低密度偶校碼(Low-Density Parity-Check Codes)是一種錯誤更正碼，其被證實在足夠碼長並使用疊代信息傳遞解碼時，會有接近薛農極限(Shannon limit)的性能。由於類循環低密度偶校碼(Quasi-Cyclic Low-Density Parity-Check Codes)的偶校矩陣擁有良好的代數結構，使得硬體實現及分析上的複雜度皆能大幅降低，在行動通訊領域上因而備受關注。
對於由屏蔽(masking)方法所建構的非均等錯誤保護(Unequal Error Protection)類循環低密度偶校碼，其碼字元(codeword bit)依據偶校矩陣分成多個部分，各自有著不同的糾錯能力，藉此加強保護較重要的信息。傳統上，低密度偶校碼在置信傳播(Belief Propagation)解碼下的漸近性能常利用密度演化(Density Evolution)分析，藉由使用信息密度的高斯近似(Gaussian Approximation)，其運算複雜度可大幅降低。然而，過往的分析方式並沒有針對非均等錯誤保護能力的特性做調整。
基於偶校矩陣之結構，本論文提出適用於分析非均等錯誤保護類循環低密度偶校碼之性能的方法。我們利用高斯近似推導出信息傳遞公式，計算各個保護層級的解碼閾值(threshold)，並分析解碼收斂行為。模擬結果驗證了所提方法能夠良好地預測各區段的錯誤保護能力，免除實作上的高複雜度。

Low-density parity-check (LDPC) codes have been demonstrated to approach the Shannon-limit with iterative message-passing decoding and long code lengths. Quasi-cyclic LDPC (QC-LDPC) codes with algebraic structures have attracted great interest because they can be encoded, decoded, and analyzed efficiently. For QC-LDPC codes with unequal error protection (UEP) properties, the codeword bits are divided into several parts, and each part possesses a different error-correcting capability. The asymptotic performance of LDPC codes under belief propagation decoding is usually analyzed by using density evolution, and Gaussian approximation for message densities under density evolution can be used to simplify the analysis. However, conventional analysis methods cannot be directly applied to LDPC codes with UEP properties.
In this thesis, we propose methods to analyze the behavior of UEP QC-LDPC codes by specifying the structures of parity-check matrices. Formulas for detailed representations are derived using density evolution with Gaussian approximation. Furthermore, we calculate the decoding thresholds for different protection levels and analyze the decoding convergence behavior by using the proposed analysis tool.
Simulation results verify that the proposed methods can well predict the UEP capability for various codes.

Abstract i

Contents ii

1 Introduction 2

2 Low-Density Parity-Check Codes and Decoding 5
2.1 Definition to LDPC Codes 5
2.2 Sum-Product Algorithm 7
2.3 Shuffled Decoding Algorithm 11
2.4 Density Evolution with Gaussian Approximation 12
2.4.1 Gaussian Approximation for Regular LDPC Codes 15
2.4.2 Gaussian Approximation for Irregular LDPC Codes 16

3 Preliminaries for UEP QC-LDPC Codes 19
3.1 Quasi-Cyclic Low-Density Parity-Check Codes 19
3.2 UEP QC-LDPC Codes via Masking 20
3.3 Adaptive Decoding Scheme for UEP QC-LDPC Codes 26

4 Proposed Analysis for UEP QC-LDPC Codes 29
4.1 Density Evolution for UEP QC-LDPC Codes 30
4.2 Gaussian Approximation for UEP QC-LDPC Codes 33
4.3 Threshold Prediction 36
4.4 Decoding Convergence Analysis 60

5 Conclusion 69

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