Punctured Turbo Code Ensembles

We analyze the asymptotic performance of punctured turbo codes. The analysis is based on the union bound on the word error probability of maximum likelihood decoding for a punctured turbo code ensembles averaged over all possible puncturing patterns and interleavers. By using special probabilistic puncturing, we prove that, for a given mother turbo code ensemble [C] with a finite noise threshold c[C] 0 , if the asymptotic puncturing rate λ satisfies log λ < −c[C] 0 , there exists a finite noise threshold c[CP ] 0 for the punctured turbo code ensemble which is bounded by a function of c[C] 0 and λ. Based on this result, we prove that, on any binary-input memoryless channel whose Bhattacharyya noise distance is greater than c[CP ] 0 , the average ML decoding word error probability of the punctured turbo code ensemble approaches zero at least as fast as n−β , where β is the well known “interleaver gain” exponent. This enables us to answer an important question in the practice of HARQ schemes, namely up to which puncturing rate “good” turbo codes give rise to“good” punctured codes. I. CODE AND CHANNEL MODELS A. Turbo Code Ensembles and Random Puncturing The general structure of a punctured parallel turbo code is shown in Figure 1. The punctured turbo code is obtained by