Butterfly structure in trellis-coded modulation

Trellis coded modulation was designed to improve error performance of synchronous data links without sacrificing data rate, or expanding bandwidth. The codes are interpreted as binary convolutional codes with a mapping of coded bits into multilevel/phase signals to increase free Euclidean distance. The technique referred to as "mapping by set partitioning" is claimed as an optimal assignment. The rules for assigning channel signals first proposed by Ungerboeck assure maximum free Euclidean distance. However, by taking advantage of a butterfly structure in the trellis and modifying Ungerboeck's rules, the multiplicity of error events with free Euclidean distance can be minimized. In this paper, the relationship between butterfly and signal assignment is revealed and a modified rule is proposed for optimality of the trellis assignment.