Proß, Wolfgang

Quint, Franz

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   Information Technology and Electrical  Engineering - Devices and Systems, Materials  and Technologies for the Future        Faculty of Electrical Engineering and Information Technology    Startseite / Index:  http://www.db-thueringen.de/servlets/DocumentServlet?id=14089 54. IWK Internationales Wissenschaftliches Kolloquium International Scientific Colloquium 07 - 10 September 2009 PROCEEDINGS Impressum  Herausgeber: Der Rektor der Technischen Universität llmenau  Univ.-Prof. Dr. rer. nat. habil. Dr. h. c. Prof. h. c. Peter Scharff  Redaktion: Referat Marketing   Andrea Schneider   Fakultät für Elektrotechnik und Informationstechnik  Univ.-Prof. Dr.-Ing. Frank Berger  Redaktionsschluss: 17. August 2009  Technische Realisierung (USB-Flash-Ausgabe):  Institut für Medientechnik an der TU Ilmenau  Dipl.-Ing. Christian Weigel  Dipl.-Ing. Helge Drumm  Technische Realisierung (Online-Ausgabe):  Universitätsbibliothek Ilmenau    Postfach 10 05 65  98684 Ilmenau  Verlag:   Verlag ISLE, Betriebsstätte des ISLE e.V.  Werner-von-Siemens-Str. 16  98693 llmenau   © Technische Universität llmenau (Thür.) 2009   Diese Publikationen und alle in ihr enthaltenen Beiträge und Abbildungen sind urheberrechtlich geschützt.   ISBN (USB-Flash-Ausgabe): 978-3-938843-45-1 ISBN (Druckausgabe der Kurzfassungen): 978-3-938843-44-4  Startseite / Index: http://www.db-thueringen.de/servlets/DocumentServlet?id=14089  COMPARATIVE STUDY OF A CDMA2000 TURBO CODE  AND A LINEAR TIME ENCODABLE PEG LDPC CODE OVER GF(q)  Wolfgang Proß, Franz Quint  University of Applied Sciences Karlsruhe Department of Electrical Engineering and Information Technology   ABSTRACT  In this paper we compare a CDMA2000 Turbo code and two linear-time encodable irregular PEG LDPC codes, a binary one and one defined over the Galois field GF(64). An overview over the construction methods of the Shannon limit approaching codes is given. For each code the simulation results are depicted in terms of Bit Error Rate (BER) where several decoding iterations have been applied. Thereby the simulated codes share the relatively short binary code word length of  and a code rate of . For a channel model the additive white Gaussian noise channel (AWGNC) has been used and the decoding was done with the belief propagation (BP)-decoding algorithm.  Index Terms – Turbo codes, LDCP codes, progressive edge-growth (PEG), GF(q), nonbinary, irregular, linear-time encodable codes, girth 1. INTRODUCTION In 1948 Claude E. Shannon published A mathematical theory of communication [1], where the definition of the Shannon limit stems from. This limit provides a lower bound on the signal to noise ratio. With the introduction of Turbo codes in 1993 by Berrou, Galvieux and Thitimajshima [2] a concatenated coding scheme was proposed that yielded a near Shannon limit decoding. Low-Density Parity-Check (LDPC) codes had already been published in 1962 by Robert Gallager [3] but then have been forgotten for three decades. When they were rediscovered by MacKay and Neal in 1995 [4], a real competitor to Turbo codes arose. This competition has become fierce even for short code word length , due to various improvements for the LDPC code’s underlying parity-check matrix that have been published since then. Most notably progressive edge growth (PEG) LDPC codes presented by Hu, Eleftheriou and Arnold in [5] appear to be outstanding.  2. SIMULATIONS The simulations in this paper are executed using the all-zero codeword which is always a valid codeword of any linear code. The codeword is BPSK modulated and then applied to the additive white Gaussian noise channel (AWGNC). Thereby white Gaussian noise gets added depending on the signal to noise ratio (SNR) which in channel coding simulations is usually defined as  (energy per bit) divided by  (spectral noise density). The variance of the Gaussian amplitude distribution is      (1)  and thus is not only dependent on  but also on the code rate . This way it is possible to compare codes that exhibit different code rates. With  being a sent bit, the conditional probability of a bit  being received by the decoder is then distributed as follows:   ,   (2)  For a decoder that processes soft-decision values in the log-domain, a convenient format is the log likelihood ratio (LLR) which for the AWGNC is expressed as   .   (3)  To depict the error correcting capabilities of a code, the bit error ratio (BER) is plotted on the y-axis of a graph while the according -values are plotted on the x-axis. The BER is obtained by dividing all errors occurring in a decoded codeword by the length of the code word  where  for all simulations throughout this paper. © 2009 - 54th Internationales Wissenschaftliches Kolloquium3. TURBO CODES Turbo codes in general are defined by a serial or parallel concatenation of several channel-codes and an appropriate decoder that processes soft decisions in an iterative way. Thereby the Turbo decoder constantly refines the decoding result by an exchange of data (called extrinsic data) among the component decoders. The name Turbo code stems from the similarity of the decoder’s functional principle to a turbocharger.  3.1. Turbo Encoder Here a Turbo encoder of rate  (Figure 1) is established as recommended in the CDMA2000 standard [6]. It consists of a parallel concatenation of two 8-state convolutional encoders, where one of these encodes the information sequence  (also called the systematic part) and the other one a random-interleaved version  of  with . Each convolutional encoder is described by the generator polynomial in octal notation:       (4)  The output of the first component encoder is denoted as  (parity part one) and the coded bit of the second one as . By use of an appropriate puncturing pattern  and  get compacted into . A Turbo code word then comprises of the systematic part and the packed parity part so that .                Figure 1: Turbo encoder r=1/2 3.2. Turbo decoder The Turbo decoder shown in Figure 2 consists of two BCJR component decoders that exchange extrinsic data in an iterative process. The BCJR decoder was presented in 1974 by Bahl, Cocke, Jelinek, and Raviv [7]. It describes a soft input/soft output decoder that maximizes the a posteriori probability  (MAP) with the BPSK modulated bit . Using Bayes’ theorem [8] the LLR is derived which in case of the AWGNC is . Thus the decoder is called a log MAP decoder.            Figure 2: Turbo decoder 3.3. Turbo code simulations As mentioned before, the simulation was executing using a code word of length . The result is seen in Figure 3.  Figure 3: BER of Turbo code; n=1008; r=1/2 4. LDPC CODES LDPC codes are asymptotically optimal regarding the Shannon-limit. To achieve this optimum, the code-length has to be very large and thus the decoding complexity gets extremely high.  4.1. Parity-check matrix & Tanner graph The name of LDPC codes stems from their underlying sparse Parity-Check Matrix (PCM) which exhibits a low density of nonzero elements. In the following example (Figure 4) a PCM is shown together with their corresponding Parity-check equations. 1,00E-041,00E-031,00E-021,00E-011,00E+000 0,5 1 1,5 21 iteration 3 iterations8 iterations 25 iterations80 iterations 100 iterationsuncoded    D D D D D D   Dec1           Dec2  =      Figure 4: PCM example  With the help of the Gaussian elimination any PCM can be transformed to       (5)  with  being the identity matrix. From this the generator matrix      (6)  is derived. A codeword is then obtained by multiplying the information word  with the generator-matrix .  Alternatively to the matrix representation, LDPC codes can be represented by a Tanner graph [9]. This bipartite graph consists of symbol-nodes and check-nodes connected via edges. Thereby check-nodes depict the rows, symbol-nodes the columns and edges the nonzero-entries of the PCM respectively. In Figure 5 the Tanner graph corresponding to the PCM in Figure 4 is shown. The black edges adjacent to symbol-node  correspond to the bold nonzero entries in the PCM.               Figure 5:  Tanner graph example 4.2. Cycles & girth Any set of consecutive edges that connect node  with node  is called a path. If such a path connects a node  with itself, it is called a cycle of . In Figure 5 a cycle of length 4 is shown. Local girth  refers to the length of the shortest cycle a symbol-node  participates in. Global girth  is defined as  and so equals the length of the shortest cycle in the graph [5]. A low global girth has a harmful impact on the decoding performance which is thus mainly dependent on the construction of the PCM or the Tanner graph. This is the reason for optimizing the construction method in reference to the resulting decoding performance. 