Acknowledgments First, I would like to express my sincere thanks to my supervisors, Dr. Marc Andre Armand and Dr. Mehul Motani, for their invaluable advice and patient guidance throughout the course of my project and thesis. Their knowledge and experiences in the field of coding theory are insurmountable. I would also like to thank my friend Hu Wenguang for the numerous and fruitful discussions which helped me to solve specific problems in the thesis. Besides, special thanks to Ye Jiangyang and Yu Yiding for sharing some enjoyable moments during the course of my research. My gratitude also goes to Department of Electrical and Computer Engineering, National University of Singapore, who provides the research facilities to conduct the research work. Finally, I am grateful to my family, without whose love, encouragement and support this thesis would not have been possible. i Table of Contents Acknowledgements …………………………………………………… i Summary ……………………………………………………………… .ii List of Figures ……………………………………………………… iv List of Tables ……………………………………………………….……v List of Abbreviations ………………………………………… ……vi Chapter I Introduction 1.1 Error Control Coding 1.2 Accomplishments and Contributions 1.3 Thesis outline Chapter II List Decoding 2.1 Introduction .10 2.2 RS codes and GRS codes 11 2.3 Sudan I Approach 15 2.4 Sudan II Approach 17 2.5 the KV Approach 20 2.6 Gröbner basis Interpolation Algorithm .25 2.7 Simulation Results 29 2.8 Summary .30 Chapter III Adaptive List Decoding 3.1 Introduction .31 ii 3.2 Adaptive List Decoder 32 3.3 Modified Adaptive List Decoder 38 3.4 Puncturing Effects .41 3.5 Simulation Results 41 3.6 Summary .53 Chapter IV Turbo Codes 4.1 Introduction .56 4.2 MAP Decoder .56 4.3 Turbo Decoding Scheme .61 4.4 Summary .63 Chapter V Iterative List Decoding Scheme 5.1 Introduction .64 5.2 Modified Symbol-Based MAP Decoder .65 5.3 Extrinsic Information from the KV List Decoder .67 5.4 Iterative List Decoding Scheme 69 5.5 Adaptive Iterative List Decoder 71 5.6 Simulation Results 71 5.7 Summary .75 Chapter VI Conclusion 6.1 Summary of Thesis .77 6.2 Future work .79 References iii LIST OF FIGURES Figure 1.1: Block diagram of a digital communication system Figure 2.1: The KVA Structure ……… ………… ….…………………………………… ……… 19 Figure 2.2: FER performance for (15,9,7) RS code with the KVA ……… .…… ….27 Figure 3.1: ALD Structure ……………………………………… ………………30 Figure 3.2: ALD Flow Chart … …………………………………….… .…………32 Figure 3.3: Average list size for ALD …………………… ……………………… 34 Figure 3.4: The distribution of list size for the ALD ………… …………………….35 Figure 3.5: ALD Decoding Radius …………….………… …………………….36 Figure 3.6: Average list size for the ALD and MALD ………………………… 38 Figure 3.7: The distributions of list size for the MALD ………………………… 38 Figure 3.8: ALD vs KVD for (15,9,7) RS codes ……………………………… .… .43 Figure 3.9: MALD vs KVD for (15,9,7) RS codes ………………….……… .… .44 Figure 3.10: Puncturing Scheme for (15,5,11) RS codes .….…………………….….47 Figure 3.11: ALD for Punctured (15,9,7) RS codes …………………………… .50 Figure 3.12: MALD for Punctured (15,9,7) RS codes …………………….… .50 Figure 3.13: MALD for (15,9,7) RS codes ………………………………………….51 Figure 4.1: An Example of Trellis ………………………………………………… .58 Figure 4.2: Turbo Decoding Scheme ………………………………………………61 Figure 5.1: Iterative List Encoder ………………………………………… .………69 Figure 5.2: SISO decoder …………………………………………………………….69 iv Figure 5.3: The ILD scheme ………………….…………………………………….70 Figure 5.4: Generalized List Decoder .….…………… ……………………………72 Figure 5.5: The AILD Type I Flow Chart ……………………………………….…73 Figure 5.6: The AILD Type II Flow Chart ……………………………………….…73 Figure 5.7: ILD performance for (15,9,7) RS codes ……………………………… .74 Figure 5.8: AILD performance for (15,9,7) RS codes …………………….…… .75 v LIST OF TABLES Table 2.1: GF(8) Table …………………………………………………………….11 Table 2.2: The Greedy Iterative Algorithm ………………………………………….21 Table 2.3: Gröbner basis Interpolation Algorithm ……………………………… .26 Table 3.1: Stopping Criterion for the Adaptive List Decoder …………………….…33 Table 3.2: The Modified Adaptive List Decoder …………………………………… 39 Table 4.1: Decoding Algorithm for MAP Decoder1 ……………………………… .63 Table 4.2: Decoding Algorithm for MAP Decoder2 ……………………………… .63 Table 5.1: Decoding Algorithm for SISO Decoder1 ……………………………… .71 Table 5.2: Decoding Algorithm for SISO Decoder2 ……………………………… .72 vi LIST OF ABBREVIATIONS RS Codes Reed-Solomon Codes GRS Codes Generalized Reed-Solomon Codes MDS Codes Maximum-distance separable Codes AG Codes Algebraic Geometry Codes AWGN Channel Additive White Guassian Channel RSC Encoder Recursive Systematic Convolutional Encoder SISO Soft In Soft Out MAP Maximum a Posteriori Probability APP A posteriori probability ML Maximum Likelihood LAPP Log a posteriori probability KVA Ralf Koetter and Alexander Vardy algorithm KVD Ralf Koetter and Alexander Vardy decoder ALD Adaptive list decoder ILD Iterative list decoder MALD Modified adaptive list decoder MMAPD Modified symbol-based MAP decoder AILD Adaptive iterative list decoder PCC Parallel concatenated code vii Summary Error correcting codes are designed