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Burst Correction Coding from Low-Density Parity-Check Codes

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Burst Correction Coding from Low-Density Parity-Check Codes BURST CORRECTION CODING FROM LOW-DENSITY PARITY-CHECK CODES by Wai Han Fong A Dissertation Submitted to the Graduate Faculty of George Mason University In Partial fulfillment of The Requirements for the Degree of Doctor of Philosophy Electrical and Computer Engineering Committee: Dr. Shih-Chun Chang, Dissertation Director Dr. Shu Lin, Dissertation Co-Director Dr. Geir Agnarsson, Committee Member Dr. Bernd-Peter Paris, Committee Member Dr. Brian Mark, Committee Member Dr. Monson Hayes, Department Chair Dr. Kenneth S. Ball, Dean, The Volgenau School of Engineering Date: Fall Semester 2015 George Mason University Fairfax, VA Burst Correction Coding from Low-Density Parity-Check Codes A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at George Mason University By Wai Han Fong Master of Science George Washington University, 1992 Bachelor of Science George Washington University, 1984 Director: Dr. Shih-Chun Chang, Associate Professor Department of Electrical and Computer Engineering George Mason University Fairfax, VA Co-Director: Dr. Shu Lin, Professor Department of Electrical and Computer Engineering University of California Davis, CA Fall Semester 2015 George Mason University Fairfax, Virginia Copyright c 2015 by Wai Han Fong All Rights Reserved ii Dedication I dedicate this dissertation to my beloved parents, Pung Han and Soo Chun; my wife, Ellen; children, Corey and Devon; and my sisters, Rose and Agnes. iii Acknowledgments I would like to express my sincere gratitude to my two Co-Directors: Shih-Chun Chang and Shu Lin without whom this would not be possible. In addition, I would especially like to thank Shu Lin for his encouragement, his creativity and his generosity, all of which are an inspiration to me. A special thanks to my entire family who provided much moral support, particularly my cousin, Manfai Fong. I would also like to thank the following people for their support and encouragement: Pen-Shu Yeh, Victor Sank, Shannon Rodriguez, Wing Lee, Mae Huang, Brian Gosselin, David Fisher, Pamela Barrett, Jeff Jaso, and Manuel Vega. And thanks to Qin Huang who collaborated in Section 6.3 and provided simulation results for Section 6.4. And special thanks to my many influential teachers at GMU. Finally, at my high school, much gratitude goes to my math teacher Charles Baumgartner. iv Table of Contents Page List of Figures . viii Abstract . xi 1 Introduction . .1 1.1 Background . .1 1.2 Motivation . .2 1.3 Summary of Research . .3 2 Communication Systems Background . .6 2.1 Channel Characteristics . .8 2.1.1 Random Error/Erasure . .8 2.1.2 Burst Error/Erasure . 11 2.2 Linear Block Codes for Error/Erasure Correction . 12 2.2.1 Cyclic Codes . 13 2.2.2 Quasi-Cyclic Codes . 14 3 Low-Density Parity-Check Codes . 15 3.1 Introduction . 15 3.2 Technical Description . 17 3.3 LDPC Code Construction . 21 3.3.1 Mackay Codes . 22 3.4 Structured LDPC Codes . 22 3.4.1 Euclidean Geometry Codes . 22 3.4.2 Product Codes . 24 3.5 Sum Product Algorithm . 26 3.6 Min-Sum Decoder . 28 3.7 Erasure Correcting Algorithm . 29 4 Burst Erasure Correction Ensemble Analysis for Random LDPC Codes . 30 4.1 Introduction . 30 4.2 Zero-Span Characterization for Random LDPC Codes . 33 4.2.1 Background . 33 v 4.2.2 Recursive Erasure Decoding . 33 4.2.3 Zero-Span Definition . 34 4.3 Statistical Analysis for the General Ensemble of Random LDPC Codes . 39 4.3.1 Moments of the Zero-Covering Span PMF . 44 4.3.2 Asymptotic Burst Erasure Correction Performance . 51 4.3.3 Probability Mass Function of the Correctible Profile γ ........ 52 4.4 Truncated General Ensemble Analysis . 57 4.4.1 The Truncated Correctible Profile γ .................. 62 4.4.2 Regions of Analysis . 66 5 Expurgated Ensemble of Random LDPC Codes for Burst Erasures . 75 5.1 Statistical Analysis . 