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A linear algebra approach to minimal convolutional encoders

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Author

Summary, in English

The authors review the work of G.D. Forney, Jr., on the algebraic structure of convolutional encoders upon which some new results regarding minimal convolutional encoders rest. An example is given of a basic convolutional encoding matrix whose number of abstract states is minimal over all equivalent encoding matrices. However, this encoding matrix can be realized with a minimal number of memory elements neither in controller canonical form nor in observer canonical form. Thus, this encoding matrix is not minimal according to Forney's definition of a minimal encoder. To resolve this difficulty, the following three minimality criteria are introduced: minimal-basic encoding matrix, minimal encoding matrix, and minimal encoder. It is shown that all minimal-basic encoding matrices are minimal and that there exist minimal encoding matrices that are not minimal-basic. Several equivalent conditions are given for an encoding matrix to be minimal. It is proven that the constraint lengths of two equivalent minimal-basic encoding matrices are equal one by one up to a rearrangement. All results are proven using only elementary linear algebra

Publishing year

1993

Language

English

Pages

1219-1233

Publication/Series

IEEE Transactions on Information Theory

Volume

39

Issue

4

Document type

Journal article

Publisher

IEEE - Institute of Electrical and Electronics Engineers Inc.

Topic

  • Electrical Engineering, Electronic Engineering, Information Engineering

Status

Published

ISBN/ISSN/Other

  • ISSN: 0018-9448