отличительных признаков развития современных систем радиосвязи является так называемая концепция "когнитивного радио", которая основывается на идее подстройки одного или нескольких параметров сигнала в зависимости от качества радиолинии. В таких "когнитивных" системах неизвестным параметром сигнала может выступать помехоустойчивый код, поэтому для перспективных систем "когнитивного радио" важное значение имеет разработка методов "слепой" оценки параметров помехоустойчивого кодирования. Среди используемых на практике помехоустойчивых кодов особое место занимают коды, допускающие эффективную итерационную процедуру декодирования, особенно коды с малой плотностью проверок на четность, LDPC (англ. low-density parity check codes). Высокая исправляющая способность LDPC-кодов обусловила их включение во многие стандарты радиовещания, использующие адаптивную систему кодирования и модуляции, среди которых Wi-Fi, DVB-S2(x) и DVB-T2. В связи с этим большое количество публикаций посвящено вопросу распознавания LDPC по выборке кодовых слов, принятых из канала связи, которые по сути представляют собой различные методы решения искаженных систем линейных уравнений [1-5]. В данной статье нами будет исследован следующий частный случай: мы будем рассматривать LDPC-код со сравнительно плотными проверочными соотношениями, который используется в достаточно хорошем, но низкоскоростном канале связи. Для идентификации кода в таких условиях переборные алгоритмы и алгоритмы на основе парадокса "дней рождения" не подходят ввиду их слишком большой вычислительной сложности, а для применения алгоритмов декодирования по информационным множествам недостаточно кодовых слов. Для решения поставленной задачи в статье предлагается новый метод на основе информационных множеств и показывается, что для достаточно хороших каналов проверочные уравнения даже сравнительно большого веса могут быть найдены достаточно легко. Несмотря на то, что полученные результаты не ограничиваются двоичным случаем, в настоящей статье рассматриваются только системы булевых уравнений, ввиду того что в подавляющем большинстве современных систем связи применяются двоичные низкоплотностные коды.
Ключевые слова: распознавание, помехоустойчивые коды, LDPC, системы линейных уравнений.
Литература
1. Cluzeau M., Finiasz M. Recovering length and synchronization from a noisy intercepted bitstream, 2009.
2. Chabot C. Recognition of a code in a noisy environment. In IEEE Conference, ISIT'07. С. 2210-2215.
3. Cluzeau M. Block code reconstruction using iterative decoding techniques // In IEEE Conference, ISIT'06, Seattle, USA IEEE Press, 2006. С. 2269-2273.
4. Sicot G. and Houcke. S. Blind detection of interleaver parameters. In ICCASP05, 2005.
5. Valembois A. Detection and recognition of a binary linear code Discrete Applied Mathematics, 111(1-2):199-218, July 2001.
6. May A., Meurer A and Thomae E. Decoding Random Linear Codes in 0(20.054n) In Asiacrypt 2011. Springer, 2011.
7. Балакин Г.В. Введение в теорию случайных систем уравнений // Труды по дискретной математике. М.: ТВП, 1997. T. 1. C. 1-18.
Информация об авторах:
Руменко Никита Юрьевич, Московский технический университет связи и информатики, аспирант, Москва, Россия. Костюк Александр Владимирович, Московский технический университет связи и информатики, к.т.н., старший научный сотрудник, Москва, Россия
