Parallel concatenated convolutional codes from linear systems theory viewpoint

Abstract

The aim of this work is to characterize two models of concatenated convolutional codes based on the theory of linear systems. The problem we consider can be viewed as the study of composite linear system from the classical control theory or as the interconnection from the behavioral system viewpoint. In this paper we provide an input-state-output representation of both models and introduce some conditions for such representations to be both controllable and observable. We also introduce a lower bound on their free distances and the column distances.

Minimal State-Space Representation of Convolutional Product Codes

“…where A ∈ F m×m , B ∈ F m×k , C ∈ F (n−k)×m and D ∈ F (n−k)×k . This input-state-output representation has been thoroughly studied by many authors [9,10,18,19,21,27,33,35,48], and the codewords are the finite support input-output sequences {v t } t≥0 corresponding to finite support state sequences {x t } t≥0 .…”

“…The blocks are not encoded independently and previously encoded data (matrices in this case) in the sequence have an effect over the next encoded node. Because of this, convolutional codes have memory and can be viewed as linear systems over a finite field (see, for instance, [5,[9][10][11][12][13][14][15][16][17][18]). A description of convolutional codes can be provided by a time-invariant discrete linear system called discrete-time state-space system in control theory (see [19][20][21]).…”

Erasure decoding of convolutional codes using first-order representations

“…It is known that convolutional codes are closely related to discrete-time linear systems over finite fields, in fact each convolutional code has a so-called input-state-output (ISO) representation via such a linear system [17,18]. This correspondence was also used in [4][5][6] to study concatenated convolutional codes. Moreover, the connection between linear systems and convolutional codes was investigated in a more general setup in [22], where multidimensional codes and systems over finite rings were considered.…”

Erasure decoding of convolutional codes using first order representations

“…It is known that convolutional codes are closely related to discrete-time linear systems over finite fields, in fact each convolutional code has a so-called input-state-output (ISO) representation via such a linear system [13,14]. This correspondence was also used in [4,5,6] to study concatenated convolutional codes. Moreover, the connection between linear systems and convolutional codes was investigated in a more general setup in [17], where multidimensional codes and systems over finite rings were considered.…”