Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) Codes for Deep Space and High Data Rate Applications
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1 Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) Codes for Deep Space and High Data Rate Applications Nikoleta Andreadou, Fotini-Niovi Pavlidou Dept. of Electrical & Computer Engineering Aristotle University of Thessaloniki Thessaloniki, Greece Stylianos Papaharalabos, P. Takis Mathiopoulos Institute for Space Applications and Remote Sensing National Observatory of Athens Athens, Greece Abstract In this paper we investigate and compare the performance of a selected class of Low-Density Parity-Check (LDPC) codes, i.e. Quasi-Cyclic (QC) LDPC codes, against the currently used turbo codes for deep space and high data rate applications. This is useful in future updates of the channel code option used for such applications by the Consultative Committee for Space Data Systems (CCSDS). A wide range of different code rates is examined and the obtained Bit Error Rate (BER) performance versus the Bit Energy to Noise Spectral Density ratio (E b /N ) is reported. Both low and high code rates are being studied, and specifically the /2, /3, /4 and /5 as well as the 2/3, 3/4 and 4/5 code rates are taken under consideration. The role of the iteration rounds in the decoding procedure of LDPC codes is also examined showing that QC-LDPC codes outperform the currently used CCSDS turbo codes. Another advantage of QC- LDPC codes is their simple encoding procedure, which reduces the encoding complexity. Since deep space applications are considered, the signals are transmitted over the Additive White Gaussian Noise (AWGN) channel, with the specific block size of 736 bits. Keywords Quasi-Cyclic Low-Density Parity-Check (QC- LDPC) Codes, Turbo Codes, Deep Space Communications. I. INTRODUCTION The evolution in the telecommunications area over the last years has led to an increased interest in using advanced error control coding techniques. Turbo codes, due to their enhanced Bit Error Rate (BER) performance, are considered as a serious candidate for modern communication systems []. Another coding scheme is Low-Density Parity-Check (LDPC) codes, which were first introduced by Gallager in 963 [2]. Although LDPC codes had been neglected for many years, they have become the target of research and investigation over the last decade, after the development of practical decoding implementation technologies. Their main advantage is that their performance can be similar to that reached by turbo codes [3], however LDPC codes can allow parallel decoding This research work was carried out in the context of IST SatNEx-II FP6 Project (IST-27393) architectures, thus achieving higher throughputs as compared to turbo codes [4]. LDPC codes are described by their parity check matrix and the way this is constructed. LDPC codes are divided into two categories, namely regular and irregular codes, with the latter ones experiencing a better BER performance in general. In addition, the modification introduced by the construction of the parity check matrix results in random codes [5] and also structured codes [6 8]. The former category describes LDPC codes when their parity check matrix is designed by a random computer-based procedure. On the other hand, the second category implies that this matrix has been constructed based on combinatorial methods. Such examples are the Quasi Cyclic (QC) LDPC codes. Their main advantage against randomly constructed codes is that they involve easier implementation in terms of the encoding procedure [7]. The main feature of QC-LDPC codes is that their parity check matrix consists of circulant submatrices, which could be either based on on the identity matrix [8, 9] or a smaller random matrix [6]. Permutation vectors could also be used in order to create the circulant submatrices. Their encoding procedure can be accomplished by using a series of shift registers, while the complexity is proportional to the number of parity bits or the total code length [7]. In this paper, a thorough study is carried out in terms of achievable BER under different coding schemes, such as turbo codes and LDPC codes, with low code rates as well as high code rates suitable for deep space and high data rate applications. In particular, QC-LDPC codes are employed, in order to reduce encoding complexity. The corresponding performance evaluation results show that QC-LDPC codes outperform the currently used Consultative Committee for Space Data Systems (CCSDS) turbo codes for a given number of iteration rounds. This performance evaluation study is useful in future updates of the channel code option used in deep space communications by the CCSDS [] /9/$ IEEE IWSSC 29
2 The rest of the paper is organized as follows. Section II describes the turbo codes currently used in deep space communications. The next Section III addresses the proposed LDPC code construction, the coding and decoding techniques used for such applications. In Section IV various performance evaluation results by means of computer simulations and appropriate comparisons are depicted, while the conclusions are drawn in Section V. II. TURBO CODES USED IN DEEP SPACE APPLICATIONS First, state-of-the-art channel coding techniques have traditionally been used in deep space communications, as the received signal power in such applications is very low, due to the extremely large propagation distances. Thus, the use of powerful error-correcting codes is the enabler key, in order to provide with large coding gains and improve the link budget computations. For this, turbo codes have been used by the CCSDS since 999 []. In particular, turbo codes