Distance spectra and upper bounds on error probability for trellis codes

The problem of estimating error probability for trellis codes is considered. The set of all squared Euclidean distances between code sequences is presented as a countable set. This representation is used for calculating the generating functions for upper-bounding error probability and bit error probability for trellis codes satisfying some symmetry conditions. The generating functions of squared Euclidean distances (distance spectra) are obtained by inversion of a matrix of order 2^ν. It is shown that the generating functions are defined in terms of one formal variable for QAM and uniform AM, and in terms of q/4 formal variables for q-ary PSK, q=2^m, where m⩾2 is an integer. For small ν, the generating functions may be found in closed form. For larger ν, a numerical technique for obtaining some initial terms of the power series expansion is proposed. This algorithm is based on the recurrent matrix equations and the Chinese remainder theorem