to solve the problem of reliable transmission of information over a noisy channel. A fundamental algorithm in coding theory is to decode the original message effectively even some symbols of the received words are distorted by the channel. Traditionally, decoding algorithms have been constrained to output a unique codeword. However, list decoding proposed by Elias and Wozencraft, generates a list of all candidate codewords that differ from the received word in a certain number of the positions. This thesis is intended to investigate the adaptive and iterative list decoding algorithm for Reed Solomon codes to achieve better performance. Research consists of two major parts. The first part presents the adaptive list decoder (ALD), which can complement any existing list decoding algorithm. In this thesis, we use the list decoder proposed by Ralf Koetter and Alexander Vardy (KV list decoder) as an example. We compare the output of the KV list decoder with the hard-decision received word. If the number of ⎢ d − 1⎥ , we mismatch symbols exceeds n − t for an [n, k + 1, d ] RS code, where t = ⎢ ⎣ ⎦⎥ increase the list size by a predetermined step and decode the received word again. Once the output meets this stopping criterion, the adaptive list decoder will cease decoding and generate the desired codeword. In addition, we propose a modified adaptive list decoder (MALD), which differs from the original decoder by introducing a new stopping criterion. The MALD not only increases the list size but also compares two decoded results from two consecutive iterations. After we generate the output of its component KV list decoder by increasing the maximum list size, we compare it with the previous result. If these two decoded results match, we stop iterations and generate output. Otherwise, we continue increasing the list size and find the codewords. Part two of the thesis presents an iterative list decoding algorithm (ILD). In this algorithm, a serial concatenation of the modified symbol-based MAP decoder and the KV list decoder serves as a core decoder. We calculate the symbol reliability information from the core decoder and feed it back to the other decoder. Furthermore, we propose a scheme to combine the adaptive list decoder and iterative list decoder (AILD) together to achieve better performance. Chapter I Introduction 1.1 Error Control Coding The objective of data transmission is to transfer data from an information source through a physical channel to a destination reliably. A typical communication system can be represented by a block diagram shown in Figure 1.1. Figure 1.1 Block diagram of a digital communication system Error control codes relate to protection of digital information against the errors that occur during data transmission or storage. Data, which enters the communication system from information source, first passes a source encoder, which converts the source information into an information sequence. A channel encoder transforms the information sequence into a coded sequence called a codeword. It is a new, longer sequence that contains redundancy in the form of parity-check symbols. After that, the Fig. 5.5 The AILD Type I Flow Chart Fig. 5.6 The AILD Type II Flow Chart 73 5.6 Simulation Results Fig. 5.7 ILD performance for (15,9,7) RS codes The performance of ILD for a (15,9,7) RS code over AWGN channel is simulated and compared against that of the MALD and KVD with maximum list size 5. Not surprisingly, our ILD perform much better than the latter. In Fig. 5.7, we plot the performance of the ILD and the MALD for a (15,9,7) RS code over GF (16) . In this example, we use 16-QAM as the modulation scheme. From this figure, we can observe that the ILD has a much better performance than the MALD. For example, the ILD indicates about 1.3dB coding gain over the ALD at a BER of 10 −3 and 1.5dB code gain at BER of 10 −4 . 74 Fig. 5.8 AILD for (15,9,7) RS codes Fig. 5.8 shows the performance of the AILD. In this simulation, we use the (15,9,7) Reed Solomon codes. We can observe that the AILD does not outperform much over the original ILD as we expect. Both Type I and Type II AILD just achieve about 0.3 dB coding gain over the original algorithm at BER of 10 −3 and Type I AILD almost has the same performance at BER of 10 −4 with ILD. This is because the majority of output generated by the inner ILD satisfies the stopping criteria of the outer ALD. Thus, in most cases, we not increase the list size of the inner ILD. Furthermore, we can draw a conclusion that the two performance curvers converge for large Eb/No. 5.7 Summary In this chapter, we illustrate an approach for applying iterations to the soft-decision list 75 RS decoding such as the KVD. Our SISO decoding architecture consists of a MMAPD and a soft-decision list decoder. The MMAPD generates the soft information, which can be used in the KVD. The KVD takes the symbol reliabilities as input and outputs a list of candidate codewords. From the list of the candidate, we calculate the extrinsic information, which can be used by the MMAPD. Then we can apply the turbo decoding structure to the SISO decoders. Simulation results show that the ILD can improve the overall BER performance. Also we present the AILD and the simulation results in this chapter. 