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.