Convolutional Codes and Matrix Control Theory

A general description of convolutional codes is provided by a set of linear equations called discrete-time state-space equations in control theory. We discuss how the notions of controllability, observability and minimality in control, are applied to convolutional code design. The notion of output-observability is used to ensure minimality of a convolutional code. Without concatenation, searching for good trellis codes requires examining one code within each group of range-equivalent encoders (i.e. equivalent in Forney's sense [1]). So it is su cient to restrict attention within a set of canonical encoders. However, turbo code constituent encoders are optimized under speci c input to output mappings. We describe this class of encoders as \inputHamming weight equivalent" and identify the canonical structures for the constituent encoder search space. In many cases of practical interest, the optimal structure for these constituent encoders connects the memory elements in a single row. Tables of codes optimized for e ective free distance are also provided.