have demonstrated an additional coding gain of approximately 2 db with respect to the former channel code solution consisting of a Reed-Solomon (RS) code concatenated with a Convolutional Code (CC) through a bit interleaver []. Fig. depicts the turbo encoder used by the CCSDS standard. It is a Parallel Concatenated Convolutional Code (PCCC) that makes use of two Recursive Systematic Convolutional (RSC) codes, each one having 6-states and code rate equal to /4. The different code rate options in CCSDS standard, i.e. /2, /3, /4 and /6, are obtained through appropriate interconnections of the parity bits produced by the two RSC codes, as explicitly shown in Fig.. The interleaver permutation, been designed by Berrou, follows a deterministic procedure in order to reduce as much as possible the memory storage requirements. There are four frame length options in CCSDS standard, i.e. 784, 3568, 736, and 892 bits. For trellis termination, an additional output sequence of bits is produced, depending on the code rate, in order to flash the constituent RSC encoders to the zero state. III. PROPOSED LDPC CODING TECHNIQUE A. Characteristics of LDPC codes LDPC codes belong to the category of linear block codes meaning that a codeword (c) of n bits is produced by the original uncoded word (u) of k bits. Their parity check matrix describes this class of codes, which is sparse and the majority of its digits are zero. A codeword c is valid if the subsequent parity check equation is true: H * c T = () The resultant code rate k/n defines the size of the parity check matrix, which is specified as (n k) x n. Uniform and non-uniform row and column weights as regards to the parity check matrix, result in regular and irregular LDPC codes, respectively [2]. A typical characteristic of LDPC codes is that they can be described by a bipartite graph. Two groups of nodes, namely the check nodes and bit nodes constitute this bipartite graph. The size of the parity check matrix plays an important role in determining the number of these nodes, meaning that if its size is m x n, then the graph will consist of m check nodes and n bit nodes, respectively. One row in the parity check matrix is represented by one check node, whereas each bit node corresponds to one codeword bit [2]. The presence of an ace in each row of the parity check matrix means that the two corresponding nodes are connected to each other in the bipartite graph [3]. For example, if there is an ace in the third column and fifth row of the matrix, then the third bit node will be connected to the fifth check node. Having an irregular code implies that not every bit node is connected to the same number of check nodes. In Fig. 2 an example of a parity check matrix and its equivalent bipartite graph are demonstrated. H = Fig.. Turbo encoder used by CCSDS standard. Fig. 2. A parity check matrix (H) and its corresponding bipartite graph.
3 B. Encoding Procedure of QC-LDPC Codes This class of LDPC codes is characterised by a parity check matrix H, which consists of square blocks, as already has been mentioned. The square blocks could either be the zero matrix or circulant permutation matrices. Eq. (2) illustrates a permutation matrix P of size q x q. K P = M M M M K Let P i stand for the circulant permutation matrix, which is derived from the identity matrix I after the latter one is shifted to the right by i times ( i q). The zero matrix is defined as P, whereas the resulting parity check matrix H of size (j q) x (k q) is denoted in Eq. (3): a a a 2 ( k ) a k P P L P P a2 a a 22 2( k ) a2 k P P L P P H = M M M M M aj aj2 aj( k ) ajk P P P P L where a il Є {,,, q, } [8]. Parameters k and j are related to the resultant overall code rate R by: (2) (3) R j / k (4) Special care should be taken so that q is set to a prime number and the equation q k j is not validated, otherwise the coding system would be inadequate. By selecting the parity check matrix in such a way results in reduced memory needed for storage, since after positioning the aces in the first row of each block matrix P, then the rest aces can easily be located. Depending on how the blocks are defined, we can obtain both regular and irregular QC-LDPC codes. In case of regular codes, the code rate is larger than j / k, as shown in Eq. (4) [8]. In this paper we focus on QC-LDPC codes having a parity check matrix in the form of the following Eq. (5): I I I K I K I I P K P K P Hqjk (,,) I P P M M M K M K M K I K P ( j 2) ( k 2) 2( j 3) 2( k 3) = K L ( j )( k j) (5) It is noticeable from Eq. (5) that this matrix is in upper triangular form leading to irregular codes. The encoding procedure is also facilitated, while the overall code rate is -j/k. C. Decoding Procedure In general an iterative algorithm, the so-called Message Passing Algorithm or Belief Propagation Algorithm, describes accurately the decoding procedure followed by LDPC codes [4]. This algorithm is better comprehended via the code s bipartite graph. Although its main structure is similar, several versions have been presented in the open technical literature [4], [5]. According to this algorithm, each bit node sends a message to the check nodes with which it is connected, during each iteration round. This message is an estimation on the exact value of the corresponding codeword bit this node represents. Afterwards, all the messages received at each check node are updated so that other messages are sent back to the neighbouring bit nodes, meaning that one iteration round is completed. Subsequently, the messages are further processed and the bit nodes send information back to check nodes, so that the algorithm is repeated. The messages exchanged between connected nodes entail extrinsic