Coding for Parallel Channels: Gallager Bounds for Binary Linear Codes with Applications to Repeat-Accumulate Codes and Variations

摘要

This paper is focused on the performance analysis of binary linear blockcodes (or ensembles) whose transmission takes place over independent andmemoryless parallel channels. New upper bounds on the maximum-likelihood (ML)decoding error probability are derived. These bounds are applied to variousensembles of turbo-like codes, focusing especially on repeat-accumulate codesand their recent variations which possess low encoding and decoding complexityand exhibit remarkable performance under iterative decoding. The framework ofthe second version of the Duman and Salehi (DS2) bounds is generalized to thecase of parallel channels, along with the derivation of their optimized tiltingmeasures. The connection between the generalized DS2 and the 1961 Gallagerbounds, addressed by Divsalar and by Sason and Shamai for a single channel, isexplored in the case of an arbitrary number of independent parallel channels.The generalization of the DS2 bound for parallel channels enables to re-derivespecific bounds which were originally derived by Liu et al. as special cases ofthe Gallager bound. In the asymptotic case where we let the block length tendto infinity, the new bounds are used to obtain improved inner bounds on theattainable channel regions under ML decoding. The tightness of the new boundsfor independent parallel channels is exemplified for structured ensembles ofturbo-like codes. The improved bounds with their optimized tilting measuresshow, irrespectively of the block length of the codes, an improvement over theunion bound and other previously reported bounds for independent parallelchannels; this improvement is especially pronounced for moderate to large blocklengths.