On the performance evaluation of trellis codes

Generating function techniques for analyzing the error event and the bit-error probabilities for trellis codes are considered. The conventional state diagram approach for linear codes where the number of states is equal to the number of trellis states cannot be applied directly to arbitrary trellis codes, and instead, a state diagram where the number of states is equal to the square of the number of trellis states must be used. It is shown that for an interesting class of trellis codes a modified generating function can be defined for which the number of states {em is equal to} the number of trellis states. The class of codes considered includes trellis codes of rate based upon set partitioning whenever the first partition breaks the signal constellation into two subsets which have the same "configuration matrix," i.e., the same ordered set of mutual distances. The complexity of calculating this modified generating function is the same as for the ordinary generating function of a convolutional code with the same number of trellis states. Bounds on the performance of some interesting codes are given based upon this method.