76 Chapter VI Conclusion This chapter attempts to sum up the results and discussions that were put forward in the previous chapter, touching the value of what we have done in contributing to the area of the adaptive and iterative list decoding. In addition, the possible future research area is also included in the final part of this chapter. 6.1 Summary of Thesis This thesis has been intended to investigate ALD and ILD algorithm for Reed Solomon codes to achieve high performance with modest increase in computational complexity. In Chapter 2, we briefly reviewed the list decoding algorithms including Sudan I, Sudan II and the KVA. Furthermore, the Gröbner basis Interpolation approach is presented in this chapter. The Reed-Solomon list decoding algorithm has obvious importance to coding theory and practice. Through analysis and simulation, we provided the comparison among these three list decoding approach. 77 In chapter 3, we presented the ALD and examined the benefits of implementing this algorithm. Obviously, this structure can be used in any list decoder including the Sudan’s and the KVD. By adjusting its list size based on the input, the ALD continue to decode the received codewords until the stopping criterion is satisfied by the output generated by the decoder. Once the output satisfies the stopping criterion, the ALD will stop decoding and output the codeword. Furthermore in this chapter, we proposed the MALD in order to achieve better performance than the original one. The difference between the modified adaptive list decoder and the original one is the stopping criterion. The original algorithm keeps the stopping criterion unchanged throughout the whole procedure, while the MALD compares the two codewords generated by two consecutive iterations so as to decrease the computational complexity and increase the BER performance. Furthermore, the effect of puncturing scheme is discussed in this chapter. In chapter 4, we have given a brief discussion of turbo codes. We introduced the turbo encoder structure and its components such as RSC encoders and interleaver. Then MAP decoding algorithm was detailed studied and the derivation of α , β and γ were also included in this section. Finally, the turbo decoding principle was described at the end of this chapter. In chapter 5, an ILD for RS codes was devised by applying the turbo decoding principle and concatenating the MMAPD and the KVD. This decoding algorithm 78 provides a good BER performance. In this scheme, the SISO decoder consists of the MMAPD and the KVD. After we calculate the extrinsic information from the KVD, we feed it back to the rest SISO decoder. In addition, we combine the adaptive and iterative list decoders together to exploit the structrues. 6.2 Future work Although we managed to reveal and propose the adaptive and iterative list decoding algorithm in the thesis, we still highlight some work which remains to be done in the near future. We focus on the following parts: • Apply the adaptive and iterative list decoding algorithm to the Algebraic-Geometric (AG) codes. Algebraic-Geometric codes are a class of algebraic codes that include Reed-Solomon codes as a special case. The major drawback of RS codes is that RS codes require an alphabet size at least as large as the block length. This requirement is not suitable for many applications. However, Algebraic-Geometric codes can be defined over a fixed and small alphabet. Thus, they overcome the drawbacks of the RS codes. We can regard the Algebraic-Geometric codes as an extension of the Reed Solomon codes. Then the idea that applies the adaptive and iterative list decoding algorithm to the Algebraic-Geometric codes comes out. • Try to find another way to calculate the extrinsic information from the KVD. In this thesis, we use the algorithm by Cheng to compute the extrinsic information. But this approach is not the only solution to this question. Maybe we can calculate the probability of corresponding position from the 79 output and regard it as the soft information of the KVD. • Try to discuss the computational complexity of ILD. Since it is using Turbo code structure, it is quite difficult to discuss the computational complexity of iterative decoding algorithm. In conclusion, we want to point out the quest for better and better codes that approach Shannon capacity in the noise channel has led to many breakthroughs in coding theory. List decoding algorithm offers an opportunity and the potential of achieving Shannon capacity in the noise channel. From this thesis, the adaptive and iterative algorithm to list decoder discussed in the above chapters, we may regard them as an efficient algorithm to extend the list decoding algorithm. 