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

Abstract

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.

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

“…Here we focus on the state space realizations of the special class of 2D periodic convolutional codes introduced in the previous section by means of 2D periodic Roesser models. In general, this is a nontrivial matter, mainly due to the fact that a 2D periodic state space realization cannot be obtained by independently realizing each of the invariant polynomial operators G ij = V j H i (Aleixo and Rocha, 2017). However, in this paper we show that under certain conditions this problem does not arise, i.e., combining independent realizations of the invariant operators G ij does yield a 2D periodic realization of the corresponding 2D periodic convolutional code.…”

“…Lifted 2D realization. Following the ideas of Aleixo and Rocha (2017;2018), consider the (2,2)-periodic 2D separable Roesser state space system Σ 2D p given by (11) and define the lifted versions of the horizontal and vertical states as…”

“…Although some results have been obtained in the context of time-invariant 2D convolutional codes (Pinho et al, 2014;Fornasini et al, 2015) and 1D periodic convolutional codes (Climent et al, 2009), this problem remains unexplored in the context of periodic 2D convolutional codes. Here we aim at deriving state Roesser 2D state space representations (Aleixo and Rocha, 2017;Kaczorek, 2001) from a (2,2)-periodic two-dimensional generator matrix. This is, in general, a highly nontrivial problem and one needs to assume additional conditions to be able to build minimal state space representations.…”

On the State-Space Realization of 2-Periodic Image Behaviors

“…Let Σ 0 and Σ 1 be two state-space realizations (of the same dimension) of two time-invariant image representations M 0 z −1 and M 1 z −1 . In the following example we show that the 2-periodic system Σ p obtained from Σ 0 and Σ 1 is not, in general, a state-space realization of the associated 2-periodic image behavior (see also [13]). Example 6.…”

Periodic state-space representations of periodic convolutional codes

“…Let Σ 0 and Σ 1 be two state-space realizations (of the same dimension) of two encoders G 0 (z) and G 1 (z). In the following example we show that the 2-periodic system Σp obtained from Σ 0 and Σ 1 is not, in general, a state-space realization of φ G 0 G 1 (see also [1]).…”