Introduction

In Chapters 2–6 we introduced turbo, LDPC and RA codes and their iterative decoding algorithms. Simulation results show that these codes can perform extremely closely to Shannon's capacity limit with practical implementation complexity. In this chapter we analyze the performance of iterative decoders, determine how close they can in fact get to the Shannon limit and consider how the design of the codes will impact on this performance.

Ideally, for a given code and decoder we would like to know for which channel noise levels the message-passing decoder will be able to correct the errors and for which it will not. Unfortunately this is still an open problem. Instead, we will consider the set, or ensemble, of all possible codes with certain parameters (for example, a certain degree distribution) rather than a particular choice of code having those parameters.

For example, a turbo code ensemble is defined by its component encoders and consists of the set of codes generated by all possible interleaver permutations while an LDPC code ensemble is specified by the degree distribution of the Tanner graph nodes and consists of the set of codes generated by all possible permutations of the Tanner graph edges.

When very long codes are considered, the extrinsic LLRs passed between the component decoders can be assumed to be independent and identically distributed. Under this assumption the expected iterative decoding performance of a particular ensemble can be determined by tracking the evolution of these probability density functions through the iterative decoding process, a technique called density evolution.