Codes on Graphs and Analysis of Iterative Algorithms for Reconstructing Sparse Signals and Decoding of Check-Hybrid GLDPC Codes
نویسنده
چکیده
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.1. Block Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.1.1. Linear Block Codes . . . . . . . . . . . . . . . . . . 22 1.2. Communication Systems . . . . . . . . . . . . . . . . . . . . . . 22 1.2.1. Channel models . . . . . . . . . . . . . . . . . . . . 23 1.2.2. Decoding methods for block codes . . . . . . . . . . 24 1.3. Two families of linear block codes . . . . . . . . . . . . . . . . . 25 1.3.1. Hamming codes . . . . . . . . . . . . . . . . . . . . 25 1.3.2. BCH codes . . . . . . . . . . . . . . . . . . . . . . . 26 1.4. Low-Density Parity-Check (LDPC) codes . . . . . . . . . . . . . 27 1.4.1. Quasi-cyclic (QC)-LDPC codes . . . . . . . . . . . . 29 2. An Introduction to Compressed Sensing . . . . . . . . . . . . . . 31 2.1. Historical Background . . . . . . . . . . . . . . . . . . . . . . . 31 2.2. Primary reconstruction methods for sparse signals . . . . . . . . 32 2.2.1. `2-norm minimization: . . . . . . . . . . . . . . . . . 32 2.2.2. `0-norm minimization: . . . . . . . . . . . . . . . . . 32 2.2.3. `1-norm minimization: . . . . . . . . . . . . . . . . . 33 2.3. Iterative reconstruction algorithms . . . . . . . . . . . . . . . . 34 2.3.1. Message Passing Algorithms . . . . . . . . . . . . . . 35 2.3.2. Verification Algorithm . . . . . . . . . . . . . . . . . 37 2.3.3. Iterative Thresholding . . . . . . . . . . . . . . . . . 38 2.4. Measurement matrices based on LDPC codes . . . . . . . . . . 40 3. Interval-Passing Algorithm: Reconstruction Analysis and Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.1. Interval-Passing Algorithm . . . . . . . . . . . . . . . . . . . . . 42 3.1.1. Description of the Algorithm . . . . . . . . . . . . . 43 3.1.2. Numerical Results . . . . . . . . . . . . . . . . . . . 46 3.2. Reconstruction Analysis . . . . . . . . . . . . . . . . . . . . . . 47 3.3. Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . 55 TABLE OF CONTENTS—Continued 9 4. LDPC decoders, Generalized Low-Density Parity-Check (GLDPC) and Check-Hybrid GLDPC (CH-GLDPC) Codes . . . . . . . . . . 59 4.1. LDPC decoders and trapping sets . . . . . . . . . . . . . . . . . 59 4.1.1. Parallel Bit Flipping (PBF) Algorithm . . . . . . . . 59 4.1.2. Gallager B decoding algorithm . . . . . . . . . . . . 60 4.1.3. Trapping sets . . . . . . . . . . . . . . . . . . . . . . 60 4.2. GLDPC codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.2.1. Definition of a GLDPC code . . . . . . . . . . . . . 63 4.2.2. Bounds on parameters of a GLDPC code . . . . . . 64 4.2.3. Previous work on GLDPC codes . . . . . . . . . . . 66 4.3. CH-GLDPC codes . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.3.1. Definition and previous work on CH-GLDPC codes 67 5. Check-Hybrid Generalized LDPC (CH-GLDPC) Codes: Systematic Elimination of Trapping Sets and Guaranteed Error Correction Capability . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.1. Eliminating the trapping sets using super checks . . . . . . . . . 70 5.2. Critical sets and the splitting number . . . . . . . . . . . . . . . 73 5.2.1. Critical sets and minimal size of critical sets . . . . 74 5.2.2. Upper bounds on the splitting number . . . . . . . . 78 5.3. Guaranteed Error Correction Capability . . . . . . . . . . . . . 83 5.4. Splitting numbers of (4, ρ, 6) LDPC codes and trapping sets elimination using the Gallager B decoding algorithm . . . . . . . . 87 5.4.1. Elimination of trapping sets by super checks in (4, ρ, 6) LDPC codes . . . . . . . . . . . . . . . . . . . . . . 88 5.4.2. Elimination of trapping sets by super checks using Gallager B decoding algorithm . . . . . . . . . . . . 92 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
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