information only, which is the main feature of the algorithm. As a result, the message sent by a check node to one bit node is based on the information received by the check node from all the other bit nodes except for this particular bit node. The situation is similar when it comes to messages sent from bit nodes to check nodes. At the end of each iteration round and after the bit nodes messages are processed, a codeword is produced. Whether or not the resultant codeword is correct this is checked at the end of each iteration round. The algorithm stops either after a predefined number of iterations are carried out or in case the correct codeword is already found during the algorithm s execution [4]. From this point, it is obvious that the number of iteration rounds plays an important role in the system s performance. In case the number of iteration rounds is small, it is likely that the correct codeword is not obtained. On the other hand, a greater number of iteration rounds, leads to an increased system s complexity. The nature of the messages sent by the nodes determines the difference between the algorithm versions. For instance, the messages can be either log-likelihood ratios [5] or not [4]. In this paper, the messages sent at each iteration round of the decoding procedure are in the probability domain [4]. It should be also mentioned that irregular codes offer more protection to the information bits, due to the appropriate construction of their parity check matrix. In particular, the bit nodes that are connected to a greater number of check nodes tend to update their values quicker than the ones that are only connected to a limited number of check nodes. Because of the way encoding is performed, these bit nodes are the ones that contain the useful information, meaning that the information bits are the first ones to be corrected as long as the iterative algorithm proceeds. IV. PERFORMANCE EVALUATION RESULTS In this section BER performance evaluation results are illustrated by the means of computer simulations. It is assumed
4 Binary Phase Shift Keying (BPSK) modulation and the AWGN channel. In principle, regarding the LDPC codes the lower the code rate used the larger the parity check matrix becomes. Thus, more computer resources are required. In addition, due to the matrix calculations and to the numerous components the systems consists of, the total simulation time has been large. Consequently, there had to be a trade off between the number of transmitted data blocks and the minimum possible BER value the results approach. Moreover, the maximum number of iteration rounds used in the decoding procedure was set to 5, since it was observed that there was no noticeable difference at the resulting decoded words when more iteration rounds were applied. The simulations were run with a block size of 736 bits. The number of iteration rounds was afterwards set to 5 in order to observe how a single parameter can affect the system s performance. It should also be noted here that the iteration rounds concerning the turbo decoder were set to. Both high and low code rates have been examined. Fig. 4 shows the system s performance when the higher code rates are being employed into the system. In particular, 3/4 and 4/5 code rates are used both for LDPC and turbo codes. On the other hand, the system s performance regarding the lower code rates, i.e. /3, /4 and /5, is illustrated in Fig. 5 and Fig. 6, respectively. The lower code rates are the ones that give the most interesting results regarding the system s effectiveness. Fig. 4. Bit error rate versus Eb/N for QC LDPC codes of code rates 3/4 and 4/5 respectively. Fig. 3. Bit error rate versus Eb/N for QC LDPC and turbo codes of code rates /2 and 2/3 respectively. In Fig. 3 the BER performance is shown when QC-LDPC and turbo codes of /2 and 2/3 code rate are used. It is noticeable that the two code rates follow the same trend throughout the whole E b /N range. It is also clear that for higher E b /N rates, there is a sharp decrease concerning the BER of LDPC for both code rates. Two different curves are displayed for the LDPC code. In the first one the number of iteration rounds is set to 5, while in the second one it is set to 5. It is obvious that by changing one vital parameter, the system s performance can be highly altered. A smaller number of iteration rounds means that the system fails to correct all the errors occurring during the signal s transmission, whereas a larger number of iteration rounds leads to a better system s performance. However, the main aspect observed from this graph is that LDPC codes outperform turbo codes, while the difference in their performance is significant especially for lower E b / N values. This confirms the theory that LDPC codes are a promising coding technique and can result in a high system s performance. It is also clear that when fewer iteration rounds are defined in the decoding procedure, the LDPC coding scheme seems to perform close to turbo codes. Fig. 5. Bit error rate versus Eb/N for QC LDPC and turbo codes of code rates /3 and /4 respectively. Correspondingly to the case of /2 and 2/3 code rate, the curves in Figs. 4, 5 and 6 experience the same inclination. Furthermore, it is noticeable that the deep fall in the BER plot occurs for lower E b /N rates in Fig. 3 and higher ones in Fig. 4. This is explicable by the fact that for higher code rates, the bits redundancy is decreased at the expense of the signal s quality. Thus, at the presence of fewer coded bits the protection offered to the original data bits is of inferior quality. As a result, the /5 code rate experiences the best performance, as it can be shown from Fig. 6.
