Periodic state-space representations of periodic convolutional codes

Napp, Diego; Pereira, Ricardo; Pinto, Raquel; Rocha, Paula

doi:10.1007/s12095-018-0313-6

Cited by 7 publications

(6 citation statements)

References 18 publications

(23 reference statements)

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“…When the encoders given in the previous theorem are column reduced, then the realization of the corresponding encoding map is minimal, as stated next (Napp et al, 2019).…”

Section: State-space Realizations Of 1d Periodic Convolumentioning

confidence: 98%

“…However, in the next theorem (Napp et al, 2019) we provide a sufficient condition for a periodic state-space system to be a realization of a periodic encoding map.…”

Section: State-space Realizations Of 1d Periodic Convolumentioning

confidence: 99%

“…After Costello (1974) conjectured that time-varying convolutional codes can achieve better properties than time-invariant ones, many researchers have investigated such codes. The research in this area has focused on finding concrete encoders that yield 1D periodic convolutional codes with good distance properties (Mooser, 1983;Palazzo, 1993;Felstrom and Zigangirov, 1999;Guardia, 2019) and on state space representations of time-varying systems (Napp et al, 2019;Climent et al, 2009;Aleixo et al, 2011;Kuijper and Willems, 1997).…”

Section: Introductionmentioning

confidence: 99%

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Realization of 2D (2,2)–Periodic Encoders by Means of 2D Periodic Separable Roesser Models

Napp

Pereira

Pinto

et al. 2019

International Journal of Applied Mathematics and Computer Science

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It is well known that convolutional codes are linear systems when they are defined over a finite field. A fundamental issue in the implementation of convolutional codes is to obtain a minimal state representation of the code. Compared with the literature on one-dimensional (1D) time-invariant convolutional codes, there exist relatively few results on the realization problem for time-varying 1D convolutional codes and even fewer if the convolutional codes are two-dimensional (2D). In this paper we consider 2D periodic convolutional codes and address the minimal state space realization problem for this class of codes. This is, in general, a highly nontrivial problem. Here, we focus on separable Roesser models and show that in this case it is possible to derive, under weak conditions, concrete formulas for obtaining a 2D Roesser state space representation. Moreover, we study minimality and present necessary conditions for these representations to be minimal. Our results immediately lead to constructive algorithms to build these representations.

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“…When the encoders given in the previous theorem are column reduced, then the realization of the corresponding encoding map is minimal, as stated next (Napp et al, 2019).…”

Section: State-space Realizations Of 1d Periodic Convolumentioning

confidence: 98%

“…However, in the next theorem (Napp et al, 2019) we provide a sufficient condition for a periodic state-space system to be a realization of a periodic encoding map.…”

Section: State-space Realizations Of 1d Periodic Convolumentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Realization of 2D (2,2)–Periodic Encoders by Means of 2D Periodic Separable Roesser Models

Napp

Pereira

Pinto

et al. 2019

International Journal of Applied Mathematics and Computer Science

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“…In this paper we propose to investigate turbo codes from a system-theoretical point of view, an approach that has been extensively used to study convolutional codes [9,23,25,32]. In particular, we focus on the crucial notion of maximum effective free distance which has not been carefully investigated within this approach, see [1,24] for constructions of codes with maximum column distance.…”

Section: Introductionmentioning

confidence: 99%

1/n Turbo codes from linear system point of view

Herranz

Napp

Perea

2020

RACSAM

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The performance of turbo codes at the error floor region is largely determined by the effective free distance, which corresponds to the minimum Hamming weight among all codewords sequences generated by input sequences of weight two. In this paper, we study turbo codes of dimension one obtained from the concatenation of two equal codes and present an upper bound on the effective free distance of a turbo code with these parameters defined over any finite field. We do that making use of the so-called (A, B, C, D) statespace representations of convolutional codes and restrict to the case where A is invertible. A particular construction, from a linear systems point of view, of a recursive systematic convolutional code of rate 1/n so that the effective free distance of the corresponding turbo code attains this upper bound is also presented.

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“…If this is the state description of a convolutional code, it is not obvious whether such periodic representations can be also described by periodic encoding maps and vice versa. Some partial results in this regard were presented in [23]. Also related is the thread of research in periodic behaviors [26,30,28,31], where periodicity is defined in term of the trajectories of the system (behavior) following the ideas of Willems [32].…”

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confidence: 99%

State-Space Realizations of Periodic Convolutional Codes

Fornasini¹,

Napp²,

Pereira³

et al. 2022

SIAM J. Control Optim.

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Convolutional codes are discrete linear systems over a finite field and can be defined as \BbbF [d]-modules, where \BbbF [d] is the ring of polynomials with coefficient in a finite field \BbbF . 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 \BbbF [d 2 ]-modules and present explicit representation of the set of equivalent encoders. We investigate their statespace 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.

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Periodic state-space representations of periodic convolutional codes

Cited by 7 publications

References 18 publications

Realization of 2D (2,2)–Periodic Encoders by Means of 2D Periodic Separable Roesser Models

Realization of 2D (2,2)–Periodic Encoders by Means of 2D Periodic Separable Roesser Models

1/n Turbo codes from linear system point of view

State-Space Realizations of Periodic Convolutional Codes

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