75 5.1.1 Effective Arrival Rate . 75 5.1.2 Updated Equations . 80 5.2 Data Results . 81 6 Multiple Burst Error Correction Coding . 91 6.1 LDPC Codes Expanded . 91 6.1.1 OSMLD Algorithm . 92 6.1.2 Superposition Construction of LDPC Codes . 94 6.2 Multiple Phased-Burst Correcting LDPC Codes Based on CPM Constituent Codes ....................................... 95 6.3 Binary MPBC Codes . 96 6.4 Performances over the AWGN Channels and the BEC . 99 6.5 Multiple Phased-Burst Error Correction Bound . 103 6.6 Single Burst Correction Bound . 117 7 Multiple Burst Erasures Correction Coding . 120 7.1 Product Codes for Erasure Correction . 120 7.1.1 Parity-Check Matrix Structure . 124 7.2 Analysis of LDPC Product Codes . 126 7.2.1 Multiple Burst Erasure Correcting Capability . 128 7.3 Decoding Algorithms . 131 7.4 Product Code Multiple Burst Erasure Decoding Algorithm . 131 7.4.1 Examples of LDPC Product Codes for Burst Erasure Correction . 132 7.5 Product Code LDPC Decoding Algorithm for AWGN . 135 7.5.1 Two-Stage LDPC Product Code Decoder . 135 7.6 Product LDPC Code Examples and Results . 137 vi 8 Conclusion . 143 Bibliography . 145 vii List of Figures Figure Page 2.1 Communications Channel . .7 2.2 Simplified Communications Channel . .7 2.3 Binary Input Additive White Gaussian Noise Channel . .9 2.4 Binary Erasure Channel . 10 2.5 Binary Symmetric Channel . 10 3.1 Tanner Graph Example . 18 4.1 (240,160) Randomly Generated LDPC Code Forward Zero-Covering Span Profile and Forward Correctible Profile . 39 4.2 (240,160) Randomly Generated LDPC Code Backward Zero-Covering Span Analysis and Backward Correctible Profile . 40 4.3 (240,160) LDPC Code Scatter Plot . 40 4.4 (240,160) LDPC δF Histogram . 41 4.5 EG (255,175) LDPC Code Forward Span Analysis . 41 4.6 EG (255,175) LDPC Code Backward Span Analysis . 42 4.7 EG (255,175) LDPC Code Scatter Plot . 42 4.8 Forward Zero-Covering Span Ensemble PMF for Randomly Generated 80 × 2000 LDPC Matrices of Constant Column Weight . 45 4.9 Mean and Standard Deviation of Zero-Spans for Various Block Sizes . 50 4.10 Asymptotic Mean of Zero-Covering Spans . 51 4.11 Example LDPC code N = 100; 000 . 56 4.12 Zero-Covering Span Mean . 61 4.13 Standard Deviation of Zero-Covering Spans . 61 4.14 Mean of Correctible profile PMF of Zero-Spans for N=100 . 63 4.15 Mean of Correctible profile PMF of Zero-Spans for N=1,000 . 64 4.16 Mean of Correctible profile PMF of Zero-Spans for N=10,000 . 65 4.17 Example LDPC code for N = 100 . 66 4.18 α Factor . 68 viii 4.19 A Comparison of Analytical Thresholds . 73 5.1 The Effective Arrival Rate . 78 5.2 Row Weight Distribution Analysis . 79 5.3 Example LDPC code for N = 1; 000 . 82 5.4 Zero-covering span profile PMF of LDPC Matrices N=100 . 82 5.5 Gamma PMF with Block Size N=100 . 83 5.6 Zero-covering span profile PMF of LDPC Matrices N=1,000 . 84 5.7 Gamma PMF with Block Size N=1,000 . 84 5.8 Zero-covering span profile PMF of LDPC Matrices N=10,000 . 85 5.9 Gamma PMF with Block Size N=10,000 . 85 5.10 Correctible profile PMF of Zero-Spans for Expurgated Ensemble N=100 . 87 5.11 Correctible profile PMF of Zero-Spans for Expurgated Ensemble N=1000 . 88 5.12 Correctible profile PMF of Zero-Spans for Expurgated Ensemble N=10,000 88 6.1 Simulation Results of binary QC (4340, 3485) LDPC Code over the AWGN Channels . 101 6.2 Convergence of binary QC (4340, 3485) LDPC Code over AWGN Channels 101 6.3 Simulation Results of binary QC (4340, 3485) LDPC Code over the BEC Channel . 102 6.4 Simulation Results of binary QC (37800,34035) LDPC Code over the AWGN Channels . 102 6.5 MPBC Bound with a gap restriction for a Codelength of 10 Subblocks and 100 bits per Subblock . 112 6.6 MPBC without a gap restriction for a Codelength of 10 Subblocks and 100 bits per Subblock . 114 6.7 MPBC Bounds with and without a gap restriction . 115 7.1 Product Code Array . 122 7.2 BEC Performance of EG(63,37)×EG(63,37) . 133 7.3 BEC Performance of SPC(63,62)×EG(255,175) . ..
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