Concatenated convolutional Codes: Analysis of control properties under linear systems theory point of view M. I. Garc´ ıa-Planas * , El M. Soudi † , L. E. Um ‡ * Matem` atica Aplicada I, Universitat Polit` ecnica de Catalunya, C. Miner´ ıa 647, Esc. C, 1-3, 08038 Barcelona, Spain (e-mail: maria.isabel.garcia@upc.edu) † Laboratoire de Recherche Math´ ematiques, Informatique et Applications Universit´ e Mohammed V, Avenue des Nations Unies, Agdal, BP: 554, Rabat, Morocco (e-mail: emsouidi@yahoo.com) † Universit´ e Mohammed V, Rabat, Morocco (e-mail: laurainlee@yahoo.fr) Abstract—In this paper we consider two models of concatenated convolutional codes from the perspective of linear systems theory. We present an input-state-output representation of these models and we study the conditions for control properties as controllability, observability as well as output observability. Index Terms—Convolutional codes, linear systems, output-observability 1 Introduction In coding theory, concatenated codes form a class of error-correcting codes that are obtained by com- bining an inner code and an outer code. They were conceived in 1966 by Dave Forney as a solution to the problem of finding a code that has both exponentially decreasing error probability with increasing block length and polynomial-time decoding complexity. Concatenated codes became widely used in space communications in the 1970s. More Concretely t he concatenation of convolutional codes is used for deep-space transmissions, includ- ing also bar codes, the ISBN code for books, and the ones used for credit cards or identity cards. It is well known that the code with the correction capacities that best fit the reliability of the physical devices is used in each instance of information processing. One of these classes of codes are Turbo Codes (which combine two convolutional codes), they are used in mobile telecommunications standards and its variation for internet access. In this paper we study two kinds of concatenated convolutional codes (serial and parallel) using linear systems theory. It is well known that convolutional codes can be described using a quite more general theory, the linear systems theory over finite fields (see [19], [20], [21] for example). Following the work initiated in [14], the aim of this article is to give a input-output representation of a concatenated (serial and parallel) convolutional, and deduce conditions for control properties as controllability, observability as well output observ- ability. The control properties are relied to the minimality of strict equivalent encoders. 2 Preliminaries Throughout the paper, we denote by F = GF (q) the Galois field of q elements and ¯ F the algebraic closure of F. A convolutional code C of rate k/n and degree δ ,