State space representation of SISO periodic behaviors

Coding Theory and Applications

2017

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Abstract. In this paper we study periodically time-varying convolutional codes by means of input-state-output representations. Using these representations we investigate under which conditions a given time-invariant convolutional code can be transformed into an equivalent periodic time-varying one. The relation between these two classes of convolutional codes is studied for period 2. We illustrate the ideas presented in this paper by constructing a periodic time-varying convolutional code from a time-invariant one. The resulting periodic code has larger free distance than any time-invariant convolutional code with equivalent parameters.

Section: Constructing Periodically Time-varying Convolutional Codesmentioning

confidence: 99%

Section: Constructing Periodically Time-varying Convolutional Codesmentioning

confidence: 99%

See 1 more Smart Citation

A State Space Approach to Periodic Convolutional Codes

Napp

Pereira

Coding Theory and Applications

2017

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“…In particular, following the ideas of [4], we propose a method to obtain a 2D periodic Roesser model representation which consists in first determining an invariant input/output system associated with the original periodic one, then making (if possible) an invariant Roesser model realization, and, finally, obtaining a 2D periodic Roesser model from the invariant one. For the sake of simplicity, we shall focus on single-input/single-output (SISO) (2, 2)-periodic 2D systems, i.e., systems whose coefficients periodically vary with period 2 both in the horizontal and in the vertical direction.…”

Section: Introductionmentioning

confidence: 99%

“…Following the ideas of [4], an alternative procedure is to obtain an invariant formulation of the original 2D (P, Q)-periodic behavior, determine (if possible) a 2D invariant Roesser model representation of the obtained invariant behavior, and finally try to obtain a 2D (P, Q)-periodic Roesser model representation from the invariant one.…”

mentioning

confidence: 99%

Roesser model representation of 2D periodic behaviors: the (2,2)-periodic SISO case

Aleixo

2017 10th International Workshop on Multidimensional (nD) Systems (nDS)

2017

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In this paper we consider 2D single-input/singleoutput behaviors described by linear partial difference equations with (2,2)-periodically varying coefficients, and present a method to obtain 2D Roesser state-space representations (or realizations) for such behaviors. Since these cannot be obtained by separately realizing each shiftinvariant system resulting from "freezing" the varying coefficients, we propose a method based on the realization of an invariant input/output behavior obtained by suitably "lifting" the trajectories of the original periodic behavior.

A low‐dimensional realization algorithm for periodic input/output behavioral systems

Aleixo

Math Methods in App Sciences

2023

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The state‐space realization of linear systems is of utmost importance in linear systems theory. After the realization problem for the time‐invariant case was solved, particular attention was paid to the case of linear periodic systems. Recently, such systems have regained importance, for instance, in the context of coding theory where periodic convolutional encoders play an important role. The majority of the contributions within this area only concern the realization of transfer functions (or impulse responses), thus excluding the case of input/output linear systems without coprime representations. By the end of the eighties of the last century, Jan C. Willems suggested an approach (nowadays known as the behavioral approach) that considers a wider class of systems and allows to overcome this drawback. According to this approach, the central object in a system is its behavior which consists of all the signals that satisfy the system laws (also called system trajectories). Consequently, the behavior of a system with an input/output representation that is not coprime contains more trajectories than the set of input/output signals defined by the system transfer function. Our work takes this fact into account. In previous research conducted by the authors, some results on the realization of periodic behavioral systems were obtained, which, despite their importance, are conservative with respect to the dimension of the obtained realizations. In this paper, we revisit the behavioral periodic realization problem and give further insight into this question. This allows to derive new results and set up an algorithm to compute low‐dimensional state‐space realizations.