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Cai, M.A. Armand, M. Motani, "Adaptive and iterative soft-decision list decoding", To appear in Proceedings of 2004 IEEE International Symposium on Information Theory, Chicago, USA. [2] B. Low, F. Cai, M.A. Armand and M. Motani, "Distributed multi-code assignments for DS-CDMA ad hoc networks," Proceedings of 7th International Symposium on Digital Signal Processing and Communication Systems, 8-11 Dec 2003, Gold Coast, Australia. 88 [...]... keeping the decoding complexity low is to generate a list of candidate codewords from the received symbols, and then choose the candidate codeword with the highest reliability as the output This is the idea behind list decoding algorithm 1.1.3 List Decoding In [12], Guruswami and Sudan proposed a new list decoding algorithm, which corrects 5 up to ⎡ n − n(n − d ) − 1⎤ , where n is the block length and d... positions.” While the list decoding algorithm is list all the possible codewords that differ from the received vector at most e positions” [6] 10 List decoding was indicated by Elias [10] in the 1950’s However, no efficient decoding algorithm had been advocated for any known error-correcting code until recently and later M Sudan and Guruswami [12], who proposed new approaches to list decoding Their approach... performance curve of the iterative list decoder and compared with original list decoder to observe how much improvement they have over the AWGN channel • Proposed the scheme to combine ALD and ILD together 1.3 Thesis outline Chapter 2 is devoted to the list decoding algorithm In this chapter, the Sudan I approach, the Sudan II approach and the KVA are investigated We also compare advantages and disadvantages... their work, Ralf Kotter and Alexander Vardy [17] presented a soft-decision algorithm for list decoding, which significantly improves the performance of RS codes Definition 2.1 (List Decoding) Input: Received codewords r and positive integer e Output: A list of all codewords c1 , c2 , , cm that differ from given code r in at most e positions First, we introduce Reed Solomon codes and Generalized Reed Solomon... for decoding both BCH codes and RS codes [36] This Euclidean decoding algorithm is simple in concept and easy to implement Decoding of RS codes can also be implemented in the frequency domain The first decoding algorithm in the frequency domain is proposed by Gore [11], and later significantly improved by Blahut [3] 14 The Berlekamp algorithm and the Euclidean algorithm are known as algebraic decoding. .. accomplishments and contributions, which are elaborated throughout this thesis, can be briefly listed as follows: • Briefly surveyed current research work on list decoding algorithm including Sudan I [35], Sudan II approach [12] and KVA Furthermore, Gröbner basis, which was proposed by Henry O’Keeffe and Patrick Fitzpatrick, is included to implement interpolation algorithm • Studied in detail, adaptive list decoder... theoretical Shannon bound For a bit-error probability of 10−5 and code rate R = 1/ 2 , the authors present an impressive Eb / N 0 ratio of 0.7 dB Turbo coding introduces some new concepts such as the use of iterative decoding and random interleaving to achieve remarkable results The decoding algorithm adopted is 4 a soft decision iterative decoding algorithm, which minimizes the error probability using... (α 4 ) 2 (α 4 )1 1⎤ ⎥ 1⎥ 1⎥ ⎥ 1⎥ ⎦ 2.2.2 Hard-decision Decoding of RS codes Hard-decision decoding of RS codes is to determine both the location and the values of the symbol errors Berlekamp’s iterative decoding algorithm [39] was the first efficient decoding algorithm for both binary and non-binary BCH codes In 1975, Sugiyama, Kasahara, Hirasawa and Namekawa showed that the Euclidean algorithm for finding... comparisons of our ALD and MALD are put forward In Chapter 4, basic concepts and structure about Turbo codes are introduced including the turbo encoder structure, recursive systematic encoder (RSC) and the interleaver In this chapter, we also present the MAP decoder and the turbo decoding principle Their derivations and application will be given to aid in the understanding of turbo decoding structure In... decoder (ALD) By adaptively adjusting the list size, we achieve a significantly better performance over the KVD with only a marginal increase in computational complexity • Modified the adaptive list decoder (MALD) using a different stopping criterion, which reduces computational complexity yet improves performance • Discussed the effect of the puncturing scheme on the list decoding algorithm and explained . KVA Ralf Koetter and Alexander Vardy algorithm KVD Ralf Koetter and Alexander Vardy decoder ALD Adaptive list decoder ILD Iterative list decoder MALD Modified adaptive list decoder MMAPD. Simulation Results 29 2.8 Summary 30 Chapter III Adaptive List Decoding 3.1 Introduction 31 ii 3.2 Adaptive List Decoder 32 3.3 Modified Adaptive List Decoder 38 3.4 Puncturing Effects 41 3.5. investigate the adaptive and iterative list decoding algorithm for Reed Solomon codes to achieve better performance. Research consists of two major parts. The first part presents the adaptive list