Minimal and canonical rational generator matrices for convolutional codes

Author

Forney, G. David, Jr. ; Johannesson, Rolf ; Wan, Zhe-Xian

Author_Institution

Motorola Inc., Mansfield, MA, USA

Volume

42

Issue

6

fYear

1996

fDate

11/1/1996 12:00:00 AM

Firstpage

1865

Lastpage

1880

Abstract

A full-rank K×n matrix G(D) over the rational functions F(D) generates a rate R=k/n convolutional code C. G(D) is minimal if it can be realized with as few memory elements as any encoder for C, and G(D) is canonical if it has a minimal realization in controller canonical form. We show that G(D) is minimal if and only if for all rational input sequences u(D), the span of u(D)G(D) covers the span of u(D). Alternatively, G(D) is minimal if and only if G(D) is globally zero-free, or globally invertible. We show that G(D) is canonical if and only if G(D) is minimal and also globally orthogonal, in the valuation-theoretic sense of Monna (1970)

Keywords

convolutional codes; matrix algebra; canonical rational generator matrices; convolutional codes; encoder; full-rank matrix; globally invertible matrix; globally orthogonal matrix; globally zero-free matrix; memory elements; minimal generator matrices; rational functions; rational input sequences; span; valuation theory; Convolutional codes; Cost accounting; Helium; Information theory; Iterative decoding; Modems; Polynomials; Sections; Terminology; Testing;

fLanguage

English

Journal_Title

Information Theory, IEEE Transactions on

Publisher

ieee

ISSN

0018-9448

Type

jour

DOI

10.1109/18.556681

Filename

556681