Minimal transmitter complexity design of analytically described trellis codes

G. Ungerboeck´s (1982) design rules for a class of bandlimited codes called trellis codes are reviewed. His design of the trellis is based on a set partitioning of the signal constellation, and he realized these trellis codes by a convolutional encoder followed by a mapping rule from the coder output to modulation symbols. R. Calderbank and J.E. Mazo (1984) showed how to realize trellis codes for one-dimensional signal sets in a single-step, easily derived, nonlinear transformation with memory on a sliding block of source symbols. The design rules that give a signal (state) specification in a trellis that yields the Calderbank-Mazo transformation with the smallest number of terms are presented. This gives a minimal transmitter complexity design. It is shown how to realize the Ungerboeck from the Calderbank-Mazo form, and as a result a step-by-step, search-free design procedure for trellis codes is presented. Two additional design rules are presented and applied to two examples by analytically designing two trellis codes. A simple procedure for converting an analytic code expression to a convolutional encoder realization is discussed. The analytic designs of a 4-D code and a 2-D code are presented