On dualizing trellis-based APP decoding algorithms

The trellis of a finite Abelian group code is locally (i.e., trellis section by trellis section) related to the trellis of the corresponding dual group code which allows one to express the basic operations of the a posteriori probability (APP) decoding algorithm (defined on a single trellis section of the primal trellis) in terms of the corresponding dual trellis section. Using this local approach, any algorithm employing the same type of operations as the APP algorithm can, thus, be dualized, even if the global dual code does not exist (e.g., nongroup codes represented by a group trellis). Given this, the complexity advantage of the dual approach for high-rate codes can be generalized to a broader class of APP decoding algorithms, including suboptimum algorithms approximating the true APP, which may be more attractive in practical applications due to their reduced complexity. Moreover, the local approach opens the way for mixed approaches where the operations of the APP algorithm are not exclusively performed on the primal or dual trellis. This is inevitable if the code does not possess a trellis consisting solely of group trellis sections as, e.g., for certain terminated group or ring codes. The complexity reduction offered by applying dualization is evaluated. As examples, we give a dual implementation of a suboptimum APP decoding algorithm for tailbiting convolutional codes, as well as dual implementations of APP algorithms of the sliding-window type. Moreover, we evaluate their performance for decoding usual tailbiting codes or convolutional codes, respectively, as well as their performance as component decoders in iteratively decoded parallel concatenated schemes.