Book contents
- Frontmatter
- Contents
- Preface
- 1 Coding and Capacity
- 2 Finite Fields, Vector Spaces, Finite Geometries, and Graphs
- 3 Linear Block Codes
- 4 Convolutional Codes
- 5 Low-Density Parity-Check Codes
- 6 Computer-Based Design of LDPC Codes
- 7 Turbo Codes
- 8 Ensemble Enumerators for Turbo and LDPC Codes
- 9 Ensemble Decoding Thresholds for LDPC and Turbo Codes
- 10 Finite-Geometry LDPC Codes
- 11 Constructions of LDPC Codes Based on Finite Fields
- 12 LDPC Codes Based on Combinatorial Designs, Graphs, and Superposition
- 13 LDPC Codes for Binary Erasure Channels
- 14 Nonbinary LDPC Codes
- 15 LDPC Code Applications and Advanced Topics
- Index
9 - Ensemble Decoding Thresholds for LDPC and Turbo Codes
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- 1 Coding and Capacity
- 2 Finite Fields, Vector Spaces, Finite Geometries, and Graphs
- 3 Linear Block Codes
- 4 Convolutional Codes
- 5 Low-Density Parity-Check Codes
- 6 Computer-Based Design of LDPC Codes
- 7 Turbo Codes
- 8 Ensemble Enumerators for Turbo and LDPC Codes
- 9 Ensemble Decoding Thresholds for LDPC and Turbo Codes
- 10 Finite-Geometry LDPC Codes
- 11 Constructions of LDPC Codes Based on Finite Fields
- 12 LDPC Codes Based on Combinatorial Designs, Graphs, and Superposition
- 13 LDPC Codes for Binary Erasure Channels
- 14 Nonbinary LDPC Codes
- 15 LDPC Code Applications and Advanced Topics
- Index
Summary
The previous chapter examined code properties responsible for the floor region (high-SNR region) of LDPC and turbo codes. Specifically, the emphasis was on the computation of various weight enumerators for LDPC and turbo code ensembles because a poor weight spectrum leads to poor performance in a code's floor region both for iterative decoders and for maximum-likelihood decoders. That chapter also introduced ensemble enumerators for trapping sets and stopping sets, both of which can lead to poor floor performance in iterative decoders. In this chapter we examine the iterative decoding performance of LDPC and turbo code ensembles in the complementary low-SNR region, the “waterfall” region. We show that the iterative decoding of long LDPC and turbo codes displays a threshold effect in which communication is reliable beyond this threshold and unreliable below it. The threshold is a function of code ensemble properties and the tools introduced in this chapter allow the designer to predict the decoding threshold and its gap from Shannon's limit. The ensemble properties for LDPC codes are the degree distributions which are typically the design targets for the code-design techniques presented in subsequent chapters. The ensemble properties for turbo codes are the selected constituent codes. Our development borrows heavily from the references listed at the end of the chapter and our focus is on the binary-input AWGN channel. The references and the problems consider other channels.
Density Evolution for Regular LDPC Codes
We first summarize the main results of [1] with regard to iterative (message passing) decoding of long, regular LDPC codes.
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- Information
- Channel CodesClassical and Modern, pp. 388 - 429Publisher: Cambridge University PressPrint publication year: 2009