4.3. Regular & Irregular LDPC codes The PCM of regular LDPC codes possesses exactly  nonzero elements in each column and  in each row and thus all check-nodes and symbol-nodes share the same number of adjacent edges respectively. When Gallager introduced LDPC codes in [3] he also proposed a pseudorandom construction of a regular PCM. Though he suggested avoiding 4-cycles, he did not provide any advice on how to do so. Figure 4 shows the PCM of a  Gallager code and Figure 5 the appropriate Tanner graph. In contrast to regular LDPC codes, irregular codes exhibit several row and column weights. They are described through the use of the symbol-node degree distribution   ,   (7)  where  is the maximum number of edges connected to a symbol-node in the graph and  is the fraction of symbol-nodes connected to  check-nodes. Since it is a distribution it follows:  .    (8) 4.4. PEG LDPC codes In [5] a construction method is described which is based on Tanner graphs and maximizes the global girth. This is done by progressively establishing edges between the symbol- and check-nodes. For each placement of an ), the check-node  to get connected to the current symbol-node  is chosen in a way that the local girth  is maximized. This way the global girth  is maximized as well because . There are three different situations when choosing a check-node  in order to establish an edge ): 1. If it is the first edge to get connected to a symbol-node  choose the check-node having the lowest check-node degree (fewest connected edges) under the current graph settings. 2. If there are still check-nodes that are not already connected to the current graph  choose one of them. 3. If neither of the two former cases are true  establish a PEG-tree with  as a root of that tree. Then choose a check-node of the bottom-layer. As an example for a PEG-tree the creation process with symbol-node  as a root is depicted in Figure 6.  symbol- nodes check- nodes                4-cycle The encoding time of LDPC codes usually increases with . Hu, Eleftheriou and Arnold also propose an improved version of irregular PEG LDPC codes in [5]. It is based on a partitioning of the PCM so that   , with  . (9)  The parity bits can then be calculated according to    (10) where .  Figure 6: PEG-tree for  4.5. Nonbinary LDPC codes By an increase of a binary PCM’s column weight, the Hamming weight spectrum and hence the decoding performance gets improved. The drawback is that if the PCM possesses more nonzero entries, the number of cycles increases which results in a degradation of the codes error correction capabilities. When moving to GF(q) the mean column weight increases while the number of cycles in the nonbinary Tanner graph remains the same [10]. The construction methods to attain a nonbinary PCM do not differ from those of binary LDPC codes. But in contrast to the latter, the PCM of nonbinary LDPC codes possesses elements defined over the Galois field . Thereby the nonzero entries in  are generated through the use of a primitive polynomial  where . It is also essential to realize calculations required during the decoding process in the Galois field . They are based on a polynomial representation of the elements. A  symbol is represented by  bit, whereas the exponents of the corresponding polynomial stand for the indices of the several bits and the coefficients for their value.  4.6. LDPC code simulations Here a binary irregular PEG LDPC code that is linear-time encodable is simulated. The length of the codeword is  and the symbol-node degree distribution is , which is the same as the one in [5]. The belief propagation (BP) decoder applied for this simulation works in the log domain as is described in [11]. The simulation results can be seen in Figure 7.  