Minimal State-Space Realizations of Convolutional Codes

Pinho, Telma; Pinto, Raquel; Rocha, Paula

doi:10.1007/978-3-319-10380-8_2

Cited by 3 publications

(3 citation statements)

References 4 publications

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“…is a minimal encoder of C , i.e., it has minimal Roesser McMillan degree among all encoders of C , [10,9], in the sequel we denote this minimal degree by µ(C ).…”

Section: Minimal Syndrome Former Realizations Of a Special Class Of Cmentioning

confidence: 99%

Minimal Realizations of Syndrome Formers of a Special Class of 2D Codes

Fornasini

Pinho

Pinto

et al. 2015

Coding Theory and Applications

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In this paper we consider a special class of 2D convolutional codes (composition codes) with encoders G(d 1 , d 2 ) that can be decomposed as the product of two 1D encoders, i.e.,are prime we provide constructions of syndrome formers of the code, directly from G 1 (d 1 ) and G 2 (d 2 ). Moreover we investigate the minimality of 2D state-space realization by means of a separable Roesser model of syndrome formers of composition codes, where G 2 (d 2 ) is a quasi-systematic encoder.

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“…is a minimal encoder of C , i.e., it has minimal Roesser McMillan degree among all encoders of C , [10,9], in the sequel we denote this minimal degree by µ(C ).…”

Section: Minimal Syndrome Former Realizations Of a Special Class Of Cmentioning

confidence: 99%

Minimal Realizations of Syndrome Formers of a Special Class of 2D Codes

Fornasini

Pinho

Pinto

et al. 2015

Coding Theory and Applications

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“…However, the general problem of a minimal state space realization of multidimensional systems has not been solved even for 2-dimensional systems. Nevertheless, for some special cases, minimal state space realization methods have been derived; see the works of Zerz (2000), Pinho et al (2014), Pinho (2016), Napp et al (2010), Galkowski (1996; and the references therein.…”

Section: Introductionmentioning

confidence: 99%

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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“…A characterization of minimal encoders for general 2D convolutional codes of rate 1/n was obtained in [9,10]. However, the generalization of the results in [9,10] for 2D convolutional codes of rate k/n, with k > 1, appears to be very difficult. However the problem becomes easier to handle if we restrict our study to the classes of composition codes, and in particular to those which admit an encoder…”

Section: Introductionmentioning

confidence: 99%

Composition codes

Rocha¹,

Pinto²,

Pinho³

et al. 2016

AMC

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In this paper we introduce a special class of 2D convolutional codes, called composition codes, which admit encoders G(d 1 , d 2) that can be decomposed as the product of two 1D encoders, i.e., G(d 1 , d 2) = G 2 (d 2)G 1 (d 1). Taking into account this decomposition, we obtain syndrome formers of the code directly from G 1 (d 1) and G 2 (d 2), in case G 1 (d 1) and G 2 (d 2) 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(d 1 , d 2) = G 2 (d 2)G 1 (d 1) with G 2 (d 2) a systematic 1D encoder. Finally, we investigate the minimality of 2D separable Roesser state-space realizations for syndrome formers of these codes.

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Minimal State-Space Realizations of Convolutional Codes

Cited by 3 publications

References 4 publications

Minimal Realizations of Syndrome Formers of a Special Class of 2D Codes

Minimal Realizations of Syndrome Formers of a Special Class of 2D Codes

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

Composition codes

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