5 Fig. 6. Bit error rate versus Eb/N for QC LDPC and turbo codes of code rate /5. The most important thing to notice when examining the curves is that LDPC codes clearly outperform turbo codes for all code rates when a large number of iteration rounds are set. It is also apparent that when fewer iteration rounds are present, the system s performance tends to become similar for LDPC and turbo codes. V. CONCLUSIONS The applicability of LDPC codes for future deep space and high data rate communication systems was demonstrated. Several code rates were examined and it was shown that LDPC codes outperform turbo codes, when a large number of iteration rounds was set. The two coding schemes experience a similar performance, when fewer iteration rounds were used. In addition, as opposed to the currently used turbo codes, QC- LDPC codes offer some implementation advantages in both the encoding and decoding side. That is, the use of a series of shift registers for linear time encoding operation and the possibility of a parallel decoding implementation. REFERENCES [] C. Berrou, A. Glavieux, and P. Thitimajhima, Near Shannon limit error correcting coding and decoding: Turbo codes, IEEE Int. Conf. Commun. (ICC), pp. 64-7, Geneva, Switzerland, May, 993. [2] R. G. Gallager, Low Density Parity Check Codes. Cambridge, MA: MIT Press, 963. [3] T. J. Richardson, and R. L. Urbanke, Efficient Encoding of Low- Density Parity-Check Codes, IEEE Trans. Information Theory, vol. 47, no. 2, pp , Feb. 2. [4] A. J. Blanksby, and C. J. Howland,: A 96-mW Gb/s 24-b, rate /2 low-density parity-check code decoder', IEEE Journal of Solid-State Circuits, Vol. 37, No. 3, pp , 22. [5] R. M. Tanner, A recursive approach to low complexity codes, IEEE Trans. Inf. Theory, vol. IT-27, no. 9, pp , Sep. 98. [6] R. Echard, and C.Shih-Chun, The π-rotation low-density parity check codes, IEEE Globecom 2, vol. 2, pp Nov. 2. [7] Z. Li, L. Chen, L. Zeng, S.Lin, and W. H. Fong, Efficient Encoding of Quasi-Cyclic Low-Density Parity-Check Codes, IEEE Transactions on Communications, vol. 54, no., Jan. 26, pp [8] S. Myung, K. Yang, and J. Kim, Quasi-Cyclic LDPC Codes for Fast Encoding, IEEE Transactions on Information Theory, vol. 5, no. 8, pp , Aug. 25. [9] E. Eleftheriou and S. Olcer, Low-density parity-check codes for multilevel modulation, in Proc. IEEE Int. Symp. Information Theory (ISIT22), Lausanne, Switzerland, Jun./Jul. 22, p [] TM Synchronization and Channel Coding, CCSDS 3. B-, Sep. 23. [] G. P. Calzolari, F. Chiaraluce, R. Garello, and E. Vassallo, Turbo Code Applications on Telemetry and Deep Space Communications, in Turbo Code Applications: A Journey From a Paper to Realization, Chapter 3, (Ed.) S. Sripmanwat, Springer, 25. [2] H. Zhong and T. Zhang, Block-LDPC: A Practical LDPC Coding System Design Approach, IEEE Trans. Circuits and Systems I: Regular Papers, vol. 52, no. 4, pp , Apr. 25. [3] J. Hou, P. H. Siegel, and L. B. Milstein, Performance Analysis and Code Optimization of Low Density Parity-Check Codes on Rayleigh Fading Channels, IEEE J. Select. Areas Commun., vol. 9, no. 5, pp , May 2. [4] Robert H. Morelos - Zaragoza, The Art of Error Correcting Coding, Second Edition, Wiley 26. [5] H. Nakagawa, D. Umehara, S. Denno and Y. Morihiro, A Decoding for Low Density Parity Check Codes over Impulsive Noise Channels, IEEE ISPLC, pp , 25.
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