Figure 7: BER of linear-time encodable PEG LDPC code; n=1008; r=1/2 1,00E-051,00E-041,00E-031,00E-021,00E-011,00E+000 0,5 1 1,5 21 iteration 3 iterations8 iterations 25 iterations80 iterations 100 iterationsuncodedTanner- graph PEG- tree                1 2 3 4 5              1 2 3 4  5   For constructing the nonbinary LDPC code, GF  has been chosen with  which is taken from [7]. The code word length is  in symbols and  in bits. The decoding was done with a FFT-based BP-decoder as described in [12].  Figure 8: BER of linear-time encodable PEG LDPC code defined over GF(64); n=168; r=1/2 5. COMPARISON In Figure 9 the BER of the following codes is showed: 1. CDMA2000 Turbo code, , , 100 decoding iterations 2. Irregular linear-time encodable PEG LDPC code; , , 100 decoding iterations 3. Irregular linear-time encodable PEG LDPC code over GF(64), , , , 100 decoding iterations  As seen in Figure 9 the LDPC code defined over GF(64) clearly beats the Turbo code in terms of BER. Furthermore it is important to point out that LDPC-decoders are fully parallelizable and should therefore offer extremely high-speed applications. Moreover the parity-check matrix (PCM) of LDPC codes can be used to check after each iteration if a valid codeword has been found and thus the number of computations can be strongly reduced. In contrast to Turbo codes the minimum distance is higher and thus the LDPC decoder can detect its own errors.  Figure 9: Comparison of Turbo code and LDPC code 6. CONCLUSION The PEG algorithm offers an effective construction method for high girth LDPC codes that are competitive to Turbo codes. Especially when moving to higher order Galois fields GF(q) irregular PEG LDPC codes beat the applied Turbo code even for a short code word length. As a result of this comparison we construct a near Shannon limit coding scheme for 2D-Data Matrix code applications using the explored nonbinary linear-time encoding PEG LDPC code. This leads to better results in terms of BER and computational burden. 7. ACKNOWLEDGEMENT This work is part of the project MERSES and has been supported by the European Union through its European regional development fund (ERDF) and by the German state Baden-Württemberg.  [1] C.E. Shannon, "A mathematical theory of communication", Bell System Technical Journal, vol. 27, no. 3, p. 4. [2] C. Berrou, A. Glavieux, and P. Thitimajshima, Eds., Near Shannon limit error-correcting coding and decoding: Turbo-codes, 1993. [3] R. Gallager, "Low-density parity-check codes", Information Theory, IRE Transactions on, vol. 8, no. 1, pp. 21-28. 1,00E-051,00E-041,00E-031,00E-021,00E-011,00E+000 0,5 1 1,5 21 iteration 3 iterations8 iterations 25 iterations80 iterations 100 iterationsuncoded1,00E-051,00E-041,00E-031,00E-021,00E-011,00E+000 0,5 1 1,5 2uncoded LDPCTurbo code GF(64) LDPC  [4] MacKay D.J., and Neal R.M., Good Codes Based on Very Sparse Matrices: Cryptography and Coding. 5th IMA Conf.(Cirencester, UK), LNCS 1025: Berlin: Springer, 1995. [5] X.Y. Hu, E. Eleftheriou, and D.M. Arnold, "Regular and irregular progressive edge-growth Tanner graphs", IEEE Transactions on Information Theory, vol. 51, no. 1, pp. 386-398. [6] PN T.I., 4694 (TIA/E1A/IS-2000-2-A): Physical Layer Standard for cdma2000 Spread Spectrum Systems, Draft Baseline Version: TIA Standard 3GPP2 C. P, 1999.  [7] L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, "Optimal decoding of linear codes for minimizing symbol error rate (Corresp.)", IEEE Transactions on Information Theory, vol. 20, no. 2, pp. 284-287. [8] T. Bayes, and T. Bayes, "An essay towards solving a problem in the doctrine of chances. 1763", MD Comput, vol. 8, no. 3, pp. 157-71. [9] R. Tanner, "A recursive approach to low complexity codes", IEEE Transactions on Information Theory, vol. 27, no. 5, pp. 533-547. [10] M.C. Davey, "Error-correction using low-density parity-check codes", Univ. of Cambridge PhD dissertation. [11] W.E. Ryan, "An Introduction to LDPC Codes", The University of Arizona, 2003. [12] D. Declercq, and M. Fossorier, "Decoding algorithms for nonbinary LDPC codes over GF (q)", IEEE Transactions on Communications, vol. 55, no. 4, p. 633.   

Comparative study of a CDMA2000 turbo code and a linear time encodable PEG LDPC code over GF(q)

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Comparative study of a CDMA2000 turbo code and a linear time encodable PEG LDPC code over GF(q)

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