State-Space Realizations of Periodic Convolutional Codes

Abstract

Convolutional codes are discrete linear systems over a finite field and can be defined as F[d]-modules, where F[d] is the ring of polynomials with coefficient in a finite field F. In this paper we study the algebraic properties of periodic convolutional codes of period 2 and their representation by means of input-state-output representations. We show that they can be described as F[d2]-modules and present explicit representation of the set of equivalent encoders. We investigate their state-space representation and present two different but equivalent types of state-space realizations for these codes. These novel representations can be implemented by realizing two linear time-invariant systems separately and switching the input (or the output) that is entering (or leaving) the system. We investigate their minimality and provide necessary and also sufficient conditions in terms of the reachability and observability properties of the two linear systems involved. The ideas presented here can be easily generalized for codes with period larger than 2.