Directed graphs, 2D state models, and characteristic polynomials of irreducible matrix pairs

AbstractThe definition and main properties of a 2D digraph, namely a directed graph with two kinds of arcs, are introduced. Under the assumption of strong connectedness, the analysis of its paths and cycles is performed, based on an integer matrix whose rows represent the compositions of all circuits, and on the corresponding row module. Natural constraints on the composition of the paths connecting each pair of vertices lead to the definition of a 2D strongly connected digraph. For a 2D digraph of this kind the set of vertices can be partitioned into disjoint 2D-imprimitivity classes, whose number and composition are strictly related to the structure of the row module. Irreducible matrix pairs, i.e. pairs endowed with a 2D strongly connected digraph, are subsequently discussed. Equivalent descriptions of irreducibility, naturally extending those available for a single irreducible matrix, are obtained. These refer to the free evolution of the 2D state models described by the pairs and to their characteristic polynomials. Finally, primitivity is viewed as a special case of irreducibility, and completely characterized in terms of 2D-digraphs, characteristic polynomials, and 2D system dynamics

State-Space Realizations of Periodic Convolutional Codes

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.This work was supported by Portuguese funds through the Center for Research and Development in Mathematics and Applications (CIDMA) and the Portuguese Foundation for Science and Technology (FCT-Fundaçao para a Ciência e a Tecnologia) within project UIDB/04106/2020. It was also partially supported by Base Funding (UIDB/00147/2020) and Programmatic Funding (UIDP/00147/2020) of the Systems and Technologies Center - SYSTEC - funded by national funds through the FCT/MCTES (PIDDAC). The work of the second author was partially supported by Spanish grants PID2019-108668GB-I00 of the Ministerio de Ciencia e Innovación of the Gobierno de España and VIGROB-287 of the Universitat d'Alacant

Composition codes

In this paper we introduce a special class of 2D convolutional codes, called composition codes, which admit encoders G(d1,d2) that can be decomposed as the product of two 1D encoders, i.e., G(d1,d2)=G2(d2)G1(d1). Taking into account this decomposition, we obtain syndrome formers of the code directly from G1(d1) andG2(d2), in case G1(d1) andG2(d2) are right prime. Moreover we consider 2D state-space realizations by means of a separable Roesser model of the encoders and syndrome formers of a composition code and we investigate the minimality of such realizations. In particular, we obtain minimal realizations for composition codes which admit an encoder G(d1,d2)=G2(d2)G1(d1) withG2(d2) a systematic 1D encoder. Finally, we investigate the minimality of 2D separable Roesser state-space realizations for syndrome formers of these codes.publishe

On the structure of positive behaviours

Positive systems in the behavioural approach are introduced as sets of non-negative trajectories that satisfy a closure condition with respect to linear combinations witn non-negative coefficients. Completeness and finite menory are discussed and compared with the analogous properties of linear shift invariant behaviours

Code Decomposition in the Analysis of a Convolutional Code

A convolutional code can be decomposed into smaller codes if it admits decoupled encoders. In this paper, we show that if a code can be decomposed into smaller codes (subcodes) its column distances are the minimum of the column distances of its subcodes. Moreover, the j-th column distance of a convolutional code C is equal to the j-th column distance of the convolutional codes generated by the truncation of the canonical encoders of C to matrices which entries have degree smaller or equal than j. We show that if one of such codes can be decomposed into smaller codes, so can be all the other codes

BILATERAL CONVOLUTIONAL CODES OVER A FINITE FIELD

In this communication we introduce convolutional codes constituted by bilateral sequences over a finite field and analyze the structure of their encoders

Matrix fraction descriptions in convolutional coding

AbstractIn this paper, polynomial matrix fraction descriptions (MFDs) are used as a tool for investigating the structure of a (linear) convolutional code and the family of its encoders and syndrome formers. As static feedback and precompensation allow to obtain all minimal encoders (in particular, polynomial encoders and decoupled encoders) of a given code, a simple parametrization of their MFDs is provided. All minimal syndrome formers, by a duality argument, are obtained by resorting to output injection and postcompensation. Decoupled encoders are finally discussed as well as the possibility of representing a convolutional code as a direct sum of smaller ones