| Trellis codes, which are used in all modern high-speed computer modems, are usually described as a combination of a binary linear convolutional code and a mapping from coded output bits to signal points in a Euclidean signal space. This representation obscures the relationship between the algebraic structure of the binary code and the geometric structure of the signal space code.;In this dissertation, we present a representation of trellis codes as dynamical systems in which the output alphabet is a group of isometries (distance-preserving transformations) of Euclidean space. A system over a group of isometries is "converted" into a trellis code by applying each sequence of isometries in the code componentwise to an initial signal space sequence x; the set of all such sequences is a trellis code in an infinite-dimensional Euclidean space.;A code with such a representation is "geometrically uniform," and therefore has a number of strong symmetry properties. Foremost among them is the uniform error property: when the code is used over an additive white Gaussian noise channel with maximum likelihood decoding, the probability of error is independent of the transmitted code sequence. This property greatly simplifies performance analysis.;Not all trellis codes admit such a representation; however, most good ones do. We show that the apparently nonlinear 8-state, two-dimensional Wei code used in V.32 modems has such a representation, as does the 16-state, four-dimensional Wei code used in higher-speed modems. These representations are novel and unexpected, and show that geometrical uniformity is a broader and subtler concept than was previously recognized.;We develop a structure theory for group systems that parallels the theory of ordinary linear systems over fields. Using only elementary group theory, we are able to develop such fundamental constructs of linear system theory as state spaces and minimal input-output realizations. Remarkably, these basic constructs depend only on additive (group) and not multiplicative (field) algebraic structure. |
